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stringlengths 33
2.6k
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stringlengths 1
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stringclasses 2
values | llama8b_solve_rate
float64 0.06
0.59
|
|---|---|---|---|
Determine the number of integers \( 2 \leq n \leq 2016 \) such that \( n^n - 1 \) is divisible by \( 2, 3, 5, \) and \( 7 \).
|
9
|
olympiads
| 0.234375 |
Given positive integers \(a\) and \(b\) that satisfy the equation \(\frac{1}{a} - \frac{1}{b} = \frac{1}{2018}\), determine the number of pairs \((a, b)\) of positive integers that satisfy this equation.
|
4
|
olympiads
| 0.0625 |
Let $a$ , $b$ , $c$ , $d$ , $e$ be positive reals satisfying \begin{align*} a + b &= c a + b + c &= d a + b + c + d &= e.\end{align*} If $c=5$ , compute $a+b+c+d+e$ .
|
40
|
aops_forum
| 0.296875 |
Vanya and Petya decided to mow the lawn for a football field. Vanya alone could do it in 5 hours, and Petya alone in 6 hours. They started at 11 o'clock and stopped mowing at the same time when their parents called them, but Petya took an hour for lunch, and Vanya took a two-hour lunch break. One-tenth of the lawn remained unmown. At what time did their parents call the boys?
|
15 \text{ часов}
|
olympiads
| 0.125 |
What is the maximum number of closed intervals that can be given on the number line such that among any three intervals, there are two that intersect, but the intersection of any four intervals is empty?
|
6
|
olympiads
| 0.15625 |
The external angles located on the hypotenuse of a right triangle are in the ratio of 9:11. What are the angles of the triangle?
|
58.5^
\circ, 31.5^
\circ, 90^
\circ
|
olympiads
| 0.109375 |
Calculate the area of the parallelogram formed by the vectors \( a \) and \( b \).
\[ a = 2p - 3q \]
\[ b = 5p + q \]
\[ |p| = 2 \]
\[ |q| = 3 \]
\[ \angle(p, q) = \frac{\pi}{2} \]
|
102
|
olympiads
| 0.3125 |
For some positive integers \( m \) and \( n \), \( 2^{m} - 2^{n} = 1792 \). Determine the value of \( m^{2} + n^{2} \).
|
185
|
olympiads
| 0.078125 |
Each diagonal of a convex pentagon divides the pentagon into a quadrilateral and a triangle of unit area. Find the area of the pentagon.
|
5
|
olympiads
| 0.125 |
A crystal, while forming, uniformly increases its mass. Observing the formation of two crystals, it was noted that the first one increased its mass in 3 months by the same amount as the second one in 7 months. However, at the end of a year, it turned out that the first crystal increased its initial mass by 4%, and the second by 5%. Find the ratio of the initial masses of these crystals.
|
35:12
|
olympiads
| 0.46875 |
The cost of 60 copies of the first volume and 75 copies of the second volume is 2700 rubles. In reality, the total payment for all these books was only 2370 rubles because a discount was applied: 15% off the first volume and 10% off the second volume. Find the original price of these books.
|
20
|
olympiads
| 0.078125 |
Let \( H \) be the orthocenter of isosceles \(\triangle ABC\). If the base \( BC \) remains unchanged, as the distance from the vertex \( A \) to the base \( BC \) decreases, does the product \( S_{\triangle ABC} \cdot S_{\triangle HBC} \) decrease, increase, or remain unchanged?
|
unchanged
|
olympiads
| 0.171875 |
Screws are sold in packs of $10$ and $12$ . Harry and Sam independently go to the hardware store, and by coincidence each of them buys exactly $k$ screws. However, the number of packs of screws Harry buys is different than the number of packs Sam buys. What is the smallest possible value of $k$ ?
|
60
|
aops_forum
| 0.34375 |
If \( a, b, \) and \( c \) are positive real numbers such that
\[ ab + a + b = bc + b + c = ca + c + a = 35, \]
find the value of \( (a+1)(b+1)(c+1) \).
|
216
|
olympiads
| 0.3125 |
A dibrominated alkane contains 85.1% bromine. Determine the formula of the dibromoalkane.
|
\mathrm{C}_2\mathrm{H}_4\mathrm{Br}_2
|
olympiads
| 0.09375 |
$ABCD$ is circumscribed in a circle $k$ , such that $[ACB]=s$ , $[ACD]=t$ , $s<t$ . Determine the smallest value of $\frac{4s^2+t^2}{5st}$ and when this minimum is achieved.
|
\frac{4}{5}
|
aops_forum
| 0.15625 |
For real numbers $B,M,$ and $T,$ we have $B^2+M^2+T^2 =2022$ and $B+M+T =72.$ Compute the sum of the minimum and maximum possible values of $T.$
|
48
|
aops_forum
| 0.109375 |
Mark the number 1 at the endpoints of a diameter of a circle. Divide each resulting semicircular arc into equal parts and label the division points with the sum of the numbers at their endpoints. Repeat this process, dividing each arc into equal parts and labeling new points with the sum of the numbers at their endpoints, for $n$ steps. Find the sum of all the numbers marked on the circle after $n$ steps.
|
2 \times 3^n
|
olympiads
| 0.078125 |
The output of a factory increased 4 times over four years. By what average percentage did the output increase each year compared to the previous year?
|
41.4\%
|
olympiads
| 0.171875 |
Bryan Ai has the following 8 numbers written from left to right on a sheet of paper: $$ \textbf{1 4 1 2 0 7 0 8} $$ Now in each of the 7 gaps between adjacent numbers, Bryan Ai wants to place one of ` $+$ ', ` $-$ ', or ` $\times$ ' inside that gap.
Now, Bryan Ai wonders, if he picks a random placement out of the $3^7$ possible placements, what's the expected value of the expression (order of operations apply)?
|
0
|
aops_forum
| 0.1875 |
Initially, there are three piles on the table, consisting of 100, 101, and 102 stones, respectively. Ilya and Kostya are playing the following game. On each turn, a player can take one stone from any pile, except the pile from which they took a stone on their previous turn (during their first turn, each player can take a stone from any pile). The players take turns, and Ilya starts. The player who cannot make a move loses. Which player can guarantee a win, regardless of how the opponent plays?
|
Ilya wins
|
olympiads
| 0.109375 |
Write the canonical equations of the straight line given as the intersection of two planes (using their general equations):
$$
\left\{\begin{array}{l}
2x + 3y + z - 8 = 0 \\
x - 2y - 2z + 1 = 0
\end{array}\right.
$$
|
\frac{x-3}{-4}=\frac{y}{5}=\frac{z-2}{-7}
|
olympiads
| 0.28125 |
The perimeter of a rhombus is 2 m, and the lengths of its diagonals are in the ratio 3:4. Find the area of the rhombus.
|
0.24 \, \text{m}^2
|
olympiads
| 0.53125 |
The sequence \(\left\{\alpha_{n}\right\}\) is an arithmetic sequence with a common difference \(\beta\). The sequence \(\left\{\sin \alpha_{n}\right\}\) is a geometric sequence with a common ratio \(q\). Given that \(\alpha_{1}, \beta \in \mathbf{R}\), find \(q = \).
|
1 \text{ or } -1
|
olympiads
| 0.140625 |
Four pirates, Jack, Jimmy, Tom, and Sanji, share a total of 280 gold coins. Jack says: "I received 11 fewer coins than Jimmy, 15 more than Tom, and 20 fewer than Sanji." How many coins did Sanji receive?
|
86
|
olympiads
| 0.34375 |
Let $a$ and $b$ be positive integers. When $a^{2}+b^{2}$ is divided by $a+b,$ the quotient is $q$ and the remainder is $r.$ Find all pairs $(a,b)$ such that $q^{2}+r=1977$ .
|
(a, b) = (50, 37) or (a, b) = (37, 50)
|
aops_forum
| 0.078125 |
Let \( p, q, r \) be the three sides of triangle \( PQR \). If \( p^{4} + q^{4} + r^{4} = 2r^{2}(p^{2} + q^{2}) \), find \( a \), where \( a = \cos^{2} R \) and \( R \) denotes the angle opposite \( r \).
|
\frac{1}{2}
|
olympiads
| 0.0625 |
1994 students line up in a row. They count off from 1 to 3 from the front to the back of the line. Then, they count off from 1 to 4 from the back to the front of the line. How many people in the row count off as 1 in both counts?
|
166
|
olympiads
| 0.328125 |
On the board, there is an integer written. Two players, $A$ and $B$, take turns playing, starting with $A$. Each player, on their turn, replaces the existing number with the result of performing one of these two operations: subtracting 1 or dividing by 2, as long as the result is a positive integer. The player who reaches the number 1 wins. Determine, with reasoning, the smallest even number that requires $A$ to play at least 2015 times to win (excluding $B$'s turns).
|
6 \cdot 2^{1007} - 4
|
olympiads
| 0.078125 |
In honor of a holiday, 1% of the soldiers in the regiment received new uniforms. The soldiers are arranged in a rectangle such that those in new uniforms are in at least 30% of the columns and at least 40% of the rows. What is the smallest possible number of soldiers in the regiment?
|
1200
|
olympiads
| 0.09375 |
It is known that, for all positive integers $k,$
\[1^{2}+2^{2}+3^{2}+\cdots+k^{2}=\frac{k(k+1)(2k+1)}{6}. \]Find the smallest positive integer $k$ such that $1^{2}+2^{2}+3^{2}+\cdots+k^{2}$ is a multiple of $200.$
|
112
|
aops_forum
| 0.078125 |
I place \( n \) pairs of socks (thus \( 2n \) socks) in a line such that the left sock is to the right of the right sock for each pair. How many different ways can I place my socks in this manner?
|
\frac{(2n)!}{2^n}
|
olympiads
| 0.296875 |
Xiaoming and Xiaohong are running on a 600-meter circular track. They start from the same point at the same time, running in opposite directions. The time interval between their first and second encounters is 50 seconds. Given that Xiaohong's speed is 2 meters per second slower than Xiaoming's, find Xiaoming's speed in meters per second.
|
7
|
olympiads
| 0.375 |
As shown in the figure, in the right triangle \( \triangle ABC \), the hypotenuse \( AB = 12 \ \text{cm} \). From vertex \( N \) of the square \( ACNM \), draw \( NN_1 \perp \) the extension of \( BA \) meeting it at \( N_1 \). From vertex \( P \) of the square \( BPQC \), draw \( PP_1 \perp \) the extension of \( AB \) meeting it at \( P_1 \). Then \( NN_1 + PP_1 = \quad \text{cm} \).
|
12 ext{ cm}
|
olympiads
| 0.140625 |
For each real number \( x \), \( f(x) \) is defined to be the minimum of the values of \( 2x + 3 \), \( 3x - 2 \), and \( 25 - x \). What is the maximum value of \( f(x) \)?
|
\frac{53}{3}
|
olympiads
| 0.109375 |
Vasily Ivanovich took a loan of 750,000 rubles from a major bank to purchase a car. According to the loan terms, he will repay the initial loan amount plus 37,500 rubles in interest after 5 months. Determine the annual interest rate of the loan (in %), assuming a year consists of 360 days and each month has 30 days.
|
12
|
olympiads
| 0.421875 |
You are repeatedly flipping a fair coin. What is the expected number of flips until the first time that your previous 2012 flips are 'HTHT...HT'?
|
\frac{2^{2014} - 4}{3}
|
olympiads
| 0.125 |
Represent the polynomial \((x+1)(x+2)(x+3)(x+4) + 1\) as a perfect square.
|
\left(x^2 + 5x + 5\right)^2
|
olympiads
| 0.234375 |
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