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 | 40th IMO 1999 shortlist Problem C3 A chameleon repeatedly rests and then catches a fly. The first rest is for a period of 1 minute. The rest before catching the fly 2n is the same as the rest before catching fly n. The rest before catching fly 2n+1 is 1 minute more than the rest before catching fly 2n. How many flies d... | 510,312,462 | 138 | 11 |
math | 143. For which $n \in \boldsymbol{N}$ does the equality
$$
\sqrt[n]{17 \sqrt{5}+38}+\sqrt[n]{17 \sqrt{5}-38}=\sqrt{20} ?
$$ | 3 | 59 | 1 |
math | 1. Given positive integers $a, b, c (a<b<c)$ form a geometric sequence, and
$$
\log _{2016} a+\log _{2016} b+\log _{2016} c=3 \text {. }
$$
Then the maximum value of $a+b+c$ is $\qquad$ . | 4066273 | 79 | 7 |
math | Problem 11.2. At the zoo, oranges, bananas, and coconuts were brought for feeding three monkeys, and there were equal amounts of all three types of fruit. The first monkey was fed only oranges and bananas, with bananas being $40 \%$ more than oranges. The second monkey was fed only bananas and coconuts, with coconuts b... | \frac{1}{2} | 143 | 7 |
math | Solve the following equation in the set of natural numbers:
$$
x^{3}-y^{3}=x y+61
$$ | 6,5 | 29 | 3 |
math | 7. Let $\triangle A B C$ be a right triangle with right angle $C$. Let $I$ be the incenter of $A B C$, and let $M$ lie on $A C$ and $N$ on $B C$, respectively, such that $M, I, N$ are collinear and $\overline{M N}$ is parallel to $A B$. If $A B=36$ and the perimeter of $C M N$ is 48 , find the area of $A B C$. | 252 | 113 | 3 |
math | ## Task 1.
Determine all triples of real numbers $(x, y, z)$ for which
$$
\begin{aligned}
& x(x y-1)=2(y z-1) \\
& y(y z-1)=2(z x-1) \\
& z(z x-1)=2(x y-1)
\end{aligned}
$$ | (1,1,1),(-1,-1,-1),(2,2,2) | 76 | 20 |
math | 3. Let $x_{1}$ and $x_{2}$ be the roots of the equation
$$
p^{2} x^{2}+p^{3} x+1=0
$$
Determine $p$ such that the expression $x_{1}^{4}+x_{2}^{4}$ has the smallest value. | \\sqrt[8]{2} | 74 | 7 |
math | 6. Define the sequence $\left\{a_{n}\right\}: a_{n}$ is the last digit of $1+2+\cdots+n$, and $S_{n}$ is the sum of the first $n$ terms of the sequence $\left\{a_{n}\right\}$. Then $S_{2016}=$ $\qquad$ . | 7066 | 80 | 4 |
math | Task B-3.1. (20 points) Find all real solutions of the equation
$$
4 x^{3}-\sqrt{1-x^{2}}-3 x=0
$$ | x_{1}=-\frac{\sqrt{2}}{2},x_{2}=-\frac{1}{2}\sqrt{2-\sqrt{2}},x_{3}=\frac{1}{2}\sqrt{2+\sqrt{2}} | 42 | 54 |
math | 9. 120 unit cubes are put together to form a rectangular prism whose six faces are then painted. This leaves 24 unit cubes without any paint. What is the surface area of the prism? | 148 | 43 | 3 |
math | $$
\begin{array}{l}
\text { 4. If } x_{1}>x_{2}>x_{3}>x_{4}>0, \text { and the inequality } \\
\log _{\frac{x_{1}}{x_{2}}} 2014+\log _{\frac{x_{2}}{x_{3}}} 2014+\log _{\frac{x_{3}}{x_{4}}} 2014 \\
\geqslant k \log _{\frac{x_{1}}{}} 2014
\end{array}
$$
always holds, then the maximum value of the real number $k$ is $\qq... | 9 | 150 | 1 |
math | Solve the following system of equations:
$$
\frac{1}{2-x+2 y}-\frac{1}{x+2 y-1}=2,
$$
$$
\frac{1}{2-x+2 y}-\frac{1}{1-x-2 y}=4 .
$$ | \frac{11}{6},\frac{1}{12} | 64 | 16 |
math | 8. If a four-digit number $n$ contains at most two different digits among its four digits, then $n$ is called a "simple four-digit number" (such as 5555 and 3313). Then, the number of simple four-digit numbers is | 576 | 59 | 3 |
math | 13. A page in the middle of a book has been torn out, and the sum of the remaining page numbers is 1133. What are the two page numbers on the torn-out sheet? | 21,22 | 43 | 5 |
math | 3. From the 10 numbers $0,1,2,3,4,5,6,7,8,9$, choose 3 numbers such that their sum is an even number not less than 10.
The number of different ways to choose them is $\qquad$ . | 51 | 63 | 2 |
math | How many polynomials of degree exactly $5$ with real coefficients send the set $\{1, 2, 3, 4, 5, 6\}$ to a permutation of itself? | 718 | 42 | 3 |
math | 331. Find the last two digits of the number $137^{42}$. | 69 | 21 | 2 |
math | Four, (30 points) Let natural numbers $a, b, c, d$ satisfy $\frac{a}{b}+\frac{c}{d}<1$ and $a+c=20$. Find the maximum value of $\frac{a}{b}+\frac{c}{d}$.
| \frac{1385}{1386} | 64 | 13 |
math | 3. Given positive real numbers $a, b, c$ satisfying $a+b+c=1$, then the maximum value of $a+\sqrt{b}+\sqrt[3]{c}$ is
untranslated part:
将上面的文本翻译成英文，请保留源文本的换行和格式，直接输出翻译结果。
(As this is a note or instruction, it is not part of the translation task and thus not translated.) | \frac{5}{4}+\frac{2\sqrt{3}}{9} | 90 | 19 |
math |
3. The present ages in years of two brothers $A$ and $B$, and their father $C$ are three distinct positive integers $a, b$, and $c$ respectively. Suppose $\frac{b-1}{a-1}$ and $\frac{b+1}{a+1}$ are two consecutive integers, and $\frac{c-1}{b-1}$ and $\frac{c+1}{b+1}$ are two consecutive integers. If $a+b+c \leq 150$ d... | =3,b=7,=31 | 122 | 9 |
math | 6.032. $\sqrt{x+\sqrt{x+11}}+\sqrt{x-\sqrt{x+11}}=4$.
6.032. $\sqrt{x+\sqrt{x+11}}+\sqrt{x-\sqrt{x+11}}=4$. | 5 | 59 | 1 |
math | 5. A car left city A heading towards city B, covering $12 \mathrm{~km}$ in 10 minutes. At the same time, a truck left city B heading towards city A, covering 10 $\mathrm{km}$ in 12 minutes. How many kilometers apart will the car and the truck be after 2 hours and 30 minutes of driving if the distance between cities A a... | 5 | 132 | 1 |
math | ## Task 30/61
Five housewives want to buy rolls. After counting the available rolls, the baker allows himself a joke: "If each of you buys half of the currently available rolls plus half a roll, none will be left!" How many rolls did the baker have, and how many would each of the customers have received according to t... | 31 | 75 | 2 |
math | Example 2. Parts manufactured by two plants are stored in a warehouse. It is known that the production volume of the first plant exceeds the production volume of the second plant by 4 times. The probability of defect at the first plant is $p_{1}=0.05$, at the second plant - $p_{2}=0.01$. A randomly selected part turned... | 0.952 | 97 | 5 |
math | 10. In $\triangle A B C$, $A B=A C, \angle A=80^{\circ}, D$ is a point inside the triangle, and $\angle D A B=\angle D B A=10^{\circ}$, find the degree measure of $\angle A C D$.
(Problem 432 from "Mathematics Teaching") | 30 | 78 | 2 |
math | In Squareland, only squares live. Except for two exceptions, each of them has two friends, one with a perimeter $8 \mathrm{~cm}$ smaller and the other $8 \mathrm{~cm}$ larger. The average area of a Squareland square is $116 \mathrm{~cm}^{2}$. No two squares are the same, and the perimeter of the smallest square is equa... | 7,4\mathrm{~},16\mathrm{~},10\mathrm{~} | 138 | 22 |
math | Example 3. Given positive integers $a, b, c$ satisfying $a>b>c$, and $\quad\left\{\begin{array}{l}36-6(a+b+c)+(a b+b c+c a)=0, \\ 81-9(a+b+c)+(a b+b c+c a)=0 .\end{array}\right.$ Find the values of $a, b, c$. | a=10, b=4, c=1 | 87 | 12 |
math | 5. For any two points $P, Q$ on the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1(a>b>0)$, if $O P \perp O Q$, then the minimum value of $|O P| \cdot|O Q|$ is $\qquad$. | \frac{2 a^{2} b^{2}}{a^{2}+b^{2}} | 76 | 22 |
math | [ Divisibility of numbers. General properties ] [ Examples and counterexamples. Constructions ]
A five-digit number is called indivisible if it cannot be factored into the product of two three-digit numbers.
What is the largest number of consecutive indivisible five-digit numbers? | 99 | 55 | 2 |
math | What is the area of a triangle with side lengths $17$, $25$, and $26$?
[i]2019 CCA Math Bonanza Lightning Round #3.2[/i] | 204 | 44 | 3 |
math | 11.3. Solve in the set of real numbers the equation
$$
\cos \frac{\pi}{x}=-x^{6}-4 x^{5}+2 x^{4}+12 x^{3}-9 x^{2}-1
$$ | 1 | 57 | 1 |
math | 3. Find the largest natural number $n$, for which the number 999...99 (with 999 nines) is divisible by $9^{n}$.
---
The text has been translated from Macedonian to English while preserving the original formatting and structure. | 2 | 58 | 1 |
math | 2. Six natural numbers (possibly repeating) are written on the faces of a cube, such that the numbers on adjacent faces differ by more than 1. What is the smallest possible value of the sum of these six numbers? | 18 | 46 | 2 |
math | Problem 4. On each side of a cube, a natural number is written. At each vertex (corner) of the cube, the product of the three numbers written on the sides that form the vertex (corner) is written. The sum of the eight thus obtained products is 385. Determine the sum of the numbers written on the sides of the cube. | 23 | 75 | 2 |
math | 2. (15 points) Solve the system of equations: $\left\{\begin{array}{l}y^{2}+x y=15 \\ x^{2}+x y=10\end{array}\right.$ | (2,3),(-2,-3) | 51 | 10 |
math | *end Let $S=\{1,2, \cdots, 2005\}$. If any set of $n$ pairwise coprime numbers in $S$ contains at least one prime number, find the minimum value of $n$.
| 16 | 55 | 2 |
math | What are the numbers whose triples, when added with 1, result in a prime number between 70 and 110? | 24,26,32,34,36 | 28 | 14 |
math | N20 (20-1, Cuba) The last three digits of $1978^{n}$ and $1978^{m}$ are equal. Try to find positive integers $m$ and $n$, such that $m+n$ takes the minimum value (here $n>m \geqslant 1$). | 106 | 72 | 3 |
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 | 5. Find all pairs of positive numbers $(x, y)$ that satisfy the system of equations $\left\{\begin{array}{l}3 y-\sqrt{\frac{y}{x}}-6 \sqrt{x y}+2=0 \\ x^{2}+81 x^{2} y^{4}=2 y^{2} .\end{array}\right.$, Answer: $\left(\frac{1}{3} ; \frac{1}{3}\right),\left(\frac{\sqrt[4]{31}}{12} ; \frac{\sqrt[4]{31}}{3}\right)$ | (\frac{1}{3};\frac{1}{3}),(\frac{\sqrt[4]{31}}{12};\frac{\sqrt[4]{31}}{3}) | 134 | 41 |
math | Problem 11. Denote by $d(a, b)$ the number of the divisors of a positive integer $a$, which are greater than or equal to $b$. Find all positive integers $n$ such that
$$
d(3 n+1,1)+d(3 n+2,2)+\cdots+d(4 n, n)=2006
$$
Ivan Landjev | 708 | 89 | 3 |
math | In an isosceles right-angled triangle AOB, points P; Q and S are chosen on sides OB, OA, and AB respectively such that a square PQRS is formed as shown. If the lengths of OP and OQ are a and b respectively, and the area of PQRS is 2 5 that of triangle AOB, determine a : b.
[asy]
pair A = (0,3);
pair B = (0,0);
pair C ... | 2 : 1 | 305 | 4 |
math | 4. Two natural numbers add up to 2015. If one of them is divided by the other with a remainder, the quotient is 25. Find all pairs of such numbers (and prove that there are no others). | 1938,77 | 49 | 7 |
math | 2. At the Olympiad, for each solved problem, one could receive 3, 8, or 10 points. Vasya scored 45 points. What is the smallest number of problems he could have solved? (It is necessary to explain why he could not have solved fewer problems.) | 6 | 63 | 1 |
math | 1. Let $a_{0}=0, a_{1}=1, a_{n+2}=a_{n+1}+a_{n}(n=0,1,2, \cdots)$. Also let $x_{1} \in \mathbf{R}, x_{n}=\frac{a_{n-1}+a_{n-2} \cdot x_{1}}{a_{n}+a_{n-1} \cdot x_{1}}(n=2,3, \cdots)$. If $x_{2004}=\frac{1}{x_{1}}-1$, find $x_{1}$. | x_{1}=\frac{1}{2}(-1\\sqrt{5}) | 145 | 18 |
math | 1. In still water, the speed of boat A is twice the speed of boat B. Boats A and B start from points $A$ and $B$ respectively at the same time, moving towards each other, and meet at a point where the distances from $A$ and $B$ are in the ratio of 3:1. If boats A and B start from points $B$ and $A$ respectively at the ... | 5:7 | 118 | 3 |
math | Example 10. Let real numbers $x, y$ satisfy the equation $x^{3}+y^{3}=a^{3}(a>0)$. Find the range of values for $x+y$:
(1992, Taiyuan City Junior High School Mathematics Competition) | 0<x+y \leqslant \sqrt[3]{4} a | 62 | 16 |
math | ## Task A-3.2.
Determine all prime numbers $p$ for which there exist natural numbers $x$ and $y$ such that
$$
\left\{\begin{aligned}
p+1 & =2 x^{2} \\
p^{2}+1 & =2 y^{2}
\end{aligned}\right.
$$ | 7 | 74 | 1 |
math | 3.306. $\frac{\sqrt{1+\sin \alpha}+\sqrt{1-\sin \alpha}}{\sqrt{1+\sin \alpha}-\sqrt{1-\sin \alpha}}$, if a) $0^{\circ}<\alpha<90^{\circ}$; b) $90^{\circ}<\alpha<180^{\circ}$. | )\operatorname{ctg}\frac{\alpha}{2};b)\operatorname{tg}\frac{\alpha}{2} | 84 | 26 |
math | ## Task 4.
Determine all natural numbers $n$ for which there exist natural numbers $a$ and $b$ such that
$$
\left(n^{2}+2\right)^{a}=(2 n-1)^{b}
$$ | 5 | 55 | 1 |
math | Suppose $z_1, z_2 , \cdots z_n$ are $n$ complex numbers such that $min_{j \not= k} | z_{j} - z_{k} | \geq max_{1 \leq j \leq n} |z_j|$. Find the maximum possible value of $n$. Further characterise all such maximal configurations. | n = 7 | 82 | 5 |
math | Determine all non-constant polynomials $X^{n}+a_{1} X^{n-1}+\cdots+a_{n-1} X+a_{n}$, with integer coefficients, whose roots are exactly the numbers $a_{1}, \ldots, a_{n-1}, a_{n}$ (with multiplicity). | P(X)=X^{n}, P(X)=X^{n}(X^{2}+X-2), P(X)=X^{n}(X^{3}+X^{2}-X-X) | 73 | 42 |
math | 13. (2001 National High School Mathematics Competition) Let $\left\{a_{n}\right\}$ be an arithmetic sequence, and $\left\{b_{n}\right\}$ be a geometric sequence, and $b_{1}=a_{1}^{2}$, $b_{2}=a_{2}^{2}, b_{3}=a_{3}^{2}\left(a_{1}<a_{2}\right)$, and $\lim \left(b_{1}+b_{2}+\cdots+b_{n}\right)=\sqrt{2}+1$, find the first... | a_{1}=-\sqrt{2},=2\sqrt{2}-2 | 148 | 18 |
math | 7.211. $\log _{\sqrt{3}} x \cdot \sqrt{\log _{\sqrt{3}} 3-\log _{x} 9}+4=0$. | \frac{1}{3} | 43 | 7 |
math | 3.42 Find all natural numbers $n$, such that every natural number represented in decimal notation by $n-1$ digits 1 and one digit 7 is a prime number.
(31st International Mathematical Olympiad Preliminary Question, 1990) | 1,2 | 57 | 3 |
math | 13.309. There are two gold and silver alloys. In one alloy, the quantities of these metals are in the ratio $1: 2$, in the other - $2: 3$.
How many grams should be taken from each alloy to obtain 19 g of an alloy in which gold and silver are in the ratio 7:12? | 9 | 78 | 1 |
math | The sum of the first 2011 terms of a geometric series is 200. The sum of the first 4022 terms of the same series is 380. Find the sum of the first 6033 terms of the series. | 542 | 58 | 3 |
math | Three, let $x \geqslant y \geqslant z \geqslant \frac{\pi}{12}$, and $x+y+z=\frac{\pi}{2}$. Find the maximum and minimum values of the product $\cos x \sin y \cos z$.
untranslated text:
设 $x \geqslant y \geqslant z \geqslant \frac{\pi}{12}$, 且 $x+y+z=\frac{\pi}{2}$. 求乘积 $\cos x \sin y \cos z$ 的最大值和最小值.
translated t... | \frac{1}{8} | 196 | 7 |
math | [ Geometry (other) ]
A sphere with radius $3 / 2$ has its center at point $N$. From point $K$, located at a distance of $3 \sqrt{5} / 2$ from the center of the sphere, two lines $K L$ and $K M$ are drawn, touching the sphere at points $L$ and $M$ respectively. Find the volume of the pyramid $K L M N$, given that $M L=... | 1 | 100 | 1 |
math | 5. If for any $x \in[0,1]$, we have
$$
f(x)=k\left(x^{2}-x+1\right)-x^{4}(1-x)^{4} \geqslant 0 \text {, }
$$
then the minimum value of $k$ is $\qquad$ . | \frac{1}{192} | 74 | 9 |
math | Find the least possible area of a convex set in the plane that intersects both branches of the hyperbola $ xy\equal{}1$ and both branches of the hyperbola $ xy\equal{}\minus{}1.$ (A set $ S$ in the plane is called [i]convex[/i] if for any two points in $ S$ the line segment connecting them is contained in $ S.$) | 4 | 85 | 1 |
math | 4B. Three weary travelers arrived at an inn and asked for food. The innkeeper had nothing else to offer them except baked potatoes. While the potatoes were baking, the travelers fell asleep. After some time, once the potatoes were ready, the first traveler woke up, took $\frac{1}{3}$ of the potatoes, and continued to s... | 27 | 165 | 2 |
math | 3. If non-zero real numbers $a, b, c$ are the $m$-th, $n$-th, $p$-th terms of an arithmetic sequence, and also the $m$-th, $n$-th, $p$-th terms of a geometric sequence, then the value of $a^{b-c} b^{c-a} c^{a-b}$ is $\qquad$ . | 1 | 91 | 1 |
math | 12. Find positive integers $\boldsymbol{n}$ and $m, n>m \geqslant 1$, such that the last three digits of $1978^{n}$ and $1978^{m}$ are equal, and make $n+m$ as small as possible. (20th International Mathematical Olympiad Problem) | 106 | 73 | 3 |
math | 4・154 From the system of equations
$$\left\{\begin{array}{l}
a=\cos u+\cos v+\cos w \\
b=\sin u+\sin v+\sin w \\
c=\cos 2 u+\cos 2 v+\cos w \\
d=\sin 2 u+\sin 2 v+\sin 2 w
\end{array}\right.$$
eliminate \( u, v, w \). | \left(a^{2}-b^{2}-c\right)^{2}+(2 a b-d)^{2}=4\left(a^{2}+b^{2}\right) | 94 | 40 |
math | 18th Putnam 1958 Problem B4 Let S be a spherical shell radius 1. Find the average straight line distance between two points of S. [In other words S is the set of points (x, y, z) with x 2 + y 2 + z 2 = 1). Solution | \frac{4}{3} | 69 | 7 |
math | 11. (20 points) Let the sequence $\left\{a_{n}\right\}$ satisfy $a_{1}=a_{2}=1$, and $a_{n+1} a_{n-1}=a_{n}^{2}+n a_{n} a_{n-1}(n=2,3, \cdots)$.
(1) Find the general term formula for $\left\{a_{n}\right\}$;
(2) Find $\sum_{k=2}^{n} \frac{a_{k}}{(k-2) \text { ! }}$. | \frac{(n+1)!}{2^{n-1}}-2 | 131 | 16 |
math | 13. 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 | Square $ABCD$ has side length $1$; circle $\Gamma$ is centered at $A$ with radius $1$. Let $M$ be the midpoint of $BC$, and let $N$ be the point on segment $CD$ such that $MN$ is tangent to $\Gamma$. Compute $MN$.
[i]2018 CCA Math Bonanza Individual Round #11[/i] | \frac{5}{6} | 86 | 7 |
math | 15.26. Into how many parts do the planes of the faces divide the space a) of a cube; b) of a tetrahedron? | 15 | 34 | 2 |
math | .
With inspiration drawn from the rectilinear network of streets in New York, the Manhattan distance between two points $(a, b)$ and $(c, d)$ in the plane is defined to be
$$
|a-c|+|b-d| \text {. }
$$
Suppose only two distinct Manhattan distances occur between all pairs of distinct points of some point set. What is ... | 9 | 88 | 1 |
math | Example 6 (2006 National Training Team Test) Find all positive integer pairs $(a, n)$ such that $\frac{(a+1)^{n}-a^{n}}{n}$ is an integer. | (,1) | 46 | 4 |
math | B1. Both the rows and columns of an $8 \times 8$ chessboard are numbered from $1$ to $8$. On each square of the chessboard, a number of grains of wheat is placed that is equal to the product of the row number and the column number.
How many grains of wheat are there in total on the chessboard? | 1296 | 74 | 4 |
math | 23. Find all non-zero integer triples $\{a, b, c\}$, satisfying the conditions:
$$a \equiv b(\bmod |c|), \quad b \equiv c(\bmod |a|), \quad c \equiv a(\bmod |b|)$$ | \{1,1, c\}, \{-1,-1, c\}, \{1,-n, n+1\}, \{2,-(2 n+1), 2 n+3\}, \{-1,1,2\}, \{-1,2,3\}, \{1,-1,1\}, \{-1,1,1\} | 62 | 82 |
math | 4. For natural numbers $m$ and $n$, it is known that $3 n^{3}=5 m^{2}$. Find the smallest possible value of $m+n$. | 60 | 38 | 2 |
math | Find all pairs of integers $(x, y)$ for which $x^{2}+x y=y^{2}$. | (0,0) | 25 | 5 |
math | Let's determine those decimal numbers whose square ends in 76. | 24,26,74,76 | 14 | 11 |
math | Example 7: In $1^{2}, 2^{2}, 3^{2}, \cdots, 2005^{2}$, add a “+” or “-” sign before each number to make their algebraic sum the smallest non-negative number, and write out the equation. | 1 | 64 | 1 |
math | Example 6 How many ordered quadruples of integers $(a$, $b, c, d)$ satisfy $0<a<b<c<d<500, a+d=b$ $+c$ and $bc-ad=93$?
(11th American Invitational Mathematics Examination) | 870 | 59 | 3 |
math | On an $6 \times 6$ chessboard, we randomly place counters on three different squares. What is the probability that no two counters are in the same row or column? | \frac{40}{119} | 37 | 10 |
math | 3. Find the relationship between the coefficients $a, b, c$ such that the system of equations
$$
\left\{\begin{array}{l}
a x^{2}+b x+c=0, \\
b x^{2}+c x+a=0, \\
c x^{2}+a x+b=0
\end{array}\right.
$$
has a solution. | a+b+c=0 | 85 | 5 |
math | 14.4.27 ** Find all prime numbers that can be expressed both as the sum of two prime numbers and as the difference of two prime numbers. | 5 | 33 | 1 |
math |
Problem 10.1. Find all values of the real positive parameter $a$ such that the inequality $a^{\cos 2 x}+a^{2 \sin ^{2} x} \leq 2$ holds for any real $x$.
| \in[1,\frac{-1+\sqrt{5}}{2}] | 58 | 16 |
math | Problem 9.1. Vanya runs from home to school at a constant speed. If he had initially increased his speed by 2 m/s, he would have arrived at school 2.5 times faster. How many times faster would he have arrived at school if he had initially increased his speed by 4 m/s?
# | 4 | 68 | 1 |
math | 25th Swedish 1985 Problem 2 Find the smallest positive integer n such that if the first digit is moved to become the last digit, then the new number is 7n/2. | 153846 | 43 | 6 |
math | 7. If the equation $x^{2}-x y-2 y^{2}+x+a=0$ represents two straight lines, then $a=$ | \frac{2}{9} | 33 | 7 |
math | For example, the area of $\triangle ABC$ is $1$, points $D$ and $E$ are on sides $AB$ and $AC$ respectively, $BE$ and $CD$ intersect at point $P$, and the area of quadrilateral $BCED$ is twice the area of $\triangle PBC$. Find the maximum value of the area of $\triangle PDE$. | 5\sqrt{2}-7 | 80 | 7 |
math | 4. In a school chess tournament, boys and girls competed, with the number of boys being five times the number of girls. According to the tournament rules, each chess player played against every other player twice. How many players in total participated if it is known that the boys scored exactly twice as many points in... | 6 | 93 | 1 |
math | 11. Find the largest real number $c$ such that for any 2017 real numbers $x_{1}, x_{2}, \ldots, x_{2017}$, the inequality $\sum_{i=1}^{2016} x_{i} \cdot\left(x_{i}+x_{i+1}\right) \geq c \cdot x_{2017}^{2}$ always holds. | -\frac{1008}{2017} | 98 | 13 |
math | 12. Given $F_{1}$ and $F_{2}$ are the left and right foci of the ellipse $C: \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1(a>b>0)$, respectively. Point $P\left(\frac{2 \sqrt{6}}{3}, 1\right)$ lies on the ellipse $C$, and the orthocenter of $\triangle F_{1} P F_{2}$ is $H\left(\frac{2 \sqrt{6}}{3},-\frac{5}{3}\right)$.
(1) ... | 2(x-1) | 251 | 5 |
math | ## Problem Statement
Find the cosine of the angle between vectors $\overrightarrow{A B}$ and $\overrightarrow{A C}$.
$A(7 ; 0 ; 2), B(7 ; 1 ; 3), C(8 ;-1 ; 2)$ | -\frac{1}{2} | 58 | 7 |
math | 2. Given $x>0, y>0$, and satisfy
$$
\left\{\begin{array}{l}
\cos ^{2}(\pi x)+2 \sin (\pi y)=1, \\
\sin (\pi x)+\sin (\pi y)=0, \\
x^{2}-y^{2}=12 .
\end{array}\right.
$$
Then the ordered pair $(x, y)=$ $\qquad$ . | (4,2) | 97 | 5 |
math | 231*. $x^{2}+x y+y^{2}+x+y-5=0$, if one solution is known: $x=1 ; y=1$. | (1,1),(1,-3),(-3,1) | 39 | 14 |
math | Solve the equation: $$ \frac{xy}{z}+\frac{yz}{x}+\frac{zx}{y}=3$$ where $x$, $y$ and $z$ are integers | (1, 1, 1) | 42 | 10 |
math | 7. Given that two people, A and B, are playing a game, the probability of A winning is $\frac{2}{3}$, and the probability of B winning is $\frac{1}{3}$. If one of them wins two more games than the other, the game ends. Then, the expected number of games that need to be played is $\qquad$ | \frac{18}{5} | 78 | 8 |
math | ## Task Condition
Find the derivative.
$y=\frac{x^{2}+2}{2 \sqrt{1-x^{4}}}$ | \frac{2x^{3}+x}{(1-x^{4})\sqrt{1-x^{4}}} | 29 | 25 |
math | Let $n$ be a nonzero natural number, and $x_1, x_2,..., x_n$ positive real numbers that $ \frac{1}{x_1}+\frac{1}{x_2}+...+\frac{1}{x_n}= n$. Find the minimum value of the expression $x_1 +\frac{x_2^2}{2}++\frac{x_3^3}{3}+...++\frac{x_n^n}{n}$. | 1 + \frac{1}{2} + \frac{1}{3} + \cdots + \frac{1}{n} | 104 | 29 |
math | Example 7 Let $a>0, b>0$, and $\sqrt{a}(\sqrt{a}+2 \sqrt[3]{b})$ $=\sqrt[3]{b}(\sqrt{a}+6 \sqrt[3]{b})$. Then the value of $\frac{2 a^{4}+a^{3} b-128 a b^{2}-64 b^{3}+b^{4}}{a^{4}+2 a^{3} b-64 a b^{2}-128 b^{3}+2 b^{4}}$ is | \frac{1}{2} | 130 | 7 |
math | 1. In the expression $2: 2: 2: 2: 2$ place parentheses so that the result is equal to 2. Find two different ways! | 2 | 37 | 1 |
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