problem
stringlengths 33
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values | llama8b_solve_rate
float64 0.06
0.59
|
|---|---|---|---|
In triangle \(ABC\), the height \(AM\) is not less than \(BC\), and the height \(BH\) is not less than \(AC\). Find the angles of triangle \(ABC\).
|
45^
\circ, 45^
\circ, 90^
\circ
|
olympiads
| 0.09375 |
Bronquinha can mow the lawn in her backyard in 3 hours, but if she drinks Gummy fruit juice, she can do it in 2 hours. On a certain day, Bronquinha started mowing the lawn at 10 o'clock, and at a certain point, she drank the Gummy fruit juice, finishing at 12:30 PM. At what time did Bronquinha drink the Gummy fruit juice?
|
11:30 \, \text{AM}
|
olympiads
| 0.359375 |
What is the smallest positive integer $n$ such that $n=x^3+y^3$ for two different positive integer tuples $(x,y)$ ?
|
1729
|
aops_forum
| 0.125 |
Given numbers \( x, y, z \in [0, \pi] \), find the maximum value of the expression
\[
A = \sin(x - y) + \sin(y - z) + \sin(z - x)
\]
|
2
|
olympiads
| 0.09375 |
Determine all sequences of integers $\left(a_{i}\right)_{1 \leqslant i \leqslant 2016}$ such that $0 \leqslant a_{i} \leqslant 2016$ for all $i$, and for all $i, j$, $i + j$ divides $i a_{i} + j a_{j}$.
|
a_n = c \text{ for all } n \in \{1, 2, \ldots, 2016\}, \text{ where } c \text{ is a constant integer in } [0, 2016].
|
olympiads
| 0.125 |
A merchant has two types of tea: Ceylon tea at 10 rubles per pound and Indian tea at 6 rubles per pound. To increase profit, the merchant decides to mix the two types but sell the mixture at the same price of 10 rubles per pound. In what proportion should he mix them to make an additional profit of 3 rubles per pound?
|
1:3
|
olympiads
| 0.1875 |
Suppose that two curves $ C_1 : y\equal{}x^3\minus{}x, C_2 : y\equal{}(x\minus{}a)^3\minus{}(x\minus{}a)$ have two intersection points.
Find the maximum area of the figure bounded by $ C_1,\ C_2$ .
|
\frac{3}{4}
|
aops_forum
| 0.0625 |
There are 500 matches on the table. Two players take turns. On each turn, a player can take $1, 2, 4, 8, \ldots$ (any power of two) matches from the table. The player who cannot make a move loses.
Who wins with optimal play: the starting player or their partner?
|
Player 1 wins
|
olympiads
| 0.140625 |
Given any positive integer $n$, calculate the sum $\sum_{k=0}^{\infty}\left\lfloor\frac{n+2^{k}}{2^{k+1}}\right\rfloor$.
|
n
|
olympiads
| 0.0625 |
If the range of the function \( y = \lg (x^2 - ax + a) \) is \(\mathbf{R}\), what is the range of the real number \( a \)?
|
a \leq 0 \; \text{or} \; a \geq 4.
|
olympiads
| 0.078125 |
In a cycling race on a circular track, three cyclists named Vasya, Petya, and Kolya started simultaneously. Vasya completed each lap 2 seconds faster than Petya, and Petya completed each lap 3 seconds faster than Kolya. When Vasya finished the race, Petya still had one lap to go, and Kolya had two laps remaining. How many laps was the race?
|
6 \text{ laps}
|
olympiads
| 0.09375 |
In an isosceles trapezoid, the length of the midline is 5, and the diagonals are mutually perpendicular. Find the area of the trapezoid.
|
25
|
olympiads
| 0.09375 |
Laura and her grandmother Ana just discovered that last year, their ages were divisible by 8 and that next year, their ages will be divisible by 7. Grandma Ana is not yet 100 years old. What is Laura's age?
|
41
|
olympiads
| 0.09375 |
The following equations apply to the real numbers \(x, y, a\):
\[
\begin{aligned}
x + y &= a \\
x^{3} + y^{3} &= a \\
x^{5} + y^{5} &= a
\end{aligned}
\]
Determine all possible values of \(a\).
|
a = -2, -1, 0, 1, 2
|
olympiads
| 0.0625 |
Given the sequence \(\{a_n\}\) with the sum of the first \(n\) terms \(S_n\) and \(a_1 = \frac{1}{5}\), for any positive integers \(m\) and \(n\), \(a_{m+n} = a_m a_n\). If \(S_n < a\) holds for any \(n \in \mathbf{Z}_{+}\), find the minimum value of the real number \(a\).
|
\frac{1}{4}
|
olympiads
| 0.59375 |
If the line \(5x + 2y - 100 = 0\) has \(d\) points whose \(x\) and \(y\) coordinates are both positive integers, find the value of \(d\).
|
9
|
olympiads
| 0.25 |
Given that the greatest common divisor (GCD) of \(a\) and \(b\) is 4, the least common multiple (LCM) of \(a\) and \(c\) as well as the LCM of \(b\) and \(c\) are both 100, and \(a \leqslant b\), how many sets of natural numbers \(a\), \(b\), and \(c\) satisfy these conditions?
|
9
|
olympiads
| 0.078125 |
For how many numbers $n$ does $2017$ divided by $n$ have a remainder of either $1$ or $2$ ?
|
43
|
aops_forum
| 0.109375 |
Solve for \(x\):
\[ 6 - \left(1 + 4 \cdot 9^{4 - 2 \log_{\sqrt{3}} 3}\right) \log_{7} x = \log_{x} 7. \]
|
\sqrt[5]{7} \; \text{and} \; 7
|
olympiads
| 0.21875 |
From a point on a circle, two chords are drawn with lengths of 10 cm and 12 cm. Find the radius of the circle if the distance from the midpoint of the shorter chord to the longer chord is 4 cm.
|
6.25 \text{ cm}
|
olympiads
| 0.078125 |
The product of all natural numbers from 1 to \( n \) is denoted as \( n! \) (read as "n-factorial"). Which number is greater, \( 200! \) or \( 100^{200} \)?
|
100^{200}
|
olympiads
| 0.515625 |
If \(a\) is a positive multiple of 5, which gives a remainder of 1 when divided by 3, find the smallest possible value of \(a\).
|
10
|
olympiads
| 0.515625 |
Find numbers \(a, b, c\) such that for every positive integer \(n\) the equation
\[
(n+3)^{2}=a \cdot(n+2)^{2}+b \cdot(n+1)^{2}+c \cdot n^{2}
\]
holds true.
|
a = 3, b = -3, c = 1
|
olympiads
| 0.234375 |
Let $\mathbb{Z} / n \mathbb{Z}$ denote the set of integers considered modulo $n$ (hence $\mathbb{Z} / n \mathbb{Z}$ has $n$ elements). Find all positive integers $n$ for which there exists a bijective function $g: \mathbb{Z} / n \mathbb{Z} \rightarrow \mathbb{Z} / n \mathbb{Z}$, such that the 101 functions
$$
g(x), \quad g(x)+x, \quad g(x)+2x, \quad \ldots, \quad g(x)+100x
$$
are all bijections on $\mathbb{Z} / n \mathbb{Z}$.
|
101
|
olympiads
| 0.109375 |
How many ordered triples $(a, b, c)$ of odd positive integers satisfy $a + b + c = 25?$
|
78
|
aops_forum
| 0.265625 |
Calculate the area of the figure bounded by the graphs of the functions:
$$
y = x \sqrt{36 - x^{2}}, \quad y = 0, \quad (0 \leq x \leq 6)
$$
|
72
|
olympiads
| 0.3125 |
The line with equation \( y = -2x + t \) and the parabola with equation \( y = (x - 1)^2 + 1 \) intersect at point \( P \) in the first quadrant. What is the \( y \)-coordinate of \( P \)?
|
5
|
olympiads
| 0.0625 |
An InterCity (IC) train overtakes a freight train on a parallel track and then they pass each other in the opposite direction. The ratio of the speed of the IC train to the speed of the freight train is the same as the ratio of the time taken to overtake to the time taken to pass each other in opposite directions. By how many times is the speed of the IC train greater than the speed of the freight train, assuming both trains maintain constant speeds?
|
1 + \sqrt{2}
|
olympiads
| 0.0625 |
Oleg has 550 rubles and wants to buy an odd number of tulips, making sure that no color is repeated. In the store where Oleg goes, one tulip costs 49 rubles, and there are eleven different shades available. How many different ways are there for Oleg to give flowers to his mother? (The answer should be a compact expression that does not contain summation signs, ellipses, etc.)
|
1024
|
olympiads
| 0.28125 |
Count in two different ways the number of ways to choose a leader, a deputy leader, and a team from $n$ people.
|
\sum_{k=0}^{n} k(k-1)\binom{n}{k}=n(n-1) 2^{n-2}
|
olympiads
| 0.140625 |
Form an n-digit number using the digits 1, 2, and 3, such that each of 1, 2, and 3 appears at least once. Find the number of such n-digit numbers.
|
3^n - 3 \cdot 2^n + 3
|
olympiads
| 0.078125 |
Two circles touch each other internally at point K. Chord \( AB \) of the larger circle touches the smaller circle at point L, with \( A L = 10 \). Find \( B L \) given that \( A K : B K = 2 : 5 \).
|
25
|
olympiads
| 0.140625 |
Find the number which, when multiplied by 5, is equal to its square added to four.
|
x = 4 \ \text{or} \ x = 1
|
olympiads
| 0.234375 |
How many pairs of integers $a$ and $b$ are there such that $a$ and $b$ are between $1$ and $42$ and $a^9 = b^7 \mod 43$ ?
|
42
|
aops_forum
| 0.203125 |
Inside a square with side length \(a\), a semicircle is constructed on each side, using the side as the diameter. Find the area of the rosette enclosed by the arcs of the semicircles.
|
\frac{a^2 (\pi - 2)}{2}
|
olympiads
| 0.28125 |
Find the sum of every even positive integer less than 233 not divisible by 10.
|
10812
|
olympiads
| 0.0625 |
Determinants of nine digits. Nine positive digits can be arranged in the form of a 3rd-order determinant in 9! ways. Find the sum of all such determinants.
|
0
|
olympiads
| 0.3125 |
Find \(\sum_{k=0}^n \frac{(-1)^k \binom{n}{k}}{k^3 + 9k^2 + 26k + 24}\), where \(\binom{n}{k} = \frac{n!}{k!(n-k)!}\).
|
0
|
olympiads
| 0.078125 |
Come up with five different natural numbers whose product is 1000.
|
1, 2, 4, 5, 25
|
olympiads
| 0.0625 |
In a certain country, there are 50 cities. The Ministry of Aviation requires that each pair of cities be connected by a bidirectional flight by exactly one airline, and that it should be possible to travel from any city to any other (possibly with transfers) using the flights of each airline. What is the maximum number of airlines for which this is possible?
|
25
|
olympiads
| 0.234375 |
It is known that $\operatorname{tg} a$ and $\operatorname{tg} 3a$ are integers. Find all possible values of $\operatorname{tg} a$.
|
-1, 0, 1
|
olympiads
| 0.265625 |
For which positive integer values of $k$ can the product of the first $k$ prime numbers be expressed as the sum of two positive cube numbers?
|
k = 1
|
olympiads
| 0.140625 |
In a convex quadrilateral \(ABCD\) with parallel sides \(AD\) and \(BC\), a line \(L\) is drawn parallel to \(AD\) and intersects sides \(AB\) and \(CD\) at points \(M\) and \(N\), respectively. It is known that quadrilaterals \(AMND\) and \(MBNS\) are similar, and the sum of the lengths of sides \(AD\) and \(BC\) is no more than 4. Find the maximum possible length of segment \(MN\).
|
2
|
olympiads
| 0.234375 |
Let \( D \) be a point on side \( AB \) of triangle \( \triangle ABC \) and \( F \) be the intersection of \( CD \) and the median \( AM \). If \( AF = AD \), find the ratio between \( BD \) and \( FM \).
|
2
|
olympiads
| 0.171875 |
In cube \( A B C D A^{\prime} B^{\prime} C^{\prime} D^{\prime} \) with edge length 1, the points \( T \), \( P \), and \( Q \) are the centers of the faces \( A A^{\prime} B^{\prime} B \), \( A^{\prime} B^{\prime} C^{\prime} D^{\prime} \), and \( B B^{\prime} C^{\prime} C \) respectively. Find the distance from point \( P \) to the plane \( A T Q \).
|
\frac{\sqrt{3}}{3}
|
olympiads
| 0.0625 |
Let \( x \) and \( y \) both be positive numbers. Then the minimum value of \( M = \frac{4x}{x + 3y} + \frac{3y}{x} \) is ______.
|
3
|
olympiads
| 0.078125 |
Natural numbers \(a, b, c\) are such that \(\gcd(\operatorname{lcm}(a, b), c) \cdot \operatorname{lcm}(\gcd(a, b), c) = 200\).
What is the maximum value of \(\gcd(\operatorname{lcm}(a, b), c) ?\)
|
10
|
olympiads
| 0.0625 |
We draw \( n \) lines in the plane, with no two of them being parallel and no three of them being concurrent. Into how many regions is the plane divided?
|
\frac{n(n+1)}{2} + 1
|
olympiads
| 0.078125 |
Six distinguishable players are participating in a tennis tournament. Each player plays one match of tennis against every other player. The outcome of each tennis match is a win for one player and a loss for the other players; there are no ties. Suppose that whenever $A$ and $B$ are players in the tournament for which $A$ won (strictly) more matches than $B$ over the course of the tournament, it is also the case that $A$ won the match against $B$ during the tournament. In how many ways could the tournament have gone?
|
720
|
aops_forum
| 0.078125 |
What is the least positive integer by which $2^5 \cdot 3^6 \cdot 4^3 \cdot 5^3 \cdot 6^7$ should be multiplied so that, the product is a perfect square?
|
15
|
aops_forum
| 0.421875 |
In a regular tetrahedron $A B C D$ with an edge length of 1 meter, a bug starts crawling from vertex $A$. At each vertex, the bug randomly chooses one of the three edges and crawls to the end of that edge with equal probability. Let $P_{n}$ be the probability that the bug returns to vertex $A$ after crawling $n$ meters. Find $P_{4}$.
|
\frac{7}{27}
|
olympiads
| 0.09375 |
Let $M$ be the intersection point of the medians $AD$ and $BE$ of a right triangle $ABC$ ( $\angle C=90^\circ$ ). It is known that the circumcircles of triangles $AEM$ and $CDM$ are tangent. Find the angle $\angle BMC.$
|
90^
|
aops_forum
| 0.078125 |
Define $ \{ a_n \}_{n\equal{}1}$ as follows: $ a_1 \equal{} 1989^{1989}; \ a_n, n > 1,$ is the sum of the digits of $ a_{n\minus{}1}$ . What is the value of $ a_5$ ?
|
9
|
aops_forum
| 0.390625 |
The polynomial \( x^n + n x^{n-1} + a_2 x^{n-2} + \cdots + a_0 \) has \( n \) roots whose 16th powers have sum \( n \). Find the roots.
|
-1
|
olympiads
| 0.09375 |
Given that the area of the shaded region is $\frac{32}{\pi}$, and the radius of the smaller circle is three times smaller than the radius of the larger circle. What is the circumference of the smaller circle?
|
4
|
olympiads
| 0.203125 |
Define the length of the interval \(\left[x_{1}, x_{2}\right]\) as \(x_{2}-x_{1}\). If the domain of the function \(y=\left|\log _{2} x\right|\) is \([a, b]\) and the range is \([0,2]\), then the difference between the maximum and minimum lengths of the interval \([a, b]\) is \(\qquad\).
|
3
|
olympiads
| 0.171875 |
If the vector \(\vec{d}=(\vec{a} \cdot \vec{c}) \vec{b}-(\vec{a} \cdot \vec{b}) \vec{c}\), then the angle between \(\vec{a}\) and \(\vec{d}\) is what?
|
90^
|
olympiads
| 0.078125 |
When measuring a part, random errors occur that follow a normal distribution with a parameter $\sigma=10$ mm. Find the probability that the measurement is made with an error not exceeding $15$ mm.
|
0.8664
|
olympiads
| 0.125 |
There is a natural number that is a multiple of both 5 and 7 and leaves a remainder of 1 when divided by 3. What is the smallest natural number that satisfies these conditions?
|
70
|
olympiads
| 0.21875 |
The sum of five positive integers equals 11. In this equation, equal numbers are covered by the same letter, and different numbers are covered by different letters.
Given the equation: $\quad \mathbf{C}+\mathbf{y}+\mathbf{M}+\mathbf{M}+\mathbf{A}=11$.
Can you determine which number is represented by the letter M?
|
1
|
olympiads
| 0.140625 |
Let be given $a < b < c$ and $f(x) =\frac{c(x - a)(x - b)}{(c - a)(c - b)}+\frac{a(x - b)(x - c)}{(a - b)(a -c)}+\frac{b(x -c)(x - a)}{(b - c)(b - a)}$ .
Determine $f(2014)$ .
|
2014
|
aops_forum
| 0.0625 |
rainbow is the name of a bird. this bird has $n$ colors and it's colors in two consecutive days are not equal. there doesn't exist $4$ days in this bird's life like $i,j,k,l$ such that $i<j<k<l$ and the bird has the same color in days $i$ and $k$ and the same color in days $j$ and $l$ different from the colors it has in days $i$ and $k$ . what is the maximum number of days rainbow can live in terms of $n$ ?
|
2n - 1
|
aops_forum
| 0.15625 |
A cube with sides of length 3cm is painted red
and then cut into 3 x 3 x 3 = 27 cubes with sides of length 1cm.
If a denotes the number of small cubes (of 1cm x 1cm x 1cm) that
are not painted at all, b the number painted on one sides, c the
number painted on two sides, and d the number painted on three
sides, determine the value a-b-c+d.
|
-9
|
aops_forum
| 0.078125 |
Let $ABC$ be a triangle with $\angle BAC=40^\circ $ , $O$ be the center of its circumscribed circle and $G$ is its centroid. Point $D$ of line $BC$ is such that $CD=AC$ and $C$ is between $B$ and $D$ . If $AD\parallel OG$ , find $\angle ACB$ .
|
70^
|
aops_forum
| 0.0625 |
Three concentric circles have radii of 1, 2, and 3 units, respectively. Points are chosen on each of these circles such that they are the vertices of an equilateral triangle. What can be the side length of this equilateral triangle?
|
\sqrt{7}
|
olympiads
| 0.078125 |
The volume of a bottle of kvass is 1.5 liters. The first person drank half of the bottle, the second drank a third of what was left after the first, the third drank a quarter of what was left after the previous ones, and so on, with the fourteenth person drinking one-fifteenth of what remained. How much kvass is left in the bottle?
|
0.1 \, \text{liters}
|
olympiads
| 0.140625 |
In a certain class, with any distribution of 200 candies, there will be at least two students who receive the same number of candies (possibly none). What is the minimum number of students in such a class?
|
21
|
olympiads
| 0.0625 |
Factorize: $a+(a+b) x+(a+2 b) x^{2}+(a+3 b) x^{3}+3 b x^{4}+2 b x^{5}+b x^{6}$.
|
(1 + x)(1 + x^2)(a + bx + bx^2 + bx^3)
|
olympiads
| 0.171875 |
Call a number greater than 25 semi-prime if it is the sum of any two distinct prime numbers. What is the maximum number of consecutive semi-prime numbers that can be semi-prime?
|
5
|
olympiads
| 0.09375 |
Non-negative numbers \(a\) and \(b\) satisfy the equations \(a^2 + b^2 = 74\) and \(ab = 35\). What is the value of the expression \(a^2 - 12a + 54\)?
|
19
|
olympiads
| 0.5625 |
Consider the set of points that are inside or within one unit of a rectangular parallelepiped (box) that measures 3 by 4 by 5 units. Given that the volume of this set is $(m + n \pi)/p$ , where $m$ , $n$ , and $p$ are positive integers, and $n$ and $p$ are relatively prime, find $m + n + p$ .
|
505
|
aops_forum
| 0.109375 |
If the graph of the function \(y=f(x)\) is shifted 2 units to the left and 2 units downward, resulting in the graph of the function \( y=2^{x} \), then \( f(x) = \quad .\)
|
f(x) = 2^{x-2} + 2
|
olympiads
| 0.375 |
Into how many equal octagons can an $8 \times 8$ square be divided? (All cuts must follow the grid lines.)
|
2, 4, 8, \text{ or } 16
|
olympiads
| 0.34375 |
There are 33 knights, liars, and dreamers living on an island. Each resident was asked in turn, "How many knights are there among you?" Ten different answers were received, each of which was given by more than one resident. Knights always tell the truth, liars always give an incorrect number that hasn't been mentioned yet, and dreamers always give a number that is one greater than the previous answer. Was the number 40 necessarily mentioned?
|
40 must be mentioned.
|
olympiads
| 0.390625 |
Given the probability density function \( f(x)=\frac{1}{\sqrt{2 \pi}} \mathrm{e}^{-x^{2} / 2},(-\infty<x<\infty) \) of a normally distributed random variable \( X \), find the density function \( g(y) \) of the random variable \( Y = X^2 \).
|
g(y) = \frac{1}{\sqrt{2\pi y}} e^{-y / 2}, \quad 0 < y < \infty
|
olympiads
| 0.46875 |
In isosceles triangles \(ABC \quad (AB=BC)\) and \(A_{1}B_{1}C_{1} \quad (A_{1}B_{1}=B_{1}C_{1})\), the triangles are similar and \(BC : B_{1}C_{1} = 4 : 3\).
Vertex \(B_{1}\) is located on side \(AC\), vertices \(A_{1}\) and \(C_{1}\) are respectively on the extensions of side \(BA\) beyond point \(A\) and side \(CB\) beyond point \(B\), and \(A_{1}C_{1} \perp BC\). Find angle \(B\).
|
arcsin \frac{3}{5}
|
olympiads
| 0.109375 |
100 prisoners are placed in a prison. The warden tells them:
"I will give you an evening to talk with each other, and then I will place you in separate cells, and you will no longer be able to communicate. Occasionally, I will take one of you to a room that contains a lamp (initially turned off). When you leave the room, you can leave the lamp either on or off.
If at some point one of you tells me that all of you have already been in the room and it is true, then I will set you all free. If you are wrong, I will feed you all to the crocodiles. Don't worry about anyone being forgotten - if you remain silent, then everyone will eventually visit the room, and no one's visit to the room will be their last."
Devise a strategy that guarantees the prisoners' release.
|
All prisoners will be set free following this strategy.
|
olympiads
| 0.515625 |
Xiao Wang and Xiao Li have to process the same number of identical parts and start working at the same time. It is known that Xiao Wang processes 15 parts per hour and must rest for 1 hour after every 2 hours of work. Xiao Li works continuously, processing 12 parts per hour. Both of them finish at the exact same time. How many parts did Xiao Wang process?
|
60
|
olympiads
| 0.0625 |
Let \( a \in \mathbb{N}^{*}, n, m \in \mathbb{N} \). Calculate the GCD of \( a^{n}-1 \) and \( a^{m}-1 \).
|
a^{\gcd(n, m)} - 1
|
olympiads
| 0.46875 |
In the math class, $\frac{2}{3}$ of the students had a problem set, and $\frac{4}{5}$ of them brought a calculator. Among those who brought a calculator, the same proportion of students did not have a problem set as among those who did not bring a calculator. What fraction of the students had both a problem set and a calculator?
|
\frac{8}{15}
|
olympiads
| 0.09375 |
Find all non-negative integers \( x \) and \( y \) such that \( 33^x + 31 = 2^y \).
|
\{(0, 5), (1, 6)\}
|
olympiads
| 0.125 |
A $10 \times 10 \times 10$-inch wooden cube is painted red on the outside and then cut into its constituent 1-inch cubes. How many of these small cubes have at least one red face?
|
488
|
olympiads
| 0.375 |
The lengths of the sides of a triangle and its area are respectively four consecutive integers. Find the lengths of the sides of this triangle.
|
3, 4, 5
|
olympiads
| 0.09375 |
Three friends bought a square plot of land and want to divide it into three plots of equal area, as indicated in the figure, because in the corner of the plot marked by \( A \) there is a good water source. How far from the vertex \( C \) of the plot should the boundary points \( M \) and \( N \) indicated in the figure be?
|
\frac{1}{3} \text{ of the side length of the square}
|
olympiads
| 0.109375 |
Three positive integers add to 93 and have a product of 3375. The integers are in the ratio 1: k: k². What are the three integers?
|
3, 15, 75
|
olympiads
| 0.25 |
A particle starts at $(0,0,0)$ in three-dimensional space. Each second, it randomly selects one of the eight lattice points a distance of $\sqrt{3}$ from its current location and moves to that point. What is the probability that, after two seconds, the particle is a distance of $2\sqrt{2}$ from its original location?
|
\frac{3}{8}
|
aops_forum
| 0.0625 |
There are 68 coins in a bag, all having different weights. Describe how to find the heaviest coin and the lightest coin using 100 weighings on a two-pan balance scale.
Note: On a two-pan balance scale, objects are placed on the pans, and it is determined which set of objects is heavier.
|
The total number of weighings used to find the heaviest and lightest coin is 100.
|
olympiads
| 0.359375 |
Given three points: \( A_{1}, B_{2}, C_{3} \). Construct triangle \( ABC \) where the midpoint of side \( BC \) is \( A_{1} \), the foot of the altitude from \( B \) is \( B_{2} \), and the midpoint of the segment between the orthocenter and vertex \( C \) is \( C_{3} \).
|
ext{Solution Valid}
|
olympiads
| 0.234375 |
For which integers $k$ does there exist a function $f: \mathbb{N} \rightarrow \mathbb{Z}$ satisfying $f(2006) = 2007$ and:
$$
f(xy) = f(x) + f(y) + kf(\operatorname{PGCD}(x, y))
$$
for all integers $x$ and $y$?
|
0 \text{ or } -1
|
olympiads
| 0.375 |
Define a list of number with the following properties:
- The first number of the list is a one-digit natural number.
- Each number (since the second) is obtained by adding $9$ to the number before in the list.
- The number $2012$ is in that list.
Find the first number of the list.
|
5
|
aops_forum
| 0.328125 |
Find a graph with chromatic number 3 that does not contain any 3-cliques.
|
C_5 \text{ meets the requirements.
|
olympiads
| 0.109375 |
Calculate the area of the region bounded above by the lines \( x - y + 2 = 0 \) and \( y = 2 - \frac{1}{2}x \), and below by the parabola \( y = x^2 - 2x - 8 \).
|
36.33
|
olympiads
| 0.28125 |
A right angle is divided into an infinite number of square cells with a side length of one. We consider rows of cells parallel to the sides of the angle (vertical and horizontal rows). Is it possible to write a natural number in each cell so that every vertical and every horizontal row of cells contains all natural numbers exactly once?
|
Possible
|
olympiads
| 0.125 |
Mom preserves plums in jars so that the plums from one jar are enough for either 16 turnovers, or 4 cakes, or half a tray of fruit bars.
She has 4 such jars in her pantry and wants to bake one tray of fruit bars and 6 cakes. How many turnovers can she make with the remaining plums?
|
8
|
olympiads
| 0.125 |
Three good friends went fishing together; none of them caught the same number of fish. The next day they say:
- $A$: I caught the most fish, and $C$ caught the least.
- $B$: I caught the most fish, more than $A$ and $C$ combined.
- $C$: I caught the most fish, and $B$ caught only half as many as I did.
Who caught the most fish if exactly 3 out of the 6 statements made by the children are true? Can it be determined who caught the least?
|
B \text{ caught the most fish and } C \text{ caught the least fish.
|
olympiads
| 0.171875 |
Let $[x]$ denote the greatest integer less than or equal to the real number $x$. Given:
$$
a_{k}=\sum_{i=k^{2}}^{(k+1)^{2}-1} \frac{1}{i} \quad (k=1,2, \cdots)
$$
Find:
$$
\sum_{k=1}^{n}\left(\left[\frac{1}{a_{k}}\right]+\left[\frac{1}{a_{k}}+\frac{1}{2}\right]\right) .
$$
|
\frac{n(n+1)}{2}
|
olympiads
| 0.0625 |
In a store, there are 21 white shirts and 21 purple shirts hanging in a row. Find the minimum value of $k$ such that regardless of the initial order of the shirts, it is possible to remove $k$ white shirts and $k$ purple shirts so that the remaining white shirts hang consecutively and the remaining purple shirts also hang consecutively.
|
10
|
olympiads
| 0.125 |
Seven cities are connected in a circle by seven one-way flights. Add (draw arrows indicating) additional one-way flights so that it is possible to travel from any city to any other city with no more than two stops. Try to minimize the number of additional flights.
|
5
|
olympiads
| 0.09375 |
Calculate the limit of the function:
$$\lim _{x \rightarrow 0} \sqrt{x\left(2+\sin \left(\frac{1}{x}\right)\right)+4 \cos x}$$
|
2
|
olympiads
| 0.5 |
Puss in Boots caught pikes: he caught four pikes and another half of the catch. How many pikes did Puss in Boots catch?
|
8
|
olympiads
| 0.4375 |
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