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stringlengths 33
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|
|---|---|---|---|
Given a triangle \( \triangle ABC \) with area 1, and side length \( a \) opposite to angle \( A \), find the minimum value of \( a^2 + \frac{1}{\sin A} \).
|
3
|
olympiads
| 0.078125 |
Given the sets \(A=\left\{x \left\lvert\, \frac{1}{81}<3^{x-1} \leqslant 3\right., x \in \mathbf{Z}\right\}\) and \(B=\left\{x \left\lvert\, \frac{x+2}{x-3}<0\right., x \in \mathbf{N}\right\}\), find the number of elements in the set \(C=\{m \mid m=x y, x \in A, y \in B\}\).
|
7
|
olympiads
| 0.109375 |
In a class, the total numbers of boys and girls are in the ratio $4 : 3$ . On one day it was found that $8$ boys and $14$ girls were absent from the class, and that the number of boys was the square of the number of girls. What is the total number of students in the class?
|
42
|
aops_forum
| 0.140625 |
Two squares are arranged as shown in the figure. If the overlapping part is subtracted from the smaller square, $52\%$ of its area remains. If the overlapping part is subtracted from the larger square, $73\%$ of its area remains. Find the ratio of the side of the smaller square to the side of the larger square.
|
0.75
|
olympiads
| 0.15625 |
From a three-digit number, the sum of its digits was subtracted and the result was 261. Find the second digit of the original number.
|
7
|
olympiads
| 0.4375 |
How many five-digit numbers do not contain the digit 5?
|
52488
|
olympiads
| 0.546875 |
Find all pairs of positive integers \((m, n)\) such that
$$
m+n-\frac{3mn}{m+n}=\frac{2011}{3}.
$$
|
(1144, 377) \text{ or } (377, 1144)
|
olympiads
| 0.125 |
Let $f$ be a non-constant polynomial such that \[ f(x-1) + f(x) + f(x+1) = \frac {f(x)^2}{2013x} \] for all nonzero real numbers $x$ . Find the sum of all possible values of $f(1)$ .
|
6039
|
aops_forum
| 0.1875 |
What series is formed by the differences between the differences ("second differences") of the cubes of consecutive natural numbers?
|
6n + 6
|
olympiads
| 0.125 |
As shown in Figure 1, a circle centered at point \( M \) on the hyperbola \(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1\) (\(a, b > 0\)) is tangent to the \(x\)-axis at one of the foci \( F \) of the hyperbola and intersects the \(y\)-axis at points \( P \) and \( Q \). If the triangle \( \triangle MPQ \) is an equilateral triangle, what is the eccentricity of the hyperbola?
|
\sqrt{3}
|
olympiads
| 0.109375 |
A coin is flipped $20$ times. Let $p$ be the probability that each of the following sequences of flips occur exactly twice:
- one head, two tails, one head
- one head, one tails, two heads.
Given that $p$ can be expressed as $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers, compute $\gcd (m,n)$ .
|
1
|
aops_forum
| 0.59375 |
Given 27 coins, one of which is counterfeit and lighter than the real ones, how can you determine the counterfeit coin using a balance scale in three weighings?
|
Fake coin identified in three weighings
|
olympiads
| 0.203125 |
Fix a point \( O \) on the plane. Any other point \( A \) is uniquely determined by its radius vector \(\overrightarrow{OA}\), which we will denote simply as \(\vec{A}\). A fixed point \( O \) is called the initial point. We will also introduce the symbol \((ABC)\) - the simple ratio of three points \( A, B, C \) that lie on a straight line; points \( A \) and \( B \) are called the base points, and point \( C \) is the dividing point. The symbol \( (ABC) \) is a number defined by the formula:
$$
\overrightarrow{CA} = (ABC) \cdot \overrightarrow{CB}
$$
For two collinear vectors \(\vec{a}\) and \(\vec{b}\), a unique number \(\lambda\) is defined such that \(\vec{a} = \lambda \overrightarrow{b}\). We introduce the notation \(\lambda = \frac{\vec{a}}{\vec{b}}\) (the ratio of collinear vectors). In this case, we can write:
$$
(ABC) = \frac{\overrightarrow{CA}}{\overrightarrow{CB}}
$$
Derive a formula that expresses the vector \(\vec{C}\) in terms of the vectors \(\vec{A}\) and \(\vec{B}\), given that \((ABC) = k\).
|
\vec{C} = \frac{k\vec{B} - \vec{A}}{k - 1}
|
olympiads
| 0.40625 |
Determine all the possible non-negative integer values that are able to satisfy the expression: $\frac{(m^2+mn+n^2)}{(mn-1)}$
if $m$ and $n$ are non-negative integers such that $mn \neq 1$ .
|
0, 4, 7
|
aops_forum
| 0.0625 |
Anya was making pancakes, intending for the five members of her family to get an equal amount. But something went wrong: every fifth pancake Anya couldn't flip; 49% of the flipped pancakes got burnt; and $\frac{1}{6}$ of the edible pancakes Anya dropped on the floor. What percentage of the intended number of pancakes was Anya able to offer her family?
|
34 \%
|
olympiads
| 0.28125 |
The germination rate of seeds is $90\%$. What is the probability that 5 out of 7 sown seeds will germinate?
|
0.124
|
olympiads
| 0.21875 |
The sequence of integers is defined recursively: \(a_{0}=1, a_{2}=2, a_{n+2}=a_{n}+\left(a_{n+1}\right)^{2}\). What is the remainder when \(a_{14}\) is divided by 7?
|
5
|
olympiads
| 0.0625 |
Let $f (x)$ be a function mapping real numbers to real numbers. Given that $f (f (x)) =\frac{1}{3x}$ , and $f (2) =\frac19$ , find $ f\left(\frac{1}{6}\right)$ .
|
3
|
aops_forum
| 0.265625 |
A square is cut into two identical rectangles. Each of these two rectangles has a perimeter of $24 \text{ cm}$. What is the area of the original square?
|
64 \, \text{cm}^2
|
olympiads
| 0.15625 |
The following figure shows a part of the graph \( y = \sin 3x^{\circ} \). What is the \( x \)-coordinate of \( P \)?
|
60
|
olympiads
| 0.171875 |
Determine all integer numbers \( x \) and \( y \) such that
\[ \frac{1}{x} + \frac{1}{y} = \frac{1}{19} \]
|
(38, 38), (380, 20), (-342, 18), (20, 380), (18, -342)
|
olympiads
| 0.28125 |
Find the value of \( \frac{1}{\sin ^{3} \frac{x}{2} \cos ^{3} \frac{x}{2}} - 6 \cos ^{-1} x \) if it equals \( \tan^{3} \frac{x}{2} + \cot^{3} \frac{x}{2} \).
|
x = \frac{\pi}{4} + k\pi, \quad k \in \mathbb{Z}
|
olympiads
| 0.078125 |
In a plane, there are \( n \) gears arranged in such a way that the first gear meshes its teeth with the second, the second with the third, and so on. Finally, the last gear meshes with the first. Can the gears in such a system rotate?
|
при n чётном могут, при n нечётном не могут
|
olympiads
| 0.09375 |
Represent the number 2015 as a sum of three integers, one of which is prime, the second is divisible by 3, and the third belongs to the interval (400; 500) and is not divisible by 3 (it is sufficient to provide only the answer, no need to show the solution).
|
2015 = 7 + 1605 + 403
|
olympiads
| 0.078125 |
Jerry the mouse decided to give Tom the cat a birthday cake in the shape of an 8 x 8 square. He placed fish in three pieces marked with the letter "P," sausage in two pieces marked with the letter "K," and in one piece he added both fish and sausage, but did not mark this piece (all other pieces are without filling). He also informed Tom that in any 6 x 6 square, there are at least 2 pieces with fish, and in any 3 x 3 square, there is at most one piece with sausage.
What is the minimum number of pieces Tom needs to eat to ensure he gets a piece with both fish and sausage?
|
5
|
olympiads
| 0.171875 |
Determine all the positive integers with more than one digit, all distinct, such that the sum of its digits is equal to the product of its digits.
|
\{123, 132, 213, 231, 312, 321\}
|
aops_forum
| 0.078125 |
Through points \(A(0 ; 14)\) and \(B(0 ; 4)\), two parallel lines are drawn. The first line, passing through point \(A\), intersects the hyperbola \(y = \frac{1}{x}\) at points \(K\) and \(L\). The second line, passing through point \(B\), intersects the hyperbola \(y = \frac{1}{x}\) at points \(M\) and \(N\).
What is the value of \(\frac{A L - A K}{B N - B M}?\)
|
3.5
|
olympiads
| 0.078125 |
The inscribed circle of the right triangle \(ABC\) (with the right angle at \(C\)) touches the sides \(AB\), \(BC\), and \(CA\) at the points \(C_1\), \(A_1\), and \(B_1\) respectively. The altitudes of the triangle \(A_1B_1C_1\) intersect at the point \(D\). Find the distance between the points \(C\) and \(D\), given that the lengths of the legs of the triangle \(ABC\) are 3 and 4.
|
1
|
olympiads
| 0.171875 |
March 19, 2017 was a Sunday. Determine the day of the week on September 1, 2017.
|
Friday
|
olympiads
| 0.375 |
\[9^{\cos x}=9^{\sin x} \cdot 3^{\frac{2}{\cos x}}\]
Domain: \(\cos x \neq 0\).
|
x_1 = \pi n \quad \text{and} \quad x_2 = -\frac{\pi}{4} + \pi k, \quad n, k \in \mathbb{Z}
|
olympiads
| 0.0625 |
Circle $\omega_1$ and $\omega_2$ have centers $(0,6)$ and $(20,0)$ , respectively. Both circles have radius $30$ , and intersect at two points $X$ and $Y$ . The line through $X$ and $Y$ can be written in the form $y = mx+b$ . Compute $100m+b$ .
|
303
|
aops_forum
| 0.125 |
Ivan Ivanovich approached a water source with two empty canisters, one with a capacity of 10 liters and the other with a capacity of 8 liters. Water flowed from the source in two streams - one stronger, the other weaker. Ivan Ivanovich placed the canisters under the streams simultaneously, and when half of the smaller canister was filled, he switched the canisters. To his surprise, the canisters filled up at the same time. By what factor is the stronger stream's flow rate greater than the weaker stream's flow rate?
|
2
|
olympiads
| 0.078125 |
If \( f(x) \) where \( x \in \mathbf{R} \) is an even function with a period of 2, and when \( x \in [0,1] \), \( f(x) = x^{\frac{1}{1998}} \), then order \( f\left(\frac{98}{19}\right) \), \( f\left(\frac{101}{17}\right) \), and \( f\left(\frac{104}{15}\right) \) from smallest to largest.
|
f\left( \frac{101}{17} \right), f\left( \frac{98}{19} \right), f\left( \frac{104}{15} \right)
|
olympiads
| 0.40625 |
\( m \) and \( n \) are two different positive integers. Find the common complex root of the equations
\[
\begin{aligned}
x^{m+1} - x^{n} + 1 = 0 \\
x^{n+1} - x^{m} + 1 = 0
\end{aligned}
\]
|
x = \frac{1}{2} \pm \frac{\sqrt{3}}{2} i
|
olympiads
| 0.078125 |
Let $a \triangle b$ and $a \nabla b$ denote the minimum and maximum of the numbers $a$ and $b$, respectively. For example, $3 \triangle 4=3$ and $3 \nabla 4=4$. Determine the number of possible values of $6 \triangle [4 \nabla (x \triangle 5)]$ for different natural numbers $x$.
|
2
|
olympiads
| 0.484375 |
In space, there are \(n\) planes. Any 2 planes intersect in a line, any 3 planes intersect at a point, and no 4 planes intersect at a single point. How many non-overlapping regions do these \(n\) planes divide the space into?
|
\frac{n^3 + 5n + 6}{6}
|
olympiads
| 0.109375 |
Person A and person B start simultaneously from points A and B, respectively, and travel towards each other at constant speeds. They meet after 8 hours. If each person's speed is increased by 2 kilometers per hour, they would meet after 6 hours at a point 3 kilometers from the midpoint of AB. Given that person A travels faster than person B, determine the original speed of person A in kilometers per hour.
|
6.5
|
olympiads
| 0.15625 |
Write the decomposition of vector \( x \) in terms of vectors \( p, q, r \):
\[ x = \{ 3, -3, 4 \} \]
\[ p = \{ 1, 0, 2 \} \]
\[ q = \{ 0, 1, 1 \} \]
\[ r = \{ 2, -1, 4 \} \]
|
\mathbf{x} = \mathbf{p} - 2\mathbf{q} + \mathbf{r}
|
olympiads
| 0.203125 |
Several stones are placed in 5 piles. It is known that:
- There are six times as many stones in the fifth pile as in the third.
- There are twice as many stones in the second pile as in the third and fifth piles combined.
- There are three times fewer stones in the first pile than in the fifth pile, and 10 fewer stones than in the fourth pile.
- There are half as many stones in the fourth pile as in the second pile.
How many stones are there in total in these five piles?
|
60
|
olympiads
| 0.1875 |
Find all positive integers \( n \) such that every positive integer with \( n \) digits, one of which is 7 and the others are 1, is prime.
|
n = 1 \text{ and } 2
|
olympiads
| 0.3125 |
Let \(\ell\) be the line through \((0,0)\) and tangent to the curve \(y=x^{3}+x+16\). Find the slope of \(\ell\).
|
13
|
olympiads
| 0.109375 |
Points \( D \) and \( E \) are located on the diagonals \( A B_{1} \) and \( C A_{1} \) of the lateral faces of the prism \( A B C A_{1} B_{1} C_{1} \) such that the lines \( D E \) and \( B C_{1} \) are parallel. Find the ratio of the segments \( D E \) and \( B C_{1} \).
|
1:2
|
olympiads
| 0.21875 |
Create a problem using the following brief notes and solve it:
- The first quantity is 8 more
- The second quantity
- The third quantity is 3 times more
The total sum of these quantities is 108.
|
First class: 28 km, Second class: 20 km, Third class: 60 km
|
olympiads
| 0.1875 |
A student passed 31 exams over 5 years. Each subsequent year, he passed more exams than the previous year, and in his fifth year, he passed three times as many exams as in his first year. How many exams did he pass in his fourth year?
|
8
|
olympiads
| 0.078125 |
Dima calculated the factorials of all natural numbers from 80 to 99, found the reciprocals of them, and printed the resulting decimal fractions on 20 endless ribbons (for example, the last ribbon had the number \(\frac{1}{99!}=0. \underbrace{00\ldots00}_{155 \text{ zeros}} 10715 \ldots \) printed on it). Sasha wants to cut a piece from one ribbon that contains \(N\) consecutive digits without any decimal points. For what largest \(N\) can he do this so that Dima cannot determine from this piece which ribbon Sasha spoiled?
|
155
|
olympiads
| 0.09375 |
Five people stand in a line, each wearing a different hat numbered $1, 2, 3, 4, 5$. Each person can only see the hats of the people in front of them. Xiao Wang cannot see any hats; Xiao Zha can only see hat $4$; Xiao Tian does not see hat $3$, but can see hat $1$; Xiao Yan can see three hats, but not hat $3$; Xiao Wei can see hat $3$ and hat $2$. What number hat is Xiao Wei wearing?
|
5
|
olympiads
| 0.203125 |
Find the volume of an oblique triangular prism with a base that is an equilateral triangle with side length $a$, given that the lateral edge of the prism is equal to the side of the base and is inclined at an angle of $60^{\circ}$ to the plane of the base.
|
\frac{3 a^3}{8}
|
olympiads
| 0.34375 |
Compare the magnitude of the following three numbers.
$$ \sqrt[3]{\frac{25}{3}} ,\, \sqrt[3]{\frac{1148}{135}} ,\, \frac{\sqrt[3]{25}}{3} + \sqrt[3]{\frac{6}{5}}
$$
|
a < c < b
|
aops_forum
| 0.09375 |
Suppose there are 9 boxes arranged in a $3 \times 3$ square and an ample supply of black and white stones. Two players, A and B, play a stone-placing game where each player takes turns placing 3 stones in three boxes within the same row or column. The stones placed can be of any color. No mixed-color stones are allowed in any box. If a player places a white stone in a box containing black stones, one white stone and one black stone are simultaneously removed from the box. The game ends when the central and four corner boxes each have one black stone, and all other boxes are empty. At some stage in the game, if there are $x$ boxes each containing one black stone while all other boxes are empty, find all possible values of $x$.
|
3, 4, 5, 6, 9
|
olympiads
| 0.0625 |
One fair die is rolled; let \( a \) denote the number that comes up. We then roll \( a \) dice; let the sum of the resulting \( a \) numbers be \( b \). Finally, we roll \( b \) dice, and let \( c \) be the sum of the resulting \( b \) numbers. Find the expected (average) value of \( c \).
|
\frac{343}{8}
|
olympiads
| 0.359375 |
Determine all real values of \(x\) for which \(16^x - \frac{5}{2} \left(2^{2x+1}\right) + 4 = 0\).
|
0 \text{ and } 1
|
olympiads
| 0.5 |
There is a pile containing 1992 stones. Two players play the following game: on each turn, each player can take from the pile any number of stones that is a divisor of the number of stones taken by the opponent on their previous turn. The first player, on their first turn, can take any number of stones, but not all at once. The player who takes the last stone wins. Who wins with optimal play?
|
First player wins
|
olympiads
| 0.09375 |
Find the number of triples \((a, b, c)\) of positive integers such that \(a + ab + abc = 11\).
|
3
|
olympiads
| 0.3125 |
Tyler rolls two $ 4025 $ sided fair dice with sides numbered $ 1, \dots , 4025 $ . Given that the number on the first die is greater than or equal to the number on the second die, what is the probability that the number on the first die is less than or equal to $ 2012 $ ?
|
\frac{1006}{4025}
|
aops_forum
| 0.203125 |
Let $a$ , $b$ , and $c$ be real numbers such that $0\le a,b,c\le 5$ and $2a + b + c = 10$ . Over all possible values of $a$ , $b$ , and $c$ , determine the maximum possible value of $a + 2b + 3c$ .
|
25
|
aops_forum
| 0.0625 |
Find all numbers $ n $ for which there exist three (not necessarily distinct) roots of unity of order $ n $ whose sum is $
1. $
|
Any even positive integer
|
aops_forum
| 0.0625 |
Let \( S = \{1, 2, \cdots, 2009\} \). \( A \) is a 3-element subset of \( S \) such that all elements in \( A \) form an arithmetic sequence. How many such 3-element subsets \( A \) are there?
|
1008016
|
olympiads
| 0.125 |
Given the sequence \(a_{1}=1\), \(a_{2}=\frac{5}{3}\), and the recurrence relation \(a_{n+2}=\frac{5}{3}a_{n+1}-\frac{2}{3}a_{n}\) for \( n \in \mathbf{N}^{*}\), find the general term \(a_{n}\) of the sequence \(\{a_{n}\}\).
|
a_{n}= 2 - \frac{3}{2} \left(\frac{2}{3}\right)^n
|
olympiads
| 0.140625 |
A and B start from points A and B simultaneously, moving towards each other and meet at point C. If A starts 2 minutes earlier, then their meeting point is 42 meters away from point C. Given that A's speed is \( a \) meters per minute, B's speed is \( b \) meters per minute, where \( a \) and \( b \) are integers, \( a > b \), and \( b \) is not a factor of \( a \). What is the value of \( a \)?
|
21
|
olympiads
| 0.125 |
Let $n \in \mathbb{N}_{\geq 2}.$ For any real numbers $a_1,a_2,...,a_n$ denote $S_0=1$ and for $1 \leq k \leq n$ denote $$ S_k=\sum_{1 \leq i_1 < i_2 < ... <i_k \leq n}a_{i_1}a_{i_2}...a_{i_k} $$ Find the number of $n-$ tuples $(a_1,a_2,...a_n)$ such that $$ (S_n-S_{n-2}+S_{n-4}-...)^2+(S_{n-1}-S_{n-3}+S_{n-5}-...)^2=2^nS_n. $$
|
2^{n-1}
|
aops_forum
| 0.0625 |
13 children sat around a circular table and agreed that boys would lie to girls but tell the truth to each other. Conversely, girls would lie to boys but tell the truth to each other. One of the children told their right-hand neighbor, "The majority of us are boys." That child then told their right-hand neighbor, "The majority of us are girls," and the third child told their right-hand neighbor, "The majority of us are boys," and so on, continuing in this pattern until the last child told the first child, "The majority of us are boys." How many boys were at the table?
|
7
|
olympiads
| 0.265625 |
What is the sum of the coefficients of the expansion \((x + 2y - 1)^{6}\)?
|
64
|
olympiads
| 0.5 |
Two candidates, A and B, are participating in an election. Candidate A receives $n$ votes, and candidate B receives $m$ votes, where $n > m$. What is the probability that during the counting process, the cumulative vote count for candidate A always remains ahead of that for candidate B?
|
\frac{n-m}{n+m}
|
olympiads
| 0.0625 |
In a right-angled triangle \( ABC \) with \(\angle ACB = 90^\circ\), \( AC = 6 \), and \( BC = 4 \), a point \( D \) is marked on the line \( BC \) such that \( CD > BD \) and \(\angle ADC = 45^\circ\). A point \( E \) is marked on the line \( AD \) such that the perimeter of the triangle \( CBE \) is the smallest possible. Then, a point \( F \) is marked on the line \( DC \) such that the perimeter of the triangle \( AFE \) is the smallest possible. Find \( CF \).
|
3.6
|
olympiads
| 0.0625 |
In an acute-angled triangle \(ABC\), the angle \(\angle ACB = 75^\circ\), and the height drawn from the vertex of this angle is equal to 1.
Find the radius of the circumscribed circle if it is known that the perimeter of the triangle \(ABC\) is \(4 + \sqrt{6} - \sqrt{2}\).
|
\sqrt{6} - \sqrt{2}
|
olympiads
| 0.0625 |
Jia selected 13 different three-digit numbers. Yi then selected some of the three-digit numbers chosen by Jia. If Yi can use the four basic arithmetic operations to make the final result fall within the interval (3, 4), Yi wins; otherwise, Jia wins. Who has a winning strategy?
|
Yi has a winning strategy
|
olympiads
| 0.109375 |
The population of a city increases annually by $1 / 50$ of the current number of inhabitants. In how many years will the population triple?
|
55 \text{ years}
|
olympiads
| 0.0625 |
Two cubes of different sizes are glued together to form the three-dimensional shape shown in the figure. The four vertices of the smaller cube's glued face are each at one of the trisection points of the edges of the larger cube's glued face. If the edge length of the larger cube is 3, what is the surface area of this three-dimensional shape?
|
49
|
olympiads
| 0.0625 |
Let $0=x_0<x_1<\cdots<x_n=1$ .Find the largest real number $ C$ such that for any positive integer $ n $ , we have $$ \sum_{k=1}^n x^2_k (x_k - x_{k-1})>C $$
|
\frac{1}{3}
|
aops_forum
| 0.140625 |
There are 30 logs, with lengths of either 3 or 4 meters, and their total length is 100 meters. What is the minimum number of cuts needed to cut all these logs into 1-meter pieces? (Each cut is made on exactly one log).
|
70
|
olympiads
| 0.140625 |
Calculate the limit of the function:
$$\lim _{x \rightarrow 1} \frac{3-\sqrt{10-x}}{\sin 3 \pi x}$$
|
-\frac{1}{18\pi}
|
olympiads
| 0.078125 |
Let $T$ denote the $15$ -element set $\{10a+b:a,b\in\mathbb{Z},1\le a<b\le 6\}$ . Let $S$ be a subset of $T$ in which all six digits $1,2,\ldots ,6$ appear and in which no three elements together use all these six digits. Determine the largest possible size of $S$ .
|
9
|
aops_forum
| 0.0625 |
Find the angles at which the parabola \( y = x^2 - x \) intersects the x-axis.
|
\frac{3\pi}{4} \text{ and } \frac{\pi}{4}
|
olympiads
| 0.109375 |
In an arithmetic sequence, the third, fifth, and eleventh terms are distinct and form a geometric sequence. If the fourth term of the arithmetic sequence is 6, what is its 2007th term?
|
6015
|
olympiads
| 0.125 |
How many positive integers less than 2011 are multiples of 3 or 4, but not of 5?
|
804
|
olympiads
| 0.25 |
Find all functions from the positive real numbers to the positive real numbers such that:
\[ f(x) f(y) = 2 f(x + y f(x)) \]
|
f(x) = 2
|
olympiads
| 0.109375 |
Determine all odd natural numbers of the form
$$
\frac{p+q}{p-q}
$$
where \( p > q \) are prime numbers.
|
5
|
olympiads
| 0.1875 |
The USAMO is a $6$ question test. For each question, you submit a positive integer number $p$ of pages on which your solution is written. On the $i$ th page of this question, you write the fraction $i/p$ to denote that this is the $i$ th page out of $p$ for this question. When you turned in your submissions for the $2017$ USAMO, the bored proctor computed the sum of the fractions for all of the pages which you turned in. Surprisingly, this number turned out to be $2017$ . How many pages did you turn in?
|
4028
|
aops_forum
| 0.0625 |
Determine all positive integers $n$ such that the equation $x^{3}+y^{3}+z^{3}=n x^{2} y^{2} z^{2}$ has positive integer solutions $(x, y, z)$.
|
n = 1 \text{ or } n = 3
|
olympiads
| 0.078125 |
For real numbers $a$ and $b$ , define $$ f(a,b) = \sqrt{a^2+b^2+26a+86b+2018}. $$ Find the smallest possible value of the expression $$ f(a, b) + f (a,-b) + f(-a, b) + f (-a, -b). $$
|
4 \sqrt{2018}
|
aops_forum
| 0.125 |
Given $\sin \theta + \cos \theta = \frac{7}{5}$ and $\tan \theta < 1$. Find $\sin \theta$.
|
\frac{3}{5}
|
olympiads
| 0.109375 |
In the plane quadrilateral $\mathrm{ABCD}$, given $\mathrm{AB}=1, \mathrm{BC}=4, \mathrm{CD}=2, \mathrm{DA}=3$, find the value of $\overrightarrow{\mathrm{AC}} \cdot \overrightarrow{\mathrm{BD}}$.
|
10
|
olympiads
| 0.0625 |
A right rectangular prism has all edges with integer lengths. The prism has faces with areas of 30 and 13 square units. What is its volume?
|
30
|
olympiads
| 0.109375 |
Calculate the limit of the function:
\[
\lim _{x \rightarrow 1}\left(\frac{\sin (x-1)}{x-1}\right)^{\frac{\sin (x-1)}{x-1-\sin (x-1)}}
\]
|
e^{-1}
|
olympiads
| 0.109375 |
Given a function \( f(x) \) that satisfies the condition
\[ f(xy + 1) = f(x)f(y) - f(y) - x + 2, \]
what is \( f(2017) \) if it is known that \( f(0) = 1 \)?
|
2018
|
olympiads
| 0.21875 |
Find the smallest odd integer $ k$ such that: for every $ 3\minus{}$ degree polynomials $ f$ with integer coefficients, if there exist $ k$ integer $ n$ such that $ |f(n)|$ is a prime number, then $ f$ is irreducible in $ \mathbb{Z}[n]$ .
|
5
|
aops_forum
| 0.25 |
What values can be taken by the greatest common divisor of the natural numbers \( m \) and \( n \) if it is known that increasing the number \( m \) by 6 increases it by 9 times?
|
3 \text{ or } 6
|
olympiads
| 0.1875 |
Compute the number of ways to rearrange nine white cubes and eighteen black cubes into a $3\times 3\times 3$ cube such that each $1\times1\times3$ row or column contains exactly one white cube. Note that rotations are considered $\textit{distinct}$ .
|
60480
|
aops_forum
| 0.0625 |
In a square with a side length of 3, $n$ points are randomly selected. Among these points, $m$ points are at a distance less than 1 from the vertices of the square. The experimental value of the circumference of $\pi$ obtained using random simulation is $\qquad$.
|
\frac{9m}{n}
|
olympiads
| 0.078125 |
Let \( n \) be a positive integer. If \( n^m \geqslant m^n \) holds for all positive integers \( m \), what is \( n \)?
|
3
|
olympiads
| 0.09375 |
Biologists have discovered an amazing type of amoeba. Each amoeba divides into two exactly every minute. A biologist places an amoeba in a test tube, and exactly one hour later, the test tube is completely filled with amoebas. How much time will it take for the entire test tube to be filled with amoebas if two amoebas are placed in it?
|
59
|
olympiads
| 0.328125 |
Let $AD,BF$ and ${CE}$ be the altitudes of $\vartriangle ABC$ . A line passing through ${D}$ and parallel to ${AB}$ intersects the line ${EF}$ at the point ${G}$ . If ${H}$ is the orthocenter of $\vartriangle ABC$ , find the angle ${\angle{CGH}}$ .
|
\angle CGH = 90^\circ
|
aops_forum
| 0.359375 |
A bag contains 20 balls: 9 white balls, 5 red balls, and 6 black balls. If 10 balls are randomly selected from the bag, what is the probability that the number of white balls is between 3 and 7 inclusive, the number of red balls is between 2 and 5 inclusive, and the number of black balls is between 1 and 3 inclusive?
|
\frac{7}{92378}
|
olympiads
| 0.078125 |
Graph on the coordinate plane the set of points that satisfy the inequality \(\sqrt{x^{2} - 2xy} > \sqrt{1 - y^{2}}\).
|
These conditions can be graphed on the coordinate plane to show the region that satisfies the given inequality.
|
olympiads
| 0.0625 |
Use the arithmetic operators to form an expression using the numbers 1, 4, 7, and 7, such that the result equals 24.
|
(7 - 4) \times (1 + 7) = 24
|
olympiads
| 0.140625 |
In $\triangle ABC$, $\angle ABC = 60^\circ$, $O$ and $H$ are the circumcenter and orthocenter of $\triangle ABC$ respectively. Points $D$ and $E$ lie on sides $BC$ and $AB$ respectively, such that $BD = BH$, $BE = BO$, and $BO = a$. Find the area of $\triangle BDE$.
|
\frac{\sqrt{3}}{4} a^2
|
olympiads
| 0.328125 |
Consider the quadratic polynomial \( P(x) = a x^2 + b x + c \) which has distinct positive roots. Vasya wrote four numbers on the board: the roots of \( P(x) \) and the roots of the polynomial \( Q(x) = c x^2 + b x + a \). What is the smallest integer value that the sum of these four numbers can have?
|
5
|
olympiads
| 0.078125 |
Alice and Bob are playing a game with a string of 2015 beads. The player whose turn it is cuts the string between two beads. The other player decides which of the two pieces will be used for the rest of the game, and the other piece is discarded.
Alice starts, and they take turns cutting the string. The player who has to make a move when only one bead is left loses the game since no more cuts can be made.
Who has a winning strategy?
|
Bob
|
olympiads
| 0.140625 |
A piece is placed on a chessboard. Two players take turns moving the piece to an adjacent square along the side. It is not allowed to place the piece on a square where it has already been. The player who cannot make a move on their turn loses. Who wins with optimal play?
|
First
|
olympiads
| 0.078125 |
In a class exam, there are 3 multiple-choice questions, each with 4 possible answer choices. It is known that a group of students' answers have the property $P$: any two students' answers have at most 1 question with the same answer. Furthermore, if one more student is added to this group, regardless of how this student answers, property $P$ no longer holds. What is the minimum number of students in this group?
|
8
|
olympiads
| 0.078125 |
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