task_type stringclasses 1
value | problem stringlengths 23 3.94k | answer stringlengths 1 231 | problem_tokens int64 8 1.39k | answer_tokens int64 1 200 |
|---|---|---|---|---|
math | An integer minus 77, then multiplied by 8, and divided by 7, the quotient is 37, and there is a remainder. This integer is $\qquad$ | 110 | 39 | 3 |
math | 4・143 Solve the system of equations
$$\left\{\begin{array}{l}
x-y+z=1, \\
y-z+u=2, \\
z-u+v=3, \\
u-v+x=4, \\
v-x+y=5
\end{array}\right.$$ | \left\{\begin{array}{l}
x=0 \\
y=6 \\
z=7 \\
u=3 \\
v=-1
\end{array}\right.} | 64 | 38 |
math | 4. In rectangle $A B C D$, $A B=2, A D=1$, point $P$ on side $D C$ (including points $D, C$) and point $Q$ on the extension of $C B$ (including point $B$) satisfy $|\overrightarrow{D P}|=|\overrightarrow{B Q}|$, then the dot product $\overrightarrow{P A} \cdot \overrightarrow{P Q}$ of vector $\overrightarrow{P A}$ and ... | \frac{3}{4} | 125 | 7 |
math | In the 5-number lottery on September 10, 2005, the following numbers were drawn: 4, 16, 22, 48, 88. All five numbers are even, exactly four of them are divisible by 4, three by 8, and two by 16. In how many ways can five different numbers with these properties be selected from the integers ranging from 1 to 90? | 15180 | 96 | 5 |
math | 2. Solve the inequality $\frac{\sqrt{2-x}-2}{1-\sqrt{3-x}} \geq 1+\sqrt{3-x}$. | x\in[1;2) | 34 | 8 |
math | 4. Let $x$ and $y$ be positive numbers whose sum is 2. Find the maximum value of the expression $x^{2} y^{2}\left(x^{2}+y^{2}\right)$. | 2 | 48 | 1 |
math | Example 1. The vertex of the parabola $y=f(x)$ is $(-2,3)$, and the difference between the two roots of $\mathrm{f}(\mathrm{x})=0$ is 2. Find $\mathrm{f}(\mathrm{x})$. | f(x)=-3 x^{2}-12 x-9 | 59 | 14 |
math | Example 2. Find $\lim _{x \rightarrow 0} \frac{\sin x e^{2 x}-x}{5 x^{2}+x^{3}}$. | \frac{2}{5} | 38 | 7 |
math | 3. 20 different villages are located along the coast of a circular island. Each of these villages has 20 fighters, with all 400 fighters having different strengths.
Two neighboring villages $A$ and $B$ now have a competition in which each of the 20 fighters from village $A$ competes with each of the 20 fighters from v... | 290 | 172 | 3 |
math | 21. (2004 National College Entrance Examination, Tianjin Paper) From 4 boys and 2 girls, 3 people are randomly selected to participate in a speech contest. Let the random variable $\xi$ represent the number of girls among the 3 selected people.
(I) Find the distribution of $\xi$;
(II) Find the mathematical expectation ... | \frac{4}{5} | 109 | 7 |
math | ## Problem II - 3
Two spheres of radius $r$ are externally tangent. Three spheres of radius $R$ are externally tangent to each other, each tangent to the other two. Each of these spheres is also externally tangent to the first two.
Find the relationship between $R$ and $r$. | 6r | 63 | 2 |
math | Example 2 Define the function $f(x)$ on $\mathbf{R}$ that satisfies: $f(2+x)=2-f(x), f(x+3) \geqslant f(x)$. Try to find $f(x)$. | f(x)=1 | 52 | 4 |
math | 4. Given $a_{1}=1, a_{n+1}=2 a_{n}+\frac{n^{3}-2 n-2}{n^{2}+n}$. Then the general term formula of $\left\{a_{n}\right\}$ is $\qquad$ . | a_{n}=2^{n-1}+\frac{1-n^{2}}{n} | 63 | 21 |
math | 155.
From the Middle Sanctuary, one can exit through four doors $\mathrm{X}, \mathrm{Y}, \mathrm{Z}$, and $\mathrm{W}$. At least one of them leads to the Inner Sanctuary. Those who exit through another door are devoured by a fire-breathing dragon.
In the Middle Sanctuary, during the trial, there are eight priests A,... | X | 225 | 1 |
math | 13.296. Two excavator operators must complete a certain job. After the first one worked for 15 hours, the second one starts and finishes the job in 10 hours. If, working separately, the first one completed $1 / 6$ of the job, and the second one completed $1 / 4$ of the job, it would take an additional 7 hours of their ... | 20 | 109 | 2 |
math | \section*{Task 1 - 041011}
In a company where electric motors are assembled, the acquisition of a new conveyor belt system, which costs 105000 MDN, can reduce the labor costs per motor by 0.50 MDN and the overhead costs by 8800 MDN annually.
a) How many motors must be assembled annually at a minimum for the costs of ... | 52400 | 143 | 5 |
math | 11. Observe the array: $(1),(3,5),(7,9,11),(13,15,17$,
19), $\cdots \cdots$. Then 2003 is in the group. | 45 | 53 | 2 |
math | (2) Given that $S_{n}$ and $T_{n}$ are the sums of the first $n$ terms of the arithmetic sequences $\left\{a_{n}\right\}$ and $\left\{b_{n}\right\}$, respectively, and
$$
\frac{S_{n}}{T_{n}}=\frac{2 n+1}{4 n-2}(n=1,2, \cdots) \text {, then } \frac{a_{10}}{b_{3}+b_{18}}+\frac{a_{11}}{b_{6}+b_{15}}=
$$
$\qquad$ | \frac{41}{78} | 146 | 9 |
math | 1. It is known that $\sin x=2 \cos y-\frac{5}{2} \sin y, \cos x=2 \sin y-\frac{5}{2} \cos y$. Find $\sin 2 y$. | \frac{37}{40} | 51 | 9 |
math | 6. The minor axis of the ellipse $\rho=\frac{1}{2-\cos \theta}$ is equal to
保留了源文本的换行和格式。 | \frac{2}{3}\sqrt{3} | 35 | 11 |
math | Find the number of positive integers $m$ for which there exist nonnegative integers $x_0$, $x_1$ , $\dots$ , $x_{2011}$ such that
\[m^{x_0} = \sum_{k = 1}^{2011} m^{x_k}.\] | 16 | 71 | 2 |
math | Let $S=\{p/q| q\leq 2009, p/q <1257/2009, p,q \in \mathbb{N} \}$. If the maximum element of $S$ is $p_0/q_0$ in reduced form, find $p_0+q_0$. | 595 | 75 | 3 |
math | 3. Equilateral $\triangle A B C$ and square $A B D E$ share a common side $A B$, the cosine of the dihedral angle $C-A B-D$ is $\frac{\sqrt{3}}{3}, M, N$ are the midpoints of $A C, B C$ respectively, then the cosine of the angle formed by $E M, A N$ is $\qquad$ . | \frac{1}{6} | 90 | 7 |
math | 6. Let $a_{n}=1+2+\cdots+n, n \in \mathbf{N}_{+}, S_{m}=a_{1}+a_{2}+\cdots+a_{m}, m=1,2, \cdots, m$, then the number of elements in $S_{1}, S_{2}, \cdots, S_{2017}$ that are divisible by 2 but not by 4 is | 252 | 97 | 3 |
math | Task 1. Find all non-negative integers $n$ for which there exist integers $a$ and $b$ such that $n^{2}=a+b$ and $n^{3}=a^{2}+b^{2}$. | n=0,n=1,n=2 | 50 | 9 |
math | Let $2000 < N < 2100$ be an integer. Suppose the last day of year $N$ is a Tuesday while the first day of year $N+2$ is a Friday. The fourth Sunday of year $N+3$ is the $m$th day of January. What is $m$?
[i]Based on a proposal by Neelabh Deka[/i] | 23 | 87 | 2 |
math | XXXIV OM - I - Problem 1
$ A $ tosses a coin $ n $ times, $ B $ tosses it $ n+1 $ times. What is the probability that $ B $ will get more heads than $ A $? | \frac{1}{2} | 51 | 7 |
math | 13. (15 points) In the sequence $\left\{a_{n}\right\}$, $a_{n}=2^{n} a+b n-80\left(a, b \in \mathbf{Z}_{+}\right)$.
It is known that the minimum value of the sum of the first $n$ terms $S_{n}$ is obtained if and only if $n=6$, and $7 \mid a_{36}$. Find the value of $\sum_{i=1}^{12}\left|a_{i}\right|$. | 8010 | 124 | 4 |
math | ## $\mathrm{I}-1$
Let $\mathbb{R}$ denote the set of all real numbers. For each pair $(\alpha, \beta)$ of nonnegative real numbers subject to $\alpha+\beta \geq 2$, determine all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ satisfying
$$
f(x) f(y) \leq f(x y)+\alpha x+\beta y
$$
for all real numbers $x$ and $y$.... | f(z)=z+1for(\alpha,\beta)=(1,1) | 108 | 16 |
math | \section*{Exercise 2 - 061012}
How many natural numbers \(n<1000\) are there that are neither divisible by 3 nor by 5? | 533 | 42 | 3 |
math | Someone for 10 years deposits $500 \mathrm{~K}-\mathrm{t}$ in the savings bank at the beginning of each year; then for another 10 years, withdraws $500 \mathrm{~K}-\mathrm{t}$ from the savings bank at the beginning of each year. How much money will they have at the end of the 20th year? $p=4$. | 2996.4\mathrm{~K} | 91 | 12 |
math | # Problem №5
In a train, there are 18 identical cars. In some of the cars, exactly half of the seats are free, in some others, exactly one third of the seats are free, and in the rest, all seats are occupied. At the same time, in the entire train, exactly one ninth of all seats are free. In how many cars are all seats... | 13 | 82 | 2 |
math | A courier travels from location $A$ to location $B$ in 14 hours: another courier starts at the same time as the first, from a place 10 km behind $A$, and arrives at $B$ at the same time as the first. The latter covers the last $20 \mathrm{~km}$ in half an hour less time than the first. What is the distance between $A$ ... | 70 | 92 | 2 |
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 | 208. $\int x e^{x} d x$ | xe^{x}-e^{x}+C | 14 | 10 |
math | 12. (6 points) Three natural numbers, the largest is 6 more than the smallest, another is their average, and the product of the three numbers is 46332. The largest number is $\qquad$ . | 39 | 50 | 2 |
math | Mary told John her score on the American High School Mathematics Examination (AHSME), which was over 80. From this, John was able to determine the number of problems Mary solved correctly. If Mary's score had been any lower, but still over 80, John could not have determined this. What was Mary's score? (Recall that... | 119 | 146 | 3 |
math | 8.019. $\cos \frac{x}{2} \cos \frac{3 x}{2}-\sin x \sin 3 x-\sin 2 x \sin 3 x=0$. | \frac{\pi}{9}(2k+1)\quadk\in\mathbb{Z} | 45 | 22 |
math | 42nd Putnam 1981 Problem B2 What is the minimum value of (a - 1) 2 + (b/a - 1) 2 + (c/b - 1) 2 + (4/c - 1) 2 , over all real numbers a, b, c satisfying 1 ≤ a ≤ b ≤ c ≤ 4. Solution | 12-8\sqrt{2} | 81 | 9 |
math | Example 3 Find all integer pairs $(m, n)(m, n \geqslant 2)$, such that for any integer $x$ we have
$$
x^{n} \equiv x(\bmod m)
$$ | m=p_{1} p_{2} \cdots p_{k}, n=1+u\left[p_{1}-1, p_{2}-1, \cdots, p_{k}-1\right] | 50 | 47 |
math | 6. The numbers $1,2, \ldots, 2016$ are written on a board. It is allowed to erase any two numbers and replace them with their arithmetic mean. How should one proceed to ensure that the number 1000 remains on the board? | 1000 | 60 | 4 |
math | ## Task A-4.4.
Determine the number of complex solutions to the equation
$$
z^{2019}=z+\bar{z}
$$ | 2019 | 35 | 4 |
math | Example 9 Given real numbers $a, b, c, d$ satisfy $a+b+c+d=3, a^{2}+2 b^{2}+$ $3 c^{2}+6 d^{2}=5$, try to find the range of values for $a$.
untranslated text remains in its original format and lines. | [1,2] | 71 | 5 |
math | 2A. Jane read five books. From these five books, 5 sets of four books can be formed. The four books in each of these sets together had 913, 973, 873, 1011, and 1002 pages. How many pages did each of the five books have? | =182,b=191,=320,=220,e=280 | 73 | 24 |
math | There is a strip $1 \times 99$, divided into 99 cells $1 \times 1$, which are alternately painted in black and white. It is allowed to repaint simultaneously all cells of any rectangular cell $1 \times k$. What is the minimum number of repaintings required to make the entire strip monochromatic?
# | 49 | 72 | 2 |
math | 【Question 6】
Weiwei is 8 years old this year, and his father is 34 years old. In $\qquad$ years, his father's age will be three times Weiwei's age. | 5 | 46 | 1 |
math | 4・169 Solve the equation in the set of natural numbers
$$
x^{y}=y^{x}(x \neq y) .
$$ | 2,4or4,2 | 33 | 7 |
math | 3. Represent the fraction $\frac{179}{140}$ as a sum of three positive fractions with single-digit denominators. Explain your answer! | \frac{1}{4}+\frac{3}{5}+\frac{3}{7}=\frac{179}{140} | 33 | 32 |
math | Example 2 Find four consecutive integers, each of which is divisible by $2^{2}, 3^{2}, 5^{2}$, and $7^{2}$ respectively. | 29348,29349,29350,29351 | 38 | 23 |
math | Example 13 (2004 National High School Competition Question) A "level-passing game" has the following rules: at the $n$-th level, a die must be rolled $n$ times. If the sum of the points from these $n$ rolls is greater than $2^{n}$, the player passes the level. The questions are:
(1) What is the maximum number of levels... | \frac{100}{243} | 167 | 11 |
math | Find the least positive integer $m$ such that $m^2 - m + 11$ is a product of at least four not necessarily distinct primes. | 132 | 33 | 3 |
math | Example 6 Let $f(x)$ represent a quartic polynomial in $x$. If $f(1)=f(2)=f(3)=0, f(4)=6$, $f(5)=72$, then the last digit of $f(2010)$ is $\qquad$. ${ }^{3}$
(2010, International Cities Mathematics Invitational for Youth) | 2 | 85 | 1 |
math | Determine the polynomial $P\in \mathbb{R}[x]$ for which there exists $n\in \mathbb{Z}_{>0}$ such that for all $x\in \mathbb{Q}$ we have: \[P\left(x+\frac1n\right)+P\left(x-\frac1n\right)=2P(x).\]
[i]Dumitru Bușneag[/i] | P(x) = ax + b | 92 | 8 |
math | 3. We will denote $\max (A, B, C)$ as the greatest of the numbers $A, B, C$. Find the smallest value of the quantity $\max \left(x^{2}+|y|,(x+2)^{2}+|y|, x^{2}+|y-1|\right)$. | 1.5 | 72 | 3 |
math | 2. If $n$ positive real numbers $x_{1}, x_{2}, \cdots, x_{n}$ satisfy the equation
$$
\sum_{k=1}^{n}\left|\lg x_{k}\right|+\sum_{k=1}^{n}\left|\lg \frac{1}{x_{k}}\right|=\left|\sum_{k=1}^{n} \lg x_{k}\right|
$$
then the values of $x_{1}, x_{2}, \cdots, x_{n}$ are $\qquad$ | x_{1}=x_{2}=\cdots=x_{n}=1 | 124 | 16 |
math | A square pyramid with base $ABCD$ and vertex $E$ has eight edges of length 4. A plane passes through the midpoints of $\overline{AE}$, $\overline{BC}$, and $\overline{CD}$. The plane's intersection with the pyramid has an area that can be expressed as $\sqrt{p}$. Find $p$. | p = 80 | 80 | 6 |
math | Example 11 Let the function $f: \mathbf{R} \rightarrow \mathbf{R}$, for all $x, y \in \mathbf{R}$, satisfy $f\left[x^{2}+f(y)\right]=y+[f(x)]^{2}$.
Find $f(x)$. | f(x)=x | 70 | 4 |
math | Problem 2. Masha wrote the numbers $4,5,6, \ldots, 16$ on the board, and then erased one or several of them. It turned out that the remaining numbers on the board cannot be divided into several groups such that the sums of the numbers in the groups are equal. What is the greatest possible value of the sum of the remain... | 121 | 83 | 3 |
math | 4. (8 points) There are four people, A, B, C, and D. B owes A 1 yuan, C owes B 2 yuan, D owes C 3 yuan, and A owes D 4 yuan. To settle all debts among them, A needs to pay $\qquad$ yuan.
保留源文本的换行和格式，翻译结果如下：
4. (8 points) There are four people, A, B, C, and D. B owes A 1 yuan, C owes B 2 yuan, D owes C 3 yuan, and A o... | 3 | 145 | 1 |
math | 8. Given that $a_{1}, a_{2}, a_{3}, a_{4}, a_{5}$ is a permutation of $1,2,3,4,5$, and satisfies $\left|a_{i}-a_{i+1}\right| \neq 1(i=1,2,3,4)$. Then the number of permutations $a_{1}, a_{2}, a_{3}, a_{4}, a_{5}$ that meet the condition is $\qquad$. | 14 | 108 | 2 |
math | ## Task 2 - 140612
Bernd and Monika are discussing the last meeting of their Young Mathematicians group, where exactly 6 more boys were present than girls.
Bernd says that at this event, exactly 2 of the 25 circle participants were missing.
Monika counters after some thought that this cannot be correct.
Who is righ... | Monika | 78 | 2 |
math | 23. Find all such prime numbers, so that each of them can be represented as the sum and difference of two prime numbers. | 5 | 27 | 1 |
math | 11.2. In a row, 21 numbers are written sequentially: from 2000 to 2020 inclusive. Enthusiastic numerologists Vova and Dima performed the following ritual: first, Vova erased several consecutive numbers, then Dima erased several consecutive numbers, and finally, Vova erased several consecutive numbers (at each step, the... | 2009,2010,2011 | 145 | 14 |
math | Find all pairs of positive integers $(x,y) $ for which $x^3 + y^3 = 4(x^2y + xy^2 - 5) .$ | (1, 3), (3, 1) | 37 | 12 |
math | Let $x$, $y$, $z$ be arbitrary positive numbers such that $xy+yz+zx=x+y+z$.
Prove that $$\frac{1}{x^2+y+1} + \frac{1}{y^2+z+1} + \frac{1}{z^2+x+1} \leq 1$$.
When does equality occur?
[i]Proposed by Marko Radovanovic[/i] | 1 | 95 | 1 |
math | 9. (16 points) Given that the three interior angles of $\triangle A B C$ satisfy
$$
\begin{array}{l}
\angle A+\angle C=2 \angle B, \\
\cos A+\cos C=-\frac{\sqrt{2} \cos A \cdot \cos C}{\cos B} .
\end{array}
$$
Find the value of $\cos \frac{A-C}{2}$. | \frac{\sqrt{2}}{2} | 94 | 10 |
math | 9.194. $25 \cdot 2^{x}-10^{x}+5^{x}>25$. | x\in(0;2) | 29 | 8 |
math | The Fibonacci sequence is defined recursively by $F_{n+2}=F_{n+1}+F_{n}$ for $n \in \mathbb{Z}$ and $F_{1}=F_{2}=1$. Determine the value of:
$$
\left(1-\frac{F_{2}^{2}}{F_{3}^{2}}\right)\left(1-\frac{F_{3}^{2}}{F_{4}^{2}}\right) \cdot \ldots \cdot\left(1-\frac{F_{2019}^{2}}{F_{2020}^{2}}\right)
$$ | \frac{F_{2021}}{2F_{2019}F_{2020}} | 143 | 26 |
math | ## Problem Statement
Find the derivative.
$$
y=(\tan x)^{\ln (\tan x) / 4}
$$ | (\tanx)^{(\ln(\tanx)/4)}\cdot\frac{\ln(\tanx)}{\sin2x} | 27 | 28 |
math | Task 3. Find the smallest possible value of
$$
x y+y z+z x+\frac{1}{x}+\frac{2}{y}+\frac{5}{z}
$$
for positive real numbers $x, y$ and $z$. | 3\sqrt[3]{36} | 55 | 9 |
math | 2. Find all values of $n, n \in N$, for which the sum of the first terms of the sequence $a_{k}=3 k^{2}-3 k+1, \quad k \in N, \quad$ is equal to the sum of the first $n$ terms of the sequence $b_{k}=2 k+89, k \in N$ (12 points) | 10 | 86 | 2 |
math | The value of
\[\frac{1}{20}-\frac{1}{21}+\frac{1}{20\times 21}\]
can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $m+n$.
[i]Proposed by Nathan Xiong[/i] | 211 | 76 | 3 |
math | 26. (KOR 4) Let $n$ be a positive integer and let $a, b$ be given real numbers. Determine the range of $x_{0}$ for which
$$ \sum_{i=0}^{n} x_{i}=a \quad \text { and } \quad \sum_{i=0}^{n} x_{i}^{2}=b $$
where $x_{0}, x_{1}, \ldots, x_{n}$ are real variables. | \frac{a-\sqrt{D} / 2}{n+1} \leq x_{0} \leq \frac{a+\sqrt{D} / 2}{n+1} | 108 | 44 |
math | Exercise 5
Calculate using a simple method.
1. $682+325$
$573+198$
2. $897+234$
$788+143$
3. $694+367$
$595+698$ | 1007,771,1131,931,1061,1293 | 70 | 27 |
math | ## Task 24/83
Let $p_{1}$ and $p_{2}$ be two consecutive prime numbers with $p_{1}<p_{2}$ and $f(x)$ a polynomial of degree $n$ in $x$ with integer coefficients $a_{i}(i=0 ; 1 ; 2 ; \ldots ; n)$. Determine $p_{1}$ and $p_{2}$ from $f\left(p_{1}\right)=1234$ and $f\left(p_{2}\right)=4321$. | p_{1}=2,p_{2}=3 | 121 | 10 |
math |
4. There are four basket-ball players $A, B, C, D$. Initially, the ball is with $A$. The ball is always passed from one person to a different person. In how many ways can the ball come back to $A$ after seven passes? (For example $A \rightarrow C \rightarrow B \rightarrow D \rightarrow A \rightarrow B \rightarrow C \r... | 546 | 128 | 3 |
math | 1.1. Find the greatest negative root of the equation $\sin 2 \pi x=\sqrt{3} \sin \pi x$. | -\frac{1}{6} | 30 | 7 |
math | Problem 3. A row of 101 numbers is written (the numbers are not necessarily integers). The arithmetic mean of all the numbers without the first one is 2022, the arithmetic mean of all the numbers without the last one is 2023, and the arithmetic mean of the first and last numbers is 51. What is the sum of all the writte... | 202301 | 83 | 6 |
math | The sequence $\mathrm{Az}\left(a_{i}\right)$ is defined as follows: $a_{1}=0, a_{2}=2, a_{3}=3, a_{n}=\max _{1<d<n}\left\{a_{d} \cdot a_{n-d}\right\}(n=4,5,6, \ldots)$. Determine the value of $a_{1998}$. | 3^{666} | 93 | 6 |
math | 7. Students have three types of number cards: $1$, $2$, and $3$, with many cards of each type. The teacher asks each student to take out two or three number cards to form a two-digit or three-digit number. If at least three students form the same number, then these students have at least people. | 73 | 69 | 2 |
math | 170. Find the derivatives of the following functions:
1) $y=5 \arcsin k x+3 \arccos k x$; 2) $y=\arcsin \frac{a}{x}-\operatorname{arcctg} \frac{x}{a}$;
2) $r=\operatorname{arctg} \frac{m}{\varphi}+\operatorname{arcctg}(m \operatorname{ctg} \varphi)$; $r^{\prime}(0)$ ?, $r^{\prime}(\pi)$ ? | \begin{aligned}1)&\quady^{\}=\frac{2k}{\sqrt{1-k^{2}x^{2}}}\\2)&\quady^{\}=\frac{}{^{2}+x^{2}}-\frac{}{|x|\sqrt{x^{2}-^{2}}}\\3)&\quadr^{\}(0)=0,\ | 126 | 79 |
math | $ABCDEF GH$ is a regular octagon with $10$ units side . The circle with center $A$ and radius $AC$ intersects the circle with center $D$ and radius $CD$ at point $ I$, different from $C$. What is the length of the segment $IF$? | 10 | 66 | 2 |
math | 【18】A column of soldiers 600 meters long is marching at a speed of 2 meters per second. A soldier needs to run from the end of the column to the front, and then immediately return to the end. If he runs at a speed of 3 meters per second, the total distance he travels back and forth is ( ) meters. | 720 | 75 | 3 |
math | 2.199. $\left(\frac{\sqrt[3]{m n^{2}}+\sqrt[3]{m^{2} n}}{\sqrt[3]{m^{2}}+2 \sqrt[3]{m n}+\sqrt[3]{n^{2}}}-2 \sqrt[3]{n}+\frac{m-n}{\sqrt[3]{m^{2}}-\sqrt[3]{n^{2}}}\right):(\sqrt[6]{m}+\sqrt[6]{n})$. | \sqrt[6]{}-\sqrt[6]{n} | 110 | 13 |
math | 2. Three circles are inscribed in a corner - a small, a medium, and a large one. The large circle passes through the center of the medium one, and the medium one through the center of the small one. Determine the radii of the medium and large circles if the radius of the smaller one is $r$ and the distance from its cen... | \frac{}{-r};\quad\frac{^{2}r}{(-r)^{2}} | 83 | 22 |
math | G4.4 If $W=2006^{2}-2005^{2}+2004^{2}-2003^{2}+\ldots+4^{2}-3^{2}+2^{2}-1^{2}$, find the value of $W$. | 2013021 | 66 | 7 |
math | Using each of the digits $1,2,3,\ldots ,8,9$ exactly once,we form nine,not necassarily distinct,nine-digit numbers.Their sum ends in $n$ zeroes,where $n$ is a non-negative integer.Determine the maximum possible value of $n$. | 8 | 64 | 3 |
math | [ [rectangular parallelepipeds]
The diagonals of the faces of a rectangular parallelepiped are $\sqrt{3}, \sqrt{5}$, and 2. Find its volume. | \sqrt{6} | 41 | 5 |
math | We have a $6 \times 6$ square, partitioned into 36 unit squares. We select some of these unit squares and draw some of their diagonals, subject to the condition that no two diagonals we draw have any common points. What is the maximal number of diagonals that we can draw? | 18 | 65 | 2 |
math | [ Regular Pyramid ]
Find the volume of a regular hexagonal pyramid with a lateral edge $b$ and the area $Q$ of a lateral face.
# | \frac{\sqrt{3}}{2}(2b^2-\sqrt{4b^{2}-16Q^{2}})\sqrt{\sqrt{4b^{2}-16Q^{2}}-b^{2}} | 32 | 49 |
math | 3. Give all possible triples $(a, b, c)$ of positive integers with the following properties:
- $\operatorname{gcd}(a, b)=\operatorname{gcd}(a, c)=\operatorname{gcd}(b, c)=1$;
- $a$ is a divisor of $a+b+c$;
- $b$ is a divisor of $a+b+c$;
- $c$ is a divisor of $a+b+c$.
(For given numbers $x$ and $y$, $\operatorname{gcd... | (1,1,1),(1,1,2),(1,2,1),(2,1,1),(1,2,3),(1,3,2),(2,1,3),(2,3,1),(3,1,2),(3,2,1) | 132 | 61 |
math | Four, on a circular road, there are 4 middle schools arranged clockwise: $A_{1}$, $A_{2}$, $A_{3}$, $A_{4}$. They have 15, 8, 5, and 12 color TVs, respectively. To make the number of color TVs in each school the same, some schools are allowed to transfer color TVs to adjacent schools. How should the TVs be transferred ... | 10 | 112 | 2 |
math | Let $x$ and $y$ be real numbers such that $x+y=10$ and $x^{3}+y^{3}=400$, determine the value of $x^{2}+y^{2}$. | 60 | 50 | 2 |
math | In base-$2$ notation, digits are $0$ and $1$ only and the places go up in powers of $-2$. For example, $11011$ stands for $(-2)^4+(-2)^3+(-2)^1+(-2)^0$ and equals number $7$ in base $10$. If the decimal number $2019$ is expressed in base $-2$ how many non-zero digits does it contain ? | 6 | 101 | 1 |
math | Determine the number of pairs $(A, B)$ of subsets (possibly empty) of $\{1,2, \ldots, 10\}$ such that $A \cap B=\emptyset$.
N.B. If $A \neq B$ then $(A, B) \neq(B, A)$. | 59049 | 70 | 5 |
math | 8.129 Let the sequence $\left\{x_{n}\right\}$ satisfy $x_{1}=5$, and
$$x_{n+1}=x_{n}^{2}-2, n=1,2, \cdots$$
Find: $\lim _{n \rightarrow \infty} \frac{x_{n+1}}{x_{1} x_{2} \cdots x_{n}}$. | \sqrt{21} | 94 | 6 |
math | A kite is inscribed in a circle with center $O$ and radius $60$. The diagonals of the kite meet at a point $P$, and $OP$ is an integer. The minimum possible area of the kite can be expressed in the form $a\sqrt{b}$, where $a$ and $b$ are positive integers and $b$ is squarefree. Find $a+b$. | 239 | 85 | 5 |
math | Find the number of permutations $x_1, x_2, x_3, x_4, x_5$ of numbers $1, 2, 3, 4, 5$ such that the sum of five products \[x_1x_2x_3 + x_2x_3x_4 + x_3x_4x_5 + x_4x_5x_1 + x_5x_1x_2\]
is divisible by $3$. | 80 | 109 | 2 |
math | 1. A pedestrian left point $A$ for point $B$, and after some delay, a second pedestrian followed. When the first pedestrian had walked half the distance, the second had walked 15 km, and when the second pedestrian had walked half the distance, the first had walked 24 km. Both pedestrians arrived at point $B$ simultaneo... | 40 | 87 | 2 |
math | ( JBMO 2018 )
$n$ numbers are written on a board, satisfying the following properties:
- all the numbers have 3 digits
- none of them contain a 0
- two different numbers never have the same hundreds digit, or the same tens digit, or the same units digit
- the sum of the digits of each number is 9
What is the maximum... | 5 | 88 | 1 |
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