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 | 5. Find all solutions to the equation $2017^{x}-2016^{x}=1$.
Translate the above text into English, please keep the original text's line breaks and format, and output the translation result directly. | 1 | 51 | 1 |
math | Find all quadruples of real numbers $(a, b, c, d)$ satisfying the system of equations
$$
\left\{\begin{array}{l}
(b+c+d)^{2010}=3 a \\
(a+c+d)^{2010}=3 b \\
(a+b+d)^{2010}=3 c \\
(a+b+c)^{2010}=3 d
\end{array}\right.
$$ | (0,0,0,0) \text{ and } \left(\frac{1}{3}, \frac{1}{3}, \frac{1}{3}, \frac{1}{3}\right) | 94 | 46 |
math | How many ordered pairs of integers $(m,n)$ are there such that $m$ and $n$ are the legs of a right triangle with an area equal to a prime number not exceeding $80$? | 87 | 43 | 2 |
math | Triangles $\triangle ABC$ and $\triangle A'B'C'$ lie in the coordinate plane with vertices $A(0,0)$, $B(0,12)$, $C(16,0)$, $A'(24,18)$, $B'(36,18)$, and $C'(24,2)$. A rotation of $m$ degrees clockwise around the point $(x,y)$, where $0<m<180$, will transform $\triangle ABC$ to $\triangle A'B'C'$. Find $m+x+y$. | 108 | 120 | 3 |
math | ## Problem Statement
Calculate the limit of the function:
$$
\lim _{x \rightarrow 0}\left(\frac{1+x^{2} \cdot 2^{x}}{1+x^{2} \cdot 5^{x}}\right)^{\frac{1}{\sin ^{3} x}}
$$ | \frac{2}{5} | 69 | 7 |
math | 5. Find the largest natural number that cannot be represented as the sum of two composite numbers.
ANSWER: 11 | 11 | 25 | 2 |
math | \section*{Exercise 1 - 341021}
a) How many different distributions of the numbers \(1,2, \ldots, 6\) on the six side faces of a cube are there in total?
b) How many different distributions among these satisfy the additional condition that for each pair of opposite side faces, the numbers on these two faces sum to 7?
... | 2 | 111 | 1 |
math | Find the largest possible value in the real numbers of the term $$\frac{3x^2 + 16xy + 15y^2}{x^2 + y^2}$$ with $x^2 + y^2 \ne 0$. | 19 | 56 | 2 |
math | $$
\begin{array}{l}
1 \times 2 - 3 \times 4 + 5 \times 6 - 7 \times 8 + \cdots + \\
2009 \times 2010 - 2011 \times 2012 \\
= \quad .
\end{array}
$$ | -2025078 | 76 | 8 |
math | 3. Let non-zero real numbers $a, b$ satisfy $a^{2}+b^{2}=25$. If the function $y=\frac{a x+b}{x^{2}+1}$ has a maximum value $y_{1}$ and a minimum value $y_{2}$, then $y_{1}-y_{2}=$ $\qquad$. | 5 | 79 | 1 |
math | 9.1 $\quad A_{x}^{2} C_{x}^{x-1}=48$.
Translate the above text into English, keeping the original text's line breaks and format, and output the translation result directly.
9.1 $\quad A_{x}^{2} C_{x}^{x-1}=48$. | 4 | 73 | 1 |
math | 8. Given complex numbers $z_{1}, z_{2}, z_{3}$ satisfy $\left|z_{1}\right|=\left|z_{2}\right|=\left|z_{3}\right|=1,\left|z_{1}+z_{2}+z_{3}\right|=r$, where $r$ is a given real number, then the real part of $\frac{z_{1}}{z_{2}}+\frac{z_{2}}{z_{3}}+\frac{z_{3}}{z_{1}}$ is $\qquad$ (expressed in terms of $r$). | \frac{r^{2}-3}{2} | 135 | 11 |
math | Problem 2. A group of adventurers is showing off their loot. It is known that exactly 9 adventurers have rubies; exactly 8 have emeralds; exactly 2 have sapphires; exactly 11 have diamonds. Moreover, it is known that
- if an adventurer has diamonds, then they have either rubies or sapphires (but not both at the same t... | 17 | 125 | 2 |
math | 10.5. An increasing geometric progression consists of four different positive numbers, three of which form an arithmetic progression. What can the denominator of this progression be? Provide all possible answers and prove that there are no others. | \frac{1+\sqrt{5}}{2} | 45 | 12 |
math | ## Problem Statement
Calculate the limit of the function:
$$
\lim _{x \rightarrow 0} \frac{\cos (1+x)}{\left(2+\sin \left(\frac{1}{x}\right)\right) \ln (1+x)+2}
$$ | \frac{\cos1}{2} | 59 | 8 |
math | Find values of the parameter $u$ for which the expression
\[y = \frac{ \tan(x-u) + \tan(x) + \tan(x+u)}{ \tan(x-u)\tan(x)\tan(x+u)}\]
does not depend on $x.$ | u = k\pi \pm \frac{\pi}{3} | 60 | 15 |
math | 1. Vasya thought of a four-digit number and for each pair of its adjacent digits, he wrote down their product on the board. After that, he erased one product, and the numbers 20 and 21 remained on the board. What is the smallest number Vasya could have thought of? | 3745 | 65 | 4 |
math | 6.1. We will call a natural number interesting if all its digits, except the first and last, are less than the arithmetic mean of the two adjacent digits. Find the largest interesting number. | 96433469 | 40 | 8 |
math | 26. Compute the second-order determinants:
a) $\left|\begin{array}{rr}2 & 5 \\ -3 & -4\end{array}\right|$;
b) $\left|\begin{array}{ll}a^{2} & a b \\ a b & b^{2}\end{array}\right|$. | 7 | 72 | 1 |
math | $2 \cdot 105$ Let $k, m, n$ be integers, and satisfy $1<n \leqslant m-1 \leqslant k$. Try to find the maximum order of the subset $s$ of the set $\{1,2, \cdots, k\}$, such that in $s$, the sum of any $n$ different elements is not equal to $m$. | k-[\frac{}{n}-\frac{n-1}{2}] | 91 | 16 |
math | Let's write numbers in place of the letters in the equation
$$
\overline{a b c}^{2}=\overline{a d e f f}
$$
such that the equation is true. (The overline indicates that the letters represent digits in a decimal number system. Identical letters represent identical digits, and identical digits are represented by identi... | 138^{2}=19044 | 77 | 11 |
math | 46. (9th grade) Find the sum
$$
\frac{1}{2 \cdot 5}+\frac{1}{5 \cdot 8}+\frac{1}{8 \cdot 11}+\ldots+\frac{1}{(3 n-1)(3 n+2)}
$$ | \frac{n}{2(3n+2)} | 68 | 11 |
math | $3+$ [Properties and characteristics of an isosceles triangle.]
On each side of a square, one point was taken. It turned out that these points are the vertices of a rectangle, the sides of which are parallel to the diagonals of the square. Find the perimeter of the rectangle if the diagonal of the square is 6.
# | 12 | 71 | 2 |
math | 5. On the edge $S B$ of a regular quadrilateral pyramid $S A B C D$ with vertex $S$, a point $L$ is marked such that $B L: L S=2: 5$. The point $L$ is the vertex of a right circular cone, on the circumference of the base of which lie three vertices of the pyramid $S A B C D$.
a) Find the ratio $A S: C D$.
b) Suppose ... | \frac{125\pi}{\sqrt{21}} | 128 | 15 |
math | Carl, James, Saif, and Ted play several games of two-player For The Win on the Art of Problem Solving website. If, among these games, Carl wins $5$ and loses $0,$ James wins $4$ and loses $2,$ Saif wins $1$ and loses $6,$ and Ted wins $4,$ how many games does Ted lose? | 6 | 77 | 1 |
math | Carmen selects four different numbers from the set $\{1, 2, 3, 4, 5, 6, 7\}$ whose sum is 11. If $l$ is the largest of these four numbers, what is the value of $l$? | 5 | 61 | 1 |
math | 108. Unique Square. What square is equal to the product of four consecutive odd numbers? | 9 | 20 | 1 |
math | 18.1. $[10.2$ (15 points)] A wooden parallelepiped, all sides of which are expressed in whole centimeters, was painted red, and then sawn parallel to the faces into cubes with a side of 1 cm. It turned out that one third of the resulting cubes have at least one red face, while the remaining two thirds have all faces un... | 18 | 111 | 2 |
math | 6. find all functions $f: \mathbb{R}_{>0} \rightarrow \mathbb{R}_{>0}$ which fulfill the following equation for all $x>y>z>0$:
$$
f(x-y+z)=f(x)+f(y)+f(z)-x y-y z+x z
$$
## 1st solution | f()=\frac{^{2}}{2} | 74 | 11 |
math | 【Example 3】Let $n$ be a positive integer. A particle starts at the origin $A(0,0)$ and ends at $B(n, n)$, making non-decreasing moves. The entire path lies below the "diagonal" and touches it at most. Try to find the number of such paths. | \frac{C_{2n}^{n}}{n+1} | 68 | 16 |
math | 5th Putnam 1942 Problem A1 ABCD is a square side 2a with vertices in that order. It rotates in the first quadrant with A remaining on the positive x-axis and B on the positive y-axis. Find the locus of its center. Solution | (,)to(\sqrt{2},\sqrt{2}) | 57 | 13 |
math | 3. In the tournament, each participant was supposed to play exactly one game with each of the remaining participants, but two participants dropped out during the tournament, having played only 4 games each. In the end, the total number of games played turned out to be 62. How many participants were there in total? | 13 | 65 | 2 |
math | A uniform ladder of mass $m$ and length $\mathcal{L}$ is resting on a wall. A man of mass $m$ climbs up the ladder and is in perfect equilibrium with the ladder when he is $\frac{2}{3}\mathcal{L}$ the way up the ladder. The ladder makes an angle of $ \theta = 30^\circ $ with the horizontal floor. If the coefficient of ... | \mu \approx 0.678 | 140 | 11 |
math | Let $ABC$ be an isosceles triangle with $AB = AC = 4$ and $BC = 5$. Two circles centered at $B$ and $C$ each have radius $2$, and the line through the midpoint of $\overline{BC}$ perpendicular to $\overline{AC}$ intersects the two circles in four different points. If the greatest possible distance between any two of th... | 451 | 162 | 3 |
math | $[$ Mathematical logic (other).]
About mushrooms. In the basket, there are 30 mushrooms. Among any 12 of them, there is at least one russula, and among any 20 mushrooms, there is at least one boletus. How many russulas and how many boletuses are in the basket?
# | 19 | 71 | 2 |
math | Let $a$ ,$b$ and $c$ be distinct real numbers.
$a)$ Determine value of $ \frac{1+ab }{a-b} \cdot \frac{1+bc }{b-c} + \frac{1+bc }{b-c} \cdot \frac{1+ca }{c-a} + \frac{1+ca }{c-a} \cdot \frac{1+ab}{a-b} $
$b)$ Determine value of $ \frac{1-ab }{a-b} \cdot \frac{1-bc }{b-c} + \frac{1-bc }{b-c} \cdot \frac{1-ca }{c-a} + ... | \frac{3}{2} | 257 | 7 |
math | 1. Let $x, y, z$ be real numbers, $3 x, 4 y, 5 z$ form a geometric sequence, and $\frac{1}{x}, \frac{1}{y}, \frac{1}{z}$ form an arithmetic sequence, then the value of $\frac{x}{z}+\frac{z}{x}$ is . $\qquad$ | \frac{34}{15} | 81 | 9 |
math | Problem 13. Determine the length and width of a rectangular plot of land that will maximize its area given a fixed perimeter. | \frac{p}{4} | 26 | 7 |
math | Example 6 If the solution set of the inequality $(m-3) x^{2}-2 m x-8>0$ with respect to $x$ is an open interval, and the length $l$ of the interval satisfies $l \in[1,2]$, find the range of real number $m$ (Note: The length $l$ of an open interval $(a, b)$ is $l=b-a$). | (-\infty,-15]\cup[\frac{7}{3},\frac{33}{14}] | 91 | 25 |
math | 5. The number of non-empty subsets of the set $\{1,2, \cdots, 2016\}$ whose elements sum to an odd number is $\qquad$. | 2^{2015} | 40 | 7 |
math | 2. In a class, there are two types of students: one type always lies, and the other type never lies. Each student knows what type the other students are. During a gathering today, each student has to state what type the other students are, and all students together said "liar" 240 times. At a similar gathering yesterda... | 22 | 107 | 2 |
math | 13. Given the vectors $\boldsymbol{a}=(1, x), \boldsymbol{b}=\left(x^{2}+x,-x\right)$, and a constant $m$ such that $m \leqslant-2$, find the range of $x$ that satisfies $\boldsymbol{a} \cdot \boldsymbol{b}+2>$ $m\left(\frac{2}{\boldsymbol{a} \cdot \boldsymbol{b}}+1\right)$. | (,-2)\cup(0,+\infty) | 110 | 12 |
math |
2. Find the set of all real values of $a$ for which the real polynomial equation $P(x)=x^{2}-2 a x+b=0$ has real roots given that $P(0) \cdot P(1) \cdot P(2) \neq 0$ and $P(0), P(1), P(2)$ form a geometric progression.
| [\frac{2-\sqrt{2}}{2},\frac{2+\sqrt{2}}{2}] | 82 | 24 |
math | If for the real numbers $x, y, z, k$ the following conditions are valid, $x \neq y \neq z \neq x$ and $x^{3}+y^{3}+k\left(x^{2}+y^{2}\right)=y^{3}+z^{3}+k\left(y^{2}+z^{2}\right)=z^{3}+x^{3}+k\left(z^{2}+x^{2}\right)=2008$, find the product $x y z$. | x y z=1004 | 122 | 8 |
math | 8.278. $4 \sin ^{3} x \cos 3 x+4 \cos ^{3} x \sin 3 x=3 \sin 2 x$.
8.278. $4 \sin ^{3} x \cos 3 x + 4 \cos ^{3} x \sin 3 x = 3 \sin 2 x$. | x_{1}=\frac{\pi}{6}(2n+1),n\inZ;x_{2}=\pik,k\inZ | 87 | 31 |
math | 4.2. (Belgium, 79). Find the sum of all $7!$ numbers that can be obtained by all possible permutations of the digits in the number 1234567. | 22399997760 | 45 | 11 |
math | The seventy-sixth question: The function $G(n): N \rightarrow N$ satisfies $G(0)=0, G(n)=n-G(G(n-1))$, find all $G(n)$ that satisfy the condition. | G(n)=[\frac{\sqrt{5}-1}{2}(n+1)] | 47 | 19 |
math | 10. When less than 100 students perform a group dance, there are two combinations: one is a group of 5 in the middle, with the rest forming groups of 8 around the outer circle; the other is a group of 8 in the middle, with the rest forming groups of 5 around the outer circle. How many students are there at most? | 93 | 78 | 2 |
math | 7. (4 points) Given a triangle $A B C$, where $A B=2, B C=8, A C=8$. From point $B$, a bisector is drawn which intersects the circumcircle of this triangle at point $D$. Find the value of $D I$, where $I$ is the center of the inscribed circle of triangle $A B C$. | \frac{16}{3} | 81 | 8 |
math | Example 3 Let $a_{i}=0,1(i=1,2, \cdots, n)$. Try to find the number of sequences $\left(a_{1}, a_{2}, \cdots, a_{n}\right)$ that satisfy
$$
a_{1} \leqslant a_{2}, a_{2} \geqslant a_{3}, a_{3} \leqslant a_{4}, a_{4} \geqslant a_{5}, \cdots \cdots .
$$ | f_{n}=\frac{1}{\sqrt{5}}\left[\left(\frac{1+\sqrt{5}}{2}\right)^{n+2}-\left(\frac{1-\sqrt{5}}{2}\right)^{n+2}\right] | 117 | 60 |
math | 26. Let $S=\{1,2,3,4, \ldots, 100000\}$. Find the least possible value of $k$ such that any subset $A$ of $S$ with $|A|=2010$ contains two distinct numbers $a$ and $b$ with $|a-b| \leq k$. | 49 | 82 | 2 |
math | 26. For what values of $x$ is the inequality
$$
\frac{4 x^{2}}{(1-\sqrt{1+2 x})^{2}}<2 x+9 ?
$$ | -\frac{1}{2}\leqslantx<0,0<x<5\frac{5}{8} | 44 | 26 |
math | 2. Through the right focus of the hyperbola $x^{2}-\frac{y^{2}}{2}=1$, a line $l$ intersects the hyperbola at points $A$ and $B$. If a real number $\lambda$ makes $|A B|=\lambda$, and there are exactly 3 such lines $l$, then $\lambda=$ $\qquad$. | 4 | 82 | 1 |
math | What is the largest even integer that cannot be written as the sum of two odd composite numbers? | 38 | 19 | 2 |
math | Example 12 (Olympiad Training Question from "Intermediate Mathematics" Issue 2, 2003) Find the range of real numbers $a$ such that the inequality $\sin 2 \theta-(2 \sqrt{2}+\sqrt{2} a) \cdot \sin \left(\theta+\frac{\pi}{4}\right)-\frac{2 \sqrt{2}}{\cos \left(\theta-\frac{\pi}{4}\right)}>-3-2 a$ holds for all $\theta \i... | >3 | 127 | 2 |
math | 8. Given real numbers $a, b$ satisfy
$$
a+\lg a=10, b+10^{b}=10 \text {. }
$$
Then $\lg (a+b)=$ $\qquad$ | 1 | 49 | 1 |
math | 5. In the sequence $\left\{a_{n}\right\}$,
$$
a_{0}=a\left(a \in \mathbf{Z}_{+}\right), a_{n+1}=\frac{a_{n}^{2}}{a_{n}+1} \text {. }
$$
When $0 \leqslant n \leqslant \frac{a}{2}+1$, the greatest integer not exceeding $a_{n}$ is $\qquad$ | a-n | 108 | 2 |
math | For each integer $n \ge 3$ solve in real numbers the system of equations:
$$\begin{cases} x_1^3 = x_2 + x_3 + 1 \\...\\x_{n-1}^3 = x_n+ x_1 + 1\\x_{n}^3 = x_1+ x_2 + 1 \end{cases}$$ | x_i = -1 | 87 | 6 |
math | Determine all integers $ n > 3$ for which there exist $ n$ points $ A_{1},\cdots ,A_{n}$ in the plane, no three collinear, and real numbers $ r_{1},\cdots ,r_{n}$ such that for $ 1\leq i < j < k\leq n$, the area of $ \triangle A_{i}A_{j}A_{k}$ is $ r_{i} \plus{} r_{j} \plus{} r_{k}$. | n = 4 | 114 | 5 |
math | 7.6. A rectangular chocolate bar is divided by grooves into $1995 \times 1995$ squares. Two players, A and B, play a game according to the following rules: Each player, on their turn, divides one rectangular piece of chocolate into two smaller rectangles (only along the grooves) and can immediately eat one of the resul... | B | 108 | 1 |
math | Find all positive integers $ n$ such that $ 8^n \plus{} n$ is divisible by $ 2^n \plus{} n$. | 1, 2, 4, 6 | 30 | 10 |
math | 2. Write all two-digit numbers that can be written using the digits 3, 4, and 9. How many such numbers are there? | 9 | 31 | 1 |
math | 6. Two people take turns throwing dice, each throwing two at a time. The first person to get a sum greater than 6 on the two dice wins; otherwise, the other person throws. What is the probability that the first person to throw wins? $\qquad$ | \frac{12}{17} | 56 | 9 |
math | Solve the following equation:
$$
\sqrt[4]{16+x}+\sqrt[4]{16-x}=4
$$ | 0 | 29 | 1 |
math | 66. a) $(x+y)\left(x^{2}-x y+y^{2}\right)$; b) $(x+3)\left(x^{2}-3 x+9\right)$; c) $(x-1)\left(x^{2}+x+1\right)$; d) $(2 x-3)\left(4 x^{2}+6 x+9\right)$ | )x^{3}+y^{3};b)x^{3}+27;)x^{3}-1;)8x^{3}-27 | 86 | 31 |
math | 19. (3 points) At Dongfanghong Primary School, the flag-raising time in 2012 varies by date, with the following rules: from January 1 to January 10, it is fixed at 7:13 AM; from January 11 to June 6, it gradually advances from 7:13 AM to 4:46 AM, with each day advancing by 1 minute; from June 7 to June 21, it is fixed ... | 6:13 | 146 | 4 |
math | 5. Given that the complex number $z$ satisfies $|z|=1$. Then the minimum value of $\left|z^{2}-2 z+5\right|$ is $\qquad$
Translate the above text into English, please retain the original text's line breaks and format, and output the translation result directly. | \frac{8\sqrt{5}}{5} | 66 | 12 |
math | \section*{Problem 4 - V01004}
Determine the unknowns from:
\[
2^{x} \cdot 2^{y}=2^{22} \quad(1) \quad ; \quad x-y=4
\] | 13,9 | 57 | 4 |
math | 15. Let $0<\theta<\pi$, find the maximum value of $\sin \frac{\theta}{2}(1+\cos \theta)$. | \frac{4\sqrt{3}}{9} | 34 | 12 |
math | 8. If the inequality $\log _{\frac{1}{a}}\left(\sqrt{x^{2}+a x+5}+1\right) \cdot \log _{5}\left(x^{2}+a x+6\right)+$ $\log _{3} a \geqslant 0$ has exactly one solution for $x$, then the range of values for $a$ is $\qquad$ | 2 | 93 | 1 |
math | 7. (5 points) This year is 2017, the sum of the digits of the year is 10, then within this century, the sum of all years whose digit sum is 10
保留源文本的换行和格式，翻译结果如下：
7. (5 points) This year is 2017, the sum of the digits of the year is 10, then within this century, the sum of all years whose digit sum is 10 | 18396 | 104 | 5 |
math |
3. Find the magnitudes of the interior angles $\alpha, \beta, \gamma$ of a triangle which satisfy
$$
\begin{aligned}
& 2 \sin \beta \sin (\alpha+\beta)-\cos \alpha=1 \\
& 2 \sin \gamma \sin (\beta+\gamma)-\cos \beta=0
\end{aligned}
$$
| \alpha=120,\beta=\gamma=30 | 82 | 13 |
math |
Problem 3. Determine all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that
$$
f\left(x^{3}+y^{3}+x y\right)=x^{2} f(x)+y^{2} f(y)+f(x y)
$$
for all $x, y \in \mathbb{R}$.
| f(x)=xf(1) | 83 | 7 |
math | When the polynomial $f(x)=x^{4}+a x^{3}+b x^{2}+c x+d$ is divided by each of $x-4, x-3, x+3$, and $x+4$, the remainder in each case is 102. Determine all values of $x$ for which $f(x)=246$. | 0,5,-5 | 80 | 5 |
math | A right cylinder is given with a height of $20$ and a circular base of radius $5$. A vertical planar cut is made into this base of radius $5$. A vertical planar cut, perpendicular to the base, is made into this cylinder, splitting the cylinder into two pieces. Suppose the area the cut leaves behind on one of the piece... | 625 | 116 | 3 |
math | 7. Given real numbers $a_{1}, a_{2}, \cdots, a_{18}$ satisfy:
$$
a_{1}=0,\left|a_{k+1}-a_{k}\right|=1(k=1,2, \cdots, 17) \text {. }
$$
Then the probability that $a_{18}=13$ is $\qquad$ | \frac{17}{16384} | 86 | 12 |
math | 9.030. $\frac{1}{x+2}<\frac{3}{x-3}$. | x\in(-\frac{9}{2};-2)\cup(3;\infty) | 25 | 21 |
math | 9. (10 points) Given that location $C$ is the midpoint between locations $A$ and $B$. At 7:00 AM, car A departs from $A$ heading towards $B$, while car B and car C depart from $B$ and $C$ respectively, heading towards $A$. When car A and car C meet, car B has completed exactly $\frac{3}{8}$ of the total distance. Car C... | 336 | 154 | 3 |
math | Solve the equation $\left(x^{2}-x+1\right)^{4}-10 x^{2}\left(x^{2}-x+1\right)^{2}+9 x^{4}=0$. | -1,1,2-\sqrt{3},2+\sqrt{3} | 47 | 17 |
math | How many different ways are there to write 2004 as a sum of one or more positive integers which are all "aproximately equal" to each other? Two numbers are called aproximately equal if their difference is at most 1. The order of terms does not matter: two ways which only differ in the order of terms are not considered ... | 2004 | 74 | 4 |
math | 6. Given the system of equations $\left\{\begin{array}{l}\frac{x}{a}+\frac{y}{b}=1, \\ x^{2}+y^{2}=50\end{array}\right.$ has only integer solutions. Then the number of real pairs $(a, b)$ that satisfy the condition is $\qquad$ . | 60 | 76 | 2 |
math | 113 Given the function $f(x)=\sqrt{x+2}+k$, and there exist $a, b(a<b)$ such that the range of $f(x)$ on $[a, b]$ is $[a, b]$, find the range of the real number $k$. | k\in(-\frac{9}{4},-2] | 63 | 14 |
math | [ $\left[\begin{array}{ll}\text { Irreducible fractions }\end{array}\right]$
What number should be subtracted from the numerator of the fraction ${ }^{537} / 463$ and added to the denominator to obtain $1 / 9$ after simplification? | 437 | 66 | 3 |
math | $A$, $B$, $C$, and $D$ are points on a circle, and segments $\overline{AC}$ and $\overline{BD}$ intersect at $P$, such that $AP=8$, $PC=1$, and $BD=6$. Find $BP$, given that $BP<DP$. | 2 | 69 | 1 |
math | 1. Let $A$ be a set of three distinct real numbers, and let the set $B=\{x+y \mid x 、 y \in A, x \neq y\}$. If
$$
B=\left\{\log _{2} 6, \log _{2} 10, \log _{2} 15\right\} \text {, }
$$
then the set $A=$ $\qquad$ | {1,\log_{2}3,\log_{2}5} | 100 | 15 |
math | 5. Solve the inequality
$$
\frac{\left(2 \cdot 3^{-\log _{x} 2}-6\right) \sqrt{2-\sqrt{2 \log _{x} 2+3}}}{2+\sqrt{\log _{x} 2+2}}>\frac{\left(3^{-\log _{x} 2}-3\right) \sqrt{2-\sqrt{2 \log _{x} 2+3}}}{\sqrt{\log _{x} 2+2}-1}
$$ | x\in(0;1/2)\cup(1/2;1/\sqrt[3]{4}]\cup(4;+\infty) | 122 | 33 |
math | 11. The solution set of the equation $16 \sin \pi x \cdot \cos \pi x=16 x+\frac{1}{x}$ is $\qquad$ . | {\frac{1}{4},-\frac{1}{4}} | 41 | 14 |
math | 142. Given a regular $n$-sided prism. The area of the base is $S$. Two planes intersect all the lateral edges of the prism in such a way that the volume of the part of the prism between the planes is $V$. Find the sum of the lengths of the segments of the lateral edges of the prism, enclosed between the planes, if it i... | \frac{nV}{S} | 92 | 7 |
math | 14. If $a, b, c \in \mathbf{R}^{+}$, and satisfy $\frac{k u b c}{a+b+c} \geqslant(a+b)^{2}+(a+b+4 c)^{2}$. Find the minimum value of $k$. | 100 | 65 | 3 |
math | 3. Let $0^{\circ}<\alpha<90^{\circ}$. If $1+\sqrt{3} \operatorname{tg}(60-\alpha)$ $=\frac{1}{\sin \alpha}$, then $\alpha$ equals $\qquad$. | 30^{\circ} \text{ or } 50^{\circ} | 59 | 18 |
math | Example 8 Let $f(a, b, c)=\frac{1}{\sqrt{1+2 a}}+\frac{1}{\sqrt{1+2 b}}+\frac{1}{\sqrt{1+2 c}}$, where $a, b, c>0$ and $abc=1$, find the minimum value of the constant $\lambda$ such that $f(a, b, c)<\lambda$ always holds. | 2 | 94 | 1 |
math | ## Task 30/79
Solve the following system of Diophantine equations
$$
\begin{array}{r}
135 x_{1}+100 x_{2}-x_{3}=-4 \\
97 x_{1}+132 x_{2}-x_{4}=20 \\
7 x_{1}+193 x_{2}-x_{4}=0
\end{array}
$$
with the conditions $x_{1} ; x_{2} ; x_{3} ; x_{4} \in N, x_{2} \leq 30$ and form the triples $\left(x_{1} ; x_{2}, x_{3}\right... | (7,10,1949)(7,10,1979) | 176 | 21 |
math | 33. Let $L$ denote the minimum value of the quotient of a 3-digit number formed by three distinct digits divided by the sum of its digits. Determine $\lfloor 10 L\rfloor$. | 105 | 45 | 3 |
math | Eight, let $x, y, z \in [0,1]$, and
$$
|x-y| \leqslant \frac{1}{2}, |y-z| \leqslant \frac{1}{2}, |z-x| \leqslant \frac{1}{2} \text{. }
$$
Find the minimum and maximum values of $W=x+y+z-xy-yz-zx$.
(Liu Kangning, An Zhenping) | \frac{5}{6} | 105 | 7 |
math | Example 4.10. Solve the differential equation:
$$
y^{\prime}+y \cdot \tan x=\cos ^{2} x
$$ | \cosx(\sinx+C) | 35 | 8 |
math | 17. Helping the collective farm with the harvest, 5th Grade A class collected 45715 kg of potatoes over 5 days. How many students were there in total, and how much did each student collect on average per day, if the average daily output of one student is a whole number of kilograms? | 41 | 67 | 2 |
math | Two people, $A$ and $B$, play the following game: $A$ start choosing a positive integrer number and then, each player in it's turn, say a number due to the following rule:
If the last number said was odd, the player add $7$ to this number;
If the last number said was even, the player divide it by $2$.
The winner... | 1, 2, 4, 7, 8, 14 | 110 | 17 |
math | [ [Volume helps solve the problem]
Does a triangular pyramid exist whose heights are 1, 2, 3, and 6?
# | No | 30 | 1 |
math | What are the bounds between which $x^{2}+2 x y$ varies, if $x^{2}+y^{2}=1$? | [\frac{1}{2}-\frac{\sqrt{5}}{2},\frac{1}{2}+\frac{\sqrt{5}}{2}] | 32 | 34 |
math | 2. Determine all integer solutions \(a\) and \(b\) for which
\[
7a + 14b = 5a^2 + 5ab + 5b^2
\] | (-1,3),(0,0),(1,2) | 44 | 13 |
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