task_type stringclasses 4
values | problem stringlengths 21 5.23k | answer stringlengths 1 8.29k | problem_tokens int64 11 1.16k | answer_tokens int64 1 2.04k |
|---|---|---|---|---|
math | A hare is running a 2024-meter race. At the start, it pushes off with its left foot and alternates regularly between its left foot, right foot, and both feet throughout the race. When the hare pushes off with its left foot, it jumps $35 \mathrm{dm}$, when it pushes off with its right foot, it jumps $15 \mathrm{dm}$, an... | 548 | 138 | 3 |
math | 2. Given $f(x)=\lg (x+1)-\frac{1}{2} \log _{3} x$. Then the set
$$
M=\left\{n \mid f\left(n^{2}-8 n-2018\right) \geqslant 0, n \in \mathbf{Z}\right\}
$$
the number of subsets of $M$ is $\qquad$. | 1 | 95 | 1 |
math | 9. [6] Compute the side length of the largest cube contained in the region
$$
\left\{(x, y, z): x^{2}+y^{2}+z^{2} \leq 25 \text { and } x \geq 0\right\}
$$
of three-dimensional space. | \frac{5\sqrt{6}}{3} | 72 | 12 |
math | 35. Find the largest integer $N$ such that both $N+496$ and $N+224$ are perfect squares. | 4265 | 32 | 4 |
math | Problem 4. For an infinite sequence of numbers $x_{1}, x_{2}, x_{3}, \ldots$, for all natural $n \geq 4$, the relation $x_{n}=x_{n-1} \cdot x_{n-3}$ holds. It is known that $x_{1}=1, x_{2}=1, x_{3}=-1$. Find $x_{2022}$. | 1 | 95 | 1 |
math | 1. Let the function $y=\sin x \cdot \cos x(\sin x+\cos x)$, where $x \in\left[-\frac{\pi}{4}, \frac{\pi}{2}\right]$. Then the range of the function $y$ is $\qquad$ | [-\frac{\sqrt{3}}{9},\frac{\sqrt{2}}{2}] | 62 | 21 |
math | 1. (8 points) The sum of the digits of the result of the expression $999999999-88888888+7777777-666666+55555-4444+333-22+1$ is $\qquad$ . | 45 | 78 | 2 |
math | 11. Real numbers $x, y \in(1,+\infty)$, and $x y-2 x-y+1=0$, find the minimum value of $\frac{3}{2} x^{2}+y^{2}$.
| 15 | 54 | 2 |
math | Let $T$ be the answer to question $18$. Rectangle $ZOMR$ has $ZO = 2T$ and $ZR = T$. Point $B$ lies on segment $ZO$, $O'$ lies on segment $OM$, and $E$ lies on segment $RM$ such that $BR = BE = EO'$, and $\angle BEO' = 90^o$. Compute $2(ZO + O'M + ER)$.
PS. You had better calculate it in terms of $T$. | 7T | 110 | 2 |
math | # Problem 1:
Calculate: $\left[\frac{1^{2}}{2}\right]+2^{1} \cdot\left[\frac{2^{2}}{3}\right]+2^{2} \cdot\left[\frac{3^{2}}{4}\right]+\ldots+2^{n-1} \cdot\left[\frac{n^{2}}{n+1}\right]$. | n\cdot2^{n}-2^{n+1}+2 | 89 | 15 |
math | 1. Given that the domain of the function $f(x)$ is $[-1,1]$, find the domain of $f(a x)+f\left(\frac{x}{a}\right)$ (where $a>0$). | [-,]for\in(0,1);[-\frac{1}{},\frac{1}{}]for\in[1,+\infty) | 49 | 34 |
math | 10. Given the sequence $\left\{a_{n}\right\}, a_{1}=1, a_{n+1}$ $=\frac{\sqrt{3} a_{n}+1}{\sqrt{3}-a_{n}}$, then $\sum_{n=1}^{2022} a_{n}=$ $\qquad$ | 0 | 77 | 1 |
math | Ex. 118. A circle with center on side $AB$ of triangle $ABC$ touches sides $AC$ and $BC$. Find the radius of the circle, given that it is expressed as an integer, and sides $AC$ and $BC$ are equal to 5 and 3. | 1 | 64 | 1 |
math | Example 1. Reduce the general equations of a line to canonical form
\[
\left\{\begin{array}{l}
2 x-3 y-3 z-9=0 \\
x-2 y+z+3=0
\end{array}\right.
\] | \frac{x}{9}=\frac{y}{5}=\frac{z+3}{1} | 59 | 22 |
math | 1. Find the value of the expression $\left(\left(\frac{3}{a-b}+\frac{3 a}{a^{3}-b^{3}} \cdot \frac{a^{2}+a b+b^{2}}{a+b}\right)\right.$ ? $\left.\frac{2 a+b}{a^{2}+2 a b+b^{2}}\right) \cdot \frac{3}{a+b}$ when $a=2023, b=2020$ | 3 | 109 | 1 |
math | Problem 6. Calculate $2 \operatorname{arctg} 4+\arcsin \frac{8}{17}$. | \pi | 30 | 2 |
math | Problem 2. On each of the two gardens, Grandpa planted the same number of turnips. If Granddaughter enters the garden, she pulls out exactly $1 / 3$ of the turnips present at that moment. If Doggy enters, she pulls out $1 / 7$ of the turnips, and if Mousey enters, she pulls out only $1 / 12$ of the turnips. By the end ... | Yes | 120 | 1 |
math | Example 5: On the ground, there are 10 birds pecking, and among any 5 birds, at least 4 birds are on the same circumference. What is the maximum number of birds on the circumference that has the most birds? (6th China Mathematical Olympiad) | 9 | 60 | 1 |
math | 3 [ Shortest path on the surface ]
A sphere of radius 2 is intersected by a plane at a distance of 1 from the center. Find the length of the shortest path on the surface of the sphere between the two most distant points of the intersection.
# | \frac{4\pi}{3} | 55 | 9 |
math | 8.3. On the island of knights and liars, each resident was asked about each of the others: is he a knight or a liar. In total, 42 answers of "knight" and 48 answers of "liar" were received. What is the maximum number of knights that could have been on the island? Justify your answer. (It is known that knights always te... | 6 | 93 | 1 |
math | 1) Given the vectors $\overrightarrow{O P}=\left(2 \cos \left(\frac{\pi}{2}+x\right),-1\right), \overrightarrow{O Q}=\left(-\sin \left(\frac{\pi}{2}-\right.\right.$ $x), \cos 2 x), f(x)=\overrightarrow{O P} \cdot \overrightarrow{O Q}$. If $a, b, c$ are the sides opposite to the angles $A, B, C$ of the acute triangle $\... | \frac{15}{2} | 174 | 8 |
math | 7. (10 points) Households A, B, and C plan to subscribe to newspapers, with 5 different newspapers available for selection. It is known that each household subscribes to two different newspapers, and any two households have exactly one newspaper in common. How many different subscription methods are there for the three... | 180 | 66 | 3 |
math | 1. When one of two integers was increased 1996 times, and the other was reduced 96 times, their sum did not change. What can their quotient be | 2016 | 37 | 4 |
math | 7.114. $\lg (x(x+9))+\lg \frac{x+9}{x}=0$. | -10 | 26 | 3 |
math | 39. What is the greatest value that the modulus of a complex number $Z$ can take, satisfying the equation $\left|Z+\frac{1}{z}\right|=1$? | \frac{1+\sqrt{5}}{2} | 40 | 12 |
math | Example 9 In a sequence of coin tosses, the number of times a tail is followed by a head (denoted as "tail-head"), a head is followed by a tail (denoted as "head-tail"), a head is followed by a head (denoted as "head-head"), and a tail is followed by a tail (denoted as "tail-tail") can be counted. How many different se... | 560 | 128 | 3 |
math | 17. (18 points) Given the ellipse
$$
C_{1}: \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1(a>b>0)
$$
has an eccentricity of $\frac{\sqrt{3}}{2}$, and the right focus is the center of the circle
$$
C_{2}:(x-\sqrt{3})^{2}+y^{2}=7
$$
(1) Find the equation of the ellipse $C_{1}$;
(2) If the line $l$ intersects the curves $C_{... | A(0,2)A(0,-2) | 174 | 12 |
math | Two shooters aim at the same target. The first one hits the target with a probability of 0.8, the second one with a probability of 0.6. The first shooter fires twice, the second one fires three times. What is the probability that the target is
a) hit by at least one of them,
b) hit once by the first and twice by the ... | 0.99744,0.13824,0.87328,0.032 | 102 | 29 |
math | 6. Let $P_{1}, P_{2}, \ldots, P_{6}$ be points in the complex plane, which are also roots of the equation $x^{6}+6 x^{3}-216=0$. Given that $P_{1} P_{2} P_{3} P_{4} P_{5} P_{6}$ is a convex hexagon, determine the area of this hexagon. | 9\sqrt{3} | 92 | 6 |
math | Let $\alpha$ be a non-zero real number. Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that
$$
x f(x+y)=(x+\alpha y) f(x)+x f(y)
$$
for all $x, y \in \mathbb{R}$.
Answer: $f(x)=c x^{2}$ for any real constant $c$ if $\alpha=2 ; f(x)=0$ otherwise. | f(x)=c x^{2} \text{ for any real constant } c \text{ if } \alpha=2 ; f(x)=0 \text{ otherwise} | 101 | 36 |
math | 11.1. Two runners, starting simultaneously at constant speeds, run on a circular track in opposite directions. One of them runs the loop in 5 minutes, while the other takes 8 minutes. Find the number of different meeting points of the runners on the track, if they ran for at least an hour.
# | 13 | 66 | 2 |
math | 1. Ivan Ivanovich approached a source with two empty cans; one held 10 liters, and the other held 8 liters. Water from the source flowed in two streams - one stronger, the other weaker. Ivan Ivanovich simultaneously placed the cans under the streams and, when half of the smaller can was filled, he switched the cans. To... | 2 | 99 | 1 |
math | Find the maximum value of the expression
$$
x \sqrt{1-y^{2}}+y \sqrt{1-x^{2}}
$$ | 1 | 30 | 1 |
math | Let $f$ be the function that associates with a positive integer the sum of its digits when it is written in base 10. For example, $f(537)=15$. Calculate $f\left(f\left(f\left(4444^{4444}\right)\right)\right)$. | 7 | 70 | 1 |
math | 3. For arbitrary real numbers $a$ and $b (a \neq 0)$, find the minimum value of the expression $\frac{1}{a^{2}}+2 a^{2}+3 b^{2}+4 a b$. | \sqrt{\frac{8}{3}} | 54 | 9 |
math | ## 8. Height
In a certain class, the average height of all twenty-two students was $163.5 \mathrm{~cm}$. Soon, one more student was enrolled, increasing the average height by $0.3 \mathrm{~cm}$, and the very next day, one student dropped out, causing the average height to increase again, this time by $0.4 \mathrm{~cm}... | 155 | 113 | 3 |
math | Roman likes magic and mathematics. The last time he cast a spell with three-digit or four-digit numbers like this:
- from a given number, he created two new numbers by splitting it between the hundreds and tens digits (for example, from the number 581, he would get 5 and 81),
- he added the new numbers and wrote down ... | 18810 | 202 | 5 |
math | 4. The average age of 5 basketball players currently on the court is 24 years and 6 months. If the coach's age is included in the calculation of the average, then the average age is 27 years. How old is the coach? | 39.5 | 54 | 4 |
math | ## Task 1.
Determine all polynomials $P$ with real coefficients such that the expression
$$
P(x+3 y)+P(3 x-y)
$$
has the same value for all real numbers $x, y$ for which $x^{2}+y^{2}=1$. | P(x)=(x^{2}-5)Q((x^{2}-5)^{2})+\frac{C}{2} | 64 | 27 |
math | We randomly choose 5 distinct positive integers less than or equal to 90. What is the floor of 10 times the expected value of the fourth largest number? | 606 | 35 | 3 |
math | The bisector of one of the acute angles of a right triangle, divided by the height dropped to the hypotenuse, has a ratio of
$1+\sqrt{2}$, measured from the vertex. Find the acute angles of the triangle. | 45,45 | 51 | 5 |
math | 7. Let $M=\{1,2,3,4,5\}$. Then the number of mappings $f: M \rightarrow M$ such that
$$
f(f(x))=f(x)
$$
is $\qquad$ | 196 | 52 | 3 |
math | $$
\left(\sqrt[3]{9-\sqrt{17}}-\sqrt[3]{\frac{1}{8} \sqrt{17}-1 \frac{1}{8}}\right) \cdot \sqrt[3]{3+\frac{1}{3} \sqrt{17}}=?
$$ | 2\cdot\sqrt[3]{9} | 69 | 10 |
math | Find all positive integers $x$ and $y$ such that $3^{x}+7=2^{y}$. | (x,y)=(2,4) | 26 | 7 |
math | The kindergarten received cards for reading lessons: some have "MA" written on them, and others have "NYA". Each child took three cards and started forming words from them. It turned out that 20 children can form the word "MAMA", 30 children can form the word "NYANYA", and 40 children can form the word "MANYA". How man... | 10 | 90 | 2 |
math | Find the number of sequences $(a_n)_{n=1}^\infty$ of integers satisfying $a_n \ne -1$ and
\[a_{n+2} =\frac{a_n + 2006}{a_{n+1} + 1}\]
for each $n \in \mathbb{N}$. | a_i \in \{a, b\} \text{ where } ab = 2006 \text{ and } a_n = a_{n+2} \text{ for all } n | 75 | 44 |
math | Find all the real numbers $ N$ that satisfy these requirements:
1. Only two of the digits of $ N$ are distinct from $ 0$, and one of them is $ 3$.
2. $ N$ is a perfect square. | 36 \times 100^n | 51 | 11 |
math | ## Task $7 / 63$
Ten buckets of the same size and appearance are filled with coins that look identical. In nine of these buckets, each coin weighs 10 g, while in one bucket they weigh 11 g.
How can one determine with a single weighing which bucket contains the coins weighing 11 g? | G-550 | 69 | 5 |
math | In a sports club, 100 overweight people are training, weighing from 1 to 100 kg. What is the smallest number of teams they can be divided into so that no team has two overweight people, one of whom weighs twice as much as the other?
# | 2 | 58 | 1 |
math | In tetrahedron $ABCD$, edge $AB$ has length 3 cm. The area of face $ABC$ is 15 $\text{cm}^2$ and the area of face $ABD$ is 12 $\text{cm}^2$. These two faces meet each other at a $30^\circ$ angle. Find the volume of the tetrahedron in $\text{cm}^3$. | 20 | 96 | 2 |
math | 12th VMO 1974 Problem A2 (1) How many positive integers n are such that n is divisible by 8 and n+1 is divisible by 25? (2) How many positive integers n are such that n is divisible by 21 and n+1 is divisible by 165? (3) Find all integers n such that n is divisible by 9, n+1 is divisible by 25 and n+2 is divisible by 4... | 24\mod200,none,774\mod900 | 106 | 18 |
math | [ Numerical inequalities. Comparing numbers.]
Sasha wrote several two-digit numbers on the board in ascending order, and then replaced identical digits with identical letters, and different digits with different letters. He ended up with (in the same order)
$$
\text { AC, AR, YR, YK, OK, OM, UM, UZ, IZ, IA }
$$
Rest... | A=5,Zh=4,I=9,K=2,M=3,O=7,P=1,C=0,Y=8,Ya=6 | 84 | 32 |
math | [ Coordinate method in space ] [
The edge of the cube $E F G H E 1 F 1 G 1 H 1$ is 2. Points $A$ and $B$ are taken on the edges $E H$ and $H H 1$, respectively, such that $\frac{E A}{A H}=2, \frac{B H}{B H 1}=\frac{1}{2}$. A plane is drawn through points $A, B$, and $G 1$. Find the distance from point $E$ to this plan... | 2\sqrt{\frac{2}{11}} | 122 | 11 |
math | Suppose that
$$
\left(1+\frac{1}{2}\right)\left(1+\frac{1}{3}\right)\left(1+\frac{1}{4}\right) \cdots\left(1+\frac{1}{k}\right)\left(1+\frac{1}{k+1}\right)=2014
$$
for some positive integer $k$. (There are $k$ factors in the product.) What is the value of $k$ ? | 4026 | 107 | 4 |
math | \section*{Problem 4 - 081014}
Some students from grades 9 and 10 of a school participated in a chess tournament. Each participant played exactly one game with every other participant. A win earned one point, a draw earned half a point. Although exactly 10 times as many students from grade 10 as from grade 9 participat... | =1,P_{}=10 | 123 | 8 |
math | 2. If the complex number $\frac{z+\frac{1}{3}}{z-\frac{1}{3}}$ is purely imaginary, then $|z|=$ | \frac{1}{3} | 37 | 7 |
math | ## Task A-4.2.
A Gaussian integer is a complex number whose real and imaginary parts are integers. Determine the largest natural number $n$ for which there exists a set of $n$ Gaussian integers such that the squares of their absolute values are consecutive natural numbers. | 3 | 56 | 1 |
math | ## Problem Statement
Calculate the limit of the numerical sequence:
$$
\lim _{n \rightarrow \infty}\left(\frac{n^{3}+n+1}{n^{3}+2}\right)^{2 n^{2}}
$$ | e^2 | 53 | 3 |
math | $2 \cdot 46$ Insert "+" or "-" between $1,2,3 \cdots, 1989$, what is the smallest non-negative number that the sum expression can obtain? | 1 | 43 | 1 |
math | Fix integers $n\ge k\ge 2$. We call a collection of integral valued coins $n-diverse$ if no value occurs in it more than $n$ times. Given such a collection, a number $S$ is $n-reachable$ if that collection contains $n$ coins whose sum of values equals $S$. Find the least positive integer $D$ such that for any $n$-diver... | n+k-1 | 127 | 4 |
math | Our school's ball last year allocated 10% of its net income to the acquisition of specialized clubs, and the remaining portion exactly covered the rental fee for the sports field. This year, we cannot issue more tickets, and the rental fee remains unchanged, so the share for the clubs could only be increased by raising... | 12.5 | 88 | 4 |
math | ## Task 16/63
The following "card trick" is to be mathematically justified:
A player is asked to draw a card from a deck of 32 cards and place it face down. Let's assume, for example (without regard to the suit), that this is a Seven.
The player then places additional cards on the drawn one, counting from the value ... | 12n+r-32 | 184 | 7 |
math | In the middle of the school year, $40\%$ of Poolesville magnet students decided to transfer to the Blair magnet, and $5\%$ of the original Blair magnet students transferred to the Poolesville magnet. If the Blair magnet grew from $400$ students to $480$ students, how many students does the Poolesville magnet have after... | 170 | 87 | 3 |
math | Let $A = (0,0)$ and $B = (b,2)$ be points on the coordinate plane. Let $ABCDEF$ be a convex equilateral hexagon such that $\angle FAB = 120^\circ,$ $\overline{AB}\parallel \overline{DE},$ $\overline{BC}\parallel \overline{EF,}$ $\overline{CD}\parallel \overline{FA},$ and the y-coordinates of its vertices are distinct e... | 51 | 173 | 2 |
math | For the give functions in $\mathbb{N}$:
[b](a)[/b] Euler's $\phi$ function ($\phi(n)$- the number of natural numbers smaller than $n$ and coprime with $n$);
[b](b)[/b] the $\sigma$ function such that the $\sigma(n)$ is the sum of natural divisors of $n$.
solve the equation $\phi(\sigma(2^x))=2^x$. | 1 | 98 | 1 |
math | 3B. Determine all pairs of integers $(x, y)$ that satisfy the equation
$$
x y + 3 y = x^2 + 6 x + 12
$$ | (-4,-4),(-2,4),(0,4),(-6,-4) | 40 | 19 |
math | A fair coin is to be tossed $10$ times. Let $i/j$, in lowest terms, be the probability that heads never occur on consecutive tosses. Find $i+j$. | 73 | 39 | 2 |
math | 5. Let $p(x)=2 x^{3}-3 x^{2}+1$. How many squares of integers are among the numbers $p(1), p(2), \ldots$, $p(2016) ?$
# | 32 | 53 | 2 |
math | 1. [5] David, Delong, and Justin each showed up to a problem writing session at a random time during the session. If David arrived before Delong, what is the probability that he also arrived before Justin? | \frac{2}{3} | 46 | 7 |
math | 1. If the real number $x$ satisfies $\lg \sin x+\lg \cos x=-1$, then $\lg (\sin x+\cos x)=$ | \frac{1}{2}(\lg12-1) | 34 | 14 |
math | 2. Let the sequence $\left\{a_{n}\right\}$ have the sum of the first $n$ terms $S_{n}=2 a_{n}-1(n=1,2, \cdots)$, and the sequence $\left\{b_{n}\right\}$ satisfies $b_{1}=3, b_{k+1}=$ $a_{k}+b_{k}(k=1,2, \cdots)$. Find the sum of the first $n$ terms of the sequence $\left\{b_{n}\right\}$.
(1996 National High School Comp... | 2^{n}+2n-1 | 134 | 9 |
math | 3.282. $\frac{\sin 8 \alpha+\sin 9 \alpha+\sin 10 \alpha+\sin 11 \alpha}{\cos 8 \alpha+\cos 9 \alpha+\cos 10 \alpha+\cos 11 \alpha} \times$
$\times \frac{\cos 8 \alpha-\cos 9 \alpha-\cos 10 \alpha+\cos 11 \alpha}{\sin 8 \alpha-\sin 9 \alpha-\sin 10 \alpha+\sin 11 \alpha}$. | 1 | 124 | 1 |
math | The first term of a certain arithmetic and geometric progression is 5; the second term of the arithmetic progression is 2 less than the second term of the geometric progression; the third term of the geometric progression is equal to the sixth term of the arithmetic progression. What progressions satisfy the conditions... | thegeometricprogressions:5,15,45,\ldots;5,10,20,\ldots\quadthearithmeticprogressions:5,13,21,\ldots;5,8,11,\ldots | 61 | 55 |
math | Example 3 Let $x_{i} \geqslant 0(i=1,2, \cdots, 7)$, and satisfy $x_{1}+x_{2}+\cdots+x_{7}=a$ (a constant), denote
$$
A=\max \left\{x_{1}+x_{2}+x_{3}, x_{2}+x_{3}+x_{4}, \cdots, x_{5}+x_{6}+x_{7}\right\} \text {. }
$$
Try to find $A_{\min }$. | \frac{a}{3} | 131 | 7 |
math | One, (Full marks 20 points) Given that positive integers $p, q$ are both prime numbers, and $7 p+q$ and $p q+11$ are also prime numbers. Calculate the value of $\left(p^{2}+q^{p}\right)\left(q^{2}+p^{q}\right)$. | 221 | 74 | 3 |
math | 5. Let the function
$$
f(x)=x \log _{2} x+(a-x) \log _{2}(a-x)
$$
be symmetric about the line $x=\frac{1}{2}$. Then for any real numbers $x_{i} \in(0,1)(1 \leqslant i \leqslant 4)$ satisfying $\sum_{i=1}^{4} x_{i}=1$, the minimum value of $s=\sum_{i=1}^{4} x_{i} \log _{2} x_{i}$ is . $\qquad$ | -2 | 134 | 2 |
math | 3. The lengths of the diagonals of a rhombus and the length of its side form a geometric progression. Find the sine of the angle between the side of the rhombus and its larger diagonal, given that it is greater than $1 / 2$. | \sqrt{\frac{\sqrt{17}-1}{8}} | 55 | 14 |
math | ## Problem Statement
Calculate the definite integral:
$$
\int_{-3}^{0}\left(x^{2}+6 x+9\right) \sin 2 x \, d x
$$ | -\frac{17+\cos6}{4} | 44 | 11 |
math | Find the largest value of the expression $\frac{p}{R}\left( 1- \frac{r}{3R}\right)$ , where $p,R, r$ is, respectively, the perimeter, the radius of the circumscribed circle and the radius of the inscribed circle of a triangle. | \frac{5\sqrt{3}}{2} | 65 | 12 |
math | 9. Given that $\left\{a_{n}\right\}$ is a geometric sequence with the first term 1 and common ratio 2, and $\left\{b_{n}\right\}$ is an arithmetic sequence with the first term 2 and common difference 5, the numbers that appear in both sequences are arranged in ascending order to form the sequence $\left\{x_{n}\right\}$... | 2^{397} | 100 | 6 |
math | Condition of the problem
Find the derivative.
$$
y=\sqrt{x} \ln (\sqrt{x}+\sqrt{x+a})-\sqrt{x+a}
$$ | \frac{1}{2\sqrt{x}}\cdot\ln(\sqrt{x}+\sqrt{x+}) | 33 | 23 |
math | ## 49. Math Puzzle 6/69
A train, with a distance of $240 \mathrm{~m}$ between the first and last axle, travels at a speed of $72 \frac{\mathrm{km}}{\mathrm{h}}$ over a $36 \mathrm{~m}$ long bridge.
How much time elapses from the moment the first axle of the train reaches the beginning of the bridge until the last axl... | 13.8\mathrm{~} | 103 | 9 |
math | Problem 7.4. On Monday, 5 people in the class received fives in math, on Tuesday, 8 people received fives, on Wednesday - 6 people, on Thursday - 4 people, on Friday - 9 people. No student received fives on two consecutive days. What is the minimum number of students that could have been in the class | 14 | 76 | 2 |
math | Example 2 Given $f(x)=\frac{2 x}{1+x}$. Find
$$
\begin{array}{l}
f(i)+f(2)+\cdots+f(100)+f\left(\frac{1}{2}\right) \\
+f\left(\frac{2}{2}\right)+\cdots+f\left(\frac{100}{2}\right)+\cdots+f\left(\frac{1}{100}\right) \\
+f\left(\frac{2}{100}\right)+\cdots+f\left(\frac{100}{100}\right)=
\end{array}
$$ | 10000 | 145 | 5 |
math | 1) Find a $4$-digit number $\overline{PERU}$ such that $\overline{PERU}=(P+E+R+U)^U$. Also prove that there is only one number satisfying this property.
| 4913 | 48 | 4 |
math | 13. 4.3 * Given that $A$ is any point on the ellipse $x^{2}+4 y^{2}=4$, and $B$ is any point on the circle $x^{2}+(y-2)^{2}=\frac{1}{3}$. Find the maximum and minimum values of $|A B|$. | |AB|_{\max}=\frac{2\sqrt{21}}{3}+\frac{\sqrt{3}}{3},|AB|_{\}=1-\frac{\sqrt{3}}{3} | 76 | 47 |
math | For any integer $n \geq 2$, we define $A_{n}$ to be the number of positive integers $m$ with the following property: the distance from $n$ to the nearest non-negative multiple of $m$ is equal to the distance from $n^{3}$ to the nearest non-negative multiple of $m$. Find all integers $n \geq 2$ for which $A_{n}$ is odd.... | n \text{ is an even square} | 114 | 9 |
math | 2. Equation
$$
x^{2}-31 x+220=2^{x}\left(31-2 x-2^{x}\right)
$$
The sum of the squares of all real roots is $\qquad$ | 25 | 52 | 2 |
math | Example 2 (to item $6^{\circ}$). Solve the system
$$
\left\{\begin{aligned}
3 x_{1}-x_{2}+3 x_{3} & =5 \\
2 x_{1}-x_{2}+4 x_{3} & =5 \\
x_{1}+2 x_{2}-3 x_{3} & =0
\end{aligned}\right.
$$ | {\begin{pmatrix}x_{1}=1\\x_{2}=1\\x_{3}=1\end{pmatrix}.} | 92 | 30 |
math | 5. Let $E$ be a moving point inside square $ABCD$. It is known that the minimum value of the sum of the distances from $E$ to points $A$, $B$, and $C$ is $\sqrt{2}+\sqrt{6}$. Try to find the side length of this square. | 2 | 67 | 1 |
math | 9 Given three sets, $A=\{x \mid x \in \mathbf{R}$ and $x>1\}, B=\left\{y \left\lvert\, y=\lg \frac{2}{x+1}\right., x \in \mathrm{A}\right\}$, $C=\left\{z \left\lvert\, z=\frac{m^{2} x-1}{m x+1}\right., x \in \mathrm{A}\right\}$. If $C \subseteq B$, then the range of values for $m$ is $\qquad$. | (-\infty,-1]\cup{0} | 133 | 11 |
math | Example 3 (China Girls Mathematical Olympiad) Find all positive real numbers $a$ such that there exists a positive integer $n$ and $n$ pairwise disjoint infinite sets $A_{1}, A_{2}, \cdots, A_{n}$ satisfying $A_{1} \cup A_{2} \cup \cdots \cup A_{n}=\mathbf{Z}$, and for any two numbers $b>c$ in each $A_{i}$, we have $b-... | 2 | 115 | 1 |
math | Example 10 (1) Find the domain of the function $y=\frac{\sqrt{x^{2}-4}}{\log _{2}\left(x^{2}+2 x-3\right)}$;
(2) Given that the domain of the function $f(x)$ is $[-1,1]$. Find the domain of $f(a x)+f\left(\frac{x}{a}\right)$, where | (-\infty,-1-\sqrt{5})\cup(-1-\sqrt{5},-3)\cup[2,+\infty)(forpart1),[-,](when0<<1)or[-\frac{1}{},\frac{1}{}](when\geqslant1 | 90 | 65 |
math | Example 9. Find the integral $\int \sqrt{x^{2}+8 x+25} d x$. | \frac{x+4}{2}\sqrt{x^{2}+8x+25}+\frac{9}{2}\ln|x+4+\sqrt{x^{2}+8x+25}|+C | 25 | 46 |
math | 7. For any $x, y \in [0,1]$, the function
$$
f(x, y)=x \sqrt{1-y}+y \sqrt{1-x}
$$
has a maximum value of $\qquad$ . | 1 | 53 | 1 |
math | Example 4 (1992 "Friendship Cup" International Mathematics Competition Question) Find the largest natural number $x$, such that for every natural number $y$, $x$ divides $7^{y}+12 y-1$. | 18 | 51 | 2 |
math | ## Task 6 - V00606
What is the fraction with a single-digit denominator that is greater than $\frac{7}{9}$ and less than $\frac{8}{9}$? | \frac{5}{6} | 43 | 7 |
math | In a quarry, 50 stone blocks have been carved out. The stones can be arranged in such a way that in the row - starting from the second one - each stone weighs $2 \mathrm{~kg}$ more than the one in front of it. The weight of the first stone is $370 \mathrm{~kg}$. Can all the stone blocks be transported with $7 \mathrm{d... | 3016 | 107 | 4 |
math | Let $S$ be a $k$-element set.
[i](a)[/i] Find the number of mappings $f : S \to S$ such that
\[\text{(i) } f(x) \neq x \text{ for } x \in S, \quad \text{(ii) } f(f(x)) = x \text{ for }x \in S.\]
[i](b)[/i] The same with the condition $\text{(i)}$ left out. | \sum_{i=0}^{\left\lfloor \frac{k}{2} \right\rfloor} \binom{k}{2i} \times \frac{(2i)!}{i! \times 2^i} | 109 | 51 |
math | 7.4. Winnie-the-Pooh and Tigger are climbing two identical fir trees. Winnie-the-Pooh climbs up at half the speed of Tigger, but he descends three times faster than Tigger. Winnie-the-Pooh and Tigger started and finished at the same time. How many times faster does Tigger climb up than he descends? | 1.5 | 78 | 3 |
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