Unnamed: 0 int64 0 40.3k | problem stringlengths 10 5.15k | ground_truth stringlengths 1 1.22k | solved_percentage float64 0 100 |
|---|---|---|---|
13,600 | A herd of 183 elephants could drink the lake in 1 day, and a herd of 37 elephants could do it in 5 days.
In how many days will one elephant drink the lake? | 365 | 20.3125 |
13,601 | The intercept on the x-axis, the intercept on the y-axis, and the slope of the line 4x-5y-20=0 are respectively. | \dfrac{4}{5} | 8.59375 |
13,602 | In the given configuration, triangle $ABC$ has a right angle at $C$, with $AC=4$ and $BC=3$. Triangle $ABE$ has a right angle at $A$ where $AE=5$. The line through $E$ parallel to $\overline{AC}$ meets $\overline{BC}$ extended at $D$. Calculate the ratio $\frac{ED}{EB}$. | \frac{4}{5} | 7.03125 |
13,603 | Given an arithmetic sequence {a_n} with the sum of its first n terms denoted as S_n, and given that a_1008 > 0 and a_1007 + a_1008 < 0, find the positive integer value(s) of n that satisfy S_nS_{n+1} < 0. | 2014 | 33.59375 |
13,604 |
Given the sets
$$
\begin{array}{l}
A=\left\{(x, y) \mid x=m, y=-3m+2, m \in \mathbf{Z}_{+}\right\}, \\
B=\left\{(x, y) \mid x=n, y=a\left(a^{2}-n+1\right), n \in \mathbf{Z}_{+}\right\},
\end{array}
$$
find the total number of integers $a$ such that $A \cap B \neq \varnothing$. | 10 | 5.46875 |
13,605 | Find the maximum value of the expression \( (\sin 3x + \sin 2y + \sin z)(\cos 3x + \cos 2y + \cos z) \). | 4.5 | 0.78125 |
13,606 | Last year, Isabella took 8 math tests and received 8 different scores, each an integer between 91 and 100, inclusive. After each test, she noted that the average of her test scores was an integer. Her score on the seventh test was 97. What was her score on the eighth test? | 96 | 3.125 |
13,607 | On the hypotenuse of a right triangle, a square is constructed externally. Find the distance from the center of this square to the vertex of the right angle, given the legs of the triangle are 3 and 5. | \sqrt{8.5} | 0 |
13,608 | Find the largest real \( k \) such that if \( a, b, c, d \) are positive integers such that \( a + b = c + d \), \( 2ab = cd \) and \( a \geq b \), then \(\frac{a}{b} \geq k\). | 3 + 2\sqrt{2} | 43.75 |
13,609 | Let \( x \) and \( y \) be positive real numbers, and \( x + y = 1 \). Find the minimum value of \( \frac{x^2}{x+2} + \frac{y^2}{y+1} \). | 1/4 | 60.15625 |
13,610 | A computer can apply three operations to a number: "increase by 2," "increase by 3," "multiply by 2." The computer starts with the number 1 and is made to go through all possible combinations of 6 operations (each combination is applied to the initial number 1). After how many of these combinations will the computer en... | 486 | 62.5 |
13,611 | In triangle \(ABC\), a circle is constructed with diameter \(AC\), which intersects side \(AB\) at point \(M\) and side \(BC\) at point \(N\). Given that \(AC = 2\), \(AB = 3\), and \(\frac{AM}{MB} = \frac{2}{3}\), find \(AN\). | \frac{24}{\sqrt{145}} | 0 |
13,612 | Given a right triangular prism $ABC-A_{1}B_{1}C_{1}$ whose side edge length is equal to the base edge length, find the sine value of the angle formed by $AB_{1}$ and the side face $ACC_{1}A_{1}$. | \frac{\sqrt{6}}{4} | 1.5625 |
13,613 | The ellipse $x^2 + 9y^2 = 9$ and the hyperbola $x^2 - m(y+3)^2 = 1$ are tangent. Compute $m$. | \frac{8}{9} | 0.78125 |
13,614 | In $\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are $a$, $b$, $c$ respectively, and they satisfy $(3b-c)\cos A - a\cos C = 0$.
(1) Find $\cos A$;
(2) If $a = 2\sqrt{3}$ and the area of $\triangle ABC$ is $S_{\triangle ABC} = 3\sqrt{2}$, determine the shape of $\triangle ABC$ and explain the reason;
(3) I... | 4\sqrt{2} | 49.21875 |
13,615 | There are $N$ natural numbers written on a board, where $N \geq 5$. It is known that the sum of all the numbers is 80, and the sum of any five of them is no more than 19. What is the smallest possible value of $N$? | 26 | 2.34375 |
13,616 | Given two plane vectors $\boldsymbol{\alpha}$ and $\boldsymbol{\beta}$ that satisfy
\[ |\boldsymbol{\alpha} + 2\boldsymbol{\beta}| = 3 \]
\[ |2\boldsymbol{\alpha} + 3\boldsymbol{\beta}| = 4, \]
find the minimum value of $\boldsymbol{\alpha} \cdot \boldsymbol{\beta}$. | -170 | 0 |
13,617 | Assume an even function $f(x)$ satisfies $f(x+6) = f(x) + f(3)$ for any $x \in \mathbb{R}$, and $f(x) = 5x$ when $x \in (-3, -2)$. Calculate $f(201.2)$. | -16 | 41.40625 |
13,618 | There is a strip of paper with three types of scale lines that divide the strip into 6 parts, 10 parts, and 12 parts along its length. If the strip is cut along all the scale lines, into how many parts is the strip divided? | 20 | 1.5625 |
13,619 | Two people are tossing a coin: one tossed it 10 times, and the other tossed it 11 times.
What is the probability that the second person's coin landed on heads more times than the first person's coin? | \frac{1}{2} | 92.1875 |
13,620 | Let three non-identical complex numbers \( z_1, z_2, z_3 \) satisfy the equation \( 4z_1^2 + 5z_2^2 + 5z_3^2 = 4z_1z_2 + 6z_2z_3 + 4z_3z_1 \). Denote the lengths of the sides of the triangle in the complex plane, with vertices at \( z_1, z_2, z_3 \), from smallest to largest as \( a, b, c \). Find the ratio \( a : b : ... | 2:\sqrt{5}:\sqrt{5} | 1.5625 |
13,621 |
Over two days, 100 bankers collected funds to fight a new virus. Each banker contributed a whole number of thousands of rubles, not exceeding 200. Contributions on the first day did not exceed 100 thousand, and contributions on the second day were greater than this amount. Additionally, no pair of all 100 contribution... | 10050 | 3.90625 |
13,622 |
In a math competition with problems $A$, $B$, and $C$, there are 39 participants, each of whom answered at least one question correctly. Among those who answered problem $A$ correctly, the number of participants who answered only problem $A$ is 5 more than those who also answered other problems. Among those who did no... | 23 | 10.9375 |
13,623 | Given vectors $\vec{a}$ and $\vec{b}$ with magnitudes $|\vec{a}|=2$ and $|\vec{b}|=\sqrt{3}$, respectively, and the equation $( \vec{a}+2\vec{b}) \cdot ( \vec{b}-3\vec{a})=9$:
(1) Find the dot product $\vec{a} \cdot \vec{b}$.
(2) In triangle $ABC$, with $\vec{AB}=\vec{a}$ and $\vec{AC}=\vec{b}$, find the length of si... | -\sqrt{3} | 3.125 |
13,624 | On grid paper, a step-like right triangle was drawn with legs equal to 6 cells. Then all the grid lines inside the triangle were traced. What is the maximum number of rectangles that can be found in this drawing? | 126 | 7.03125 |
13,625 | The numbers \(a, b, c, d\) belong to the interval \([-6, 6]\). Find the maximum value of the expression \(a + 2b + c + 2d - ab - bc - cd - da\). | 156 | 97.65625 |
13,626 | In the diagram, $\triangle ABE$, $\triangle BCE$ and $\triangle CDE$ are right-angled triangles with $\angle AEB=\angle BEC = \angle CED = 45^\circ$ and $AE=32$. Find the length of $CE.$ | 16 | 58.59375 |
13,627 | The measure of each exterior angle of a regular polygon is \(20^\circ\). What is the sum of the measures of the interior angles and the total number of diagonals of this polygon? | 135 | 25 |
13,628 | In the convex quadrilateral \( MNLQ \), the angles at vertices \( N \) and \( L \) are right angles, and \(\operatorname{tg} \angle QMN = \frac{2}{3}\). Find the diagonal \( NQ \), given that the side \( LQ \) is half the length of side \( MN \) and is 2 units longer than side \( LN \). | 2\sqrt{13} | 13.28125 |
13,629 | Three identical rods each have a piece broken off at a random point. What is the probability that the three resulting pieces can form a triangle?
| 1/2 | 11.71875 |
13,630 | \(5^{-2 \log_{0.04}\left(3-4x^{2}\right)} + 1.5 \log_{\frac{1}{8}} 4^{x} = 0\) | \frac{3}{4} | 10.15625 |
13,631 | Given the set $M=\{x|2x^{2}-3x-2=0\}$ and the set $N=\{x|ax=1\}$. If $N \subset M$, what is the value of $a$? | \frac{1}{2} | 14.84375 |
13,632 | A confectionery factory received 5 spools of ribbon, each 60 meters long, for packaging cakes. How many cuts are needed to obtain pieces of ribbon, each 1 meter 50 centimeters long? | 195 | 0.78125 |
13,633 | A wheel is rolled without slipping through $15$ laps on a circular race course with radius $7$ . The wheel is perfectly circular and has radius $5$ . After the three laps, how many revolutions around its axis has the wheel been turned through? | 21 | 99.21875 |
13,634 | How many such pairs of numbers \((n, k)\) are there, for which \(n > k\) and the difference between the internal angles of regular polygons with \(n\) and \(k\) sides is \(1^{\circ}\)? | 52 | 7.03125 |
13,635 | There are 196 students numbered from 1 to 196 arranged in a line. Students at odd-numbered positions (1, 3, 5, ...) leave the line. The remaining students are renumbered starting from 1 in order. Then, again, students at odd-numbered positions leave the line. This process repeats until only one student remains. What wa... | 128 | 100 |
13,636 | Given the set \( A = \{1,2,3,4\} \), Ander randomly selects a number from \( A \) every second (with replacement). The selection stops when the sum of the last two selected numbers is a prime number. What is the probability that the last number selected is "1"? | \frac{15}{44} | 0 |
13,637 | Determine the radius of the sphere that touches the faces of the unit cube passing through vertex $A$ and the edges passing through vertex $B$. | 2 - \sqrt{2} | 1.5625 |
13,638 | In a plane, a right angle is given. A circle with a center located outside of this angle is tangent to the bisector of the right angle. It intersects one side of the right angle at points \(A\) and \(B\) and the extension of the other side at points \(C\) and \(D\). Given that \(AB = \sqrt{7}\) and \(CD = 1\), find the... | 1.5 | 0.78125 |
13,639 | Given an integer \( n > 4 \), the coefficients of the terms \( x^{n-4} \) and \( xy \) in the expansion of \( (x + 2 \sqrt{y} - 1)^n \) are equal. Find the value of \( n \). | 51 | 8.59375 |
13,640 | How many decreasing sequences $a_1, a_2, \ldots, a_{2019}$ of positive integers are there such that $a_1\le 2019^2$ and $a_n + n$ is even for each $1 \le n \le 2019$ ? | \binom{2037171}{2019} | 0 |
13,641 | Huahua is writing letters to Yuanyuan with a pen. When she finishes the 3rd pen refill, she is working on the 4th letter; when she finishes the 5th letter, the 4th pen refill is not yet used up. If Huahua uses the same amount of ink for each letter, how many pen refills does she need to write 16 letters? | 13 | 3.90625 |
13,642 | One side of a rectangle (the width) was increased by 10%, and the other side (the length) by 20%.
a) Could the perimeter increase by more than 20% in this case?
b) Find the ratio of the sides of the original rectangle if it is known that the perimeter of the new rectangle is 18% greater than the perimeter of the origi... | 1:4 | 8.59375 |
13,643 | The product $1! \cdot 2! \cdot 3! \cdots \cdots \cdot 99! \cdot 100!$ has ___ consecutive zeros at the end. | 1124 | 99.21875 |
13,644 |
Evaluate the limit as \( n \) approaches infinity:
$$
\lim _{n \rightarrow \infty} \frac{(1+2n)^{3} - 8n^{5}}{(1+2n)^{2} + 4n^{2}}
$$ | -1 | 0.78125 |
13,645 | Compute the definite integral:
$$
\int_{\pi / 2}^{\pi} 2^{4} \cdot \sin ^{6} x \cos ^{2} x \, dx
$$ | \frac{5\pi}{16} | 42.1875 |
13,646 | On a section of the map, three roads form a right triangle. When motorcyclists were asked about the distance between $A$ and $B$, one of them responded that after traveling from $A$ to $B$, then to $C$, and back to $A$, his odometer showed 60 km. The second motorcyclist added that he knew by chance that $C$ was 12 km f... | 22.5 | 13.28125 |
13,647 | Calculate the definite integral:
$$
\int_{0}^{\pi} 2^{4} \cdot \sin^{2}\left(\frac{x}{2}\right) \cos^{6}\left(\frac{x}{2}\right) \, dx
$$ | \frac{5 \pi}{8} | 57.03125 |
13,648 | A, B, C, and D obtained the top 4 positions in the school (no ties). They made the following statements:
- A: "I am neither first nor second."
- B: "I am neither second nor third."
- C: "My position is adjacent to B."
- D: "My position is adjacent to C."
Given that A, B, C, and D are all honest students, determine the... | 4123 | 14.0625 |
13,649 | If the odd function \( y=f(x) \) defined on \( \mathbf{R} \) is symmetrical about the line \( x=1 \), and when \( 0 < x \leqslant 1 \), \( f(x)=\log_{3}x \), find the sum of all real roots of the equation \( f(x)=-\frac{1}{3}+f(0) \) in the interval \( (0,10) \). | 30 | 6.25 |
13,650 | Suppose you have an equilateral triangle divided into 9 smaller equilateral triangles with the bottom side horizontal. Starting from the top corner labeled \( A \), you must walk to the bottom right corner labeled \( B \), and are only allowed to take steps along the edges down to the left, down to the right, or horizo... | 22 | 0 |
13,651 | 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 | 0.78125 |
13,652 | In an \(8 \times 8\) table, some cells are black, and the rest are white. In each white cell, the total number of black cells located in the same row or column is written. Nothing is written in the black cells. What is the maximum possible value of the sum of the numbers in the entire table? | 256 | 34.375 |
13,653 | Given squares $ABCD$ and $EFGH$ are congruent, $AB=12$, and $H$ is located at vertex $D$ of square $ABCD$. Calculate the total area of the region in the plane covered by these squares. | 252 | 4.6875 |
13,654 | For a real number \( x \), find the maximum value of
\[
\frac{x^6}{x^{12} + 3x^8 - 6x^6 + 12x^4 + 36}
\] | \frac{1}{18} | 3.125 |
13,655 | Given sets $A=\{x,\frac{y}{x},1\}$ and $B=\{{x}^{2},x+y,0\}$, if $A=B$, then $x^{2023}+y^{2024}=\_\_\_\_\_\_.$ | -1 | 54.6875 |
13,656 | Equilateral triangle \( \triangle ABC \) and square \( ABDE \) have a common side \( AB \). The cosine of the dihedral angle \( C-ABD \) is \(\frac{\sqrt{3}}{3}\). If \( M \) and \( N \) are the midpoints of \( AC \) and \( BC \) respectively, then the cosine of the angle between \( EM \) and \( AN \) is \(\qquad\). | \frac{1}{6} | 0.78125 |
13,657 | How many $9$-digit palindromes can be formed using the digits $1$, $1$, $2$, $2$, $2$, $4$, $4$, $5$, $5$? | 36 | 4.6875 |
13,658 | In triangle $XYZ$, side $y = 7$, side $z = 3$, and $\cos(Y - Z) = \frac{17}{32}$. Find the length of side $x$. | \sqrt{41} | 0 |
13,659 | In a store, we paid with a 1000 forint bill. On the receipt, the amount to be paid and the change were composed of the same digits but in a different order. What is the sum of the digits? | 14 | 70.3125 |
13,660 | The diagonal of an isosceles trapezoid bisects its obtuse angle. The shorter base of the trapezoid is 3 cm, and the perimeter is 42 cm. Find the area of the trapezoid. | 96 | 2.34375 |
13,661 | In \(\triangle ABC\), \(AC = AB = 25\) and \(BC = 40\). From \(D\), perpendiculars are drawn to meet \(AC\) at \(E\) and \(AB\) at \(F\), calculate the value of \(DE + DF\). | 24 | 33.59375 |
13,662 | In the triangle \(ABC\), let \(l\) be the bisector of the external angle at \(C\). The line through the midpoint \(O\) of the segment \(AB\), parallel to \(l\), meets the line \(AC\) at \(E\). Determine \(|CE|\), if \(|AC| = 7\) and \(|CB| = 4\). | 11/2 | 1.5625 |
13,663 | Given that the complex numbers \( z_{1}, z_{2}, z_{3} \) satisfy \( \frac{z_{3}-z_{1}}{z_{2}-z_{1}} = a \mathrm{i} \) where \( a \) is a non-zero real number (\( a \in \mathbf{R}, a \neq 0 \)), find the angle between the vectors \( \overrightarrow{Z_{1} Z_{2}} \) and \( \overrightarrow{Z_{1} Z_{3}} \). | \frac{\pi}{2} | 65.625 |
13,664 | Let \( a_{1}, a_{2}, \cdots, a_{105} \) be a permutation of \( 1, 2, \cdots, 105 \), satisfying the condition that for any \( m \in \{3, 5, 7\} \), for all \( n \) such that \( 1 \leqslant n < n+m \leqslant 105 \), we have \( m \mid (a_{n+m}-a_{n}) \). How many such distinct permutations exist? (Provide the answer as a... | 3628800 | 3.90625 |
13,665 | Given that vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ satisfy $|\overrightarrow{a}|=\sqrt{2}$, $|\overrightarrow{b}|=2$, and $\overrightarrow{a}\bot (\overrightarrow{a}-\overrightarrow{b})$, calculate the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$. | \frac{\pi}{4} | 93.75 |
13,666 | Determine the increment of the argument and the function \( y = x^2 \) if the argument \( x \) changes from 2 to 2.5. | 2.25 | 21.875 |
13,667 | 10 chatterboxes sat in a circle. Initially, one of them told one joke, the next one clockwise told two jokes, the next one three jokes, and so on in a circle until one of them told 100 jokes at once. Then the chatterboxes got tired, and the next one clockwise told 99 jokes, the next one 98 jokes, and so on in a circle ... | 1000 | 10.15625 |
13,668 | Find the number of different recommendation plans for the high school given that 3 male and 2 female students are selected as candidates, where both Russian and Japanese exams must include male participants, and 2 spots are available for Russian, 2 for Japanese, and 1 for Spanish. | 24 | 3.90625 |
13,669 | A bacterium develops on a $100 \times 100$ grid. It can contaminate a new cell if and only if two adjacent cells are already contaminated. What is the minimal number of initially contaminated cells required for the bacterium to be able to spread everywhere on the grid? | 100 | 24.21875 |
13,670 | Points \( A, B, C, D \) are marked on a sheet of paper. A recognition device can perform two types of operations with absolute accuracy: a) measure the distance between two given points in centimeters; b) compare two given numbers. What is the minimum number of operations this device needs to perform to definitively de... | 10 | 22.65625 |
13,671 | Riquinho distributed $R \$ 1000.00$ among his friends: Antônio, Bernardo, and Carlos in the following manner: he successively gave 1 real to Antônio, 2 reais to Bernardo, 3 reais to Carlos, 4 reais to Antônio, 5 reais to Bernardo, and so on. How much did Bernardo receive? | 345 | 0 |
13,672 | Given a cube of side length $8$ and balls of clay of radius $1.5$, determine the maximum number of balls that can completely fit inside the cube when the balls are reshaped but not compressed. | 36 | 52.34375 |
13,673 | The punch machines from before the flood punch some or even all of the nine numbered fields of a ticket. The inspectors request from the machine setter that the machine should not punch the same fields if someone places their ticket in reverse, instead of the prescribed orientation. How many such settings are possible ... | 448 | 1.5625 |
13,674 | Consider integers \( \{1, 2, \ldots, 10\} \). A particle is initially at 1. It moves to an adjacent integer in the next step. What is the expected number of steps it will take to reach 10 for the first time? | 90 | 0.78125 |
13,675 | In a box, 10 smaller boxes are placed. Some of the boxes are empty, and some contain another 10 smaller boxes each. Out of all the boxes, exactly 6 contain smaller boxes. How many empty boxes are there? | 55 | 0 |
13,676 | Given that \( x_{i}=\frac{i}{101} \), find the value of \( S=\sum_{i=0}^{101} \frac{x_{i}^{3}}{3 x_{i}^{2}-3 x_{i}+1} \). | 51 | 83.59375 |
13,677 | Let \( a < b < c < d < e \) be real numbers. Among the 10 sums of the pairs of these numbers, the least three are 32, 36, and 37, while the largest two are 48 and 51. Find all possible values of \( e \). | 27.5 | 17.1875 |
13,678 | Given the three interior angles \( A, B, C \) of \(\triangle ABC\) satisfy \( A = 3B = 9C \), find the value of \( \cos A \cos B + \cos B \cos C + \cos C \cos A \). | -1/4 | 0 |
13,679 | A positive integer \( n \) cannot be divided by \( 2 \) or \( 3 \), and there do not exist non-negative integers \( a \) and \( b \) such that \( |2^a - 3^b| = n \). Find the smallest value of \( n \). | 35 | 64.0625 |
13,680 | In a class, there are 15 boys and 15 girls. On Women's Day, some boys called some girls to congratulate them (no boy called the same girl more than once). It turned out that the children can be uniquely divided into 15 pairs, such that each pair consists of a boy and a girl whom he called. What is the maximum number of... | 120 | 0 |
13,681 | In \(\triangle ABC\), \(DC = 2BD\), \(\angle ABC = 45^\circ\), and \(\angle ADC = 60^\circ\). Find \(\angle ACB\) in degrees. | 75 | 26.5625 |
13,682 | Suppose that $a$ and $b$ are positive integers such that $a$ has $4$ factors and $b$ has $a$ factors. If $b$ is divisible by $a$, then what is the least possible value of $b$? | 24 | 2.34375 |
13,683 | Line $\ell$ passes through $A$ and into the interior of the equilateral triangle $ABC$ . $D$ and $E$ are the orthogonal projections of $B$ and $C$ onto $\ell$ respectively. If $DE=1$ and $2BD=CE$ , then the area of $ABC$ can be expressed as $m\sqrt n$ , where $m$ and $n$ are positive integers... | 10 | 1.5625 |
13,684 | Given $y=f(x)+x^2$ is an odd function, and $f(1)=1$, if $g(x)=f(x)+2$, then $g(-1)=$ . | -1 | 99.21875 |
13,685 | In a WeChat group, there are five people playing the red envelope game: A, B, C, D, and E. There are 4 red envelopes, each person can grab at most one, and all red envelopes must be grabbed. Among the 4 red envelopes, there are two worth 2 yuan, one worth 3 yuan, and one worth 4 yuan (red envelopes with the same amount... | 36 | 62.5 |
13,686 | In a relay race from Moscow to Petushki, two teams of 20 people each participated. Each team divided the distance into 20 segments (not necessarily equal) and assigned them among the participants so that each person ran exactly one segment (each participant's speed is constant, but the speeds of different participants ... | 38 | 0.78125 |
13,687 | A company has recruited 8 new employees, who are to be evenly distributed between two sub-departments, A and B. There are restrictions that the two translators cannot be in the same department, and the three computer programmers cannot all be in the same department. How many different distribution plans are possible? | 36 | 1.5625 |
13,688 | Let $\mathcal{O}$ be a regular octahedron. How many lines are there such that a rotation of at most $180^{\circ}$ around these lines maps $\mathcal{O}$ onto itself? | 13 | 47.65625 |
13,689 | The sequence \(\{a_n\}\) is defined such that \(a_1 = \frac{\pi}{6}\) and \(a_{n+1} = \arctan \left(\sec a_n\right)\) for \( n \in \mathbf{N}^{*}\). Find the positive integer \(m\) such that
\[
\sin a_1 \cdot \sin a_2 \cdots \cdot \sin a_m = \frac{1}{100}.
\] | 3333 | 1.5625 |
13,690 | A table can seat 6 people. Two tables joined together can seat 10 people. Three tables joined together can seat 14 people. Following this pattern, if 10 tables are arranged in two rows with 5 tables in each row, how many people can sit? | 44 | 77.34375 |
13,691 | Natural numbers \( m \) and \( n \) are such that \( m > n \), \( m \) is not divisible by \( n \), and \( m \) has the same remainder when divided by \( n \) as \( m + n \) has when divided by \( m - n \).
Find the ratio \( m : n \). | 5/2 | 1.5625 |
13,692 |
Solve the equation:
$$
\begin{gathered}
\frac{10}{x+10}+\frac{10 \cdot 9}{(x+10)(x+9)}+\frac{10 \cdot 9 \cdot 8}{(x+10)(x+9)(x+8)}+\cdots+ \\
+\frac{10 \cdot 9 \ldots 2 \cdot 1}{(x+10)(x+9) \ldots(x+1)}=11
\end{gathered}
$$ | -\frac{1}{11} | 10.15625 |
13,693 | Define a set \( \mathcal{T} \) of distinct positive integers such that for every integer \( y \) in \( \mathcal{T}, \) the geometric mean of the values obtained by omitting \( y \) from \( \mathcal{T} \) remains a positive integer. In addition, assume that 1 is a member of \( \mathcal{T} \) and the largest element is 2... | 15 | 7.8125 |
13,694 | When studying the operation of a new type of cyclic thermal engine, it was found that during part of the period it receives heat, and the absolute power of heat supply is expressed by the law:
\[ P_{1}(t)=P_{0} \frac{\sin (\omega t)}{100+\sin (t^{2})}, \quad 0<t<\frac{\pi}{\omega}. \]
The gas performs work, developin... | 1/3 | 22.65625 |
13,695 | Let \(a, b, c, d\) be positive integers such that \(a^5 =\) | 757 | 2.34375 |
13,696 | Buses leave Moscow for Voronezh every hour, at 00 minutes. Buses leave Voronezh for Moscow every hour, at 30 minutes. The trip between cities takes 8 hours. How many buses from Voronezh will a bus leaving Moscow meet on its way? | 16 | 10.9375 |
13,697 | Find the number of natural numbers \( k \) not exceeding 353500 such that \( k^{2} + k \) is divisible by 505. | 2800 | 54.6875 |
13,698 | Let \(\triangle ABC\) be an equilateral triangle with height 13, and let \(O\) be its center. Point \(X\) is chosen at random from all points inside \(\triangle ABC\). Given that the circle of radius 1 centered at \(X\) lies entirely inside \(\triangle ABC\), what is the probability that this circle contains \(O\)? | \frac{\sqrt{3} \pi}{121} | 11.71875 |
13,699 | Calculate the definite integral:
$$
\int_{\frac{\pi}{2}}^{2 \operatorname{arctg} 2} \frac{d x}{\sin ^{2} x(1+\cos x)}
$$ | 29/24 | 19.53125 |
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