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 | 2. Positive numbers $a, b, c$ are such that $a+b+c=3$. Find the minimum value of the expression
$$
A=\frac{a^{3}+b^{3}}{8 a b+9-c^{2}}+\frac{b^{3}+c^{3}}{8 b c+9-a^{2}}+\frac{c^{3}+a^{3}}{8 c a+9-b^{2}}
$$ | \frac{3}{8} | 99 | 7 |
math | 8,"
The angles of triangle $ABC$ satisfy the equation
$$
\cos ^{2} A+\cos ^{2} B+\cos ^{2} C=1
$$
Find the area of this triangle if the radii of the inscribed and circumscribed circles are $\sqrt{3}$ and $3 \sqrt{2}$, respectively. | 6\sqrt{6}+3 | 78 | 8 |
math | 1. Determine all integers $n$ for which the equation $x^{2}+n x+n+5=0$ has only integer solutions. | -5,-3,7,9 | 31 | 8 |
math | 6.71 Determine all pairs of positive integers $(n, p)$ satisfying:
$p$ is a prime, $n \leqslant 2 p$, and $(p-1)^{n}+1$ is divisible by $n^{p-1}$.
(40th International Mathematical Olympiad, 1999) | (n,p)=(1,p),(2,2),(3,3) | 72 | 14 |
math | 1. Let $x, y$ be real numbers, and $2 x^{2}+x y+2 y^{2}=2$. Then the range of values for $x^{2}-5 x y+y^{2}$ is $\qquad$ . | \left[-\frac{6}{5}, \frac{14}{3}\right] | 54 | 20 |
math | 5 . Find the number of divisors of 1984, as well as the sum of all its divisors. | n=14, S=4064 | 26 | 11 |
math | Solve the following inequality:
$$
\frac{x-a}{x+a}<\frac{x+a}{x-a}
$$ | -<x<0\quador\quad<x | 25 | 11 |
math | 17.1. [9.6 (15 points), 10.3 (15 points)] Find the prime factorization of the smallest natural number that has exactly 2020 distinct natural divisors. | 2^{100}\cdot3^{4}\cdot5\cdot7 | 47 | 16 |
math | The number 76 has an interesting property: the last two digits of $76^{2}=5776$ are 76 again.
a) Are there any other two-digit numbers with this property?
b) Find all such three-digit numbers $A$ such that the last three digits of $A^{2}$ form the number $A$.
c) Does there exist an infinite sequence of digits $a_{1}... | )25,76;b)376,625;)exists | 197 | 17 |
math | Example 7.13 Find the number of first-class circular permutations made from 2 $a$s, 2 $b$s, 2 $c$s. | 16 | 34 | 2 |
math | 15. Let $M$ be a set composed of a finite number of positive integers
$$
\text { such that, } \begin{aligned}
M & =\bigcup_{i=1}^{20} A_{i}=\bigcup_{i=1}^{20} B_{i}, \text { where, } \\
A_{i} & \neq \varnothing, B_{i} \neq \varnothing(i=1,2, \cdots, 20),
\end{aligned}
$$
and satisfies:
(1) For any $1 \leqslant i<j \le... | 180 | 289 | 3 |
math | 6.214. $\left\{\begin{array}{l}x+y+z=0, \\ 2 x+3 y+z=0, \\ (x+1)^{2}+(y+2)^{2}+(z+3)^{2}=14 .\end{array}\right.$ | (0;0;0),(2;-1;-1) | 68 | 13 |
math | Find the smallest positive integer that cannot be expressed in the form $\frac{2^a - 2^b}{2^c - 2^d}$, where $a$, $ b$, $c$, $d$ are non-negative integers.
| 11 | 53 | 2 |
math | ## [ Rectangles and Squares. Properties and Characteristics Method of Coordinates
On the plane, a square $A B C D$ is given. Find the minimum of the ratio $\frac{O A+O C}{O B+O D}$, where $O-$ is an arbitrary point on the plane. | \frac{1}{\sqrt{2}} | 65 | 10 |
math | 3. In the Cartesian coordinate system $x O y$, $F_{1}$ and $F_{2}$ are the left and right foci of the hyperbola $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{4}=1(a>0)$, respectively. The line $l$ passes through the right vertex $A$ of the hyperbola and the point $B(0,2)$. If the sum of the distances from points $F_{1}$ and $F_{2}... | (2\sqrt{2},0) | 144 | 9 |
math | Example 1 Factorize the polynomial $f(x)=x^{8}+x^{7}+1$ over the integers.
(1978 National High School League Question) | f(x)=(x^{2}+x+1)\cdot(x^{6}-x^{4}+x^{3}-x+1) | 38 | 30 |
math | 6. Three targets. A shooter shoots at three targets until all are hit. The probability of hitting a target with one shot is $p$.
a) (from 7th grade. 2 points). Find the probability that exactly 5 shots will be required.
b) (from 8th grade. 2 points). Find the expected number of shots. | )6p^{3}(1-p)^{2};b)\frac{3}{p} | 75 | 20 |
math | (Cauchy)
Find all functions $f: \mathbb{Z} \mapsto \mathbb{Z}$ such that for all $x, y \in \mathbb{Z}$,
$$
f(x+y)=f(x)+f(y)
$$ | f(x)= | 56 | 3 |
math | G4.1 If $f(x)=\frac{4^{x}}{4^{x}+2}$ and $P=f\left(\frac{1}{1001}\right)+f\left(\frac{2}{1001}\right)+\cdots+f\left(\frac{1000}{1001}\right)$, find the value of $P$. | 500 | 86 | 3 |
math | 7. Line segments $A B, C D$ are between two parallel planes $\alpha$ and $\beta$, $A C \subset \alpha, B D \subset \beta, A B \perp \alpha, A C=B D=5$, $A B=12, C D=13, E, F$ divide $A B, C D$ in the ratio $1: 2$. Find the length of line segment $E F$. | \frac{5}{3}\sqrt{7} | 97 | 11 |
math | $\underline{\text { Ans } A .}$
Numbers $1,2,3, \ldots, N$ are written in a row in such an order that if a number $i$ is written somewhere (not in the first place), then at least one of the numbers $i+1$ and $i-1$ will be found somewhere to its left. In how many ways can this be done? | 2^{N-1} | 86 | 6 |
math | 7. (10 points) Lei Ke bought a new book and liked it very much. On the first day, he read $\frac{1}{5}$ of the book plus 12 pages. On the second day, he read $\frac{1}{4}$ of the remaining pages plus 15 pages. On the third day, he read $\frac{1}{3}$ of the remaining pages plus 18 pages. At this point, there were still ... | 190 | 121 | 3 |
math | Let $ f(x)$ be a polynomial and $ C$ be a real number.
Find the $ f(x)$ and $ C$ such that $ \int_0^x f(y)dy\plus{}\int_0^1 (x\plus{}y)^2f(y)dy\equal{}x^2\plus{}C$. | f(x) = \frac{3x - 1}{2}, \quad C = \frac{5}{24} | 71 | 28 |
math | 17. Matthew writes a list of all three-digit squares backwards. For example, in his list Matthew writes the three-digit square ' 625 ' as '526'. Norma looks at Matthew's list and notices that some of the numbers are prime numbers. What is the mean of those prime numbers in Matthew's list? | 447 | 69 | 3 |
math | Example 11 Find the range of the function $f(x)=\sqrt{2 x-1}-\sqrt{x+3}$. | [-\frac{\sqrt{14}}{2},+\infty) | 29 | 16 |
math | Which positive integers $m$ are such that $k^m - 1$ is divisible by $2^m$ for all odd numbers $k \ge 3$?
| 1, 2, 4 | 37 | 7 |
math | $PQ$ is a diameter of a circle. $PR$ and $QS$ are chords with intersection at $T$. If $\angle PTQ= \theta$, determine the ratio of the area of $\triangle QTP$ to the area of $\triangle SRT$ (i.e. area of $\triangle QTP$/area of $\triangle SRT$) in terms of trigonometric functions of $\theta$ | \frac{1}{\cos^2(\theta)} | 86 | 12 |
math | 9. (16 points) Given the sequence $\left\{a_{n}\right\}$ with the general term formula $a_{n}=\frac{1}{\sqrt{5}}\left(\left(\frac{1+\sqrt{5}}{2}\right)^{n}-\left(\frac{1-\sqrt{5}}{2}\right)^{n}\right)\left(n \in \mathbf{Z}_{+}\right)$. Let $S_{n}=\mathrm{C}_{n}^{1} a_{1}+\mathrm{C}_{n}^{2} a_{2}+\cdots+\mathrm{C}_{n}^{... | n=4k(k=1,2,\cdots) | 168 | 13 |
math | 9. For what values of $\alpha$ is the equality
$$
\sqrt{\operatorname{tg}^{2} \alpha-\sin ^{2} \alpha}=\operatorname{tg} \alpha \cdot \sin \alpha ?
$$ | (4k-1)\frac{\pi}{2}<\alpha<(4k+1)\frac{\pi}{2};\quadk=0,\1,\2,\ldots | 53 | 38 |
math | 9.1. Petya had a large wooden cube with dimensions $30 \times 30 \times 30$ cm $^{3}$. Petya decided to paint the entire cube, which took 100 grams of paint. Later, Petya needed smaller cubes, so he cut the large cube with 6 cuts, parallel to the faces of the cube (2 cuts parallel to each pair of faces), into 27 cubes ... | 200 | 145 | 3 |
math | Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that
$$
f(x) f(y) + f(x+y) = xy
$$ | f(x)=\x-1 | 40 | 7 |
math | A regular tetrahedron $SABC$ of volume $V$ is given. The midpoints $D$ and $E$ are taken on $SA$ and $SB$ respectively and the point $F$ is taken on the edge $SC$ such that $SF: FC = 1: 3$. Find the volume of the pentahedron $FDEABC$. | \frac{15}{16}V | 80 | 10 |
math | 15. [9] Vijay chooses three distinct integers $a, b, c$ from the set $\{1,2,3,4,5,6,7,8,9,10,11\}$. If $k$ is the minimum value taken on by the polynomial $a(x-b)(x-c)$ over all real numbers $x$, and $l$ is the minimum value taken on by the polynomial $a(x-b)(x+c)$ over all real numbers $x$, compute the maximum possibl... | 990 | 116 | 3 |
math | 7.060. $\sqrt{2^{x} \cdot \sqrt[3]{4^{x} \cdot 0.125^{1 / x}}}=4 \sqrt[3]{2}$. | -\frac{1}{5};3 | 47 | 8 |
math | 1. $\frac{1}{1-\log _{2} 20}+\frac{1}{1-\log _{5} 50}$ The value is $\qquad$ | -1 | 41 | 2 |
math | Let ABCD be a unit square. Draw a quadrant of a circle with A as centre and B;D
as end points of the arc. Similarly, draw a quadrant of a circle with B as centre and
A;C as end points of the arc. Inscribe a circle ? touching the arc AC internally, the
arc BD internally and also touching the side AB. Find the radius of ... | \frac{3}{8} | 84 | 7 |
math | Example 18 (25th All-Soviet Union Mathematical Olympiad) Find the integer solutions of the system of equations $\left\{\begin{array}{l}x z-2 y t=3, \\ x t+y z=1 .\end{array}\right.$ | (1,0,3,1),(-1,0,-3,-1),(3,1,1,0),(-3,-1,-1,0) | 60 | 35 |
math | At a fish market, there are 10 stalls, all selling the same 10 types of fish. All the fish are caught in either the North Sea or the Mediterranean Sea, and each stall has only one sea of origin for each type of fish. A number, $k$, of customers buy one fish from each stall such that they have one fish of each type. Fur... | 2^{10}-10 | 122 | 7 |
math | 1.a) Find 11 consecutive natural numbers whose sum is 99.
b) A four-digit natural number has its first two digits identical, and the units digit is 5. This number is divided by a two-digit number, and the remainder is 98. Find the dividend, divisor, and quotient.
Prof. Ana Marcela Popa | 4,5,6,\ldots,14 | 73 | 11 |
math | (1) 2009 Kansai University entrance exam
Calculate $ \int \frac{e^{\minus{}2x}}{1\plus{}e^{\minus{}x}}\ dx$.
(2) 2009 Rikkyo University entrance exam/Science
Evaluate $ \int_0^ 1 \frac{2x^3}{1\plus{}x^2}\ dx$. | 1 - \ln 2 | 91 | 6 |
math | 5. The lengths of the adjacent sides of a rectangle are $\sqrt{404} \mathrm{~cm}$ and $\sqrt{909} \mathrm{~cm}$. Determine the perimeter and the area of the circle circumscribed around a square whose area is equal to the area of the given rectangle. | 2\sqrt{303}\pi\mathrm{~},303\pi\mathrm{~}^{2} | 67 | 27 |
math | $4-$ [ GCD and LCM. Mutual simplicity ]
Find integer solutions of the equation $x^{2} y=10000 x+y$.
# | (-9,-1125),(-3,-3750),(0,0),(3,3750),(9,1125) | 36 | 34 |
math | 5.1. Construct a rectangle, where each side is greater than 1, using six rectangles $7 \times 1, 6 \times 1, 5 \times 1, 4 \times 1, 3 \times 1, 2 \times 1$ and a square $1 \times 1$. | 7\times4 | 72 | 4 |
math | 357. Find \( f^{\prime}(1 / 5) \) if \( f(x)=\operatorname{arctg} 5 x+x^{2} \). | 2.9 | 40 | 3 |
math | 4. find all pairs $(a, b)$ of integers with different divisors such that
$$
a^{2}+a=b^{3}+b
$$ | (,b)=(1,1),(-2,1),(-1,0),(5,3) | 35 | 22 |
math | 1. In a regular quadrilateral pyramid $P-ABCD$, all four lateral faces are equilateral triangles. Let the dihedral angle between a lateral face and the base be $\theta$, then $\tan \theta=$ $\qquad$ | \sqrt{2} | 50 | 5 |
math | 6. Into each row of a $9 \times 9$ grid, Nigel writes the digits $1,2,3,4,5,6,7,8,9$ in order, starting at one of the digits and returning to 1 after 9 : for example, one row might contain $7,8,9,1,2,3,4,5,6$. The grid is gorgeous if each nine-digit number read along a row or column or along the diagonal from the top-l... | 9^{8} | 145 | 4 |
math | Example 6. Find $\lim _{x \rightarrow+\infty} \frac{6 x^{2}+5 x+4}{3 x^{2}+7 x-2}$. | 2 | 42 | 1 |
math | 1) Two nonnegative real numbers $x, y$ have constant sum $a$. Find the minimum value of $x^m + y^m$, where m is a given positive integer.
2) Let $m, n$ be positive integers and $k$ a positive real number. Consider nonnegative real numbers $x_1, x_2, . . . , x_n$ having constant sum $k$. Prove that the minimum value of ... | \frac{k^m}{n^{m-1}} | 126 | 12 |
math | 3. The company conducted a survey among its employees - which social networks they use: VKontakte or Odnoklassniki. Some employees said they use VKontakte, some - Odnoklassniki, some said they use both social networks, and 40 employees said they do not use social networks. Among all those who use social networks, 75% u... | 540 | 125 | 3 |
math | 1. Factorize $f(x, y, z)=(y+z)(z+x)(x+y)+x y z$.
untranslated text remains the same as requested. | f(x,y,z)=(x+y+z)(xy+yz+zx) | 34 | 15 |
math | 4. 3. 11 $\star \star$ Find the maximum and minimum values of the function $y=\sqrt{3 x+4}+\sqrt{3-4 x}$. | \frac{5}{2} | 41 | 7 |
math | 35 Let $[x]$ denote the greatest integer not exceeding $x$. Then, when $0 \leqslant x \leqslant 10$, the number of all different integers represented by $f(x)=[x]+[2 x]+[3 x]+[4 x]$ is $\qquad$ | 61 | 68 | 2 |
math | Example 12 (Problem from the 4th China Girls Mathematical Olympiad) Solve the system of equations
$$
\left\{\begin{array}{l}
5\left(x+\frac{1}{x}\right)=12\left(y+\frac{1}{y}\right)=13\left(z+\frac{1}{z}\right) \\
x y+y z+z x=1
\end{array}\right.
$$ | (\frac{1}{5},\frac{2}{3},1)(-\frac{1}{5},-\frac{2}{3},-1) | 94 | 33 |
math | A6 (5-1, Czechoslovakia) Find all real solutions of the equation $\sqrt{x^{2}-p}+2 \sqrt{x^{2}-1}=x$, where $p$ is a real parameter. | \frac{4-p}{\sqrt{8(2-p)}} | 48 | 14 |
math | 8.96 Find all sequences of natural numbers $a_{1}, a_{2}, a_{3}, \cdots$ that satisfy the following two conditions:
(1) $a_{n} \leqslant n \sqrt{n}, n=1,2,3, \cdots$
(2) For any different $m, n$,
$$m-n \mid a_{m}-a_{n}$$ | a_{n}=1 \text{, for any } n \geqslant 1 \text{; } a_{n}=n, \text{ for any } n \geqslant 1 | 90 | 45 |
math | Let $r$ be a positive real number. Denote by $[r]$ the integer part of $r$ and by $\{r\}$ the fractional part of $r$. For example, if $r=32.86$, then $\{r\}=0.86$ and $[r]=32$. What is the sum of all positive numbers $r$ satisfying $25\{r\}+[r]=125$? | 2837 | 98 | 4 |
math | Example 3 Real numbers $x, y, z$ satisfy
$$
x+y+z=4(\sqrt{x-5}+\sqrt{y-4}+\sqrt{z-3}) \text {. }
$$
Then $x=$ $\qquad$ ,$y=$ $\qquad$ ,$z=$ $\qquad$ .
(10th Five Sheep Cup Junior High School Mathematics Competition) | x=9, y=8, z=7 | 83 | 11 |
math | We have 30 padlocks and for each one, we have a key that does not open any of the other padlocks. Someone randomly drops the keys into the closed padlocks, one into each. We break open two padlocks. What is the probability that we can open the rest without breaking any more padlocks? | \frac{1}{15} | 66 | 8 |
math | For which pairs of numbers $x, y$ does the following equality hold:
$$
3 \sin x-4 \cos x=4 y^{2}+4 y+6
$$ | -\arccos\frac{4}{5}+(2k+1)\pi,-\frac{1}{2} | 40 | 26 |
math | 13. Given the sequence $\left\{a_{n}\right\}$ satisfies:
$$
a_{1}=a, a_{n+1}=\frac{5 a_{n}-8}{a_{n}-1}\left(n \in \mathbf{Z}_{+}\right) \text {. }
$$
(1) If $a=3$, prove that $\left\{\frac{a_{n}-2}{a_{n}-4}\right\}$ is a geometric sequence, and find the general term formula of the sequence $\left\{a_{n}\right\}$;
(2) I... | \in(3,+\infty) | 155 | 9 |
math | 4. Given a sequence of positive terms $\left\{a_{n}\right\}$ with the sum of the first $n$ terms being $S_{n}$. If $\left\{a_{n}\right\}$ and $\left\{\sqrt{S_{n}}\right\}$ are both arithmetic sequences with the same common difference, then $S_{n}=$ $\qquad$ | \frac{n^{2}}{4} | 84 | 9 |
math | 4. 47. Let $a, b$ be real numbers, and $x^{4}+a x^{3}+b x^{2}+a x+1=0$ has at least one real root. Try to find the minimum value of $a^{2}+b^{2}$. | \frac{4}{5} | 67 | 7 |
math | 12. Person A and Person B start from points A and B respectively at the same time and travel towards each other at a constant speed, meeting after 8 hours. If both increase their speed by 2 kilometers per hour, they will meet 6 hours later at a point 3 kilometers away from the midpoint of AB. Given that A travels faste... | 6.5 | 91 | 3 |
math | In how many ways can six marbles be placed in the squares of a $6$-by-$6$ grid such that no two marbles lie in the same row or column? | 720 | 38 | 3 |
math | 6. Find the minimum value of $f(x)=\sqrt{10 x^{2}+28 x+26}+\sqrt{10 x^{2}-18 x+13}$. | \frac{\sqrt{1885}}{5} | 45 | 13 |
math | 30.2. Given that $x+2$ and $x-3$ are factors of $p(x)=a x^{3}+a x^{2}+b x+12$, what is the remainder when $p(x)$ is divided by $x-1$ ? | 18 | 61 | 2 |
math | 2. $[\mathbf{1 5}]$ Consider the following two-player game. Player 1 starts with a number, $N$. He then subtracts a proper divisor of $N$ from $N$ and gives the result to player 2 (a proper divisor of $N$ is a positive divisor of $N$ that is not equal to 1 or $N$ ). Player 2 does the same thing with the number she gets... | Allevenexceptforoddpowersof2 | 147 | 9 |
math | 5-4. Andrei, Boris, Vladimir, and Dmitry each made two statements. For each boy, one of his statements turned out to be true, and the other - false.
Andrei: "Boris is not the tallest among us four." "Vladimir is the shortest among us four."
Boris: "Andrei is the oldest in the room." "Andrei is the shortest in the roo... | Vladimir | 156 | 3 |
math | 11. (6 points) Natural numbers $1,2,3, \cdots$ are written down consecutively to form a number $123456789101112 \cdots$. When a certain number is reached, the formed number is exactly divisible by 72 for the first time. This number is $\qquad$ _. $\qquad$ | 36 | 84 | 2 |
math | Solve following system equations:
\[\left\{ \begin{array}{c}
3x+4y=26\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\
\sqrt{x^2+y^2-4x+2y+5}+\sqrt{x^2+y^2-20x-10y+125}=10\ \end{array}
\right.\ \ \]
| (x, y) = (6, 2) | 151 | 11 |
math | 8.4. There are two types of five-digit numbers as follows:
(1) The sum of the digits is 36, and it is an even number;
(2) The sum of the digits is 38, and it is an odd number.
Try to determine: which type of number is more? Explain your reasoning. | There\ are\ more\ even\ numbers\ whose\ sum\ of\ digits\ equals\ 36. | 70 | 24 |
math | ## problem statement
Find the cosine of the angle between vectors $\overrightarrow{A B}$ and $\overrightarrow{A C}$.
$A(-4 ; 3 ; 0), B(0 ; 1 ; 3), C(-2 ; 4 ;-2)$ | 0 | 58 | 1 |
math | Susan plays a game in which she rolls two fair standard six-sided dice with sides labeled one
through six. She wins if the number on one of the dice is three times the number on the other
die. If Susan plays this game three times, compute the probability that she wins at least once. | \frac{217}{729} | 61 | 11 |
math | 3.272. $\frac{\sqrt{\tan \alpha}+\sqrt{\cot \alpha}}{\sqrt{\tan \alpha}-\sqrt{\cot \alpha}}, 0<\alpha<\frac{\pi}{2}$ and $\alpha \neq \frac{\pi}{4}$. | \cot(\alpha-\frac{\pi}{4}) | 63 | 11 |
math | One, (14 points) Solve the system of equations
$$
\left\{\begin{array}{l}
x+y+\frac{9}{x}+\frac{4}{y}=10, \\
\left(x^{2}+9\right)\left(y^{2}+4\right)=24 x y .
\end{array}\right.
$$ | \left\{\begin{array}{l}x=3, \\ y=2 .\end{array}\right.} | 79 | 27 |
math | 13. An Arabic Tale. A Flock of Pigeons
$>\Delta \theta \Pi \mathrm{V} \oplus \theta\mathbf{v} \Pi \square \nabla \square \Lambda$
$\oplus \Lambda \nabla \theta \Pi \oplus \mathbf{V V} \varnothing \odot$
ロจ৫ఠ<>VIVOO flew up to a tall tree.
Some of the pigeons perched on the branches, while others settled under the t... | 7 | 202 | 1 |
math | Let $\mathbb{Z}_{>0}$ denote the set of positive integers. For any positive integer $k$, a function $f: \mathbb{Z}_{>0} \to \mathbb{Z}_{>0}$ is called [i]$k$-good[/i] if $\gcd(f(m) + n, f(n) + m) \le k$ for all $m \neq n$. Find all $k$ such that there exists a $k$-good function.
[i]Proposed by James Rickards, Canada[/... | k \ge 2 | 119 | 6 |
math | 3. The product of the areas of the six faces of a rectangular prism is 14641, then the volume of the rectangular prism is
保留源文本的换行和格式，直接输出翻译结果。 | 11 | 46 | 2 |
math |
A3. Let $A$ and $B$ be two non-empty subsets of $X=\{1,2, \ldots, 11\}$ with $A \cup B=X$. Let $P_{A}$ be the product of all elements of $A$ and let $P_{B}$ be the product of all elements of $B$. Find the minimum and maximum possible value of $P_{A}+P_{B}$ and find all possible equality cases.
| 12636 | 101 | 5 |
math | How to determine the function $\ln z$ for a complex argument $z$?
# | w_{k}=\ln|z|+i(\varphi+2k\pi)(k\in{Z}) | 18 | 26 |
math | Which is the positive integer that is a three-digit number in both the decimal and octal number systems, and the sum of its digits is fourteen in both cases? | 455 | 33 | 3 |
math | $6 \cdot 103$ Find the smallest positive integer $n$ (where $n>1$) such that the quadratic mean of the first $n$ natural numbers is an integer. Here, the quadratic mean of $n$ numbers $a_{1}, a_{2}, \cdots, a_{n}$ is given by
$$\left(\frac{a_{1}^{2}+a_{2}^{2}+\cdots+a_{n}^{2}}{n}\right)^{\frac{1}{2}} .$$ | 337 | 118 | 3 |
math | 4. Let $f(x)=\frac{1}{x^{3}+3 x^{2}+2 x}$. Determine the smallest positive integer $n$ such that
$$
f(1)+f(2)+f(3)+\cdots+f(n)>\frac{503}{2014}
$$ | 44 | 71 | 2 |
math | K2) Let $n$ be a natural number. A volleyball team consisting of $n$ women and $n$ men lines up for a game. Each team member occupies one of the positions $1,2, \ldots, 2 n$, whereby exactly positions 1 and $n+1$ are outside the court. During the game, all team members rotate, switching from position $i$ to position $i... | (n!)^{2}\cdot2^{n} | 156 | 10 |
math | 7. Given real numbers $a>b>0$. Then the maximum value of the function
$$
f(x)=\frac{x}{\sqrt{a-x^{2}}-\sqrt{b-x^{2}}}
$$
is . $\qquad$ | \frac{\sqrt{}}{-b} | 53 | 9 |
math | 10. (5 points) The sum of three numbers, A, B, and C, is 2017. A is 3 less than twice B, and B is 20 more than three times C. Then A is $\qquad$ . | 1213 | 56 | 4 |
math | 5. Given the vector $\boldsymbol{a}=(\cos \theta, \sin \theta)$, vector $\boldsymbol{b}=(\sqrt{3},-1)$, then the maximum value of $|2 \boldsymbol{a}-\boldsymbol{b}|$ is | 4 | 62 | 1 |
math | 6th CanMO 1974 Problem 4 What is the maximum possible value for the sum of the absolute values of the differences between each pair of n non-negative real numbers which do not exceed 1? | \lfloor\frac{n^2}{4}\rfloor | 44 | 13 |
math | 11. (2002 National High School Competition Question) Given that $f(x)$ is a function defined on $\mathbf{R}$, $f(1)=1$, and for any $x \in \mathbf{R}$, $f(x+5) \geqslant f(x)+5$, $f(x+1) \leqslant f(x)+1$. If $g(x)=f(x)+1-x$, then $g(2002)=$ $\qquad$ . | 1 | 111 | 1 |
math | 10. Satisfy $0 \leqslant k_{i} \leqslant 20(i=1,2,3,4)$, and $k_{1}+k_{3}=k_{2}+k_{4}$ of the ordered integer tuples $\left(k_{1}, k_{2}, k_{3}, k_{4}\right)$ the number is $\qquad$ . | 6181 | 89 | 4 |
math | ## Task B-1.2.
Let $n$ be a natural number and let
$$
A=0.1^{n} \cdot 0.01^{n} \cdot 0.001^{n} \cdot 0.0001^{n} \cdot \ldots \cdot 0 . \underbrace{00 \ldots 0}_{199 \text { zeros }} 1^{n} .
$$
The number $A^{-1}$ has 140701 digits. Determine the number $n$. | 7 | 123 | 1 |
math | Find all pair of positive integers $(x, y)$ satisfying the equation
\[x^2 + y^2 - 5 \cdot x \cdot y + 5 = 0.\] | (3, 1) | 40 | 7 |
math | Example 7. Find $\int \sin ^{5} x d x$. | -\cosx+\frac{2}{3}\cos^{3}x-\frac{1}{5}\cos^{5}x+C | 17 | 28 |
math | [ Law of Sines ]
In triangle $ABC$, it is known that $\angle A=\alpha, \angle C=\beta, AB=a$; $AD$ is the angle bisector. Find $BD$.
# | \cdot\sin\alpha/2/\sin(\alpha/2+\beta) | 46 | 17 |
math | Problem 7. In an isosceles triangle with a perimeter of 60 cm, the point of intersection of the medians lies on the inscribed circle. Find the sides of the triangle. | 25,25,10 | 42 | 8 |
math | $5.57 \sin ^{3} x(1+\operatorname{ctg} x)+\cos ^{3} x(1+\operatorname{tg} x)=2 \sqrt{\sin x \cos x}$. | \frac{\pi}{4}+2\pik,k\inZ | 51 | 16 |
math | 9. (4th IMO problem) Solve the inequality $\sqrt{3-x}-\sqrt{x+1}>\frac{1}{2}$.
| {x\lvert\,-1\leqslantx<1-\frac{\sqrt{31}}{8}.} | 31 | 27 |
math | ## Problem Statement
Calculate the limit of the numerical sequence:
$\lim _{n \rightarrow \infty} \frac{\sqrt{n^{8}+6}-\sqrt{n-6}}{\sqrt[8]{n^{8}+6}+\sqrt{n-6}}$ | \infty | 59 | 3 |
math | 1. Find the sum of all even natural numbers $n$ for which the number of divisors (including 1 and $n$ itself) is equal to $\frac{n}{2}$. (For example, the number 12 has 6 divisors: $1,2,3,4,6,12$.) | 20 | 70 | 2 |
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