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 | Example 5 Given $P(x)=x^{5}+a_{1} x^{4}+a_{2} x^{3}+$ $a_{3} x^{2}+a_{4} x+a_{5}$, and when $k=1,2,3,4$, $P(k)$ $=k \times 1$ 997. Then $P(10)-P(-5)=$ $\qquad$ | 75315 | 97 | 5 |
math | 2. In 8 "Y" grade, there are quite a few underachievers, but Vovochka performs the worst of all. The pedagogical council decided that either Vovochka must correct his failing grades by the end of the quarter, or he will be expelled. If Vovochka corrects his grades, then 24% of the class will be underachievers, but if h... | 28 | 118 | 2 |
math | Example 11 Let $y=\frac{x}{1+x}=f(x)$, and $f(1)$ represents the value of $y$ when $x=1$, i.e., $f(1)=\frac{1}{1+1}=\frac{1}{2}$; $f\left(\frac{1}{2}\right)$ represents the value of $y$ when $x=\frac{1}{2}$, i.e., $f\left(\frac{1}{2}\right)=$ $\frac{\frac{1}{2}}{1+\frac{1}{2}}=\frac{1}{3} ; \cdots \cdots$. Try to find
... | 2006 | 248 | 4 |
math | Using the digits $1,2,3,4,5,6,7,8,9$ to form all possible four-digit numbers without repeating any digit, the sum of these numbers is $\qquad$. | 16798320 | 44 | 8 |
math | 13.302. Two points move uniformly in the same direction along a circle 60 m in circumference. One of them completes a full revolution 5 seconds faster than the other. At the same time, the points coincide every 1 minute. Determine the speeds of the points. | 3 | 60 | 1 |
math | 6. Given non-negative real numbers $a, b, c, x, y, z$ satisfy
$$
a+b+c=x+y+z=1 \text {. }
$$
Find the maximum value of $\left(a-x^{2}\right)\left(b-y^{2}\right)\left(c-z^{2}\right)$. | \frac{1}{16} | 69 | 8 |
math | Solve the following system of equations:
$$
\begin{aligned}
x+y & =2 a \ldots \\
x y\left(x^{2}+y^{2}\right) & =2 b^{4} \ldots
\end{aligned}
$$
What is the condition for obtaining real solutions?
Num. values: $\quad a=10, \quad b^{4}=9375$. | 15,5or5,15 | 89 | 9 |
math | 5. Find all solutions of the equation $(x+1)(x+2)(x+3)(x+4)=120$ in the set of complex
(8) numbers. | x_{1}=-6,x_{2}=1,x_{3}=\frac{-5-i\sqrt{39}}{2},x_{4}=\frac{-5+i\sqrt{39}}{2} | 40 | 47 |
math | 2. For $x, y, z \in(0,1]$ find the maximum value of the expression
$$
A=\frac{\sqrt{8 x^{4}+y}+\sqrt{8 y^{4}+z}+\sqrt{8 z^{4}+x}-3}{x+y+z}
$$ | 2 | 69 | 1 |
math | 8. Solve the system
$$
\left\{\begin{array}{l}
5 x^{2}+3 y^{2}+3 x y+2 x z-y z-10 y+5=0 \\
49 x^{2}+65 y^{2}+49 z^{2}-14 x y-98 x z+14 y z-182 x-102 y+182 z+233=0
\end{array}\right.
$$ | (0,1,-2),(\frac{2}{7},1,-\frac{12}{7}) | 112 | 24 |
math | 3. If real numbers $x, y, z$ satisfy $x^{2}+y^{2}+z^{2}=1$, find the maximum value of $\left(x^{2}-y z\right)\left(y^{2}-z x\right)\left(z^{2}-x y\right)$.
(2016, Korean Mathematical Olympiad) | \frac{1}{8} | 78 | 7 |
math | 3. Arrange $1,2, \cdots, k$ in a row so that each number is strictly greater than all the numbers preceding it, or strictly less than all the numbers preceding it. Let the number of different arrangements be $a_{k}(k=1,2, \cdots)$. Then $a_{n}=$ $\qquad$ | 2^{n-1} | 75 | 6 |
math | 5. In a regular tetrahedron \( P-ABC \), \( AB=1, AP=2 \). A plane \(\alpha\) passing through \( AB \) bisects its volume. Then the cosine of the angle formed by the edge \( PC \) and the plane \(\alpha\) is \(\qquad\) | \frac{3\sqrt{5}}{10} | 70 | 13 |
math | Task 1. Electronic clocks show the time: hours, minutes, and seconds, for example, 18:00:00. One day, two digits on the clock went out, and only the digits $2,0,2,2$ (in that exact order) remained. Name the latest moment in the day when this could have happened.
[3 points
(М. А. Калинин)] | 23:50:22 | 89 | 8 |
math | 16. (9th "Hope Cup" Invitational Training Question) If the function $f\left(x-\frac{1}{x}\right)=\frac{x}{x^{2}-1}-x^{2}-\frac{1}{x^{2}}(x \neq 0$, $x \neq \pm 1)$, then $f(x)=$ $\qquad$ . | -x^{2}+\frac{1}{x}-2 | 85 | 12 |
math | Let $n \ge 3$ be a fixed integer. The numbers $1,2,3, \cdots , n$ are written on a board. In every move one chooses two numbers and replaces them by their arithmetic mean. This is done until only a single number remains on the board.
Determine the least integer that can be reached at the end by an appropriate seque... | \left\lceil \frac{n+1}{2} \right\rceil | 92 | 18 |
math | 3. Determine all triples $(x, y, z)$ consisting of three distinct real numbers that satisfy the following system of equations:
$$
\begin{aligned}
x^{2}+y^{2} & =-x+3 y+z, \\
y^{2}+z^{2} & =x+3 y-z, \\
x^{2}+z^{2} & =2 x+2 y-z .
\end{aligned}
$$ | (x,y,z)=(-\frac{3}{2},\frac{5}{2},-\frac{1}{2})(x,y,z)=(0,1,-2) | 94 | 36 |
math | 9. (16 points) Let the sequence $\left\{a_{n}\right\}$ satisfy
$$
a_{1}=1, a_{n+1}=2 a_{n}+n\left(1+2^{n}\right)(n=1,2, \cdots) \text {. }
$$
Try to find the general term $a_{n}$ of the sequence. | a_{n}=2^{n-2}\left(n^{2}-n+6\right)-n-1(n \geqslant 2) | 86 | 33 |
math | 5.69 11 theater troupes participate in a performance, with some of the troupes scheduled to perform each day, while the rest join the general audience. By the end of the performance, each troupe, apart from their own performance days, must have watched at least one performance by each of the other troupes. How many day... | 6 | 83 | 1 |
math | 2. Explanation: How many divisors does the number 360 have? What is the sum of these divisors? | 24divisors,sum1170 | 26 | 9 |
math | 1B. In a goldsmith's workshop, a mixture of $9 \mathrm{~kg}$ needs to be made in which gold and silver will be in the ratio 7:11. There are two mixtures available. In the first, the quantities of gold and silver are in the ratio $4: 5$, and in the second, in the ratio $2: 5$. How much should be taken from each mixture ... | 5.85,3.15 | 100 | 9 |
math | 2. 79 In a game, scoring is as follows: answering an easy question earns 3 points, and answering a difficult question earns 7 points. Among the integers that cannot be the total score of a player, find the maximum value. | 11 | 51 | 2 |
math | 1. How many five-digit numbers of the form $\overline{37 a b c}$ are there such that each of the numbers $\overline{37 a b c}, \overline{37 b c a}$, and $\overline{37 c a b}$ is divisible by 37? | 28 | 67 | 2 |
math | ## Task Condition
Find the derivative.
$y=x^{e^{x}} \cdot x^{9}$ | y^{\}=x^{e^{x}}\cdotx^{9}\cdot(e^{x}\cdot(\lnx+\frac{1}{x})+\frac{9}{x}) | 22 | 39 |
math | Given an integer $n\ge 2$, compute $\sum_{\sigma} \textrm{sgn}(\sigma) n^{\ell(\sigma)}$, where all $n$-element permutations are considered, and where $\ell(\sigma)$ is the number of disjoint cycles in the standard decomposition of $\sigma$. | n! | 68 | 3 |
math | 4.1. A two-digit natural number $\overline{a b}$ is randomly selected from 21 to 45 (the probability of selecting each number is the same). The probability that the number $\overline{a 8573 b}$ is divisible by 6 is $n$ percent. Find $n$. | 16 | 70 | 2 |
math | 8.3. On 10 cards, different natural numbers are written, their sum is 60. Masha laid out the cards in ascending order. What can the sum of the numbers on the two middle cards be? Find all possible answers. | 11,12 | 52 | 5 |
math | 10. If real numbers $a, b, c, d, e$ satisfy $a+b+c+d+e=8, a^{2}+b^{2}+c^{2}+d^{2}+e^{2}=16$, then the sum of the maximum and minimum values of $e$ is $\qquad$ | \frac{16}{5} | 73 | 8 |
math | # Task 4. Maximum 20 points
A client of a brokerage firm deposited 12,000 rubles into a brokerage account at an exchange rate of 60 rubles per dollar, instructing the broker to invest the amount in bonds of foreign banks with a guaranteed yield of $12\%$ per year in dollars.
(a) Determine the amount in rubles that th... | 16742.4 | 186 | 7 |
math | 1. Fill in the same digit in the $\square$ below to make the equation true. $97+\square \times(19+91 \div \square)=321, \quad \square=$ $\qquad$ . | 7 | 51 | 1 |
math | The lengths of the sides of a triangle with positive area are $\log_{10} 12$, $\log_{10} 75$, and $\log_{10} n$, where $n$ is a positive integer. Find the number of possible values for $n$. | 893 | 60 | 3 |
math | Determine the number of pairs of integers $(m, n)$ such that
$$
\sqrt{n+\sqrt{2016}}+\sqrt{m-\sqrt{2016}} \in \mathbb{Q}
$$ | 1 | 50 | 1 |
math | 2. (6 points) The 50th term of the arithmetic sequence $3, 5, 7, 9 \cdots$ is
保留源文本的换行和格式，翻译结果如下：
2. (6 points) The 50th term of the arithmetic sequence $3, 5, 7, 9 \cdots$ is | 101 | 78 | 3 |
math | 2.009. $\frac{2 \sqrt{1+\frac{1}{4}\left(\sqrt{\frac{1}{t}}-\sqrt{t}\right)^{2}}}{\sqrt{1+\frac{1}{4}\left(\sqrt{\frac{1}{t}}-\sqrt{t}\right)^{2}}-\frac{1}{2}\left(\sqrt{\frac{1}{t}}-\sqrt{t}\right)}$. | \frac{1+}{} | 98 | 7 |
math | Suppose that $a$ is an integer. A sequence $x_{1}, x_{2}, x_{3}, x_{4}, \ldots$ is constructed with
- $x_{1}=a$,
- $x_{2 k}=2 x_{2 k-1}$ for every integer $k \geq 1$, and
- $x_{2 k+1}=x_{2 k}-1$ for every integer $k \geq 1$.
For example, if $a=2$, then
$$
x_{1}=2 \quad x_{2}=2 x_{1}=4 \quad x_{3}=x_{2}-1=3 \quad x_{... | 1409 | 281 | 4 |
math | For which values of $n$ is the fraction
$$
\frac{3 n^{2}+2 n+4}{n+1}
$$
irreducible? | n\not\equiv4\bmod5 | 37 | 9 |
math | 7.174. $3^{\log _{3}^{2} x}+x^{\log _{3} x}=162$.
Translate the above text into English, please retain the original text's line breaks and format, and output the translation result directly.
7.174. $3^{\log _{3}^{2} x}+x^{\log _{3} x}=162$. | \frac{1}{9};9 | 96 | 8 |
math | 8.275. $\frac{\cot 4 t}{\sin ^{2} t}+\frac{\cot t}{\sin ^{2} 4 t}=0$. | t_{1}=\frac{\pik}{5},k\neq5;t_{2}=\frac{\pi}{6}(2n+1),n\neq3+1,k,n,\inZ | 40 | 45 |
math | Task 5. Determine all pairs of prime numbers $(p, q)$ such that $p^{2}+5 p q+4 q^{2}$ is the square of an integer. | (5,11),(13,3),(7,5) | 39 | 15 |
math | 15. (26th Russian Mathematical Olympiad, 11th Grade Problem) Find all functions $f: \mathbf{R} \rightarrow \mathbf{R}$, which for all $x, y, z \in \mathbf{R}$ satisfy the inequality $f(x+y)+f(y+z)+f(z+x) \geqslant 3 f(x+2 y+3 z)$. | f(x)\equiv | 90 | 4 |
math | II. (25 points) Given a two-digit number, whose tens and units digits are $p$ and $q$ respectively, the quadratic function $y=x^{2}+q x+p$ intersects the $x$-axis at two distinct points $A$ and $B$, with the vertex at $C$, and $S_{\triangle A B C} \leqslant 1$.
(1) Find the range of $q^{2}-4 p$;
(2) Find all such two-d... | 23,65,34,86 | 114 | 11 |
math | Solve in $\mathbb{Z}$, then in $\mathbb{Q}$, the equation: $2 x^{3}+x y-7=0$ | 1,5,-1,-9,7,-97,-7,-99 | 36 | 17 |
math | 2. Let $A_{n}$ and $B_{n}$ be the sums of the first $n$ terms of the arithmetic sequences $\left\{a_{n}\right\}$ and $\left\{b_{n}\right\}$, respectively. If $\frac{A_{n}}{B_{n}}=\frac{5 n-3}{n+9}$, then $\frac{a_{8}}{b_{8}}=$ $\qquad$ . | 3 | 99 | 1 |
math | 9.5. What is the largest number of different natural numbers that can be chosen so that the sum of any three of them is a prime number? | 4 | 31 | 1 |
math | 5. There is an unlimited number of chips in six colors. What is the smallest number of chips that need to be arranged in a row so that for any two different colors, there are two adjacent chips of these colors in the row. | 18 | 48 | 2 |
math | K1) Let $n$ be a strictly positive integer. Maurice writes on the same line all the $2^{n}-1$ non-empty subsets of the set $\{1,2, \ldots, n\}$. Then, below each subset, he writes the product of its elements. Finally, he writes the inverses of the numbers present on the second line and calculates the total. What will b... | n | 323 | 1 |
math | 40. What is the smallest natural number by which the number 7 must be multiplied to obtain a number that consists only of nines? | 142857 | 29 | 6 |
math | 5. Let $x_{1}, x_{2}, \cdots, x_{7}$ be natural numbers, and $x_{1}<x_{2}$ $<\cdots<x_{6}<x_{7}$, also $x_{1}+x_{2}+\cdots+x_{7}=159$. Then the maximum value of $x_{1}+x_{2}+x_{3}$ is $\qquad$ | 61 | 95 | 2 |
math | 2. Determine the numbers $A, B$, and $C$ such that for every natural number $n \in \mathbb{N}$, the following equality holds:
$$
\frac{1}{2}+\frac{2}{2^{2}}+\frac{3}{2^{3}}+\ldots+\frac{n}{2^{n}}=\frac{A n+B}{2^{n}}+C
$$ | A=-1,B=-2,C=2 | 89 | 9 |
math | Determine all pairs $ (m,n)$ of natural numbers for which $ m^2\equal{}nk\plus{}2$ where $ k\equal{}\overline{n1}$.
EDIT. [color=#FF0000]It has been discovered the correct statement is with $ k\equal{}\overline{1n}$.[/color] | (m, n) = (11, 7) | 74 | 14 |
math | 54. Continuing her research with fractions, Tanya added 4 to the numerator of a certain fraction and 10 to the denominator. After simplifying, to her surprise, she got the original fraction. She tried the same with another fraction, but the result did not repeat. What fraction did Tanya originally have? | \frac{2}{5} | 66 | 7 |
math | ### 5.24. Investigate the function
$$
f(x)=\frac{5^{1 / x}}{1+5^{1 / x}}
$$
for continuity and determine the nature of the points of discontinuity. | 0 | 50 | 1 |
math | 13.114. The distance between two cities is $a$ km. Two motorists, setting out from these cities towards each other, will meet halfway if the first one leaves $t$ hours earlier than the second. If, however, they set out towards each other simultaneously, the meeting will occur after $2t$ hours. Determine the speed of ea... | \frac{(3-\sqrt{5})}{4} | 87 | 12 |
math | 11. In an arithmetic sequence with all terms being real numbers, the common difference is 4, and the square of the first term plus the sum of the remaining terms does not exceed 100. Such a sequence can have at most $\qquad$ terms. | 8 | 56 | 1 |
math | 15. Find the value of $\cos 20^{\circ}+\cos 60^{\circ}+\cos 100^{\circ}+\cos 140^{\circ}$. | \frac{1}{2} | 46 | 7 |
math | Example 1 Let $x, y, z > 0$, and $x+y+z=1$. Find the maximum value of
$$
f(x, y, z)=\sum \frac{x(2 y-z)}{1+x+3 y}
$$ | \frac{1}{7} | 55 | 7 |
math | How many positive integers less than $1998$ are relatively prime to $1547$? (Two integers are relatively prime if they have no common factors besides 1.) | 1487 | 39 | 4 |
math | Which is the smallest number that, when multiplied by 128, results in a number ending with 10 zeros? | 78125000 | 26 | 8 |
math | Arpon chooses a positive real number $k$. For each positive integer $n$, he places a marker at the point $(n,nk)$ in the $(x,y)$ plane. Suppose that two markers whose $x$-coordinates differ by $4$ have distance $31$. What is the distance between the markers $(7,7k)$ and $(19,19k)$? | 93 | 81 | 2 |
math | 21. Find the sum $\sum_{k=1}^{19} k\binom{19}{k}$. | 19\cdot2^{18} | 28 | 9 |
math | 4. Team A and Team B each send out 7 players to participate in a Go relay match in a pre-arranged order. Both sides start with their No. 1 players competing. The loser is eliminated, and the winner then competes with the No. 2 player of the losing side, ... until all players of one side are eliminated, and the other si... | 3432 | 98 | 4 |
math | ## Task Condition
Find the derivative.
$$
y=\frac{6^{x}(\sin 4 x \cdot \ln 6-4 \cos 4 x)}{16+\ln ^{2} 6}
$$ | 6^{x}\sin4x | 50 | 7 |
math | Broady The Boar is playing a boring board game consisting of a circle with $2021$ points on it, labeled $0$, $1$, $2$, ... $2020$ in that order clockwise. Broady is rolling $2020$-sided die which randomly produces a whole number between $1$ and $2020$, inclusive.
Broady starts at the point labelled $0$. After ... | 2021 | 222 | 4 |
math | 2.1. Trapezoid $A B C D$ with base $A D=6$ is inscribed in a circle. The tangent to the circle at point $A$ intersects lines $B D$ and $C D$ at points $M$ and $N$ respectively. Find $A N$, if $A B \perp M D$ and $A M=3$. | 12 | 82 | 2 |
math | A number of runners competed in a race. When Ammar finished, there were half as many runners who had finished before him compared to the number who finished behind him. Julia was the 10th runner to finish behind Ammar. There were twice as many runners who had finished before Julia compared to the number who finished be... | 31 | 77 | 2 |
math | Task 1. (5 points) Calculate
$$
\left(\frac{10001}{20232023}-\frac{10001}{20222022}\right) \cdot 4090506+\sqrt{4092529}
$$ | 2022 | 74 | 4 |
math | Example 2. Solve the equation $x e^{y} d x+\left(y+y x^{2}\right) d y=0$. | \ln\sqrt{1+x^{2}}-(y+1)e^{-y}=C | 30 | 19 |
math | 1. Let $x=\frac{\sqrt{3}+\sqrt{2}}{\sqrt{3}-\sqrt{2}}, y=\frac{\sqrt{3}-\sqrt{2}}{\sqrt{3}+\sqrt{2}}$. Then $x^{2}+x y$ $+y^{2}+1=$ $\qquad$ | 100 | 74 | 3 |
math | # 9. Problem 9.1*
Petya came up with four different natural numbers, wrote down all their pairwise sums on the board, and in the row below, all their sums of three. It turned out that the sum of the two largest numbers in the top row and the two smallest numbers in the bottom row (a total of four numbers) is 2017. Fin... | 806 | 111 | 3 |
math | 3. For the hyperbola $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1(a>0, b>0)$, the focal distance is $2 c$. The line $l$ passes through the points $(a, 0)$ and $(0, b)$, and the sum of the distances from the point $(1,0)$ to the line $l$ and from the point $(-1,0)$ to the line $l$ is not less than $\frac{4}{5} c$. Then th... | [\frac{\sqrt{5}}{2},\sqrt{5}] | 137 | 15 |
math | Last time, I was in a truck going from town $A$ to town $C$ via $B$. After five minutes of travel, I asked the driver how far we were from $A$. "Half as far as from B," was the answer. I complained about the slow speed of the truck, to which the driver noted that the truck was not capable of greater speed, but it would... | 26 | 159 | 2 |
math | Problem 2. Determine the integers $x, y$, for which
$$
5^{x}-\log _{2}(y+3)=3^{y} \text { and } 5^{y}-\log _{2}(x+3)=3^{x}
$$ | 1 | 60 | 1 |
math | 2. Eight numbers $a_{1}, a_{2}, a_{3}, a_{4}$ and $b_{1}, b_{2}, b_{3}, b_{4}$ satisfy the relations
$$
\left\{\begin{array}{c}
a_{1} b_{1}+a_{2} b_{3}=1 \\
a_{1} b_{2}+a_{2} b_{4}=0 \\
a_{3} b_{1}+a_{4} b_{3}=0 \\
a_{3} b_{2}+a_{4} b_{4}=1
\end{array}\right.
$$
It is known that $a_{2} b_{3}=7$. Find $a_{4} b_{4}$. | -6 | 166 | 2 |
math | Example 2 Given $2 n$ positive numbers $a_{1}, a_{2}, \cdots, a_{n}, b_{1}$,
$$
\begin{array}{l}
b_{2}, \cdots b_{n} \text { and } \\
\quad\left(a_{1}+b_{1}\right)\left(a_{2}+b_{2}\right) \cdots\left(a_{n}+b_{n}\right)=10 .
\end{array}
$$
Find the maximum value of $y=\sqrt[n]{a_{1} a_{2} \cdots a_{n}}+\sqrt[n]{b_{1} ... | \sqrt[n]{10} | 158 | 7 |
math | Call a sequence of positive integers $(a_n)_{n \ge 1}$ a "CGMO sequence" if $(a_n)_{n \ge 1}$ strictly increases, and for all integers $n \ge 2022$, $a_n$ is the smallest integer such that there exists a non-empty subset of $\{a_{1}, a_{2}, \cdots, a_{n-1} \}$ $A_n$ where $a_n \cdot \prod\limits_{a \in A_n} a$ is a per... | c_1 \cdot n^2 \le a_n \le c_2 \cdot n^2 | 205 | 23 |
math | The sequence ${a_0, a_1, a_2, ...}$ of real numbers satisfies the recursive relation $$n(n+1)a_{n+1}+(n-2)a_{n-1} = n(n-1)a_n$$ for every positive integer $n$, where $a_0 = a_1 = 1$. Calculate the sum $$\frac{a_0}{a_1} + \frac{a_1}{a_2} + ... + \frac{a_{2008}}{a_{2009}}$$. | 2009 \times 1005 | 123 | 11 |
math | 6. An aquarium in the shape of a rectangular parallelepiped has dimensions: length $2 m$, width - 600 mm, height - 60 cm. It is filled with water at a rate of 3 liters/minute. How many seconds after the start of filling will the aquarium be completely filled? (10 points) | 14400 | 71 | 5 |
math | 4. Given $\sin ^{10} x+\cos ^{10} x=\frac{11}{36}$. Then $\sin ^{14} x+\cos ^{14} x=$ $\qquad$ . | \frac{41}{216} | 52 | 10 |
math | IMO 1975 Problem B1 Let A be the sum of the decimal digits of 4444 4444 , and B be the sum of the decimal digits of A. Find the sum of the decimal digits of B. | 7 | 52 | 1 |
math | 10. Given positive real numbers $a, b$ satisfy: for real number $x$, when $1 \leqslant x \leqslant 4$, we have $a x+b-3 \leqslant 0$. Then the minimum value of $\frac{1}{a}-b$ is $\qquad$ | 1 | 72 | 1 |
math | 1. Find all injective functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that for every real number $x$ and positive integer $n$,
$$
\left|\sum_{i=1}^{n} i(f(x+i+1)-f(f(x+i)))\right|<2016
$$
(Macedonia, FYR) | f(x)=x+1 | 84 | 6 |
math | Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that for all $x, y \in \mathbb{R}$,
$$
(x-y) f(x+y)=x f(x)-y f(y) .
$$ | f()=+b | 56 | 5 |
math | # 8. Solution.
Let's find the probability of the opposite event "the groups will not be able to contact each other." Consider pairs of tourists where the first tourist is from the first group and the second is from the second group. There are a total of $6 \cdot 7=42$ such pairs. Therefore, the probability that none o... | 1-(1-p)^{42} | 113 | 9 |
math | 2. For $x, y, z \in(0,1]$ find the minimum value of the expression
$$
A=\frac{(x+2 y) \sqrt{x+y-x y}+(y+2 z) \sqrt{y+z-y z}+(z+2 x) \sqrt{z+x-z x}}{x y+y z+z x}
$$ | 3 | 79 | 1 |
math | 2. In the square $A B C D$, the bisector of $\angle C A D$ meets $C D$ at $P$ and the bisector of $\angle A B D$ meets $A C$ at $Q$.
What is the ratio of the area of triangle $A C P$ to the area of triangle $B Q A$ ? | 2:1 | 76 | 3 |
math | Problem 2. Let $a, b, c, d$ be natural numbers such that $a+b+c+d=2018$. Find the minimum value of the expression
$$
E=(a-b)^{2}+2(a-c)^{2}+3(a-d)^{2}+4(b-c)^{2}+5(b-d)^{2}+6(c-d)^{2}
$$ | 14 | 89 | 2 |
math | For which digits $a$ do exist integers $n \geq 4$ such that each digit of $\frac{n(n+1)}{2}$ equals $a \ ?$ | 5 \text{ and } 6 | 38 | 8 |
math | Example 1 Find the positive integer solutions of the equation $x^{2}+y^{2}=328$. | (18,2),(2,18) | 25 | 11 |
math | Continue the sequence $x . y, z, w$ satisfying $x y+2 y z+z w \leqslant A\left(x^{2}+y^{2}+z^{2}+w^{2}\right)$, then the minimum value of the real number $A$ is | \frac{1+\sqrt{2}}{2} | 64 | 12 |
math | # Problem 4. (3 points)
Three runners are moving along a circular track at constant equal speeds. When two runners meet, they instantly turn around and start running in opposite directions.
At some point, the first runner meets the second. Twenty minutes later, the second runner meets the third for the first time. An... | 100 | 96 | 3 |
math | $6 \cdot 30$ For a given positive integer $n$, let $p(n)$ denote the product of the non-zero digits of $n$ (if $n$ is a single digit, then $p(n)$ is equal to that digit). If
$$
S=p(1)+p(2)+p(3)+\cdots+p(999),
$$
what is the largest prime factor of $S$? | 103 | 93 | 3 |
math | Example 6 Let $f(x)=\frac{4^{x}}{4^{x}+2}$. Then, $\sum_{k=1}^{1000} f\left(\frac{k}{1001}\right)$ equals $\qquad$ - | 500 | 59 | 3 |
math | 13. (2002 National High School Mathematics Competition) Given that $f(x)$ is a function defined on $\mathbf{R}$, $f(1)=1$, and for any $x \in \mathbf{R}$, we have $f(x+5) \geqslant f(x)+5$ and $f(x+1) \leqslant f(x)+1$. If $g(x)=f(x)+1-x$, find the value of $g(2002)$. | 1 | 112 | 1 |
math | ## Task B-2.2.
The function $f(x)=x^{2}+p x+q$ takes negative values only for $x \in\langle-3,14\rangle$. How many integer values from the set $[-100,-10]$ can the function $f$ take? | 63 | 67 | 2 |
math | ## 155. Math Puzzle $4 / 78$
A new gardening season has begun. First, the Thälmann Pioneers want to fence their $50 \mathrm{~m}$ long and $20 \mathrm{~m}$ wide school garden. The $2 \mathrm{~m}$ wide gate is already in place on one of the long sides, and $3 \mathrm{~m}$ long fence panels can be delivered.
How many fe... | 46 | 125 | 2 |
math | 14. Given sets $A, B$ are both sets composed of positive integers, and $|A|=20,|B|=16$, set $A$ satisfies the following condition: if $a, b, m, n \in A$, and $a+b=m+n$, then it must be that $\{a, b\}=\{m, n\}$. Define $A+B=\{a+b \mid a \in A, b \in B\}$, try to determine the minimum value of $|A+B|$. | 200 | 116 | 3 |
math | 36. Many city residents engage in dancing, many in mathematics, and at least one in both. Those who engage only in dancing are exactly $p+1$ times more than those who engage only in mathematics, where $p-$ is some prime number. If you square the number of all mathematicians, you get the number of all dancers. How many ... | 1 | 86 | 1 |
math | ## Task A-2.1.
How many integers $n$ are there for which the inequality $x^{2}+n x+100>0$ holds for all real numbers $x$? | 39 | 44 | 2 |
math | Task 1. The houses on a street are numbered from 1 to 100. How many times does the digit 7 appear in the house numbers? List all such numbers. | 20 | 39 | 2 |
math | A postman has to deliver five letters to five different houses. Mischievously, he posts one letter through each door without looking to see if it is the correct address. In how many different ways could he do this so that exactly two of the five houses receive the correct letters?
| 20 | 58 | 2 |
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