pid
stringlengths 6
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stringlengths 28
976
| options
listlengths 0
8
| answer
stringclasses 103
values | image_1
imagewidth (px) 54
2.52k
| category
stringclasses 5
values | source
stringclasses 3
values | prompt
stringlengths 33
1.17k
|
|---|---|---|---|---|---|---|---|
Math_201
|
Using only pieces like the one shown in the diagram, Zara wants to make a complete square without gaps or overlaps.
<image_1>
What is the smallest number of pieces she can use?
|
[] |
20
|
2D Transformation
|
MathVision
|
Using only pieces like the one shown in the diagram, Zara wants to make a complete square without gaps or overlaps.
<image_1>
What is the smallest number of pieces she can use?
|
|
Math_202
|
Both of the shapes shown in the diagram are formed from the same five pieces, a $5 \mathrm{~cm}$ by $10 \mathrm{~cm}$ rectangle, two large quarter circles and two small quarter circles. What is the difference in $\mathrm{cm}$ between their perimeters? <image_1>
|
[] |
20
|
2D Transformation
|
MathVision
|
Both of the shapes shown in the diagram are formed from the same five pieces, a $5 \mathrm{~cm}$ by $10 \mathrm{~cm}$ rectangle, two large quarter circles and two small quarter circles. What is the difference in $\mathrm{cm}$ between their perimeters? <
|
|
Math_203
|
A square piece of paper of area $64 \mathrm{~cm}^{2}$ is folded twice, as shown in the diagram. What is the sum of the areas of the two shaded rectangles?
<image_1>
|
[
"$10 \\mathrm{~cm}^{2}$",
"$14 \\mathrm{~cm}^{2}$",
"$15 \\mathrm{~cm}^{2}$",
"$16 \\mathrm{~cm}^{2}$",
"$24 \\mathrm{~cm}^{2}$"
] |
D
|
2D Transformation
|
MathVision
|
A square piece of paper of area $64 \mathrm{~cm}^{2}$ is folded twice, as shown in the diagram. What is the sum of the areas of the two shaded rectangles?
<
Options:
$10 \mathrm{~cm}^{2}$
$14 \mathrm{~cm}^{2}$
$15 \mathrm{~cm}^{2}$
$16 \mathrm{~cm}^{2}$
$24 \mathrm{~cm}^{2}$
|
|
Math_204
|
What is the largest number of " $\mathrm{T}$ " shaped pieces, as shown, that can be placed on the $4 \times 5$ grid in the diagram, without any overlap of the pieces? <image_1>
|
[] |
4
|
2D Transformation
|
MathVision
|
What is the largest number of " $\mathrm{T}$ " shaped pieces, as shown, that can be placed on the $4 \times 5$ grid in the diagram, without any overlap of the pieces? <
|
|
Math_205
|
In the picture, the three strips labelled 1,2,3 have the same horizontal width $a$. These three strips connect two parallel lines. Which of these statements is true? <image_1>
|
[
"All three strips have the same area.",
"Strip 1 has the largest area.",
"Strip 2 has the largest area.",
"Strip 3 has the largest area.",
"It is impossible to say which has the largest area without knowing $a$."
] |
A
|
2D Transformation
|
MathVision
|
In the picture, the three strips labelled 1,2,3 have the same horizontal width $a$. These three strips connect two parallel lines. Which of these statements is true? <
Options:
All three strips have the same area.
Strip 1 has the largest area.
Strip 2 has the largest area.
Strip 3 has the largest area.
It is impossible to say which has the largest area without knowing $a$.
|
|
Math_206
|
A rectangular sheet of paper which measures $6 \mathrm{~cm} \times 12 \mathrm{~cm}$ is folded along its diagonal (Diagram A). The shaded areas in Diagram B are then cut off and the paper is unfolded leaving the rhombus shown in Diagram C. What is the length of the side of the rhombus? <image_1>
|
[
"$\\frac{7}{2} \\sqrt{5} \\mathrm{~cm}$",
"$7.35 \\mathrm{~cm}$",
"$7.5 \\mathrm{~cm}$",
"$7.85 \\mathrm{~cm}$",
"$8.1 \\mathrm{~cm}$"
] |
C
|
2D Transformation
|
MathVision
|
A rectangular sheet of paper which measures $6 \mathrm{~cm} \times 12 \mathrm{~cm}$ is folded along its diagonal (Diagram A). The shaded areas in Diagram B are then cut off and the paper is unfolded leaving the rhombus shown in Diagram C. What is the length of the side of the rhombus? <
Options:
$\frac{7}{2} \sqrt{5} \mathrm{~cm}$
$7.35 \mathrm{~cm}$
$7.5 \mathrm{~cm}$
$7.85 \mathrm{~cm}$
$8.1 \mathrm{~cm}$
|
|
Math_207
|
A square $P Q R S$ with sides of length 10 is rolled without slipping along a line. Initially $P$ and $Q$ are on the line and the first roll is around point $Q$ as shown in the diagram. The rolling stops when $P$ first returns to the line. What is the length of the curve that $P$ has travelled?
<image_1>
|
[
"$10 \\pi$",
"$5 \\pi+5 \\pi \\sqrt{2}$",
"$10 \\pi+5 \\pi \\sqrt{2}$",
"$5 \\pi+10 \\pi \\sqrt{2}$",
"$10 \\pi+10 \\pi \\sqrt{2}$"
] |
C
|
2D Transformation
|
MathVision
|
A square $P Q R S$ with sides of length 10 is rolled without slipping along a line. Initially $P$ and $Q$ are on the line and the first roll is around point $Q$ as shown in the diagram. The rolling stops when $P$ first returns to the line. What is the length of the curve that $P$ has travelled?
<
Options:
$10 \pi$
$5 \pi+5 \pi \sqrt{2}$
$10 \pi+5 \pi \sqrt{2}$
$5 \pi+10 \pi \sqrt{2}$
$10 \pi+10 \pi \sqrt{2}$
|
|
Math_208
|
A coin with diameter $1 \mathrm{~cm}$ rolls around the outside of a regular hexagon with edges of length $1 \mathrm{~cm}$ until it returns to its original position. In centimetres, what is the length of the path traced out by the centre of the coin? <image_1>
|
[
"$6+\\pi / 2$",
"$12+\\pi$",
"$6+\\pi$",
"$12+2 \\pi$",
"$6+2 \\pi$"
] |
C
|
2D Transformation
|
MathVision
|
A coin with diameter $1 \mathrm{~cm}$ rolls around the outside of a regular hexagon with edges of length $1 \mathrm{~cm}$ until it returns to its original position. In centimetres, what is the length of the path traced out by the centre of the coin? <
Options:
$6+\pi / 2$
$12+\pi$
$6+\pi$
$12+2 \pi$
$6+2 \pi$
|
|
Math_209
|
Six points are marked on a sheet of squared paper as shown. Which of the following shapes cannot be made by connecting some of these points using straight lines? <image_1>
|
[
"parallelogram",
"trapezium",
"right-angled triangle",
"obtuse-angled triangle",
"all the shapes $\\mathrm{A}-\\mathrm{D}$ can be made"
] |
E
|
2D Transformation
|
MathVision
|
Six points are marked on a sheet of squared paper as shown. Which of the following shapes cannot be made by connecting some of these points using straight lines? <
Options:
parallelogram
trapezium
right-angled triangle
obtuse-angled triangle
all the shapes $\mathrm{A}-\mathrm{D}$ can be made
|
|
Math_210
|
A circle of radius $4 \mathrm{~cm}$ is divided into four congruent parts by arcs of radius $2 \mathrm{~cm}$ as shown. What is the length of the perimeter of one of the parts, in $\mathrm{cm}$ ? <image_1>
|
[
"$2 \\pi$",
"$4 \\pi$",
"$6 \\pi$",
"$8 \\pi$",
"$12 \\pi$"
] |
C
|
2D Transformation
|
MathVision
|
A circle of radius $4 \mathrm{~cm}$ is divided into four congruent parts by arcs of radius $2 \mathrm{~cm}$ as shown. What is the length of the perimeter of one of the parts, in $\mathrm{cm}$ ? <
Options:
$2 \pi$
$4 \pi$
$6 \pi$
$8 \pi$
$12 \pi$
|
|
Math_211
|
A triangular piece of paper is folded along the dotted line shown in the left-hand diagram to form the heptagon shown in the right-hand diagram. The total area of the shaded parts of the heptagon is $1 \mathrm{~cm}^{2}$. The area of the original triangle is $1 \frac{1}{2}$ times the area of the heptagon. What is the area of the original triangle, in $\mathrm{cm}^{2}$ ? <image_1>
|
[] |
3
|
2D Transformation
|
MathVision
|
A triangular piece of paper is folded along the dotted line shown in the left-hand diagram to form the heptagon shown in the right-hand diagram. The total area of the shaded parts of the heptagon is $1 \mathrm{~cm}^{2}$. The area of the original triangle is $1 \frac{1}{2}$ times the area of the heptagon. What is the area of the original triangle, in $\mathrm{cm}^{2}$ ? <
|
|
Math_212
|
One of the line segments shown on the grid is the image produced by a rotation of the other line segment. Which of the points $T, U, V$, $W$ could be the centre of such a rotation? <image_1>
|
[
"only $T$",
"only $U$",
"either of $U$ and $W$",
"any of $U, V$ and $W$",
"any of $T, U, V$ and $W$"
] |
C
|
2D Transformation
|
MathVision
|
One of the line segments shown on the grid is the image produced by a rotation of the other line segment. Which of the points $T, U, V$, $W$ could be the centre of such a rotation? <
Options:
only $T$
only $U$
either of $U$ and $W$
any of $U, V$ and $W$
any of $T, U, V$ and $W$
|
|
Math_213
|
The trapezium shown in the diagram is rotated anti-clockwise by $90^{\circ}$ around the origin $O$, and then reflected in the $x$-axis. Which of the following shows the end result of these transformations? <image_1>
|
[
"A",
"B",
"C",
"D",
"E"
] |
A
|
2D Transformation
|
MathVision
|
The trapezium shown in the diagram is rotated anti-clockwise by $90^{\circ}$ around the origin $O$, and then reflected in the $x$-axis. Which of the following shows the end result of these transformations? <
Options:
A
B
C
D
E
|
|
Math_214
|
If $r, s$, and $t$ denote the lengths of the 'lines' in the picture, then which of the following inequalities is correct? <image_1>
|
[
"$r<s<t$",
"$r<t<s$",
"$s<r<t$",
"$s<t<r$",
"$t<s<r$"
] |
E
|
2D Transformation
|
MathVision
|
If $r, s$, and $t$ denote the lengths of the 'lines' in the picture, then which of the following inequalities is correct? <
Options:
$r<s<t$
$r<t<s$
$s<r<t$
$s<t<r$
$t<s<r$
|
|
Math_215
|
Which of the following diagrams shows the locus of the midpoint of the wheel when the wheel rolls along the zig-zag curve shown?
<image_1>
|
[
"A",
"B",
"C",
"D",
"E"
] |
E
|
2D Transformation
|
MathVision
|
Which of the following diagrams shows the locus of the midpoint of the wheel when the wheel rolls along the zig-zag curve shown?
<
Options:
A
B
C
D
E
|
|
Math_216
|
A circle of radius 1 rolls along a straight line from the point $K$ to the point $L$, where $K L=11 \pi$. Which of the following pictures shows the correct appearance of the circle when it reaches $L$ ?
<image_1>
|
[
"A",
"B",
"C",
"D",
"E"
] |
D
|
2D Transformation
|
MathVision
|
A circle of radius 1 rolls along a straight line from the point $K$ to the point $L$, where $K L=11 \pi$. Which of the following pictures shows the correct appearance of the circle when it reaches $L$ ?
<
Options:
A
B
C
D
E
|
|
Math_217
|
The diagram shows three congruent regular hexagons. Some diagonals have been drawn, and some regions then shaded. The total shaded areas of the hexagons are $X, Y, Z$ as shown. Which of the following statements is true? <image_1>
|
[
"$X, Y$ and $Z$ are all the same",
"$\\quad Y$ and $Z$ are equal, but $X$ is different",
"$X$ and $Z$ are equal, but $Y$ is different",
"$X$ and $Y$ are equal, but $Z$ is different",
"$X, Y, Z$ are all different"
] |
A
|
2D Transformation
|
MathVision
|
The diagram shows three congruent regular hexagons. Some diagonals have been drawn, and some regions then shaded. The total shaded areas of the hexagons are $X, Y, Z$ as shown. Which of the following statements is true? <
Options:
$X, Y$ and $Z$ are all the same
$\quad Y$ and $Z$ are equal, but $X$ is different
$X$ and $Z$ are equal, but $Y$ is different
$X$ and $Y$ are equal, but $Z$ is different
$X, Y, Z$ are all different
|
|
Math_218
|
Some shapes are drawn on a piece of paper. The teacher folds the left-hand side of the paper over the central bold line. How many of the shapes on the left-hand side will fit exactly on top of a shape on the right-hand side? <image_1>
|
[] |
3
|
2D Transformation
|
MathVision
|
Some shapes are drawn on a piece of paper. The teacher folds the left-hand side of the paper over the central bold line. How many of the shapes on the left-hand side will fit exactly on top of a shape on the right-hand side? <
|
|
Math_219
|
A ball is propelled from corner $A$ of a square snooker table of side 2 metres. After bouncing off three cushions as shown, the ball goes into a pocket at $B$. The total distance travelled by the ball is $\sqrt{k}$ metres. What is the value of $k$ ?
<image_1>
(Note that when the ball bounces off a cushion, the angle its path makes with the cushion as it approaches the point of impact is equal to the angle its path makes with the cushion as it moves away from the point of impact as shown in the diagram below.)
|
[] |
52
|
2D Transformation
|
MathVision
|
A ball is propelled from corner $A$ of a square snooker table of side 2 metres. After bouncing off three cushions as shown, the ball goes into a pocket at $B$. The total distance travelled by the ball is $\sqrt{k}$ metres. What is the value of $k$ ?
<image_1>
(Note that when the ball bounces off a cushion, the angle its path makes with the cushion as it approaches the point of impact is equal to the angle its path makes with the cushion as it moves away from the point of impact as shown in the diagram below.)
|
|
Math_220
|
An angle is drawn on a set of equally spaced parallel lines as shown. The ratio of the area of shaded region $\mathcal{C}$ to the area of shaded region $\mathcal{B}$ is $11/5$. Find the ratio of shaded region $\mathcal{D}$ to the area of shaded region $\mathcal{A}$.
<image_1>
|
[] |
408
|
2D Transformation
|
MathVision
|
An angle is drawn on a set of equally spaced parallel lines as shown. The ratio of the area of shaded region $\mathcal{C}$ to the area of shaded region $\mathcal{B}$ is $11/5$. Find the ratio of shaded region $\mathcal{D}$ to the area of shaded region $\mathcal{A}$.
<
|
|
Math_221
|
In the array of $13$ squares shown below, $8$ squares are colored red, and the remaining $5$ squares are colored blue. If one of all possible such colorings is chosen at random, the probability that the chosen colored array appears the same when rotated $90^{\circ}$ around the central square is $\frac{1}{n}$, where $n$ is a positive integer. Find $n$.
<image_1>
|
[] |
429
|
2D Transformation
|
MathVision
|
In the array of $13$ squares shown below, $8$ squares are colored red, and the remaining $5$ squares are colored blue. If one of all possible such colorings is chosen at random, the probability that the chosen colored array appears the same when rotated $90^{\circ}$ around the central square is $\frac{1}{n}$, where $n$ is a positive integer. Find $n$.
<
|
|
Math_222
|
How many of the twelve pentominoes pictured below have at least one line of symmetry?
<image_1>
|
[] |
6
|
2D Transformation
|
MathVision
|
How many of the twelve pentominoes pictured below have at least one line of symmetry?
<
|
|
Math_223
|
A white cylindrical silo has a diameter of 30 feet and a height of 80 feet. A red stripe with a horizontal width of 3 feet is painted on the silo, as shown, making two complete revolutions around it. What is the area of the stripe in square feet?
<image_1>
|
[
"$120$",
"$180$",
"$240$",
"$360$",
"$480$"
] |
C
|
2D Transformation
|
MathVision
|
A white cylindrical silo has a diameter of 30 feet and a height of 80 feet. A red stripe with a horizontal width of 3 feet is painted on the silo, as shown, making two complete revolutions around it. What is the area of the stripe in square feet?
<
Options:
$120$
$180$
$240$
$360$
$480$
|
|
Math_224
|
The letter F shown below is rotated $90^\circ$ clockwise around the origin, then reflected in the $y$-axis, and then rotated a half turn around the origin. What is the final image?
<image_1><image2>
|
[
"A",
"B",
"C",
"D",
"E"
] |
E
|
2D Transformation
|
MathVision
|
The letter F shown below is rotated $90^\circ$ clockwise around the origin, then reflected in the $y$-axis, and then rotated a half turn around the origin. What is the final image?
<image_1><image2
Options:
A
B
C
D
E
|
|
Math_225
|
In the figure below, $3$ of the $6$ disks are to be painted blue, $2$ are to be painted red, and $1$ is to be painted green. Two paintings that can be obtained from one another by a rotation or a reflection of the entire figure are considered the same. How many different paintings are possible?
<image_1>
|
[] |
12
|
2D Transformation
|
MathVision
|
In the figure below, $3$ of the $6$ disks are to be painted blue, $2$ are to be painted red, and $1$ is to be painted green. Two paintings that can be obtained from one another by a rotation or a reflection of the entire figure are considered the same. How many different paintings are possible?
<
|
|
Math_226
|
A paper triangle with sides of lengths 3, 4, and 5 inches, as shown, is folded so that point $A$ falls on point $B$. What is the length in inches of the crease?
<image_1>
|
[
"$1+\\frac{1}{2} \\sqrt{2}$",
"$\\sqrt{3}$",
"$\\frac{7}{4}$",
"$\\frac{15}{8}$",
"$2$"
] |
D
|
2D Transformation
|
MathVision
|
A paper triangle with sides of lengths 3, 4, and 5 inches, as shown, is folded so that point $A$ falls on point $B$. What is the length in inches of the crease?
<
Options:
$1+\frac{1}{2} \sqrt{2}$
$\sqrt{3}$
$\frac{7}{4}$
$\frac{15}{8}$
$2$
|
|
Math_227
|
Each square in a $5 \times 5$ grid is either filled or empty, and has up to eight adjacent neighboring squares, where neighboring squares share either a side or a corner. The grid is transformed by the following rules:
Any filled square with two or three filled neighbors remains filled.
Any empty square with exactly three filled neighbors becomes a filled square.
All other squares remain empty or become empty.
A sample transformation is shown in the figure below.
<image_1>
Suppose the $5 \times 5$ grid has a border of empty squares surrounding a $3 \times 3$ subgrid. How many initial configurations will lead to a transformed grid consisting of a single filled square in the center after a single transformation? (Rotations and reflections of the same configuration are considered different.)
|
[] |
22
|
2D Transformation
|
MathVision
|
Each square in a $5 \times 5$ grid is either filled or empty, and has up to eight adjacent neighboring squares, where neighboring squares share either a side or a corner. The grid is transformed by the following rules:
Any filled square with two or three filled neighbors remains filled.
Any empty square with exactly three filled neighbors becomes a filled square.
All other squares remain empty or become empty.
A sample transformation is shown in the figure below.
<image_1>
Suppose the $5 \times 5$ grid has a border of empty squares surrounding a $3 \times 3$ subgrid. How many initial configurations will lead to a transformed grid consisting of a single filled square in the center after a single transformation? (Rotations and reflections of the same configuration are considered different.)
|
|
Math_228
|
Let $OABC$ be a unit square in the $xy$-plane with $O(0,0),A(1,0),B(1,1)$ and $C(0,1)$. Let $u=x^2-y^2$ and $v=2xy$ be a transformation of the $xy$-plane into the $uv$-plane. The transform (or image) of the square is:
<image_1>
|
[
"A",
"B",
"C",
"D",
"E"
] |
D
|
2D Transformation
|
MathVision
|
Let $OABC$ be a unit square in the $xy$-plane with $O(0,0),A(1,0),B(1,1)$ and $C(0,1)$. Let $u=x^2-y^2$ and $v=2xy$ be a transformation of the $xy$-plane into the $uv$-plane. The transform (or image) of the square is:
<
Options:
A
B
C
D
E
|
|
Math_229
|
<image_1>
Equilateral triangle $ABP$ (see figure) with side $AB$ of length $2$ inches is placed inside square $AXYZ$ with side of length $4$ inches so that $B$ is on side $AX$. The triangle is rotated clockwise about $B$, then $P$, and so on along the sides of the square until $P$ returns to its original position. The length of the path in inches traversed by vertex $P$ is equal to
|
[
"$20\\pi/3$",
"$32\\pi/3$",
"$12\\pi$",
"$40\\pi/3$",
"$15\\pi$"
] |
D
|
2D Transformation
|
MathVision
|
<image_1>
Equilateral triangle $ABP$ (see figure) with side $AB$ of length $2$ inches is placed inside square $AXYZ$ with side of length $4$ inches so that $B$ is on side $AX$. The triangle is rotated clockwise about $B$, then $P$, and so on along the sides of the square until $P$ returns to its original position. The length of the path in inches traversed by vertex $P$ is equal to
Options:
$20\pi/3$
$32\pi/3$
$12\pi$
$40\pi/3$
$15\pi$
|
|
Math_230
|
A circle with a circumscribed and an inscribed square centered at the origin $ O$ of a rectangular coordinate system with positive $ x$ and $ y$ axes $ OX$ and $ OY$ is shown in each figure $ I$ to $ IV$ below.
<image_1>
The inequalities
\[ |x| + |y| \leq \sqrt{2(x^2 + y^2)} \leq 2\mbox{Max}(|x|, |y|)\]
are represented geometrically* by the figure numbered
|
[
"$I$",
"$II$",
"$III$",
"$IV$",
"$\\mbox{none of these}*An inequality of the form f(x, y) \\leq g(x, y), for all x and y is represented geometrically by a figure showing the containment \\{\\mbox{The set of points }(x, y)\\mbox{ such that }g(x, y) \\leq a\\} \\subset\\ \\{\\mbox{The set of points }(x, y)\\mbox{ such that }f(x, y) \\leq a\\}for a typical real number a$."
] |
B
|
2D Transformation
|
MathVision
|
A circle with a circumscribed and an inscribed square centered at the origin $ O$ of a rectangular coordinate system with positive $ x$ and $ y$ axes $ OX$ and $ OY$ is shown in each figure $ I$ to $ IV$ below.
<image_1>
The inequalities
\[ |x| + |y| \leq \sqrt{2(x^2 + y^2)} \leq 2\mbox{Max}(|x|, |y|)\]
are represented geometrically* by the figure numbered
Options:
$I$
$II$
$III$
$IV$
$\mbox{none of these}*An inequality of the form f(x, y) \leq g(x, y), for all x and y is represented geometrically by a figure showing the containment \{\mbox{The set of points }(x, y)\mbox{ such that }g(x, y) \leq a\} \subset\ \{\mbox{The set of points }(x, y)\mbox{ such that }f(x, y) \leq a\}for a typical real number a$.
|
|
Math_231
|
<image_1>
Points $A , B, C$, and $D$ are distinct and lie, in the given order, on a straight line. Line segments $AB, AC$, and $AD$ have lengths $x, y$, and $z$ , respectively. If line segments $AB$ and $CD$ may be rotated about points $B$ and $C$, respectively, so that points $A$ and $D$ coincide, to form a triangle with positive area, then which of the following three inequalities must be satisfied?
$\textbf{I. }x<\frac{z}{2}\qquad\textbf{II. }y<x+\frac{z}{2}\qquad\textbf{III. }y<\frac{z}{2}$
|
[
"$\\textbf{I. }\\text{only}$",
"$\\textbf{II. }\\text{only}$",
"$\\textbf{I. }\\text{and }\\textbf{II. }\\text{only}$",
"$\\textbf{II. }\\text{and }\\textbf{III. }\\text{only}$",
"$\\textbf{I. },\\textbf{II. },\\text{and }\\textbf{III. }$"
] |
C
|
2D Transformation
|
MathVision
|
<image_1>
Points $A , B, C$, and $D$ are distinct and lie, in the given order, on a straight line. Line segments $AB, AC$, and $AD$ have lengths $x, y$, and $z$ , respectively. If line segments $AB$ and $CD$ may be rotated about points $B$ and $C$, respectively, so that points $A$ and $D$ coincide, to form a triangle with positive area, then which of the following three inequalities must be satisfied?
$\textbf{I. }x<\frac{z}{2}\qquad\textbf{II. }y<x+\frac{z}{2}\qquad\textbf{III. }y<\frac{z}{2}$
Options:
$\textbf{I. }\text{only}$
$\textbf{II. }\text{only}$
$\textbf{I. }\text{and }\textbf{II. }\text{only}$
$\textbf{II. }\text{and }\textbf{III. }\text{only}$
$\textbf{I. },\textbf{II. },\text{and }\textbf{III. }$
|
|
Math_232
|
In one of the adjoining figures a square of side $2$ is dissected into four pieces so that $E$ and $F$ are the midpoints of opposite sides and $AG$ is perpendicular to $BF$. These four pieces can then be reassembled into a rectangle as shown in the second figure. The ratio of height to base, $XY$ / $YZ$, in this rectangle is
<image_1>
|
[
"$4$",
"$1+2\\sqrt{3}$",
"$2\\sqrt{5}$",
"$\\frac{8+4\\sqrt{3}}{3}$",
"$5$"
] |
E
|
2D Transformation
|
MathVision
|
In one of the adjoining figures a square of side $2$ is dissected into four pieces so that $E$ and $F$ are the midpoints of opposite sides and $AG$ is perpendicular to $BF$. These four pieces can then be reassembled into a rectangle as shown in the second figure. The ratio of height to base, $XY$ / $YZ$, in this rectangle is
<
Options:
$4$
$1+2\sqrt{3}$
$2\sqrt{5}$
$\frac{8+4\sqrt{3}}{3}$
$5$
|
|
Math_233
|
The sides of $\triangle ABC$ have lengths $6, 8$ and $10$. A circle with center $P$ and radius $1$ rolls around the inside of $\triangle ABC$, always remaining tangent to at least one side of the triangle. When $P$ first returns to its original position, through what distance has $P$ traveled?
<image_1>
|
[] |
12
|
2D Transformation
|
MathVision
|
The sides of $\triangle ABC$ have lengths $6, 8$ and $10$. A circle with center $P$ and radius $1$ rolls around the inside of $\triangle ABC$, always remaining tangent to at least one side of the triangle. When $P$ first returns to its original position, through what distance has $P$ traveled?
<
|
|
Math_234
|
<image_1>
Each of the sides of the five congruent rectangles is labeled with an integer, as shown above. These five rectangles are placed, without rotating or reflecting, in positions $I$ through $V$ so that the labels on coincident sides are equal.
Which of the rectangles is in position $I$?
|
[
"A",
"B",
"C",
"D",
"E"
] |
E
|
2D Transformation
|
MathVision
|
<image_1>
Each of the sides of the five congruent rectangles is labeled with an integer, as shown above. These five rectangles are placed, without rotating or reflecting, in positions $I$ through $V$ so that the labels on coincident sides are equal.
Which of the rectangles is in position $I$?
Options:
A
B
C
D
E
|
|
Math_235
|
The set $ G$ is defined by the points $ (x,y)$ with integer coordinates, $ 3\le|x|\le7$, $ 3\le|y|\le7$. How many squares of side at least $ 6$ have their four vertices in $ G$?
<image_1>
|
[] |
225
|
2D Transformation
|
MathVision
|
The set $ G$ is defined by the points $ (x,y)$ with integer coordinates, $ 3\le|x|\le7$, $ 3\le|y|\le7$. How many squares of side at least $ 6$ have their four vertices in $ G$?
<
|
|
Math_236
|
In the plane figure shown below, $3$ of the unit squares have been shaded. What is the least number of additional unit squares that must be shaded so that the resulting figure has two lines of symmetry$?$
<image_1>
|
[] |
7
|
2D Transformation
|
MathVision
|
In the plane figure shown below, $3$ of the unit squares have been shaded. What is the least number of additional unit squares that must be shaded so that the resulting figure has two lines of symmetry$?$
<
|
|
Math_237
|
Usain is walking for exercise by zigzagging across a $100$-meter by $30$-meter rectangular field, beginning at point $A$ and ending on the segment $\overline{BC}$. He wants to increase the distance walked by zigzagging as shown in the figure below $(APQRS)$. What angle $\theta=\angle PAB=\angle QPC=\angle RQB=\cdots$ will produce in a length that is $120$ meters? (Do not assume the zigzag path has exactly four segments as shown; there could be more or fewer.)
<image_1>
|
[
"$\\arccos\\frac{5}{6}$",
"$\\arccos\\frac{4}{5}$",
"$\\arccos\\frac{3}{10}$",
"$\\arcsin\\frac{4}{5}$",
"$\\arcsin\\frac{5}{6}$"
] |
A
|
2D Transformation
|
MathVision
|
Usain is walking for exercise by zigzagging across a $100$-meter by $30$-meter rectangular field, beginning at point $A$ and ending on the segment $\overline{BC}$. He wants to increase the distance walked by zigzagging as shown in the figure below $(APQRS)$. What angle $\theta=\angle PAB=\angle QPC=\angle RQB=\cdots$ will produce in a length that is $120$ meters? (Do not assume the zigzag path has exactly four segments as shown; there could be more or fewer.)
<
Options:
$\arccos\frac{5}{6}$
$\arccos\frac{4}{5}$
$\arccos\frac{3}{10}$
$\arcsin\frac{4}{5}$
$\arcsin\frac{5}{6}$
|
|
Math_238
|
<image_1>
The square in the first diagram "rolls" clockwise around the fixed regular hexagon until it reaches the bottom. In which position will the solid triangle be in diagram $4$?
|
[
"A",
"B",
"C",
"D",
"E"
] |
E
|
2D Transformation
|
MathVision
|
<image_1>
The square in the first diagram "rolls" clockwise around the fixed regular hexagon until it reaches the bottom. In which position will the solid triangle be in diagram $4$?
Options:
A
B
C
D
E
|
|
Math_239
|
Suppose a square piece of paper is folded in half vertically. The folded paper is then cut in half along the dashed line. Three rectangles are formed-a large one and two small ones. What is the ratio of the perimeter of one of the small rectangles to the perimeter of the large rectangle?
<image_1>
|
[
"$\\frac{1}{2}$",
"$\\frac{2}{3}$",
"$\\frac{3}{4}$",
"$\\frac{4}{5}$",
"$\\frac{5}{6}$"
] |
E
|
2D Transformation
|
MathVision
|
Suppose a square piece of paper is folded in half vertically. The folded paper is then cut in half along the dashed line. Three rectangles are formed-a large one and two small ones. What is the ratio of the perimeter of one of the small rectangles to the perimeter of the large rectangle?
<
Options:
$\frac{1}{2}$
$\frac{2}{3}$
$\frac{3}{4}$
$\frac{4}{5}$
$\frac{5}{6}$
|
|
Math_240
|
How many different patterns can be made by shading exactly two of the nine squares? Patterns that can be matched by flips and/or turns are not considered different. For example, the patterns shown below are not considered different.
<image_1>
|
[] |
8
|
2D Transformation
|
MathVision
|
How many different patterns can be made by shading exactly two of the nine squares? Patterns that can be matched by flips and/or turns are not considered different. For example, the patterns shown below are not considered different.
<
|
|
Math_241
|
The $16$ squares on a piece of paper are numbered as shown in the diagram. While lying on a table, the paper is folded in half four times in the following sequence:
fold the top half over the bottom half
fold the bottom half over the top half
fold the right half over the left half
fold the left half over the right half.
Which numbered square is on top after step $4$?
<image_1>
|
[] |
9
|
2D Transformation
|
MathVision
|
The $16$ squares on a piece of paper are numbered as shown in the diagram. While lying on a table, the paper is folded in half four times in the following sequence:
fold the top half over the bottom half
fold the bottom half over the top half
fold the right half over the left half
fold the left half over the right half.
Which numbered square is on top after step $4$?
<
|
|
Math_242
|
Eight $1\times 1$ square tiles are arranged as shown so their outside edges form a polygon with a perimeter of $14$ units. Two additional tiles of the same size are added to the figure so that at least one side of each tile is shared with a side of one of the squares in the original figure. Which of the following could be the perimeter of the new figure?
<image_1>
|
[
"15",
"17",
"18",
"19",
"20"
] |
C
|
2D Transformation
|
MathVision
|
Eight $1\times 1$ square tiles are arranged as shown so their outside edges form a polygon with a perimeter of $14$ units. Two additional tiles of the same size are added to the figure so that at least one side of each tile is shared with a side of one of the squares in the original figure. Which of the following could be the perimeter of the new figure?
<
Options:
15
17
18
19
20
|
|
Math_243
|
Each of the three large squares shown below is the same size. Segments that intersect the sides of the squares intersect at the midpoints of the sides. How do the shaded areas of these squares compare?
<image_1>
|
[
"$\\text{The shaded areas in all three are equal.}$",
"$\\text{Only the shaded areas of }I\\text{ and }II\\text{ are equal.}$",
"$\\text{Only the shaded areas of }I\\text{ and }III\\text{ are equal.}$",
"$\\text{Only the shaded areas of }II\\text{ and }III\\text{ are equal.}$",
"$\\text{The shaded areas of }I, II\\text{ and }III\\text{ are all different.}$"
] |
A
|
2D Transformation
|
MathVision
|
Each of the three large squares shown below is the same size. Segments that intersect the sides of the squares intersect at the midpoints of the sides. How do the shaded areas of these squares compare?
<
Options:
$\text{The shaded areas in all three are equal.}$
$\text{Only the shaded areas of }I\text{ and }II\text{ are equal.}$
$\text{Only the shaded areas of }I\text{ and }III\text{ are equal.}$
$\text{Only the shaded areas of }II\text{ and }III\text{ are equal.}$
$\text{The shaded areas of }I, II\text{ and }III\text{ are all different.}$
|
|
Math_244
|
As indicated by the diagram below, a rectangular piece of paper is folded bottom to top, then left to right, and finally, a hole is punched at X.
What does the paper look like when unfolded?
<image_1>
|
[
"A",
"B",
"C",
"D",
"E"
] |
B
|
2D Transformation
|
MathVision
|
As indicated by the diagram below, a rectangular piece of paper is folded bottom to top, then left to right, and finally, a hole is punched at X.
What does the paper look like when unfolded?
<image_1>
Options:
A
B
C
D
E
|
|
Math_245
|
Let $PQRS$ be a square piece of paper. $P$ is folded onto $R$ and then $Q$ is folded onto $S$. The area of the resulting figure is 9 square inches. Find the perimeter of square $PQRS$.
<image_1>
|
[] |
24
|
2D Transformation
|
MathVision
|
Let $PQRS$ be a square piece of paper. $P$ is folded onto $R$ and then $Q$ is folded onto $S$. The area of the resulting figure is 9 square inches. Find the perimeter of square $PQRS$.
<
|
|
Math_246
|
Each half of this figure is composed of 3 red triangles, 5 blue triangles and 8 white triangles. When the upper half is folded down over the centerline, 2 pairs of red triangles coincide, as do 3 pairs of blue triangles. There are 2 red-white pairs. How many white pairs coincide?
<image_1>
|
[] |
5
|
2D Transformation
|
MathVision
|
Each half of this figure is composed of 3 red triangles, 5 blue triangles and 8 white triangles. When the upper half is folded down over the centerline, 2 pairs of red triangles coincide, as do 3 pairs of blue triangles. There are 2 red-white pairs. How many white pairs coincide?
<
|
|
Math_247
|
Thirteen black and six white hexagonal tiles were used to create the figure below. If a new figure is created by attaching a border of white tiles with the same size and shape as the others, what will be the difference between the total number of white tiles and the total number of black tiles in the new figure?
<image_1>
|
[] |
11
|
2D Transformation
|
MathVision
|
Thirteen black and six white hexagonal tiles were used to create the figure below. If a new figure is created by attaching a border of white tiles with the same size and shape as the others, what will be the difference between the total number of white tiles and the total number of black tiles in the new figure?
<
|
|
Math_248
|
What is the minimum number of small squares that must be colored black so that a line of symmetry lies on the diagonal $ \overline{BD}$ of square $ ABCD$?
<image_1>
|
[] |
4
|
2D Transformation
|
MathVision
|
What is the minimum number of small squares that must be colored black so that a line of symmetry lies on the diagonal $ \overline{BD}$ of square $ ABCD$?
<
|
|
Math_249
|
Initially, a spinner points west. Chenille moves it clockwise $ 2 \frac{1}{4}$ revolutions and then counterclockwise $ 3 \frac{3}{4}$ revolutions. In what direction does the spinner point after the two moves?
<image_1>
|
[
"$\\text{north}$",
"$\\text{east}$",
"$\\text{south}$",
"$\\text{west}$",
"$\\text{northwest}$"
] |
B
|
2D Transformation
|
MathVision
|
Initially, a spinner points west. Chenille moves it clockwise $ 2 \frac{1}{4}$ revolutions and then counterclockwise $ 3 \frac{3}{4}$ revolutions. In what direction does the spinner point after the two moves?
<
Options:
$\text{north}$
$\text{east}$
$\text{south}$
$\text{west}$
$\text{northwest}$
|
|
Math_250
|
Tiles I, II, III and IV are translated so one tile coincides with each of the rectangles $A, B, C$ and $D$. In the final arrangement, the two numbers on any side common to two adjacent tiles must be the same. Which of the tiles is translated to Rectangle $C$?
<image_1>
|
[
"$I$",
"$II$",
"$III$",
"$IV$",
"$\\text{ cannot be determined}$"
] |
D
|
2D Transformation
|
MathVision
|
Tiles I, II, III and IV are translated so one tile coincides with each of the rectangles $A, B, C$ and $D$. In the final arrangement, the two numbers on any side common to two adjacent tiles must be the same. Which of the tiles is translated to Rectangle $C$?
<
Options:
$I$
$II$
$III$
$IV$
$\text{ cannot be determined}$
|
|
Math_251
|
What is the area of the shaded pinwheel shown in the $5\times 5$ grid?
<image_1>
|
[
"$4$",
"$6$",
"$8$",
"$10$",
"$12$"
] |
B
|
2D Transformation
|
MathVision
|
What is the area of the shaded pinwheel shown in the $5\times 5$ grid?
<
Options:
$4$
$6$
$8$
$10$
$12$
|
|
Math_252
|
The five pieces shown below can be arranged to form four of the five figures shown in the choices. Which figure cannot be formed?
<image_1>
|
[
"A",
"B",
"C",
"D",
"E"
] |
B
|
2D Transformation
|
MathVision
|
The five pieces shown below can be arranged to form four of the five figures shown in the choices. Which figure cannot be formed?
<image_1>
Options:
A
B
C
D
E
|
|
Math_253
|
Construct a square on one side of an equilateral triangle. One on non-adjacent side of the square, construct a regular pentagon, as shown. One a non-adjacent side of the pentagon, construct a hexagon. Continue to construct regular polygons in the same way, until you construct an octagon. How many sides does the resulting polygon have?
<image_1>
|
[] |
23
|
2D Transformation
|
MathVision
|
Construct a square on one side of an equilateral triangle. One on non-adjacent side of the square, construct a regular pentagon, as shown. One a non-adjacent side of the pentagon, construct a hexagon. Continue to construct regular polygons in the same way, until you construct an octagon. How many sides does the resulting polygon have?
<
|
|
Math_254
|
Each of the following four large congruent squares is subdivided into combinations of congruent triangles or rectangles and is partially bolded. What percent of the total area is partially bolded?
<image_1>
|
[
"$12\\frac{1}{2}$",
"$20$",
"$25$",
"$33 \\frac{1}{3}$",
"$37\\frac{1}{2}$"
] |
C
|
2D Transformation
|
MathVision
|
Each of the following four large congruent squares is subdivided into combinations of congruent triangles or rectangles and is partially bolded. What percent of the total area is partially bolded?
<
Options:
$12\frac{1}{2}$
$20$
$25$
$33 \frac{1}{3}$
$37\frac{1}{2}$
|
|
Math_255
|
A circle of radius 2 is cut into four congruent arcs. The four arcs are joined to form the star figure shown. What is the ratio of the area of the star figure to the area of the original circle?
<image_1>
|
[
"$\\frac{4-\\pi}\\pi$",
"$\\frac{1}{\\pi}$",
"$\\frac{\\sqrt{2}}{\\pi}$",
"$\\frac{\\pi-1}\\pi$",
"$\\frac{3}{\\pi}$"
] |
A
|
2D Transformation
|
MathVision
|
A circle of radius 2 is cut into four congruent arcs. The four arcs are joined to form the star figure shown. What is the ratio of the area of the star figure to the area of the original circle?
<
Options:
$\frac{4-\pi}\pi$
$\frac{1}{\pi}$
$\frac{\sqrt{2}}{\pi}$
$\frac{\pi-1}\pi$
$\frac{3}{\pi}$
|
|
Math_256
|
A ball with diameter 4 inches starts at point A to roll along the track shown. The track is comprised of 3 semicircular arcs whose radii are $R_1 = 100$ inches, $R_2 = 60$ inches, and $R_3 = 80$ inches, respectively. The ball always remains in contact with the track and does not slip. What is the distance the center of the ball travels over the course from A to B?
<image_1>
|
[
"$238\\pi$",
"$240\\pi$",
"$260\\pi$",
"$280\\pi$",
"$500\\pi$"
] |
A
|
2D Transformation
|
MathVision
|
A ball with diameter 4 inches starts at point A to roll along the track shown. The track is comprised of 3 semicircular arcs whose radii are $R_1 = 100$ inches, $R_2 = 60$ inches, and $R_3 = 80$ inches, respectively. The ball always remains in contact with the track and does not slip. What is the distance the center of the ball travels over the course from A to B?
<
Options:
$238\pi$
$240\pi$
$260\pi$
$280\pi$
$500\pi$
|
|
Math_257
|
There are 81 grid points (uniformly spaced) in the square shown in the diagram below, including the points on the edges. Point $P$ is the center of the square. Given that point $Q$ is randomly chosen from among the other 80 points, what is the probability that line $PQ$ is a line of symmetry for the square?
<image_1>
|
[
"$\\frac{1}{5}$",
"$\\frac{1}{4}$",
"$\\frac{2}{5}$",
"$\\frac{9}{20}$",
"$\\frac{1}{2}$"
] |
C
|
2D Transformation
|
MathVision
|
There are 81 grid points (uniformly spaced) in the square shown in the diagram below, including the points on the edges. Point $P$ is the center of the square. Given that point $Q$ is randomly chosen from among the other 80 points, what is the probability that line $PQ$ is a line of symmetry for the square?
<
Options:
$\frac{1}{5}$
$\frac{1}{4}$
$\frac{2}{5}$
$\frac{9}{20}$
$\frac{1}{2}$
|
|
Math_258
|
The letter M in the figure below is first reflected over the line $q$ and then reflected over the line $p$. What is the resulting image?
<image_1>
|
[
"A",
"B",
"C",
"D",
"E"
] |
E
|
2D Transformation
|
MathVision
|
The letter M in the figure below is first reflected over the line $q$ and then reflected over the line $p$. What is the resulting image?
<image_1>
Options:
A
B
C
D
E
|
|
Math_259
|
A square piece of paper is folded twice into four equal quarters, as shown below, then cut along the dashed line. When unfolded, the paper will match which of the following figures?
<image_1>
|
[
"A",
"B",
"C",
"D",
"E"
] |
E
|
2D Transformation
|
MathVision
|
A square piece of paper is folded twice into four equal quarters, as shown below, then cut along the dashed line. When unfolded, the paper will match which of the following figures?
<
Options:
A
B
C
D
E
|
|
Math_260
|
Suppose we have a hexagonal grid in the shape of a hexagon of side length $4$ as shown at left. Define a “chunk” to be four tiles, two of which are adjacent to the other three, and the other two of which are adjacent to just two of the others. The three possible rotations of these are shown at right.\n\nIn how many ways can we choose a chunk from the grid?\n<image_1>
|
[] |
72
|
2D Transformation
|
MathVision
|
Suppose we have a hexagonal grid in the shape of a hexagon of side length $4$ as shown at left. Define a “chunk” to be four tiles, two of which are adjacent to the other three, and the other two of which are adjacent to just two of the others. The three possible rotations of these are shown at right.\n\nIn how many ways can we choose a chunk from the grid?\n<
|
|
Math_261
|
Blahaj has two rays with a common endpoint A0 that form an angle of $1^o$. They construct a sequence of points $A_0$, $. . . $, $A_n$ such that for all $1 \le i \le n$, $|A_{i-1}A_i | = 1$, and $|A_iA_0| > |A_{i-1}A_0|$. Find the largest possible value of $n$.\n<image_1>
|
[] |
90
|
2D Transformation
|
MathVision
|
Blahaj has two rays with a common endpoint A0 that form an angle of $1^o$. They construct a sequence of points $A_0$, $. . . $, $A_n$ such that for all $1 \le i \le n$, $|A_{i-1}A_i | = 1$, and $|A_iA_0| > |A_{i-1}A_0|$. Find the largest possible value of $n$.\n<
|
|
Math_262
|
What is the maximum number of $T$-shaped polyominos (shown below) that we can put into a $6 \times 6$ grid without any overlaps. The blocks can be rotated.\n<image_1>
|
[] |
8
|
2D Transformation
|
MathVision
|
What is the maximum number of $T$-shaped polyominos (shown below) that we can put into a $6 \times 6$ grid without any overlaps. The blocks can be rotated.\n<
|
|
Math_263
|
How many lines pass through exactly two points in the following hexagonal grid?\n<image_1>
|
[] |
60
|
2D Transformation
|
MathVision
|
How many lines pass through exactly two points in the following hexagonal grid?\n<
|
|
Math_264
|
Each unit square of a $4 \times 4$ square grid is colored either red, green, or blue. Over all possible colorings of the grid, what is the maximum possible number of L-trominos that contain exactly one square of each color? (L-trominos are made up of three unit squares sharing a corner, as shown below.)\n<image_1>
|
[] |
18
|
2D Transformation
|
MathVision
|
Each unit square of a $4 \times 4$ square grid is colored either red, green, or blue. Over all possible colorings of the grid, what is the maximum possible number of L-trominos that contain exactly one square of each color? (L-trominos are made up of three unit squares sharing a corner, as shown below.)\n<
|
|
Math_265
|
Consider the L-shaped tromino below with 3 attached unit squares. It is cut into exactly two pieces of equal area by a line segment whose endpoints lie on the perimeter of the tromino. What is the longest possible length of the line segment?\n<image_1>
|
[] |
2.5
|
2D Transformation
|
MathVision
|
Consider the L-shaped tromino below with 3 attached unit squares. It is cut into exactly two pieces of equal area by a line segment whose endpoints lie on the perimeter of the tromino. What is the longest possible length of the line segment?\n<
|
|
Math_266
|
The grid below contains the $16$ points whose $x$- and $y$-coordinates are in the set $\{0,1,2,3\}$: <image_1> A square with all four of its vertices among these $16$ points has area $A$. What is the sum of all possible values of $A$?
|
[] |
21
|
2D Transformation
|
MathVision
|
The grid below contains the $16$ points whose $x$- and $y$-coordinates are in the set $\{0,1,2,3\}$: <image_1> A square with all four of its vertices among these $16$ points has area $A$. What is the sum of all possible values of $A$?
|
|
Math_267
|
How many bricks are missing in the wall?
<image_1>
|
[] |
6
|
3D Spatial Simulation
|
MathVision
|
How many bricks are missing in the wall?
<
|
|
Math_268
|
Seven sticks lay on top of each other. Stick 2 lays right at the bottom. Stick 6 lays right on top. Which stick lays exactly in the middle?
<image_1>
|
[] |
3
|
3D Spatial Simulation
|
MathVision
|
Seven sticks lay on top of each other. Stick 2 lays right at the bottom. Stick 6 lays right on top. Which stick lays exactly in the middle?
<
|
|
Math_269
|
Ingrid has 4 red, 3 blue, 2 green and 1 yellow cube. She uses them to build the following object:
<image_1>
Cubes with the same colour don't touch each other. Which colour is the cube with the question mark?
|
[
"red",
"blue",
"green",
"Yellow",
"This cannot be worked out for certain."
] |
A
|
3D Spatial Simulation
|
MathVision
|
Ingrid has 4 red, 3 blue, 2 green and 1 yellow cube. She uses them to build the following object:
<image_1>
Cubes with the same colour don't touch each other. Which colour is the cube with the question mark?
Options:
red
blue
green
Yellow
This cannot be worked out for certain.
|
|
Math_270
|
<image_1>
In the picture above we see a cube in two different positions.
The six sides of the cube look like this:
Which side is opposite to <image3>?
|
[] |
C
|
3D Spatial Simulation
|
MathVision
|
<image_1>
In the picture above we see a cube in two different positions.
The six sides of the cube look like this:
Which side is opposite to <image3>?
|
|
Math_271
|
Theodor has built this tower made up of discs. He looks at the tower from above. How many discs does he see?
<image_1>
|
[] |
3
|
3D Spatial Simulation
|
MathVision
|
Theodor has built this tower made up of discs. He looks at the tower from above. How many discs does he see?
<
|
|
Math_272
|
Five equally big square pieces of card are placed on a table on top of each other. The picture on the side is created this way. The cards are collected up from top to bottom. In which order are they collected?
<image_1>
|
[
"5-4-3-2-1",
"5-2-3-4-1",
"5-4-2-3-1",
"5-3-2-1-4",
"5-2-3-1-4"
] |
E
|
3D Spatial Simulation
|
MathVision
|
Five equally big square pieces of card are placed on a table on top of each other. The picture on the side is created this way. The cards are collected up from top to bottom. In which order are they collected?
<
Options:
5-4-3-2-1
5-2-3-4-1
5-4-2-3-1
5-3-2-1-4
5-2-3-1-4
|
|
Math_273
|
Maria made a block using white cubes and colored cubes in equal amounts. How many of the white cubes cannot be seen in the picture?
<image_1>
|
[] |
2
|
3D Spatial Simulation
|
MathVision
|
Maria made a block using white cubes and colored cubes in equal amounts. How many of the white cubes cannot be seen in the picture?
<
|
|
Math_274
|
Six figures were drawn, one on each side of a cube, as shown beside, in different positions. On the side that does not appear beside is this drawing:
<image_1>
What is the figure on the face opposite to it?
|
[
"A",
"B",
"C",
"D",
"E"
] |
B
|
3D Spatial Simulation
|
MathVision
|
Six figures were drawn, one on each side of a cube, as shown beside, in different positions. On the side that does not appear beside is this drawing:
<image_1>
What is the figure on the face opposite to it?
Options:
A
B
C
D
E
|
|
Math_275
|
Six different numbers, chosen from integers 1 to 9 , are written on the faces of a cube, one number per face. The sum of the numbers on each pair of opposite faces is always the same. Which of the following numbers could have been written on the opposite side with the number 8 ?
<image_1>
|
[] |
3
|
3D Spatial Simulation
|
MathVision
|
Six different numbers, chosen from integers 1 to 9 , are written on the faces of a cube, one number per face. The sum of the numbers on each pair of opposite faces is always the same. Which of the following numbers could have been written on the opposite side with the number 8 ?
<
|
|
Math_276
|
Four identical pieces of paper are placed as shown. Michael wants to punch a hole that goes through all four pieces. At which point should Michael punch the hole?
<image_1>
|
[
"A",
"B",
"C",
"D",
"E"
] |
D
|
3D Spatial Simulation
|
MathVision
|
Four identical pieces of paper are placed as shown. Michael wants to punch a hole that goes through all four pieces. At which point should Michael punch the hole?
<
Options:
A
B
C
D
E
|
|
Math_277
|
Hansi sticks 12 cubes together to make this figure. He always puts one drop of glue between two cubes. How many drops of glue does he need?
<image_1>
|
[] |
11
|
3D Spatial Simulation
|
MathVision
|
Hansi sticks 12 cubes together to make this figure. He always puts one drop of glue between two cubes. How many drops of glue does he need?
<
|
|
Math_278
|
A $3 \times 3 \times 3$ cube weighs 810 grams. If we drill three holes through it as shown, each of which is a $1 \times 1 \times 3$ rectangular parallelepiped, the weight of the remaining solid is:
<image_1>
|
[
"$540 \\mathrm{~g}$",
"$570 \\mathrm{~g}$",
"$600 \\mathrm{~g}$",
"$630 \\mathrm{~g}$",
"$660 \\mathrm{~g}$"
] |
C
|
3D Spatial Simulation
|
MathVision
|
A $3 \times 3 \times 3$ cube weighs 810 grams. If we drill three holes through it as shown, each of which is a $1 \times 1 \times 3$ rectangular parallelepiped, the weight of the remaining solid is:
<
Options:
$540 \mathrm{~g}$
$570 \mathrm{~g}$
$600 \mathrm{~g}$
$630 \mathrm{~g}$
$660 \mathrm{~g}$
|
|
Math_279
|
The lengths of the sides of triangle $X Y Z$ are $X Z=\sqrt{55}$, $X Y=8, Y Z=9$. Find the length of the diagonal $X A$ of the rectangular parallelepiped in the figure.
<image_1>
|
[] |
10
|
3D Spatial Simulation
|
MathVision
|
The lengths of the sides of triangle $X Y Z$ are $X Z=\sqrt{55}$, $X Y=8, Y Z=9$. Find the length of the diagonal $X A$ of the rectangular parallelepiped in the figure.
<
|
|
Math_280
|
A die is in the position shown in the picture. It can be rolled along the path of 12 squares as shown. How many times must the die go around the path in order for it to return to its initial position with all faces in the initial positions?
<image_1>
|
[] |
3
|
3D Spatial Simulation
|
MathVision
|
A die is in the position shown in the picture. It can be rolled along the path of 12 squares as shown. How many times must the die go around the path in order for it to return to its initial position with all faces in the initial positions?
<
|
|
Math_281
|
A rectangular piece of paper is wrapped around a cylinder. Then an angled straight cut is made through the points $\mathrm{X}$ and $\mathrm{Y}$ of the cylinder as shown on the left. The lower part of the piece of paper is then unrolled. Which of the following pictures could show the result?
<image_1>
|
[
"A",
"B",
"C",
"D",
"E"
] |
C
|
3D Spatial Simulation
|
MathVision
|
A rectangular piece of paper is wrapped around a cylinder. Then an angled straight cut is made through the points $\mathrm{X}$ and $\mathrm{Y}$ of the cylinder as shown on the left. The lower part of the piece of paper is then unrolled. Which of the following pictures could show the result?
<
Options:
A
B
C
D
E
|
|
Math_282
|
A rectangular piece of paper $A B C D$ with the measurements $4 \mathrm{~cm} \times 16 \mathrm{~cm}$ is folded along the line $\mathrm{MN}$ so that point $C$ coincides with point $A$ as shown. How big is the area of the quadrilateral ANMD'?
<image_1>
|
[
"$28 \\mathrm{~cm}^{2}$",
"$30 \\mathrm{~cm}^{2}$",
"$32 \\mathrm{~cm}^{2}$",
"$48 \\mathrm{~cm}^{2}$",
"$56 \\mathrm{~cm}^{2}$"
] |
C
|
3D Spatial Simulation
|
MathVision
|
A rectangular piece of paper $A B C D$ with the measurements $4 \mathrm{~cm} \times 16 \mathrm{~cm}$ is folded along the line $\mathrm{MN}$ so that point $C$ coincides with point $A$ as shown. How big is the area of the quadrilateral ANMD'?
<
Options:
$28 \mathrm{~cm}^{2}$
$30 \mathrm{~cm}^{2}$
$32 \mathrm{~cm}^{2}$
$48 \mathrm{~cm}^{2}$
$56 \mathrm{~cm}^{2}$
|
|
Math_283
|
Let $a>b$. If the ellipse shown rotates about the $x$-axis an ellipsoid $E_{x}$ with volume $\operatorname{Vol}\left(E_{x}\right)$ is obtained. If it rotates about the $y$-axis an ellipsoid $E_{y}$ with volume $\operatorname{Vol}\left(E_{y}\right)$ is obtained. Which of the following statements is true?
<image_1>
|
[
"$\\mathrm{E}_{\\mathrm{x}}=\\mathrm{E}_{\\mathrm{y}}$ and $\\operatorname{Vol}\\left(\\mathrm{E}_{\\mathrm{x}}\\right)=\\operatorname{Vol}\\left(\\mathrm{E}_{\\mathrm{y}}\\right)$",
"$\\mathrm{E}_{\\mathrm{x}}=\\mathrm{E}_{\\mathrm{y}}$ but $\\operatorname{Vol}\\left(\\mathrm{E}_{\\mathrm{x}}\\right) \\neq \\operatorname{Vol}\\left(\\mathrm{E}_{\\mathrm{y}}\\right)$",
"$\\mathrm{E}_{\\mathrm{x}} \\neq \\mathrm{E}_{\\mathrm{y}}$ and $\\operatorname{Vol}\\left(\\mathrm{E}_{\\mathrm{x}}\\right)>\\operatorname{Vol}\\left(\\mathrm{E}_{\\mathrm{y}}\\right)$",
"$\\mathrm{E}_{\\mathrm{x}} \\neq \\mathrm{E}_{\\mathrm{y}}$ and $\\operatorname{Vol}\\left(\\mathrm{E}_{\\mathrm{x}}\\right)<\\operatorname{Vol}\\left(\\mathrm{E}_{\\mathrm{y}}\\right)$",
"$\\mathrm{E}_{\\mathrm{x}} \\neq \\mathrm{E}_{\\mathrm{y}}$ but $\\operatorname{Vol}\\left(\\mathrm{E}_{\\mathrm{x}}\\right)=\\operatorname{Vol}\\left(\\mathrm{E}_{\\mathrm{y}}\\right)$"
] |
C
|
3D Spatial Simulation
|
MathVision
|
Let $a>b$. If the ellipse shown rotates about the $x$-axis an ellipsoid $E_{x}$ with volume $\operatorname{Vol}\left(E_{x}\right)$ is obtained. If it rotates about the $y$-axis an ellipsoid $E_{y}$ with volume $\operatorname{Vol}\left(E_{y}\right)$ is obtained. Which of the following statements is true?
<
Options:
$\mathrm{E}_{\mathrm{x}}=\mathrm{E}_{\mathrm{y}}$ and $\operatorname{Vol}\left(\mathrm{E}_{\mathrm{x}}\right)=\operatorname{Vol}\left(\mathrm{E}_{\mathrm{y}}\right)$
$\mathrm{E}_{\mathrm{x}}=\mathrm{E}_{\mathrm{y}}$ but $\operatorname{Vol}\left(\mathrm{E}_{\mathrm{x}}\right) \neq \operatorname{Vol}\left(\mathrm{E}_{\mathrm{y}}\right)$
$\mathrm{E}_{\mathrm{x}} \neq \mathrm{E}_{\mathrm{y}}$ and $\operatorname{Vol}\left(\mathrm{E}_{\mathrm{x}}\right)>\operatorname{Vol}\left(\mathrm{E}_{\mathrm{y}}\right)$
$\mathrm{E}_{\mathrm{x}} \neq \mathrm{E}_{\mathrm{y}}$ and $\operatorname{Vol}\left(\mathrm{E}_{\mathrm{x}}\right)<\operatorname{Vol}\left(\mathrm{E}_{\mathrm{y}}\right)$
$\mathrm{E}_{\mathrm{x}} \neq \mathrm{E}_{\mathrm{y}}$ but $\operatorname{Vol}\left(\mathrm{E}_{\mathrm{x}}\right)=\operatorname{Vol}\left(\mathrm{E}_{\mathrm{y}}\right)$
|
|
Math_284
|
If one removes some $1 \times 1 \times 1$ cubes from a $5 \times 5 \times 5$ cube, you obtain the solid shown. It consists of several equally high pillars that are built upon a common base. How many little cubes have been removed?
<image_1>
|
[] |
64
|
3D Spatial Simulation
|
MathVision
|
If one removes some $1 \times 1 \times 1$ cubes from a $5 \times 5 \times 5$ cube, you obtain the solid shown. It consists of several equally high pillars that are built upon a common base. How many little cubes have been removed?
<
|
|
Math_285
|
The curved surfaces of two identical cylinders are cut open along the vertical dotted line, as shown and then stuck together to create the curved surface of one big cylinder. What can be said about the volume of the resulting cylinder compared to the volume of one of the small cylinders?
<image_1>
|
[
"It is 2-times as big.",
"It is 3-times as big.",
"It is $\\pi$-times as big.",
"It is 4-times as big.",
"It is 8-times as big."
] |
D
|
3D Spatial Simulation
|
MathVision
|
The curved surfaces of two identical cylinders are cut open along the vertical dotted line, as shown and then stuck together to create the curved surface of one big cylinder. What can be said about the volume of the resulting cylinder compared to the volume of one of the small cylinders?
<
Options:
It is 2-times as big.
It is 3-times as big.
It is $\pi$-times as big.
It is 4-times as big.
It is 8-times as big.
|
|
Math_286
|
The vertices of a die are numbered 1 to 8, so that the sum of the four numbers on the vertices of each face are the same. The numbers 1, 4 and 6 are already indicated in the picture. Which number is in position $x$?
<image_1>
|
[] |
2
|
3D Spatial Simulation
|
MathVision
|
The vertices of a die are numbered 1 to 8, so that the sum of the four numbers on the vertices of each face are the same. The numbers 1, 4 and 6 are already indicated in the picture. Which number is in position $x$?
<
|
|
Math_287
|
In the diagram a closed polygon can be seen whose vertices are the midpoints of the edges of the die. The interior angles are as usual defined as the angle that two sides of the polygon describe in a common vertex. How big is the sum of all interior angles of the polygon?
<image_1>
|
[
"$720^{\\circ}$",
"$1080^{\\circ}$",
"$1200^{\\circ}$",
"$1440^{\\circ}$",
"$1800^{\\circ}$"
] |
B
|
3D Spatial Simulation
|
MathVision
|
In the diagram a closed polygon can be seen whose vertices are the midpoints of the edges of the die. The interior angles are as usual defined as the angle that two sides of the polygon describe in a common vertex. How big is the sum of all interior angles of the polygon?
<
Options:
$720^{\circ}$
$1080^{\circ}$
$1200^{\circ}$
$1440^{\circ}$
$1800^{\circ}$
|
|
Math_288
|
On a standard die the sum of the numbers on opposite faces is always 7. Two identical standard dice are shown in the figure. How many dots could there be on the non-visible right-hand face (marked with "?")?
<image_1>
|
[
"only 5",
"only 2",
"either 2 or 5",
"either 1, 2, 3 or 5",
"either 2,3 or 5"
] |
A
|
3D Spatial Simulation
|
MathVision
|
On a standard die the sum of the numbers on opposite faces is always 7. Two identical standard dice are shown in the figure. How many dots could there be on the non-visible right-hand face (marked with "?")?
<
Options:
only 5
only 2
either 2 or 5
either 1, 2, 3 or 5
either 2,3 or 5
|
|
Math_289
|
A rectangular piece of paper $A B C D$ is $5 \mathrm{~cm}$ wide and $50 \mathrm{~cm}$ long. The paper is white on one side and grey on the other. Christina folds the strip as shown so that the vertex $B$ coincides with $M$ the midpoint of the edge $C D$. Then she folds it so that the vertex $D$ coincides with $N$ the midpoint of the edge $A B$. How big is the area of the visible white part in the diagram?
<image_1>
|
[
"$50 \\mathrm{~cm}^{2}$",
"$60 \\mathrm{~cm}^{2}$",
"$62.5 \\mathrm{~cm}^{2}$",
"$100 \\mathrm{~cm}^{2}$",
"$125 \\mathrm{~cm}^{2}$"
] |
B
|
3D Spatial Simulation
|
MathVision
|
A rectangular piece of paper $A B C D$ is $5 \mathrm{~cm}$ wide and $50 \mathrm{~cm}$ long. The paper is white on one side and grey on the other. Christina folds the strip as shown so that the vertex $B$ coincides with $M$ the midpoint of the edge $C D$. Then she folds it so that the vertex $D$ coincides with $N$ the midpoint of the edge $A B$. How big is the area of the visible white part in the diagram?
<
Options:
$50 \mathrm{~cm}^{2}$
$60 \mathrm{~cm}^{2}$
$62.5 \mathrm{~cm}^{2}$
$100 \mathrm{~cm}^{2}$
$125 \mathrm{~cm}^{2}$
|
|
Math_290
|
A $4 \times 1 \times 1$ cuboid is made up of 2 white and 2 grey cubes as shown. Which of the following cuboids can be build entirely out of such $4 \times 1 \times 1$ cuboids?
<image_1>
|
[
"A",
"B",
"C",
"D",
"E"
] |
A
|
3D Spatial Simulation
|
MathVision
|
A $4 \times 1 \times 1$ cuboid is made up of 2 white and 2 grey cubes as shown. Which of the following cuboids can be build entirely out of such $4 \times 1 \times 1$ cuboids?
<
Options:
A
B
C
D
E
|
|
Math_291
|
The diagram shows an object made up of 12 dice glued-together. The object is dipped into some colour so that the entire outside is coloured in this new colour. How many of the small dice will have exactly four faces coloured in?
<image_1>
|
[] |
10
|
3D Spatial Simulation
|
MathVision
|
The diagram shows an object made up of 12 dice glued-together. The object is dipped into some colour so that the entire outside is coloured in this new colour. How many of the small dice will have exactly four faces coloured in?
<
|
|
Math_292
|
The faces of the brick have the areas A, B and C as shown. How big is the volume of the brick?
<image_1>
|
[
"$A B C$",
"$\\sqrt{A B C}$",
"$\\sqrt{A B+B C+C A}$",
"$\\sqrt[3]{A B C}$",
"$2(A+B+C)$"
] |
B
|
3D Spatial Simulation
|
MathVision
|
The faces of the brick have the areas A, B and C as shown. How big is the volume of the brick?
<
Options:
$A B C$
$\sqrt{A B C}$
$\sqrt{A B+B C+C A}$
$\sqrt[3]{A B C}$
$2(A+B+C)$
|
|
Math_293
|
Two dice with volumes $V$ and $W$ intersect each other as shown. $90 \%$ of the volume of the die with volume $V$ does not belong to both dice. $85 \%$ of the volume of the die with volume $W$ does not belong to both dice. What is the relationship between the volumes of the two dice?
<image_1>
|
[
"$V=\\frac{2}{3} W$",
"$V=\\frac{3}{2} W$",
"$V=\\frac{85}{90} \\mathrm{~W}$",
"$V=\\frac{90}{85} W$",
"$V=W$"
] |
B
|
3D Spatial Simulation
|
MathVision
|
Two dice with volumes $V$ and $W$ intersect each other as shown. $90 \%$ of the volume of the die with volume $V$ does not belong to both dice. $85 \%$ of the volume of the die with volume $W$ does not belong to both dice. What is the relationship between the volumes of the two dice?
<
Options:
$V=\frac{2}{3} W$
$V=\frac{3}{2} W$
$V=\frac{85}{90} \mathrm{~W}$
$V=\frac{90}{85} W$
$V=W$
|
|
Math_294
|
The five vases shown are filled with water. The filling rate is constant. For which of the five vases does the graph shown describe the height of the water $h$ as a function of the time t?
<image_1>
|
[
"A",
"B",
"C",
"D",
"E"
] |
D
|
3D Spatial Simulation
|
MathVision
|
The five vases shown are filled with water. The filling rate is constant. For which of the five vases does the graph shown describe the height of the water $h$ as a function of the time t?
<
Options:
A
B
C
D
E
|
|
Math_295
|
An octahedron is inscribed into a die with side length 1. The vertices of the octahedron are the midpoints of the faces of the die. How big is the volume of the octahedron?
<image_1>
|
[
"$\\frac{1}{3}$",
"$\\frac{1}{4}$",
"$\\frac{1}{5}$",
"$\\frac{1}{6}$",
"$\\frac{1}{8}$"
] |
D
|
3D Spatial Simulation
|
MathVision
|
An octahedron is inscribed into a die with side length 1. The vertices of the octahedron are the midpoints of the faces of the die. How big is the volume of the octahedron?
<
Options:
$\frac{1}{3}$
$\frac{1}{4}$
$\frac{1}{5}$
$\frac{1}{6}$
$\frac{1}{8}$
|
|
Math_296
|
The faces of the prism shown, are made up of two triangles and three squares. The six vertices are labelled using the numbers 1 to 6. The sum of the four numbers around each square is always the same. The numbers 1 and 5 are given in the diagram. Which number is written at vertex $X$?
<image_1>
|
[] |
2
|
3D Spatial Simulation
|
MathVision
|
The faces of the prism shown, are made up of two triangles and three squares. The six vertices are labelled using the numbers 1 to 6. The sum of the four numbers around each square is always the same. The numbers 1 and 5 are given in the diagram. Which number is written at vertex $X$?
<
|
|
Math_297
|
A cuboid-shaped container that is not filled completely contains $120 \mathrm{~m}^{3}$ of water. The depth of the water is either $2 \mathrm{~m}$ or $3 \mathrm{~m}$ or $5 \mathrm{~m}$, depending on which side the container is actually standing on (drawings not to scale). How big is the volume of the container?
<image_1>
|
[
"$160 \\mathrm{~m}^{3}$",
"$180 \\mathrm{~m}^{3}$",
"$200 \\mathrm{~m}^{3}$",
"$220 \\mathrm{~m}^{3}$",
"$240 \\mathrm{~m}^{3}$"
] |
E
|
3D Spatial Simulation
|
MathVision
|
A cuboid-shaped container that is not filled completely contains $120 \mathrm{~m}^{3}$ of water. The depth of the water is either $2 \mathrm{~m}$ or $3 \mathrm{~m}$ or $5 \mathrm{~m}$, depending on which side the container is actually standing on (drawings not to scale). How big is the volume of the container?
<
Options:
$160 \mathrm{~m}^{3}$
$180 \mathrm{~m}^{3}$
$200 \mathrm{~m}^{3}$
$220 \mathrm{~m}^{3}$
$240 \mathrm{~m}^{3}$
|
|
Math_298
|
The system shown consists of three pulleys that are connected to each other via two ropes. $P$, the end of one rope, is pulled down by $24 \mathrm{~cm}$. By how many centimeters does point $Q$ move upwards?
<image_1>
|
[] |
6
|
3D Spatial Simulation
|
MathVision
|
The system shown consists of three pulleys that are connected to each other via two ropes. $P$, the end of one rope, is pulled down by $24 \mathrm{~cm}$. By how many centimeters does point $Q$ move upwards?
<
|
|
Math_299
|
An ant walked $6 \mathrm{~m}$ every day to go from point $A$ to point $B$ in a straight line. One day Johnny put a straight cylinder of one meter high in that way. Now the ant walks on the same straight line or above it, having to go up and down the cylinder, as shown in the picture. How much does she have to walk now to go from $A$ to $B$?
<image_1>
|
[
"$8 \\mathrm{~m}$",
"$9 \\mathrm{~m}$",
"$6+\\pi \\mathrm{~m}$",
"$12-\\pi \\mathrm{~m}$",
"$10 \\mathrm{~m}$"
] |
A
|
3D Spatial Simulation
|
MathVision
|
An ant walked $6 \mathrm{~m}$ every day to go from point $A$ to point $B$ in a straight line. One day Johnny put a straight cylinder of one meter high in that way. Now the ant walks on the same straight line or above it, having to go up and down the cylinder, as shown in the picture. How much does she have to walk now to go from $A$ to $B$?
<
Options:
$8 \mathrm{~m}$
$9 \mathrm{~m}$
$6+\pi \mathrm{~m}$
$12-\pi \mathrm{~m}$
$10 \mathrm{~m}$
|
|
Math_300
|
Zilda will use six equal cubes and two different rectangular blocks to form the structure beside with eight faces. Before gluing the pieces, she will paint each one entirely and calculated that she will need 18 liters of paint (the color does not matter). How many liters of paint would she use if she painted the whole structure only after gluing the parts?
<image_1>
|
[] |
11.5
|
3D Spatial Simulation
|
MathVision
|
Zilda will use six equal cubes and two different rectangular blocks to form the structure beside with eight faces. Before gluing the pieces, she will paint each one entirely and calculated that she will need 18 liters of paint (the color does not matter). How many liters of paint would she use if she painted the whole structure only after gluing the parts?
<
|
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