problem stringlengths 33 2.6k | answer stringlengths 1 359 | source stringclasses 2
values | llama8b_solve_rate float64 0.06 0.59 |
|---|---|---|---|
Determine the sides of a triangle, if they are expressed as integers forming an arithmetic progression, and if the perimeter of the triangle is 15. | 5, 5, 5, \quad 4, 5, 6, \quad 3, 5, 7 | olympiads | 0.0625 |
Given complex numbers \( x \) and \( y \), find the maximum value of \(\frac{|3x+4y|}{\sqrt{|x|^{2} + |y|^{2} + \left|x^{2}+y^{2}\right|}}\). | \frac{5\sqrt{2}}{2} | olympiads | 0.1875 |
Given \(0 < \alpha < \pi, \pi < \beta < 2\pi\), if the equation
\[
\cos (x + \alpha) + \sin (x + \beta) + \sqrt{2} \cos x = 0
\]
holds for any \(x \in \mathbf{R}\), find the values of \(\alpha\) and \(\beta\). | \alpha = \frac{3 \pi}{4}, \beta = \frac{7 \pi}{4} | olympiads | 0.234375 |
The integers \(a\) and \(b\) are coprime (i.e., their greatest common divisor is 1), with \(a > b\). Compare the numbers:
\[
m = \left\lfloor\frac{a}{b}\right\rfloor + \left\lfloor\frac{2a}{b}\right\rfloor + \cdots + \left\lfloor\frac{(b-1)a}{b}\right\rfloor
\]
and
\[
n = \left\lfloor\frac{b}{a}\right\rfloor + \l... | \frac{(a-1)(b-1)}{2} | olympiads | 0.0625 |
Some small circles with the same radius are arranged according to a specific pattern: there are 6 small circles in the first figure, 10 small circles in the second figure, 16 small circles in the third figure, 24 small circles in the fourth figure, and so on. According to this pattern, how many small circles are there ... | 46 | olympiads | 0.234375 |
Find the remainder of the Euclidean division of \( 2018^{2019^{2020}} \) by 11. | 5 | olympiads | 0.09375 |
Three friends met at a café: sculptor Belov, violinist Chernov, and artist Ryzhov. "It's remarkable that each of us has white, black, and red hair, but none of us has hair color matching our last name," said the black-haired person. "You're right," said Belov. What is the artist's hair color? | black | olympiads | 0.15625 |
The younger sister is 18 years old this year, and the elder sister is 26 years old this year. When the sum of their ages was 20 years, the elder sister was $\qquad$ years old. | 14 | olympiads | 0.375 |
Vovochka approached a game machine, on the screen of which the number 0 was displayed. The rules of the game stated: "The screen shows the number of points. If you insert a 1-ruble coin, the number of points increases by 1. If you insert a 2-ruble coin, the number of points doubles. If you score 50 points, the machine ... | 11 \text{ rubles} | olympiads | 0.328125 |
Suppose that \( n \) is a positive integer and that \( a \) is the integer equal to \( \frac{10^{2n} - 1}{3(10^n + 1)} \). If the sum of the digits of \( a \) is 567, what is the value of \( n \)? | 189 | olympiads | 0.09375 |
The age difference between two sisters is 4 years. If the cube of the age of the first sister is reduced by the cube of the age of the second sister, the result is 988. How old are each of the sisters? | 7 \, \text{years} \, \text{(younger sister)}, \, 11 \, \text{years} \, \text{(older sister)} | olympiads | 0.453125 |
Let $[x]$ represent the greatest integer not exceeding the real number $x$. For a given positive integer $n$, calculate $\sum_{k=0}^{\left[\frac{n}{2}\right]} \mathrm{C}_{n}^{2 k} 3^{n-2 k}$. | 2 \cdot 4^{n-1} + 2^{n-1} | olympiads | 0.171875 |
A truncated pyramid has a square base with a side length of 4 units, and every lateral edge is also 4 units. The side length of the top face is 2 units. What is the greatest possible distance between any two vertices of the truncated pyramid? | \sqrt{32} | olympiads | 0.21875 |
The sum of the reciprocals of three positive integers is equal to one. What are these numbers? | (2, 4, 4), (2, 3, 6), (3, 3, 3) | olympiads | 0.0625 |
In the square \(ABCD\), points \(F\) and \(E\) are the midpoints of sides \(AB\) and \(CD\), respectively. Point \(E\) is connected to vertices \(A\) and \(B\), and point \(F\) is connected to vertices \(C\) and \(D\), as shown in the figure. Determine the area of the rhombus \(FGHE\) formed in the center, given that t... | 4 | olympiads | 0.28125 |
A college math class has $N$ teaching assistants. It takes the teaching assistants $5$ hours to grade homework assignments. One day, another teaching assistant joins them in grading and all homework assignments take only $4$ hours to grade. Assuming everyone did the same amount of work, compute the number of hour... | 20 | aops_forum | 0.5625 |
December has 31 days. If December 1st of a certain year is a Monday, what day of the week is December 19th of that year? (Use numbers 1 to 7 to represent Monday to Sunday, respectively) | 5 | olympiads | 0.3125 |
Find the least positive integer $n$ such that \[ \frac 1{\sin 45^\circ\sin 46^\circ}+\frac 1{\sin 47^\circ\sin 48^\circ}+\cdots+\frac 1{\sin 133^\circ\sin 134^\circ}=\frac 1{\sin n^\circ}. \] | 1 | aops_forum | 0.296875 |
How many real numbers \( a \) are there such that the equation \( x^2 + ax + 6a = 0 \) has only integer solutions? | 10 | olympiads | 0.09375 |
Find the intersection point of the line and the plane.
\[\frac{x-3}{1}=\frac{y-1}{-1}=\frac{z+5}{0}\]
\[x+7 y+3 z+11=0\] | (4, 0, -5) | olympiads | 0.21875 |
How many numbers from 1 to 1000 (inclusive) cannot be represented as the difference between the squares of two integers? | 250 | olympiads | 0.265625 |
Find the number of real zeros of \( x^{3} - x^{2} - x + 2 \). | 1 | olympiads | 0.40625 |
A bullet enters a plank with a thickness of 10 cm at a speed of \(200 \, \text{m/s}\), and exits the plank, having penetrated it, at a speed of 50 m/s. Find the duration of the bullet's passage through the plank, assuming the resistance of the plank to the bullet's motion is proportional to the square of its speed. | 0.001 \ \text{seconds} | olympiads | 0.125 |
Alice and Bob take turns playing the following game. The number 2 is written on the board. If the integer \( n \) is written on the board, the player whose turn it is erases it and writes an integer of the form \( n + p \), where \( p \) is a prime divisor of \( n \). The first player to write a number greater than \( ... | Alice | olympiads | 0.484375 |
Find the relationship between the coefficients of the equation \(a x^{2}+b x+c=0\) if the ratio of the roots is 2. | 2b^2 = 9ac | olympiads | 0.5625 |
Find the pairs of real numbers $(a,b)$ such that the biggest of the numbers $x=b^2-\frac{a-1}{2}$ and $y=a^2+\frac{b+1}{2}$ is less than or equal to $\frac{7}{16}$ | (a, b) = \left(\frac{1}{4}, -\frac{1}{4}\right) | aops_forum | 0.0625 |
Find such proper fractions for which, by decreasing both the numerator and the denominator by 1, the fraction becomes $\frac{1}{2}$. | \frac{2}{3}, \frac{3}{5}, \frac{4}{7}, \frac{5}{9}, \ldots | olympiads | 0.21875 |
For which positive value of \( p \) do the equations \( 3x^{2} - 4px + 9 = 0 \) and \( x^{2} - 2px + 5 = 0 \) have a common root? | p = 3 | olympiads | 0.171875 |
\(\frac{2 \sin ^{2} 4 \alpha - 1}{2 \operatorname{ctg}\left(\frac{\pi}{4} + 4 \alpha\right) \cos ^{2}\left(\frac{5 \pi}{4} - 4 \alpha\right)} = -1\). | -1 | olympiads | 0.078125 |
Kevin the Koala eats $1$ leaf on the first day of its life, $3$ leaves on the second, $5$ on the third, and in general eats $2n-1$ leaves on the $n$ th day. What is the smallest positive integer $n>1$ such that the total number of leaves Kevin has eaten his entire $n$ -day life is a perfect sixth power? | 8 | aops_forum | 0.28125 |
Find all real numbers $a$ for which there exists a function $f$ defined on the set of all real numbers which takes as its values all real numbers exactly once and satisfies the equality $$ f(f(x))=x^2f(x)+ax^2 $$ for all real $x$ . | a = 0 | aops_forum | 0.515625 |
The *cross* of a convex $n$ -gon is the quadratic mean of the lengths between the possible pairs of vertices. For example, the cross of a $3 \times 4$ rectangle is $\sqrt{ \dfrac{3^2 + 3^2 + 4^2 + 4^2 + 5^2 + 5^2}{6} } = \dfrac{5}{3} \sqrt{6}$ .
Suppose $S$ is a dodecagon ( $12$ -gon) inscribed in a unit circle.... | \frac{2\sqrt{66}}{11} | aops_forum | 0.078125 |
Maria and Joe are jogging towards each other on a long straight path. Joe is running at $10$ mph and Maria at $8$ mph. When they are $3$ miles apart, a fly begins to fly back and forth between them at a constant rate of $15$ mph, turning around instantaneously whenever it reachers one of the runners. How far, i... | \frac{5}{2} | aops_forum | 0.078125 |
Consider the graph on $1000$ vertices $v_1, v_2, ...v_{1000}$ such that for all $1 \le i < j \le 1000$ , $v_i$ is connected to $v_j$ if and only if $i$ divides $j$ . Determine the minimum number of colors that must be used to color the vertices of this graph such that no two vertices sharing an edge are th... | 10 | aops_forum | 0.0625 |
In the sequence $00$ , $01$ , $02$ , $03$ , $\cdots$ , $99$ the terms are rearranged so that each term is obtained from the previous one by increasing or decreasing one of its digits by $1$ (for example, $29$ can be followed by $19$ , $39$ , or $28$ , but not by $30$ or $20$ ). What is the maximal numb... | 50 | aops_forum | 0.15625 |
Reflect a square in sequence across each of its four sides. How many different transformations result from the sequence of four reflections? | 4 | olympiads | 0.125 |
The nine horizontal and nine vertical lines on an $8\times8$ checkerboard form $r$ rectangles, of which $s$ are squares. The number $s/r$ can be written in the form $m/n,$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$ | 125 | aops_forum | 0.40625 |
Given the equation:
\[
\frac{\left(\tan \alpha + \cos^{-1} \alpha\right)(\cos \alpha - \cot \alpha)}{(\cos \alpha + \cot \alpha)\left(\tan \alpha - \cos^{-1} \alpha\right)} = 1
\] | 1 | olympiads | 0.078125 |
Kelvin the frog lives in a pond with an infinite number of lily pads, numbered \(0,1,2,3,\) and so forth. Kelvin starts on lily pad 0 and jumps from pad to pad in the following manner: when on lily pad \(i\), he will jump to lily pad \((i+k)\) with probability \(\frac{1}{2^{k}}\) for \(k>0\). What is the probability th... | \frac{1}{2} | olympiads | 0.109375 |
A hemisphere is placed on top of a sphere with a radius of 2017. A second hemisphere is then placed on top of the first hemisphere, and a third hemisphere is placed on top of the second hemisphere. All the centers are collinear, and the three hemispheres have empty interiors and negligible width. What is the maximum he... | 6051 | olympiads | 0.203125 |
Five notebooks - blue, grey, brown, red, and yellow - were stacked in a certain order. They were then laid out on a table in two stacks, one by one, from top to bottom. As a result, the first stack contained, from bottom to top: red at the bottom, yellow in the middle, and grey on top. The second stack contained, from ... | \text{brown, red, yellow, gray, blue} | olympiads | 0.15625 |
Suppose \( x^{3} - a x^{2} + b x - 48 \) is a polynomial with three positive roots \( p, q \), and \( r \) such that \( p < q < r \). What is the minimum possible value of \( \frac{1}{p} + \frac{2}{q} + \frac{3}{r} \)? | \frac{3}{2} | olympiads | 0.078125 |
For a given rational number \(\frac{a}{b}\), construct an electrical circuit using unit resistances such that its total resistance equals \(\frac{a}{b}\). How can such a circuit be obtained by decomposing an \(a \times b\) rectangle into squares from problem \(60598\)? | Unit resistances can effectively model any rational \frac{a}{b} as shown using continued fractions. | olympiads | 0.421875 |
Find the length of the arc of the curve \( y = \arcsin(\sqrt{x}) - \sqrt{x - x^2} \) from \( x_1 = 0 \) to \( x_2 = 1 \). | L=2 | olympiads | 0.109375 |
Find the maximum value of $\lambda ,$ such that for $\forall x,y\in\mathbb R_+$ satisfying $2x-y=2x^3+y^3,x^2+\lambda y^2\leqslant 1.$ | \frac{\sqrt{5} + 1}{2} | aops_forum | 0.09375 |
How many pounds of grain need to be milled so that after paying for the work - 10% of the milling, exactly 100 pounds of flour remain?
There are no losses in the milling process. | 111 \frac{1}{9} \text{ pounds} | olympiads | 0.453125 |
Zhenya lost 20% of his weight in the spring, then gained 30% in the summer, lost 20% again in the fall, and gained 10% in the winter. Did Zhenya gain or lose weight over the year? | Zhenya has lost weight. | olympiads | 0.0625 |
A rectangle \(ADEC\) is circumscribed around a right triangle \(ABC\) with legs \(AB = 5\) and \(BC = 6\). What is the area of \(ADEC\)? | 30 | olympiads | 0.09375 |
In $\triangle ABC$, $\overrightarrow{GA} + \overrightarrow{GB} + \overrightarrow{GC} = 0$ and $\overrightarrow{GA} \cdot \overrightarrow{GB} = 0$. Find $\frac{(\tan A + \tan B) \tan C}{\tan A \tan B} = \quad$. | \frac{1}{2} | olympiads | 0.0625 |
An empty glass jar is three times lighter than the same jar filled to the brim with milk. Given that the volume occupied by the glass is ten times less than the volume occupied by the milk, compare the densities of the glass and the milk. The density of a material is defined as the ratio of the material's mass to its v... | \rho_{M} = 0.2 \rho_{CT} | olympiads | 0.125 |
Let $f$ be a monic cubic polynomial such that the sum of the coefficients of $f$ is $5$ and such that the sum of the roots of $f$ is $1$ . Find the absolute value of the sum of the cubes of the roots of $f$ . | 14 | aops_forum | 0.25 |
Point \( M \) lies on the side of a regular hexagon with side length 12. Find the sum of the distances from point \( M \) to the lines containing the remaining sides of the hexagon. | 36\sqrt{3} | olympiads | 0.328125 |
An equilateral triangle of side $n$ has been divided into little equilateral triangles of side $1$ in the usual way. We draw a path over the segments of this triangulation, in such a way that it visits exactly once each one of the $\frac{(n+1)(n+2)}{2}$ vertices. What is the minimum number of times the path can c... | n | aops_forum | 0.09375 |
Find all polynomials $ P(x)$ with real coefficients such that for every positive integer $ n$ there exists a rational $ r$ with $ P(r)=n$ . | P(x) \in \mathbb{Q}[x] | aops_forum | 0.078125 |
For what positive integers $n$ and $k$ do there exits integers $a_1,a_2,\dots,a_n$ and $b_1,b_2,\dots,b_k$ such that the products $a_ib_j$ ( $1\le i\le n,1\le j\le k$ ) give pairwise different residues modulo $nk$ ? | \gcd(n, k) = 1 | aops_forum | 0.328125 |
Of the statements "number $a$ is divisible by 2", "number $a$ is divisible by 4", "number $a$ is divisible by 12", and "number $a$ is divisible by 24", three are true and one is false. Which one is false? | Number a is divisible by 24 | olympiads | 0.203125 |
Adam has RM2010 in his bank account. He donates RM10 to charity every day. His first donation is on Monday. On what day will he donate his last RM10? | Saturday | aops_forum | 0.234375 |
Define a sequence $\{a_n\}$ by\[a_0=\frac{1}{2},\ a_{n+1}=a_{n}+\frac{a_{n}^2}{2012}, (n=0,\ 1,\ 2,\ \cdots),\] find integer $k$ such that $a_{k}<1<a_{k+1}.$ (September 29, 2012, Hohhot) | k = 2012 | aops_forum | 0.109375 |
Express \( x^{3}+y^{3}+z^{3}-3xyz \) in terms of elementary symmetric polynomials. | \sigma_1 (\sigma_1^2 - 3\sigma_2) | olympiads | 0.453125 |
Calculate the area of the parallelogram formed by the vectors $a$ and $b$.
$a = 4p + q$
$b = p - q$
$|p| = 7$
$|q| = 2$
$\angle(p, q) = \frac{\pi}{4}$ | 35\sqrt{2} | olympiads | 0.25 |
The base of an isosceles triangle is 8 cm, and the legs are 12 cm each. Find the length of the segment connecting the points where the angle bisectors of the angles at the base intersect the legs of the triangle. | 4.8 \ \text{cm} | olympiads | 0.09375 |
To build a wall, Worker A needs 10 hours if working alone, and Worker B needs 11 hours if working alone. If they work together, due to poor coordination, they lay 48 fewer bricks per hour in total. To complete the work as quickly as possible, the boss asked both Worker A and Worker B to work together, and they finished... | 1980 | olympiads | 0.453125 |
Solve the equation \( n + S(n) = 1964 \), where \( S(n) \) is the sum of the digits of the number \( n \). | 1945 | olympiads | 0.140625 |
There are 8 cards; one side of each card is blank, and the other side has the letters И, Я, Л, З, Г, О, О, О written on them. The cards are placed on the table with the blank side up, shuffled, and then turned over one by one in sequence. What is the probability that the letters will appear in the order to form the wor... | \frac{1}{6720} | olympiads | 0.21875 |
Find the intersection point of the line and the plane.
$\frac{x-5}{-2}=\frac{y-2}{0}=\frac{z+4}{-1}$
$2 x-5 y+4 z+24=0$ | (3,
2,
-5) | olympiads | 0.40625 |
Write the decomposition of the vector $x$ in terms of the vectors $p, q, r$:
$x=\{6 ; 12 ;-1\}$
$p=\{1 ; 3 ; 0\}$
$q=\{2 ;-1 ; 1\}$
$r=\{0 ;-1 ; 2\}$ | x = 4p + q - r | olympiads | 0.15625 |
In how many ways can a black and a white rook be placed on a chessboard so that they do not attack each other? | 3136 | olympiads | 0.0625 |
\[A=3\sum_{m=1}^{n^2}(\frac12-\{\sqrt{m}\})\]
where $n$ is an positive integer. Find the largest $k$ such that $n^k$ divides $[A]$ . | 1 | aops_forum | 0.125 |
How many natural numbers not exceeding 500 are not divisible by 2, 3, or 5? | 134 | olympiads | 0.375 |
Given a cube \( A B C D A_1 B_1 C_1 D_1 \), in what ratio does a point \( E \), which lies on edge \( B_1 C_1 \) and belongs to the plane passing through vertex \( A \) and the centers \( K \) and \( H \) of the faces \( A_1 B_1 C_1 D_1 \) and \( B_1 C_1 C B \), divide the edge \( B_1 C_1 \)? | 2:1 | olympiads | 0.078125 |
In two weeks three cows eat all the grass on two hectares of land, together with all the grass that regrows there during the two weeks. In four weeks, two cows eat all the grass on two hectares of land, together with all the grass that regrows there during the four weeks.
How many cows will eat all the grass on six he... | 3 | aops_forum | 0.078125 |
Let \( c_{n}=11 \ldots 1 \) be a number in which the decimal representation contains \( n \) ones. Then \( c_{n+1}=10 \cdot c_{n}+1 \). Therefore:
\[ c_{n+1}^{2}=100 \cdot c_{n}^{2} + 22 \ldots 2 \cdot 10 + 1 \]
For example,
\( c_{2}^{2}=11^{2}=(10 \cdot 1+1)^{2}=100+2 \cdot 10+1=121 \),
\( c_{3}^{2} = 111^{2} = 1... | \sqrt{c} = 11111111 | olympiads | 0.0625 |
For natural numbers $x$ and $y$ , let $(x,y)$ denote the greatest common divisor of $x$ and $y$ . How many pairs of natural numbers $x$ and $y$ with $x \le y$ satisfy the equation $xy = x + y + (x, y)$ ? | 3 | aops_forum | 0.1875 |
Let $p$ be an odd prime number. For positive integer $k$ satisfying $1\le k\le p-1$ , the number of divisors of $kp+1$ between $k$ and $p$ exclusive is $a_k$ . Find the value of $a_1+a_2+\ldots + a_{p-1}$ . | p-2 | aops_forum | 0.09375 |
Suppose that \(\log _{a} 125 = \log _{5} 3\) and \(\log _{b} 16 = \log _{4} 7\). Find the value of \(a^{\left(\log _{5} 3\right)^{2}} - b^{\left(\log _{4} 7\right)^{2}}\). | -22 | olympiads | 0.0625 |
Let there be two circles, \(K_1\) and \(K_2\), with centers \(F_1\) and \(F_2\). Denote the distance of an arbitrary point from the two circles as \(d_1\) and \(d_2\), respectively.
What is the locus of points for which the ratio \(d_1 : d_2\) is constant? | \text{The geometric loci are in general two quartic curves, which simplify to an ellipse and a hyperbola when $\lambda=1$.} | olympiads | 0.109375 |
Let $n$ be a fixed positive integer. How many ways are there to write $n$ as a sum of positive integers,
\[n = a_1 + a_2 + \cdots a_k\]
with $k$ an arbitrary positive integer and $a_1 \le a_2 \le \cdots \le a_k \le a_1 + 1$ ? For example, with $n = 4$ , there are four ways: $4$ , $2 + 2$ , $1 + 1 + 2$ , $1 ... | n | aops_forum | 0.0625 |
Plot the set of points on the plane \((x, y)\) that satisfy the equation \( |3x| + |4y| + |48 - 3x - 4y| = 48 \), and find the area of the resulting figure. | 96 | olympiads | 0.15625 |
There is a rectangular table. Two players take turns placing one euro coin on it such that the coins do not overlap. The player who cannot make a move loses. Who will win with optimal play? | При правильной игре выигрывает первый игрок. | olympiads | 0.171875 |
Calculate the area of the figure bounded by the lines given by the equations:
$$
\begin{aligned}
& \left\{\begin{array}{l}
x=3(t-\sin t) \\
y=3(1-\cos t)
\end{array}\right. \\
& y=3(0<x<6 \pi, y \geq 3)
\end{aligned}
$$ | 9\pi + 18 | olympiads | 0.15625 |
A girl always tells the truth on Thursdays and Fridays, and always lies on Mondays. On the remaining days of the week, she may either tell the truth or lie. For seven consecutive days, she was asked in which year she was born. The first six answers received were 2010, 2011, 2012, 2013, 2002, 2011. What could the answer... | 2010 \text{ or } 2011 | olympiads | 0.125 |
A sports tournament involves a team of 10 players. The regulations stipulate that there are always 8 players from the team on the field, rotating from time to time. The match lasts 45 minutes, and all 10 team members must play the same amount of time. How many minutes will each player be on the field during the game?
| 36 | olympiads | 0.078125 |
The difference between the highest common factor and the lowest common multiple of \( x \) and 18 is 120. Find the value of \( x \). | 42 | olympiads | 0.203125 |
Let $\alpha < 0 < \beta$ and consider the polynomial $f(x) = x(x-\alpha)(x-\beta)$ . Let $S$ be the set of real numbers $s$ such that $f(x) - s$ has three different real roots. For $s\in S$ , let $p(x)$ the product of the smallest and largest root of $f(x)-s$ . Determine the smallest possible value that $... | -\frac{1}{4} (\beta - \alpha)^2 | aops_forum | 0.0625 |
For which natural numbers \( n \) can a board of size \( n \times n \) be divided along the grid lines into dominos of size \( 1 \times 2 \) so that the number of vertical and horizontal dominos is equal? | n \text{ is divisible by 4} | olympiads | 0.09375 |
Given infinite sequences $a_1,a_2,a_3,\cdots$ and $b_1,b_2,b_3,\cdots$ of real numbers satisfying $\displaystyle a_{n+1}+b_{n+1}=\frac{a_n+b_n}{2}$ and $\displaystyle a_{n+1}b_{n+1}=\sqrt{a_nb_n}$ for all $n\geq1$ . Suppose $b_{2016}=1$ and $a_1>0$ . Find all possible values of $a_1$ | 2^{2015} | aops_forum | 0.15625 |
Two cars, Car A and Car B, start from point \( A \) and travel to point \( B \) at the same time. When Car A reaches point \( B \), Car B is still 15 kilometers away from point \( B \). If, starting from the midpoint between \( A \) and \( B \), both cars double their speeds, how far will Car B be from point \( B \) wh... | 15 | olympiads | 0.140625 |
If \(a, b, c, d\) are positive real numbers such that \(\frac{5a + b}{5c + d} = \frac{6a + b}{6c + d}\) and \(\frac{7a + b}{7c + d} = 9\), calculate \(\frac{9a + b}{9c + d}\). | 9 | olympiads | 0.109375 |
We draw circles over the sides of a regular pentagon as diameters. Consider the circle that encloses these 5 circles and touches them. What fraction of the area of the large circle is covered by the 5 small circles? | 0.8 | olympiads | 0.0625 |
An integer sequence \(\left\{a_{i, j}\right\}(i, j \in \mathbf{N})\), where,
\[
\begin{array}{l}
a_{1, n} = n^{n} \quad (n \in \mathbf{Z}_{+}), \\
a_{i, j} = a_{i-1, j} + a_{i-1, j+1} \quad (i, j \geq 1).
\end{array}
\]
What is the unit digit of \(a_{128,1}\)? | 4 | olympiads | 0.09375 |
Given the sequence \(\{a_n\}\) defined by \(a_0 = 0\),
\[a_{n+1} = \frac{8}{5} a_n + \frac{6}{5} \sqrt{4^n - a_n^2} \quad (n \geq 0, n \in \mathbf{N}),\]
find \(a_{10}\). | \frac{24576}{25} | olympiads | 0.125 |
What statement, which could not come from either a guilty knight or a liar, would you make in court to convince the jury of your innocence? | I am a liar. | olympiads | 0.140625 |
As shown in the figure, Rourou's vegetable garden consists of 4 square plots and 1 small rectangular pool forming a large rectangle. If the area of each plot is 20 square meters and the garden's length is 10 meters, what is the perimeter of the pool (the shaded area in the figure) in meters? | 20 \text{ meters} | olympiads | 0.078125 |
Given distinct complex numbers \( m \) and \( n \) satisfying \( m n \neq 0 \) and the set \(\left\{m^{2}, n^{2}\right\}=\{m, n\}\), find \( m + n \). | -1 | olympiads | 0.203125 |
Let $n=2^{31}3^{19}.$ How many positive integer divisors of $n^2$ are less than $n$ but do not divide $n$ ? | 589 | aops_forum | 0.078125 |
Suppose that \(a + x^2 = 2006\), \(b + x^2 = 2007\), and \(c + x^2 = 2008\), and \(abc = 3\). Find the value of:
\[
\frac{a}{bc} + \frac{b}{ca} + \frac{c}{ab} - \frac{1}{a} - \frac{1}{b} - \frac{1}{c}.
\] | 1 | olympiads | 0.0625 |
Determine the geometric mean of the following two expressions:
$$
\frac{2\left(a^{2}-a b\right)}{35 b}, \quad \frac{10 a}{7\left(a b-b^{2}\right)}
$$ | \frac{2a}{7b} | olympiads | 0.3125 |
A set of five volumes of an encyclopedia is arranged in ascending order on a shelf, i.e., from left to right, volumes 1 through 5 are lined up. We want to rearrange them in descending order, i.e., from left to right, volumes 5 through 1, but each time we are only allowed to swap the positions of two adjacent volumes. W... | 10 | olympiads | 0.09375 |
For which integer values of $x$ and $y$ do we have: $7^{x}-3 \times 2^{y}=1$? | (x, y) = (1, 1), (2, 4) | olympiads | 0.234375 |
Find all positive integers $k$ for which number $3^k+5^k$ is a power of some integer with exponent greater than $1$ . | k = 1 | aops_forum | 0.203125 |
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