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 | Problem 7.4. Masha and Olya bought many identical pens for the new school year at the store. It is known that one pen costs an integer number of rubles, more than 10. Masha bought pens for exactly 357 rubles, and Olya - for exactly 441 rubles. How many pens did they buy in total? | 38 | 79 | 2 |
math | 18. 8 (IMO 42 Preliminary) Let $\triangle A B C$ be an acute triangle, and construct isosceles $\triangle D A C$, $\triangle E A B$, and $\triangle F B C$ outside $\triangle A B C$ such that $D A=D C$, $E A=E B$, $F B=F C$, $\angle A D C=2 \angle B A C$, $\angle B E A=2 \angle A B C$, and $\angle C F B=2 \angle A C B$.... | 4 | 218 | 1 |
math | 9. (10 points) In $\overline{\mathrm{ABCD}}+\overline{\mathrm{EFG}}=2010$, different letters represent different digits, then $A+B+C+D+E+F+G$
$=$
Translate the above text into English, please retain the original text's line breaks and format, and output the translation result directly. | 30 | 78 | 2 |
math | Determine all integers $n$ for which the polynomial $P(x)=3 x^{3}-n x-n-2$ can be written as the product of two non-constant polynomials with integer coefficients. | n=38, n=-2, n=26, n=130 | 43 | 19 |
math | Kozhevnikov P.
There are 2013 cards with the digit 1 written on them, and 2013 cards with the digit 2 written on them. Vasya forms a 4026-digit number from these cards. In one move, Petya can swap two cards and pay Vasya 1 ruble. The process ends when Petya gets a number that is divisible by 11. What is the maximum am... | 5 | 117 | 1 |
math | At first, on a board, the number $1$ is written $100$ times. Every minute, we pick a number $a$ from the board, erase it, and write $a/3$ thrice instead. We say that a positive integer $n$ is [i]persistent[/i] if after any amount of time, regardless of the numbers we pick, we can find at least $n$ equal numbers on the ... | 67 | 100 | 4 |
math | Let's calculate the value of the expression
$$
\frac{1}{1+x+x y}+\frac{1}{1+y+y z}+\frac{1}{1+z+z x}
$$
given that $x y z=1$. | 1 | 52 | 1 |
math | Define a sequence $(x_{n})_{n\geq 1}$ by taking $x_{1}\in\left\{5,7\right\}$; when $k\ge 1$, $x_{k+1}\in\left\{5^{x_{k}},7^{x_{k}}\right\}$. Determine all possible last two digits of $x_{2009}$. | 25, 07, 43 | 89 | 10 |
math | Example 7. Determine the number of roots of the equation
$$
z^{6}-6 z+10=0
$$
inside the circle $|z|<1$. | 0 | 39 | 1 |
math | $\begin{aligned} & [\text { Algebraic equations and systems of equations (miscellaneous) }] \\ & {[\quad \underline{\text { Completing the square. Sums of squares }}]}\end{aligned}$
Solve the equation $a^{2}+b^{2}+c^{2}+d^{2}-a b-b c-c d-d+2 / 5=0$. | =1/5,b=2/5,=3/5,=4/5 | 87 | 19 |
math | ## Problem Statement
Find the $n$-th order derivative.
$y=a^{2 x+3}$ | ^{2x+3}\cdot2^{n}\cdot\ln^{n} | 23 | 17 |
math | Matěj and his friends went caroling. In addition to apples, walnuts, and gingerbread, each of the boys also received an orange. Jarda got one orange, Milan too. Radek, Patrik, Michal, and Dušan each got two oranges. Matěj got as many as four oranges, which was the most of all the boys. The other boys each got three ora... | 10 | 109 | 2 |
math | 9. Let $0<\theta<\pi$, find the maximum value of $y=\sin \frac{\theta}{2}(1+\cos \theta)$ | \frac{4\sqrt{3}}{9} | 34 | 12 |
math | 3-ча 1. How many real solutions does the system of two equations with three unknowns have:
$$
\left\{\begin{aligned}
x+y & =2 \\
x y-z^{2} & =1 ?
\end{aligned}\right.
$$ | 1 | 56 | 1 |
math | If $ x,y$ are positive real numbers with sum $ 2a$, prove that :
$ x^3y^3(x^2\plus{}y^2)^2 \leq 4a^{10}$
When does equality hold ?
Babis | 4a^{10} | 57 | 8 |
math | 5. A positive integer $n$ is said to be increasing if, by reversing the digits of $n$, we get an integer larger than $n$. For example, 2003 is increasing because, by reversing the digits of 2003, we get 3002, which is larger than 2003. How many four-digit positive integers are increasing?
(1 mark)
5. Let $n$ be a posit... | 4005 | 193 | 4 |
math | 3.329. $\cos ^{2}\left(\frac{5}{4} \pi-2 \alpha\right)-\cos ^{2}\left(\frac{5}{4} \pi+2 \alpha\right)$. | \sin4\alpha | 53 | 5 |
math | Patchouli is taking an exam with $k > 1$ parts, numbered Part $1, 2, \dots, k$. It is known that for $i = 1, 2, \dots, k$, Part $i$ contains $i$ multiple choice questions, each of which has $(i+1)$ answer choices. It is known that if she guesses randomly on every single question, the probability that she gets exactly o... | 2037171 | 136 | 7 |
math | 8. In the sequence $\left\{a_{n}\right\}_{n=1}^{+\infty}$, $x_{1}=1$, if for any positive integer $n$ there is $x_{n+1}=x_{n}+\left[\sqrt{x_{n}}\right]$, where the symbol $[x]$ denotes the greatest integer not exceeding $x$, then the smallest positive integer $n$ such that $x_{n} \geqslant 2019$ is $\qquad$. | 95 | 113 | 2 |
math | We are given three distinct non-zero digits. On the board, we will write all three-digit numbers that can be formed from these digits, using all three digits for each number. The sum of the written numbers is 1776.
Which three digits did we work with? Determine all solutions. | 1,2,51,3,4 | 61 | 10 |
math | ## Task B-3.2.
Niko says to Jure: "I have three numbers whose sum is 2. The sum of their squares is 6, and the sum of their cubes is 8. What do you think, if I continue to sum the $n$-th powers of these numbers, can I get a sum of $2^{2017}$ for some $n$?" What will Jure answer? Justify your response. | 2^{2017} | 97 | 7 |
math | 6. Let $A$ and $B$ be non-empty subsets of the set $\{1,2, \cdots, 10\}$, and the smallest element in set $A$ is not less than the largest element in set $B$. Then the number of such pairs $(A, B)$ is. | 9217 | 67 | 4 |
math | 3. It is known that for positive numbers $a, b$, and $c$, each of the three equations $a x^{2} +$ param $1 b x + c = 0$, $b x^{2} +$ param $1 c x + a = 0$, $c x^{2} +$ param $1 a x + b = 0$ has at least one real root.
What is the smallest possible value for the product of the roots of the second equation if the produc... | 0.24 | 379 | 4 |
math | 4. Given real numbers $x, y$ satisfy $\frac{x^{2}}{3}+y^{2}=1$. Then
$$
P=|2 x+y-4|+|4-x-2 y|
$$
the range of values for $P$ is . $\qquad$ | [2,14] | 64 | 6 |
math | ## Task 8 - V00808
Knobel Knifflig tells: I met the giant from Prague. His head and neck together are $30 \mathrm{~cm}$ long. His legs are twice as long as the head, neck, and half the torso, and the whole guy is exactly one meter taller than the head, neck, and legs together.
How tall is he? | 2.90\mathrm{~} | 85 | 9 |
math | 29 Let $f(n)=\left(C_{n}^{2}\right)^{2} \cdots\left(C_{n}^{1}\right)^{2}+\left(C_{n}^{2}\right)^{2}-\left(C_{n}^{3}\right)^{2}+\cdots+(-1)^{k}\left(C_{n}^{k}\right)^{2}+\cdots+$ $(-1)^{n}\left(C_{n}^{n}\right)^{2}$ (where $n$ is even), then $f(1990)=$ $\qquad$ . | -C_{1990}^{955} | 136 | 12 |
math | [ Trigonometric ratios in a right triangle [Area of a triangle (through the semiperimeter and the radius of the inscribed or exscribed circle) [ Two tangents drawn from one point
The area of triangle $ABC$ is $2 \sqrt{3} - 3$, and the angle $BAC$ is $60^{\circ}$. The radius of the circle touching side $BC$ and the ex... | \angleBCA=30 | 120 | 7 |
math | 3. Find such an integer $\mathrm{n}$ that for any integer $\mathrm{k} \neq 2013$, the number $n-k^{2014}$ is divisible by $2013-k$.
Answer: $2013^{2014}$. | 2013^{2014} | 63 | 10 |
math | Solve the following system of equations:
$$
\begin{aligned}
& \tan x_{i}+3 \cot x_{i}=2 \tan x_{i+1} \quad(i=1,2, \ldots, n-1) \\
& \tan x_{n}+3 \cot x_{n}=2 \tan x_{1} .
\end{aligned}
$$ | x_{i}=60+k_{i}\cdot180(k_{i}.integer)orx_{i}=-60+l_{i}\cdot180(l_{i}.integer) | 85 | 42 |
math | Example 5 Find all functions $f: \mathbf{R} \rightarrow \mathbf{R}$, such that for all $x, y \in \mathbf{R}$, we have
$$
\begin{array}{l}
f(x+y) f(x-y) \\
=(f(x)+f(y))^{2}-4 x^{2} f(y) .
\end{array}
$$ | f(x)=0 \text{ or } f(x)=x^2 | 87 | 15 |
math | In the expansion of $(x+y)^{n}$ using the binomial theorem, the second term is 240, the third term is 720, and the fourth term is 1080. Find $x, y$, and $n$.
# | 2,3,n=5 | 58 | 6 |
math | 4. (10 points) Two classes are planting trees. Class one plants 3 trees per person, and class two plants 5 trees per person, together planting a total of 115 trees. The maximum sum of the number of people in the two classes is $\qquad$ . | 37 | 61 | 2 |
math | 4. In the set of real numbers, solve the system of equations
$$
\begin{aligned}
& x^{2006} + y^{2006} + z^{2006} = 2 \\
& x^{2007} + y^{2007} + z^{2007} = 2 \\
& x^{2008} + y^{2008} + z^{2008} = 2
\end{aligned}
$$ | (x,y,z)\in{(0,1,1),(1,0,1),(1,1,0)} | 112 | 24 |
math | 11. For any positive integers $m, n$, define $r(m, n)$ as the remainder of $m \div n$ (for example, $r(8,3)$ represents the remainder of $8 \div 3$, so $r(8,3)=2$). The smallest positive integer solution that satisfies the equation $r(m, 1)+r(m, 2)+r(m, 3)+\cdots+r(m, 10)=4$ is $\qquad$. | 120 | 107 | 3 |
math | Ya. V. Yashchenko
a) In Mexico City, to limit the traffic flow, each private car is assigned two days of the week when it cannot be driven on the city's streets. A family needs to have at least ten cars available every day. What is the smallest number of cars the family can manage with, if its members can choose the p... | 14;12cars | 178 | 6 |
math | 2. Solve the inequality $\sqrt{\frac{x-4}{x+3}}-\sqrt{\frac{x+3}{x-4}}<\frac{7}{12}$. | x\in(-\infty;-12)\cup(4;+\infty) | 39 | 19 |
math | 5. Determine the sum of all four-digit numbers in which every two consecutive digits form a square of a natural number. (For example, 164 is such a three-digit number because 16 and 64 are squares of the numbers 4 and 8).
## Tasks worth 10 points: | 13462 | 65 | 5 |
math | 5.10. Calculate the limit
$$
\lim _{x \rightarrow 0} \frac{\tan 6 x}{\sin 3 x}
$$ | 2 | 36 | 1 |
math | [ [ Linear Recurrence Relations ]
Calculate the sum: $S_{n}=C_{n}^{0}-C_{n-1}^{1}+C_{n-2}^{2}-\ldots$.
# | S_{n}=0,ifn\equiv2,5(\bmod6);S_{n}=1,ifn\equiv0,1(\bmod6);S_{n}=-1,ifn\equiv3,4(\bmod6) | 48 | 55 |
math | 13.047. A musical theater announced a competition for admission to the orchestra. Initially, it was planned that the number of places for violinists, cellists, and trumpeters would be distributed in the ratio $1.6: 1: 0.4$. However, it was then decided to increase the intake, and as a result, 25% more violinists and 20... | 20 | 117 | 2 |
math | Task B-4.3. How many rational terms are there in the expansion of the binomial $(\sqrt{2013}+\sqrt[3]{2013})^{2012}$? | 336 | 46 | 3 |
math | 2. Let $[x]$ denote the greatest integer not exceeding the real number $x$. Then
$$
[\sqrt{2010+\sqrt{2009+\sqrt{\cdots+\sqrt{3+\sqrt{2}}}}}]
$$
is equal to
(there are a total of 2009 square roots). | 45 | 73 | 2 |
math | 2B. For the angles $\alpha, \beta, \gamma$ of triangle $ABC$, the following equality holds:
$$
\frac{\sin ^{2} \beta+\sin ^{2} \gamma-\sin ^{2} \alpha}{\sin \beta \sin \gamma}=1
$$
Calculate the angle $\alpha$. | \alpha=\frac{\pi}{3} | 73 | 9 |
math | 2. How many five-digit numbers exist that are not divisible by 1000, and have the first, third, and last digits even?
Otvet: 9960. | 9960 | 40 | 4 |
math | 8. determine all integers $n$ for which there are positive real numbers $0<a_{1} \leq a_{2} \leq \ldots \leq a_{n}$ exist with
$$
\sum_{k=1}^{n} a_{k}=96, \quad \sum_{k=1}^{n} a_{k}^{2}=144, \quad \sum_{k=1}^{n} a_{k}^{3}=216
$$
## Solution | 64 | 113 | 2 |
math | A [i]string of length $n$[/i] is a sequence of $n$ characters from a specified set. For example, $BCAAB$ is a string of length 5 with characters from the set $\{A,B,C\}$. A [i]substring[/i] of a given string is a string of characters that occur consecutively and in order in the given string. For example, the string $CA... | 963 | 364 | 3 |
math | 2. For any point $P$ on the ellipse $C: \frac{x^{2}}{3}+\frac{y^{2}}{2}=1$, draw a perpendicular line $P H$ from $P$ to the right directrix of the ellipse $C$ (where $H$ is the foot of the perpendicular), and extend $P H$ to point $Q$, such that $|H Q|=\lambda|P H|(\lambda \geqslant 1)$. As point $P$ moves along the el... | [\frac{\sqrt{3}}{3},1) | 136 | 12 |
math | ## Problem Statement
Calculate the limit of the function:
$\lim _{x \rightarrow \pi} \frac{\sin \left(\frac{x^{2}}{\pi}\right)}{2^{\sqrt{\sin x+1}}-2}$ | \frac{2}{\ln2} | 52 | 9 |
math | Example 1 Find the smallest positive number $\alpha$, such that there exists a positive number $\beta$, for which the inequality
$$
\sqrt{1+x}+\sqrt{1-x} \leqslant 2-\frac{x^{\alpha}}{\beta}
$$
holds for $0 \leqslant x \leqslant 1$. | \alpha=2, \beta=4 | 77 | 9 |
math | In terms of real parameter $a$ solve inequality:
$\log _{a} {x} + \mid a+\log _{a} {x} \mid \cdot \log _{\sqrt{x}} {a} \geq a\log _{x} {a}$
in set of real numbers | -\frac{1}{3} \leq a \leq 1 | 66 | 18 |
math | $2 \cdot 94$ Find all natural numbers $n$ such that
$$\min _{k \in N}\left(k^{2}+\left[\frac{n}{k^{2}}\right]\right)=1991$$
where $\left[\frac{n}{k^{2}}\right]$ denotes the greatest integer not exceeding $\frac{n}{k^{2}}$, and $N$ is the set of natural numbers. | 990208 \leqslant n \leqslant 991231 | 94 | 24 |
math | Example 6 Given that $f(x)$ is a function with domain $R$, and $f(x+2)[1-f(x)]=1+f(x), f(1)=2+\sqrt{3}$. Find $f(1989)$.
保留源文本的换行和格式，直接输出翻译结果。 | \sqrt{3}-2 | 68 | 6 |
math | Determine whether the number
$$
\frac{1}{2 \sqrt{1}+1 \sqrt{2}}+\frac{1}{3 \sqrt{2}+2 \sqrt{3}}+\frac{1}{4 \sqrt{3}+3 \sqrt{4}}+\cdots+\frac{1}{100 \sqrt{99}+99 \sqrt{100}}
$$
is rational or irrational. Explain your answer. | \frac{9}{10} | 100 | 8 |
math | 18. $A 、 n$ are natural numbers, and
$$
A=n^{2}+15 n+26
$$
is a perfect square. Then $n$ equals $\qquad$ | 23 | 46 | 2 |
math | In isosceles right-angled triangle $ABC$, $CA = CB = 1$. $P$ is an arbitrary point on the sides of $ABC$. Find the maximum of $PA \cdot PB \cdot PC$. | \frac{\sqrt{2}}{4} | 47 | 10 |
math | ## Task Condition
Find the derivative of the specified order.
$$
y=\left(2 x^{3}+1\right) \cos x, y^{V}=?
$$ | (30x^{2}-120)\cosx-(2x^{3}-120x+1)\sinx | 38 | 28 |
math | Asahanov $H . X$.
The number $x$ is such that among the four numbers $a=x-\sqrt{2}, b=x-\frac{1}{x}, c=x+\frac{1}{x}, d=x^{2}+2 \sqrt{2}$ exactly one is not an integer.
Find all such $x$. | \sqrt{2}-1 | 72 | 6 |
math | 13.196. Two hours after leaving the factory, the driver looked at the speedometer and noticed that he had only traveled 112 km. He mentally calculated that if he continued at the same speed, he would be 30 minutes late in delivering the cargo to the station. Therefore, the driver increased his speed and arrived at the ... | 56 | 112 | 2 |
math | [ Rule of product]
There are four weights and a two-pan balance without a pointer. How many different weights can be accurately weighed with these weights if
a) the weights can only be placed on one pan of the balance;
b) the weights can be placed on both pans of the balance
# | 40 | 62 | 2 |
math | 5. Can the 8 natural numbers $1,2,3,4,5,6,7,8$ be divided into two groups of 4 numbers each such that the sum of the numbers in the first group equals the sum of the numbers in the second group, and the sum of the squares of the numbers in the first group equals the sum of the squares of the numbers in the second group | {2,3,5,8},{1,4,6,7} | 83 | 17 |
math | Task B-4.4. The sum of all coefficients in the expression $(1+x)^{n}+(1+x)^{n+1}$ is equal to 1536. Determine the coefficient of $x^{6}$. | 294 | 50 | 3 |
math | Example 1. The nine-digit number in the form of $1234 \times \times \times \times \times$ composed of $1,2,3,4,5,6,7,8,9$, where the ten-thousands place is not 5, the thousands place is not 6, the hundreds place is not 7, the tens place is not 8, and the units place is not 9. How many such nine-digit numbers are there? | 44 | 102 | 2 |
math | 12. Find all pairs of sets $A, B$ such that $A, B$ satisfy:
(1) $A \cup B=\mathbf{Z}$;
(2) If $x \in A$, then $x-1 \in B$;
(3) If $x \in B, y \in B$, then $x+y \in A$.
(2002, Romania for IMO and Balkan Mathematical Olympiad Selection Test (First Round)) | A=2\mathbf{Z}, B=2\mathbf{Z}+1 | 102 | 20 |
math | Problem 3. Answer: $\frac{1}{64}$.
Translate the text above into English, please retain the original text's line breaks and format, and output the translation result directly.
Note: The provided translation is already in the requested format and directly represents the translation of the given text. However, the not... | \frac{1}{64} | 102 | 8 |
math | 626. Find $f^{\prime}(0)$, if $f(x)=e^{x} \arcsin x+\operatorname{arctg} x$. | 2 | 38 | 1 |
math | $\square$ Example 5 Let $a_{1}, a_{2}, \cdots, a_{n}$ be given non-zero real numbers. If the inequality
$$r_{1}\left(x_{1}-a_{1}\right)+r_{2}\left(x_{2}-a_{2}\right)+\cdots+r_{n}\left(x_{n}-a_{n}\right) \leqslant \sqrt[m]{x_{1}^{m}+x_{2}^{m}+\cdots+x_{n}^{m}}$$
$-\sqrt[m]{a_{1}^{m}+a_{2}^{m}+\cdots+a_{n}^{m}}$ (where $... | r_{i}=\left(\frac{a_{i}}{\sqrt[m]{a_{1}^{m}+a_{2}^{m}+\cdots+a_{n}^{m}}}\right)^{m-1}(i=1,2, \cdots, n) | 238 | 62 |
math | 7. It is known that the number $a$ satisfies the equation param1, and the number $b$ satisfies the equation param2. Find the smallest possible value of the sum $a+b$.
| param1 | param2 | |
| :---: | :---: | :--- |
| $x^{3}-3 x^{2}+5 x-17=0$ | $x^{3}-6 x^{2}+14 x+2=0$ | |
| $x^{3}+3 x^{2}+6 x-9=0$ | $x^{3}+6 x^{2}+15... | -3 | 239 | 2 |
math | 1. Solve the equation $x^{2}+3+\log _{2}\left(x^{2}-4 x+6\right)=4 x$. | 2 | 33 | 1 |
math | 11. Several identical boxes are arranged in a row. Xiao Ming distributed more than fifty identical chess pieces into the boxes, with only one box left empty. Then he went out. Xiao Guang took one chess piece from each of the boxes that had chess pieces and placed them in the empty box, and then rearranged the boxes. Wh... | 11 | 98 | 2 |
math | 3. In square $A B C D$, diagonals $A C$ and $B D$ intersect at point $O$. Point $K$ is the midpoint of side $A B$. Points $M$ and $N$ are chosen on sides $A D$ and $B C$ respectively such that rays $O K, O M$, and $O N$ divide the square into three parts of equal area. In what ratio does point $M$ divide side $A D$? | 5:1 | 102 | 3 |
math | 6. Let $a>b>0$. Then the minimum value of $a^{3}+\frac{12}{b(a-b)}$ is $\qquad$ .
| 20 | 36 | 2 |
math | 5. Real numbers $x, y, a$ satisfy $x+y=a+1$ and $xy=a^2-7a+16$. Then the maximum value of $x^2+y^2$ is $\qquad$ . | 32 | 51 | 2 |
math | 2. (10 points) YY's family of 3, plus YY's cousin, have a total age of 95 years this year. Dad is 4 years older than Mom, and YY is 3 years older than the cousin. 8 years ago, the sum of their ages was 65 years. Then Dad is $\qquad$ years old this year. | 42 | 79 | 2 |
math | From the vertex $ A$ of the equilateral triangle $ ABC$ a line is drown that intercepts the segment $ [BC]$ in the point $ E$. The point $ M \in (AE$ is such that $ M$ external to $ ABC$, $ \angle AMB \equal{} 20 ^\circ$ and $ \angle AMC \equal{} 30 ^ \circ$. What is the measure of the angle $ \angle MAB$? | 20^\circ | 97 | 4 |
math | 4. Person A and Person B agreed to have dinner at a restaurant. Since this restaurant is very popular, after A arrived, they took a waiting number and waited for B. B arrived at the restaurant a while later but didn't see A, so they also took a waiting number. While waiting, B saw A, and both of them showed their waiti... | 35 | 149 | 2 |
math | 2. $x_{1}+x_{2}+\cdots+x_{19}+2 x_{20}=3$ has $\qquad$ non-negative integer solutions. | 1349 | 39 | 4 |
math | Two cars travel along a circular track $n$ miles long, starting at the same point. One car travels $25$ miles along the track in some direction. The other car travels $3$ miles along the track in some direction. Then the two cars are once again at the same point along the track. If $n$ is a positive integer, find the s... | 89 | 84 | 2 |
math | A tank contains $m$ liters of water at $t \mathrm{C}^{\circ}$. We drain $v$ liters of water from it, and replace it with $v$ liters of water at $d \mathrm{C}^{\circ}$, then we drain $v$ liters of water again, and replace it with the same amount of water at $d \mathrm{C}^{\circ}$, and so on.
a) What will be the tempera... | 40.13\mathrm{C} | 239 | 10 |
math | 10. Given the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1(a>b>0)$ always passes through the fixed point $\left(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right)$, and the length of its major axis is in the range $[\sqrt{5}, \sqrt{6}]$. Then the range of the eccentricity is $\qquad$ | \left[\frac{\sqrt{3}}{3}, \frac{\sqrt{2}}{2}\right] | 102 | 24 |
math | Example 5 Given real numbers $x, y$ satisfy
$$
x+y=3, \frac{1}{x+y^{2}}+\frac{1}{x^{2}+y}=\frac{1}{2} \text {. }
$$
Find the value of $x^{5}+y^{5}$.
[3] | 123 | 73 | 3 |
math | 4. With one mower, it is possible to mow $2 \frac{1}{2}$ ha of meadow in 2 hours, and with another, smaller mower, $1 \frac{1}{4}$ ha can be mowed in 1 hour and 30 minutes. How much meadow area will both mowers mow together in 3 hours and 36 minutes? | 7\frac{1}{2} | 83 | 8 |
math | 12. Sarah and Hagar play a game of darts. Let $O_{0}$ be a circle of radius 1. On the $n$th turn, the player whose turn it is throws a dart and hits a point $p_{n}$ randomly selected from the points of $O_{n-1}$. The player then draws the largest circle that is centered at $p_{n}$ and contained in $O_{n-1}$, and calls ... | \frac{6\pi}{7} | 178 | 9 |
math | 6 Given that $\lambda$ is a positive real number. Find the maximum value of $\lambda$ such that for all positive real numbers $u, v, w$ satisfying the condition
$$u \sqrt{v w}+v \sqrt{w u}+w \sqrt{u v} \geqslant 1$$
we have
$$u+v+w \geqslant \lambda .$$ | \sqrt{3} | 87 | 5 |
math | Problem 81. Let \( x, y, z \) be positive real numbers satisfying \( 2xyz = 3x^2 + 4y^2 + 5z^2 \). Find the minimum of the expression \( P = 3x + 2y + z \). | 36 | 63 | 2 |
math | 1108. Find the particular integral of the equation satisfying the given initial conditions:
1) $y^{\prime \prime}+4 y^{\prime}+5 y=0 ; \quad y(0)=-3, \quad y^{\prime}(0)=0$.
2) $y^{\prime \prime \prime}+3 y^{\prime \prime}+3 y^{\prime}+y=0 ; y(0)=-1, y^{\prime}(0)=2, y^{\prime \prime}(0)=3$. | e^{-x}(3x^{2}+x-1) | 122 | 14 |
math | Let's determine those $n$-degree real-coefficient polynomials, which when divided by their derivative, result in a polynomial again. | p(x)=a_{0}(x+)^{n} | 28 | 13 |
math | 63. One bicycle for two. Two brothers had to set off on a journey and arrive at their destination simultaneously. They had only one bicycle, on which they took turns riding, and when the rider's time was up, he would dismount and leave the bicycle by the fence, then walk forward without waiting for his brother, while t... | 11\frac{1}{9} | 276 | 9 |
math | 8. In triangle $A B C$ with $A B<A C$, let $H$ be the orthocenter and $O$ be the circumcenter. Given that the midpoint of $O H$ lies on $B C, B C=1$, and the perimeter of $A B C$ is 6 , find the area of $A B C$. | \frac{6}{7} | 76 | 7 |
math | 8. (6 points) My sister is 26 years old this year, and my younger sister is 18 years old this year. When the sum of their ages was 20, my sister was $\qquad$ years old. | 14 | 51 | 2 |
math | A three-digit number's digits sum up to 20. If we subtract 16 from the number, half of the resulting difference is a three-digit number whose digits are in reverse order of the original number's digits. Which number is this? | 974 | 51 | 3 |
math | A total of $\$ 425$ was invested in three different accounts, Account A, Account B and Account C. After one year, the amount in Account A had increased by 5\%, the amount in Account B had increased by 8\%, and the amount in Account C had increased by 10\%. The increase in dollars was the same in each of the three accou... | 100 | 92 | 3 |
math | 2. Let the polynomial $f(x)$ satisfy
$$
2 f(x+1)+3 f(x-1)=10 x^{2}+11 x+32 \text {. }
$$
Then $f(x)=$ | 2 x^{2}+3 x+5 | 50 | 10 |
math | ## Problem Statement
Calculate the limit of the function:
$\lim _{x \rightarrow-1} \frac{\left(x^{2}+3 x+2\right)^{2}}{x^{3}+2 x^{2}-x-2}$ | 0 | 55 | 1 |
math | Example $\mathbf{3}$ Find the function $f: \mathbf{R}_{+} \rightarrow \mathbf{R}_{+}$, such that it satisfies $f(f(x))=6 x-f(x)$.
(30th IMO Canadian Training Problem) | f(x)=2x | 57 | 5 |
math | 8.003. $\frac{4 \operatorname{ctg} x}{1+\operatorname{ctg}^{2} x}+\sin ^{2} 2 x+1=0$.
8.003. $\frac{4 \cot x}{1+\cot^{2} x}+\sin ^{2} 2 x+1=0$. | \frac{\pi}{4}(4k-1),k\inZ | 83 | 16 |
math | 13. $A B C D$ is a trapezium with $A B / / D C$ and $A B>D C$. $E$ is a point on $A B$ such that $A E=D C$. AC meets $D E$ and $D B$ at $F$ and $G$ respectively. Find the value of $\frac{A B}{C D}$ for which $\frac{\text { Area of } \triangle D F G}{\text { Area of trapezium } A B C D}$ is maximum.
(2 marks)
$A B C D$ ... | 3 | 259 | 1 |
math | 4. Find the number of pairs of integers $(x ; y)$ that satisfy the equation $x^{2}+x y=30000000$. | 256 | 36 | 3 |
math | 9. Primary School No.1 and Primary School No.2 have the same number of students participating in the Golden Cup Competition. The schools use cars to transport the students to the examination site. Primary School No.1 uses cars that can seat 15 people each, while Primary School No.2 uses cars that can seat 13 people eac... | 184 | 175 | 3 |
math | ## Problem Statement
Based on the definition of the derivative, find $f^{\prime}(0)$:
$$
f(x)=\left\{\begin{array}{c}
2 x^{2}+x^{2} \cos \frac{1}{x}, x \neq 0 \\
0, x=0
\end{array}\right.
$$ | 0 | 77 | 1 |
math | In one of the hotels of the wellness planet Oxys, there are $2019$ saunas. The managers have decided to accommodate $k$ couples for the upcoming long weekend. We know the following about the guests: if two women know each other then their husbands also know each other, and vice versa. There are several restrictions on ... | 2018 | 188 | 4 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.