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 | [ Systems of nonlinear algebraic equations ] Higher degree equations (miscellaneous) $\quad]$
Solve the system of equations:
$x^{3}-y=6$,
$y^{3}-z=6$,
$z^{3}-x=6$. | (2,2,2) | 54 | 7 |
math | 9. Let $\alpha, \beta, \gamma$ satisfy $0<\alpha<\beta<\gamma<2 \pi$. If for any $x \in \mathbf{R}, \cos (x+\alpha)+\cos (x+$ $\beta)+\cos (x+\gamma)=0$, then $\gamma-\alpha=$ $\qquad$ . | \frac{4\pi}{3} | 77 | 9 |
math | What are the sides of a right-angled triangle whose perimeter is $40 \mathrm{~cm}$ and the radius of the inscribed circle is $3 \mathrm{~cm}$? | a_{1}=8\mathrm{~},b_{1}=15\mathrm{~};a_{2}=15\mathrm{~},b_{2}=8\mathrm{~},=17\mathrm{~} | 40 | 50 |
math | ## Task Condition
Find the derivative.
$y=\lg \ln (\operatorname{ctg} x)$ | -\frac{2}{\ln(\operatorname{ctg}x)\cdot\ln10\cdot\sin2x} | 23 | 28 |
math | Example 9 Given the equation
$$
x^{2}-11 x+30+m=0
$$
has two real roots both greater than 5. Find the range of values for $m$.
| 0<m \leqslant \frac{1}{4} | 45 | 14 |
math | IMO 1992 Problem A2 Find all functions f defined on the set of all real numbers with real values, such that f(x 2 + f(y)) = y + f(x) 2 for all x, y. | f(x)=x | 49 | 4 |
math | 5.48 Determine the real number $a$ such that the polynomials $x^{2}+a x+1$ and $x^{2}+x+a$ have at least one common root. | a=1 \text{ or } a=-2 | 44 | 11 |
math | 4.1. Mom baked four raisin buns for breakfast for her two sons. $V$ In the first three buns, she put 7, 7, 23 raisins, and some more in the fourth. It turned out that the boys ate an equal number of raisins and did not divide any bun into parts. How many raisins could Mom have put in the fourth bun? List all the option... | 9,23,37 | 89 | 7 |
math | \section*{Problem 1 - 091041}
To be determined are all pairs of natural numbers such that each pair, together with the number 41, forms a triplet for which both the sum of the three numbers of the triplet and the sum of any two numbers arbitrarily selected from the triplet are squares of natural numbers. | (,b)=(80,320)(,b)=(320,80) | 71 | 21 |
math | 4.7. Form the equation of the line passing through the given point $P_{0}(3,4,2)$ and perpendicular to the plane $8 x-4 y+5 z-4=0$ | (x-3)/8=(y-4)/(-4)=(z-2)/5 | 45 | 18 |
math | Example 6 (CMO-6 Test) Find all natural numbers $n$ such that
$$
\min _{k \in \mathbf{N}}\left(k^{2}+\left[\frac{n}{k^{2}}\right]\right)=1991(n \in \mathbf{N}) .
$$ | 1024\cdot967\leqslantn\leqslant1024\cdot967+1023 | 71 | 34 |
math | The solution set of the inequality $(m-1) x<\sqrt{4 x-x^{2}}$ with respect to $x$ is $\{x \mid 0<x<2\}$. Find the value of the real number $m$.
---
Please note that the problem statement and the solution set provided are directly translated, maintaining the original format and structure. | 2 | 77 | 1 |
math | 8. Given the function $f:\{0,1, \cdots, 2010\} \rightarrow \mathbf{N}$. If for all possible integers $x$, we have
$$
\begin{array}{l}
f(4 x+2)=f(4 x+1), \\
f(5 x+3)=f(5 x+2), \\
f(7 x+5)=f(7 x+4),
\end{array}
$$
then $f(x)$ can take at most $\qquad$ different values. | 1033 | 120 | 4 |
math | A positive integer $n\geq 4$ is called [i]interesting[/i] if there exists a complex number $z$ such that $|z|=1$ and \[1+z+z^2+z^{n-1}+z^n=0.\] Find how many interesting numbers are smaller than $2022.$ | 404 | 72 | 3 |
math | 8,9 A heptagon, three angles of which are equal to $120^{\circ}$, is inscribed in a circle. Can all its sides be of different lengths
# | No | 41 | 1 |
math | 5. A company's working hours are from 8:30 AM to 5:30 PM. During this period, the hour and minute hands of the clock overlap times. | 9 | 38 | 1 |
math | The hundreds digit of 2025 is 0, and after removing the 0, it becomes 225 (only the 0 in the hundreds place is removed), $225 \times 9=2025$, such a 4-digit number is called a "zero-clever number". Therefore, please list all the "zero-clever numbers" are $\qquad$. | 2025,4050,6075 | 84 | 14 |
math | A19 (15-3, Sweden) 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. Find the minimum value of $a^{2}+b^{2}$.
| \frac{4}{5} | 70 | 7 |
math | $f:\mathbb{R}\rightarrow \mathbb{R}$ satisfies the equation
\[f(x)f(y)-af(xy)=x+y\]
, for every real numbers $x,y$. Find all possible real values of $a$. | a = 1, -1 | 49 | 7 |
math | [The area of a triangle does not exceed half the product of two sides] Complex
What is the maximum area that a quadrilateral with side lengths of 1, 4, 7, and 8 can have? | 18 | 46 | 2 |
math | 13. (20 points) The domain of the function $f(x)$ is the set of real numbers $\mathbf{R}$, and it is known that when $x>0$, $f(x)>0$, and for any $m, n \in \mathbf{R}$, we have
$$
f(m+n)=f(m)+f(n).
$$
(1) Discuss the odd-even property and monotonicity of the function $f(x)$;
(2) Let the sets be
$$
\begin{array}{l}
A=\l... | [-\frac{13}{6},-\frac{\sqrt{15}}{3}]\cup[\frac{\sqrt{15}}{3},2] | 281 | 35 |
math | 1. Solve the equations:
(i) $\frac{1}{2^{x}}+\frac{1}{3^{x}}=\frac{1}{2^{x}+3^{x}}, x \in \mathbb{R}$.
(ii) $x+2^{x}+\log _{2} x=7, x \in(0, \infty)$. | 2 | 80 | 1 |
math | What is the largest factor of $130000$ that does not contain the digit $0$ or $5$? | 26 | 28 | 2 |
math | [ Higher degree equations (other).]
Write down the equation of which the root will be the number $\alpha=\frac{1}{2}(\sqrt[3]{5 \sqrt{2}+7}-\sqrt[3]{5 \sqrt{2}-7})$. Write the number $\alpha$ without using radicals. | \alpha=1 | 67 | 4 |
math | Let points $A,B,$ and $C$ lie on a line such that $AB=1, BC=1,$ and $AC=2.$ Let $C_1$ be the circle centered at $A$ passing through $B,$ and let $C_2$ be the circle centered at $A$ passing through $C.$ Find the area of the region outside $C_1,$ but inside $C_2.$ | 3\pi | 89 | 3 |
math | 2. Last year, 9900 umbrellas were reported to the lost and found department. Some people lost exactly one umbrella, but there were also those who lost more than one umbrella. Specifically, $4 \%$ of them lost exactly two umbrellas, $2.5 \%$ lost three umbrellas, $0.5 \%$ lost eight umbrellas, and the rest lost one umbr... | 8800 | 92 | 4 |
math | 3. For real numbers $a$ and $b$, it holds that $a^{3}=3 a b^{2}+11$ and $b^{3}=3 a^{2} b+2$. Calculate the value of the expression $a^{2}+b^{2}$. | 5 | 62 | 1 |
math | ## [
[ Systems of linear equationsFind all functions $f(x)$, defined for all real $x$ and satisfying the equation $2 f(x)+f(1$ $-x)=x^{2}$
# | f(x)=\frac{1}{3}(x^{2}+2x-1) | 45 | 20 |
math | Based on a city's rules, the buildings of a street may not have more than $9$ stories. Moreover, if the number of stories of two buildings is the same, no matter how far they are from each other, there must be a building with a higher number of stories between them. What is the maximum number of buildings that can be b... | 511 | 81 | 3 |
math | 4. Calculate the arithmetic mean of all multiples of the number 12 that are of the form $\overline{3 a 8 b}$. Which of the obtained multiples should be excluded so that the arithmetic mean of the remaining multiples is 50 greater than the arithmetic mean of all multiples? | 3084 | 61 | 4 |
math | Hello.
Suppose $a$, $b$, $c$ are real numbers such that $a+b+c = 0$ and $a^{2}+b^{2}+c^{2} = 1$.
Prove that $a^{2}b^{2}c^{2}\leq \frac{1}{54}$ and determine the cases of equality. | \frac{1}{54} | 80 | 9 |
math | 6. When four dice are rolled randomly, the probability that the product of the four numbers obtained is a perfect square is $\qquad$ .
| \frac{25}{162} | 29 | 10 |
math | 4. Given real numbers $a, b$ satisfy $\arcsin \left(1+a^{2}\right)-\arcsin (b-1)^{2} \geqslant \frac{\pi}{2}$. Then $\arccos \left(a^{2}-b^{2}\right)=$ $\qquad$ . | \pi | 73 | 2 |
math | (110. Try to 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 all positive integers. | 11 | 55 | 2 |
math | Find all pairs $(p, n)$ with $n>p$, consisting of a positive integer $n$ and a prime $p$, such that $n^{n-p}$ is an $n$-th power of a positive integer. | (4, 2) | 48 | 7 |
math | ## Problem 9
Given a number with 1998 digits which is divisible by 9 . Let $x$ be the sum of its digits, let y be the sum of the digits of $\mathrm{x}$, and $\mathrm{z}$ the sum of the digits of $\mathrm{y}$. Find $\mathrm{z}$.
| 9 | 72 | 1 |
math | \section*{Problem 4 - 211034}
Determine all real numbers \(x\) that satisfy the inequality:
\[
\frac{1-\sqrt{1-3 x^{2}}}{x}<1
\] | [-\frac{1}{\sqrt{3}};\frac{1}{2})\backslash{0} | 52 | 24 |
math | ## problem statement
Derive the equations of the tangent and normal lines to the curve at the point corresponding to the parameter value $t=t_{0}$.
$\left\{\begin{array}{l}x=\frac{t+1}{t} \\ y=\frac{t-1}{t}\end{array}\right.$
$t_{0}=-1$ | -x+2x+2 | 77 | 6 |
math | B2. An integer $n$ is called a combi-number if every pair of different digits from all possible digits $0 \mathrm{t} / \mathrm{m} 9$ appear next to each other at least once in the number. Thus, in a combi-number, the digits 3 and 5 appear next to each other somewhere. It does not matter whether they appear in the order... | 50 | 123 | 2 |
math | Example 2 Given $0<a<1$, and satisfies
$$
\left[a+\frac{1}{30}\right]+\left[a+\frac{2}{30}\right]+\cdots+\left[a+\frac{29}{30}\right]=18 \text {. }
$$
Then $[10 a]=$ $\qquad$ | 6 | 75 | 1 |
math | Example 1 (2004 National College Entrance Examination for Science) Given that there are 3 weak teams among 8 teams, the 8 teams are divided into groups $A$ and $B$ by drawing lots, with 4 teams in each group.
(1) The probability that one of the groups $A$ or $B$ has exactly two weak teams;
(2) The probability that grou... | \frac{6}{7} | 95 | 7 |
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) | 145 | 10 |
math | 5. It is known that the polynomial $f(x)=8+32 x-12 x^{2}-4 x^{3}+x^{4}$ has 4 distinct real roots $\left\{x_{1}, x_{2}, x_{3}, x_{4}\right\}$. The polynomial of the form $g(x)=b_{0}+b_{1} x+b_{2} x^{2}+b_{3} x^{3}+x^{4}$ has roots $\left\{x_{1}^{2}, x_{2}^{2}, x_{3}^{2}, x_{4}^{2}\right\}$. Find the coefficient $b_{1}$... | -1216 | 159 | 5 |
math | For any non-empty subset $A$ of $\{1, 2, \ldots , n\}$ define $f(A)$ as the largest element of $A$ minus the smallest element of $A$. Find $\sum f(A)$ where the sum is taken over all non-empty subsets of $\{1, 2, \ldots , n\}$. | (n-3)2^n + n + 3 | 76 | 13 |
math | 1. In the set of integers, solve the equation
$$
x^{2}+y^{4}+1=6^{z}
$$ | (x,y,z)={(0,0,0),(2,1,1),(-2,1,1),(2,-1,1),(-2,-1,1)} | 31 | 37 |
math | Daniel had a string that formed the perimeter of a square with area $98$. Daniel cut the string into two pieces. With one piece he formed the perimeter of a rectangle whose width and length are in the ratio $2 : 3$. With the other piece he formed the perimeter of a rectangle whose width and length are in the ratio $3 :... | 67 | 119 | 2 |
math | ## Task B-2.5.
The sum of two five-digit numbers that match in the first three digits is 123422. The last two digits of one number are 1,2, and of the other 1,0 in that order. Determine these numbers. | 6171261710 | 60 | 10 |
math | Given $a 、 b 、 c$ are real numbers, and
$$
a^{2}+b^{2}+c^{2}+2 a b=1, a b\left(a^{2}+b^{2}+c^{2}\right)=\frac{1}{8} \text {, }
$$
The roots of the quadratic equation $(a+b) x^{2}-(2 a+c) x-(a+b)=0$ are $\alpha 、 \beta$. Find the value of $2 \alpha^{3}+\beta^{-5}-\beta^{-1}$. | -1 | 129 | 2 |
math | (solved by Juliette Fournier). Let $\lambda$ be the positive root of the equation $t^{2}-1998 t-1=0$. Let the sequence $\left(x_{n}\right)$ be defined by $x_{0}=1$ and, for all $n \geqslant 0$, by:
$$
x_{n+1}=\left[\lambda x_{n}\right]
$$
where $[x]$ is the integer part of $x$. Calculate the remainder of the Euclidea... | 1000 | 129 | 4 |
math | 4. In $\triangle A B C$, the sides opposite to $\angle A$, $\angle B$, and $\angle C$ are $a$, $b$, and $c$ respectively. If $a^{2}+2\left(b^{2}+c^{2}\right)=2 \sqrt{2}$, then the maximum value of the area of $\triangle A B C$ is $\qquad$ | \frac{1}{4} | 86 | 7 |
math | Let $\mathbb{Z}^+$ denote the set of all positive integers. Find all surjective functions $f:\mathbb{Z}^+ \times \mathbb{Z}^+ \rightarrow \mathbb{Z}^+$ that satisfy all of the following conditions: for all $a,b,c \in \mathbb{Z}^+$,
(i)$f(a,b) \leq a+b$;
(ii)$f(a,f(b,c))=f(f(a,b),c)$
(iii)Both $\binom{f(a,b)}{a}$ and $\... | f(a, b) = a | b | 152 | 10 |
math | 2. (10 points) A team of 8 people completed $\frac{1}{3}$ of a project in 30 days. Then, 4 more people were added to complete the remaining part of the project. Therefore, the total time to complete the project was $\qquad$ days.
The team of 8 people completed $\frac{1}{3}$ of a project in 30 days. Then, 4 more people... | 70 | 119 | 2 |
math | 2.1. Martha-the-needlewoman has many pins in her box. The first time she took out two pins, and each subsequent time she took out $k$ more pins than the previous time. It turned out that on the tenth time she took out more than 45 pins, and on the fifteenth - less than 90. Write down all possible $k$. | 5,6 | 79 | 3 |
math | 5. If four dice are thrown at random, the probability that the smallest number shown by these four dice is exactly 3 is
$\qquad$ (Supplied by Zhao Bin) | \frac{175}{1296} | 38 | 12 |
math | 3. Let positive integers $a, b$ be such that $15a + 16b$ and $16a - 15b$ are both squares of positive integers. Find the smallest value that the smaller of these two squares can take. ${ }^{[3]}$
(1996, China Mathematical Olympiad) | 481^2 | 73 | 5 |
math | 167. To transport 60 tons of cargo from one place to another, a certain number of trucks were required. Due to the poor condition of the road, each truck had to carry 0.5 tons less than originally planned, which is why 4 additional trucks were required. How many trucks were initially required? | 20 | 67 | 2 |
math | [ Volume of a Tetrahedron and Pyramid
Given a regular quadrilateral pyramid PABCD ( $P$ - vertex) with the side of the base $a$ and the lateral edge $a$. A sphere with center at point $O$ passes through point $A$ and touches the edges $P B$ and $P D$ at their midpoints. Find the volume of the pyramid $O P C D$.
# | \frac{5a^3\sqrt{2}}{96} | 90 | 16 |
math | 4. Students from one of the schools are tasked with making 455 Christmas tree ornaments for kindergartens in the city. All students should make the same number of ornaments. More than 10, but fewer than 70 people participated in making the ornaments. How many students made the ornaments, and how many did each of them m... | 65 | 73 | 2 |
math | For which $n$ can it be achieved that among the sums of the form $\pm 1 \pm 2 \pm \ldots \pm n$, the number 100 appears? | n\geq15 | 41 | 6 |
math | There are $2022$ signs arranged in a straight line. Mark tasks Auto to color each sign with either red or blue with the following condition: for any given sequence of length $1011$ whose each term is either red or blue, Auto can always remove $1011$ signs from the line so that the remaining $1011$ signs match the given... | 2^{1011} | 104 | 7 |
math | In a jar, the ratio of the number of dimes to the number of quarters is $3: 2$. If the total value of these coins is $\$ 4$, how many dimes are in the jar?
(Each dime is worth 10 cents, each quarter is worth 25 cents, and $\$ 1$ equals 100 cents.) | 15 | 78 | 2 |
math | 6. The number of positive integers less than or equal to $10^{6}$ that are neither perfect squares, nor cubes, nor perfect fourth powers is $\qquad$ . | 998910 | 37 | 6 |
math | 6. The sequence $\left\{a_{n}\right\}$ is defined as follows:
$$
\begin{array}{l}
a_{1}=1, a_{2}=3, \\
a_{n+2}=2 a_{n+1}-a_{n}+2(n=1,2, \cdots) .
\end{array}
$$
Then the sum of its first $n$ terms is | \frac{1}{3} n\left(n^{2}+2\right) | 90 | 19 |
math | 7. Let $x_{i} \in \mathbf{R}, x_{i} \geqslant 0(i=1,2,3,4,5), \sum_{i=1}^{5} x_{i}=1$, then $\max \left\{x_{1}+x_{2}, x_{2}+x_{3}, x_{3}+x_{4}, x_{4}\right.$ $\left.+x_{5}\right\}$ the minimum value equals $\qquad$ . | \frac{1}{3} | 114 | 7 |
math | IMO 1982 Problem A1 The function f(n) is defined on the positive integers and takes non-negative integer values. f(2) = 0, f(3) > 0, f(9999) = 3333 and for all m, n: f(m+n) - f(m) - f(n) = 0 or 1. Determine f(1982). | 660 | 90 | 3 |
math | 1. Container $A$ contains 6 liters of saltwater with a concentration of $a \%$, and container $B$ contains 4 liters of saltwater with a concentration of $b \%$. After pouring 1 liter of solution from $A$ into $B$ and mixing, then pouring 1 liter of solution from $B$ back into $A$, this process is repeated $k$ times (po... | a_{k}=\frac{3+2b}{5}+(-b)\frac{2}{5}(\frac{2}{3})^{k},\quadb_{k}=\frac{3+2b}{5}-(-b)\cdot\frac{3}{5}(\frac{2}{3})^{k} | 180 | 71 |
math | 5. In the tournament, 15 volleyball teams are playing, and each team plays against all other teams only once. Since there are no draws in volleyball, there is a winner in each match. A team is considered to have performed well if it loses no more than two matches. Find the maximum possible number of teams that performe... | 5 | 69 | 1 |
math | Determine all intergers $n\geq 2$ such that $a+\sqrt{2}$ and $a^n+\sqrt{2}$ are both rational for some real number $a$ depending on $n$ | n = 2 | 46 | 5 |
math | How many ways are there to color the edges of a hexagon orange and black if we assume that two hexagons are indistinguishable if one can be rotated into the other? Note that we are saying the colorings OOBBOB and BOBBOO are distinct; we ignore flips. | 14 | 61 | 2 |
math | [Coordinate method on the plane]
Find the distance between point $A(1,7)$ and the intersection point of the lines $x-y-1=0$ and $x+3 y-12=0$.
# | \frac{\sqrt{410}}{4} | 48 | 12 |
math | 3 (Full score: 50 points) Solve the system of equations
$$
\left\{\begin{array}{l}
x-y+z-w=2, \\
x^{2}-y^{2}+z^{2}-w^{2}=6, \\
x^{3}-y^{3}+z^{3}-w^{3}=20, \\
x^{4}-y^{4}+z^{4}-w^{4}=66 .
\end{array}\right.
$$ | 3,2,1,w=0;or3,0,1,w=2;or1,2,3,w=0;or1,0,3,w=2 | 105 | 38 |
math | ## Problem Statement
Calculate the limit of the numerical sequence:
$\lim _{n \rightarrow \infty} \sqrt{n(n+1)(n+2)}\left(\sqrt{n^{3}-3}-\sqrt{n^{3}-2}\right)$ | -\frac{1}{2} | 54 | 7 |
math | For positive integers $a>b>1$, define
\[x_n = \frac {a^n-1}{b^n-1}\]
Find the least $d$ such that for any $a,b$, the sequence $x_n$ does not contain $d$ consecutive prime numbers.
[i]V. Senderov[/i] | 3 | 69 | 1 |
math | \section*{Problem 1 - 101221}
All ordered pairs \((x, y)\) of real numbers are to be determined for which the system of equations
\[
x^{2}+y^{2}=1 \quad \text { (1) } \quad ; \quad x^{6}+y^{6}=\frac{7}{16}
\]
is satisfied. | (x,y)\in{(\\frac{1}{2},\\frac{\sqrt{3}}{2}),(\\frac{1}{2},\\frac{\sqrt{3}}{2}),(\\frac{\sqrt{3}}{2},\\frac{1}{2}),(\\frac{\sqrt{} | 88 | 64 |
math | # Problem 7.3 (7 points)
A piece has fallen out of a dictionary, the first page of which is numbered 213, and the number of the last page is written with the same digits in some other order. How many pages are in the missing piece? | 100 | 58 | 3 |
math | Let $a,b,m,n$ integers greater than 1. If $a^n-1$ and $b^m+1$ are both primes, give as much info as possible on $a,b,m,n$. | m = 2^k | 45 | 7 |
math | # Task 2. (10 points)
Find the value of the parameter $p$ for which the equation $p x^{2}=|x-1|$ has exactly three solutions.
# | \frac{1}{4} | 40 | 7 |
math | 5. Find all pairs of positive integers $(m, n)$ such that
$$
m+n-\frac{3 m n}{m+n}=\frac{2011}{3}
$$ | (,n)=(1144,377)or(377,1144) | 41 | 24 |
math | 3、Buy 2 fountain pens and 3 ballpoint pens for a total of 49 yuan, with the same amount of money, you can buy 3 fountain pens and 1 ballpoint pen, then the price of 1 fountain pen is ( ) yuan. | 14 | 56 | 2 |
math | Zkov G.
A bank serves a million customers, the list of whom is known to Ostap Bender. Each has a six-digit PIN code, and different customers have different codes. In one move, Ostap Bender can choose any customer he has not yet chosen and peek at the digits of the code at any $N$ positions (he can choose different pos... | 3 | 104 | 1 |
math | 2. $42 N$ is the set of all positive integers. For a subset $S$ of $N$ and $n \in N$, define
$$
S \oplus\{n\}=\{s+n \mid s \in S\} .
$$
Additionally, define the subset $S_{k}$ as follows:
$$
S_{1}=\{1\}, S_{k}=\left\{S_{k-1} \oplus\{k\}\right\} \cup\{2 k-1\}, k=2,3,4 \cdots .
$$
(1) Find $N-\bigcup_{k=1}^{\infty} S_{k... | N-\bigcup_{k=1}^{\infty}S_{k}={2^{}\mid\inN},\,k=500 | 181 | 34 |
math | Example 23 (1993 Korean Mathematical Olympiad) Find all non-negative integers $n$, such that $2^{2^{n}}+5$ is a prime number. | 0 | 39 | 1 |
math | How many ways are there to permute the first $n$ positive integers such that in the permutation, for each value of $k \le n$, the first $k$ elements of the permutation have distinct remainder mod $k$? | 2^{n-1} | 48 | 6 |
math | 4.20 There is a $\triangle A B C, a, b, c$ are the sides opposite to $\angle A, \angle B, \angle C$ respectively. If $b$ is the arithmetic mean of $a$ and $c$, and $\operatorname{tg} \frac{B}{2}$ is the geometric mean of $\operatorname{tg} \frac{A}{2}$ and $\operatorname{tg} \frac{C}{2}$, try to form a quadratic equati... | 3 x^{2}-2 \sqrt{3} x+1=0 | 139 | 16 |
math | Find all positive integers $a,b,c$ satisfying $(a,b)=(b,c)=(c,a)=1$ and \[ \begin{cases} a^2+b\mid b^2+c\\ b^2+c\mid c^2+a \end{cases} \] and none of prime divisors of $a^2+b$ are congruent to $1$ modulo $7$ | (1, 1, 1) | 82 | 10 |
math | 4. Let $A B C D$ be a square, and let $M$ be the midpoint of side $B C$. Points $P$ and $Q$ lie on segment $A M$ such that $\angle B P D=\angle B Q D=135^{\circ}$. Given that $A P<A Q$, compute $\frac{A Q}{A P}$. | \sqrt{5} | 82 | 5 |
math | Example 7 Given $0<a<1$,
$$
\begin{array}{l}
f(x)=\log _{a}(x+1), \\
g(x)=2 \log _{a}(2 x+t),
\end{array}
$$
for $x \in[0,1]$, $f(x) \geqslant g(x)$ always holds. Find the range of the real number $t$. | t \geqslant 1 | 91 | 8 |
math | 3. (10 points) There is a pasture, 10 cows can finish the grass in 8 days; 15 cows, if one less cow starts from the second day, can finish in 5 days. Then the grass that grows on the pasture every day is enough for $\qquad$ cows to eat for one day.
| 5 | 71 | 1 |
math | ## Task A-2.1.
If $x, y, z$ and $w$ are real numbers such that
$$
x^{2}+y^{2}+z^{2}+w^{2}+x+3 y+5 z+7 w=4
$$
determine the maximum possible value of the expression $x+y+z+w$. | 2 | 78 | 1 |
math | 13th CanMO 1981 Problem 2 The circle C has radius 1 and touches the line L at P. The point X lies on C and Y is the foot of the perpendicular from X to L. Find the maximum possible value of area PXY (as X varies). Solution | \frac{3\sqrt{3}}{8} | 62 | 12 |
math | Example 7 A convex $n$-sided polygon $A_{1} A_{2} \cdots A_{n}$ is inscribed in a unit circle. Find the maximum value of the sum of the squares of all its sides and diagonals, and determine when this maximum value is achieved.
---
The above text has been translated into English while preserving the original text's li... | n^2 | 83 | 3 |
math | A regular 3-sided pyramid has a height of $m=11 \mathrm{~cm}$, and the area of one side face is $210 \mathrm{~cm}^{2}$. Let's calculate the volume of the pyramid. | 825\sqrt{3}\approx1429\mathrm{~}^{3} | 53 | 21 |
math | Solve the following system of equations:
$$
\begin{aligned}
& \left(x^{2}+y^{2}\right) \frac{x}{y}=6 \\
& \left(x^{2}-y^{2}\right) \frac{y}{x}=1
\end{aligned}
$$ | \begin{aligned}&x_{1}=\sqrt[4]{8},\quady_{1}=\sqrt[4]{2};\quadx_{2}=-\sqrt[4]{8},\quady_{2}=-\sqrt[4]{2}\\&x_{3}=\sqrt[4]{\frac{27}{4}},\quady_{3}=\sqrt[4]{\frac{3}{4}}; | 65 | 94 |
math | 6. On an infinite tape, all natural numbers with the sum of digits equal to 2018 are written in ascending order. What number is written in the 225th position? (Method Commission) | 39..998(223nines) | 45 | 13 |
math | To be factored into the product of three factors:
$$
\left(x^{2}+x y+y^{2}\right)^{2}-\left(x^{2} y^{2}+y^{2} z^{2}+z^{2} x^{2}\right)
$$ | (x^{2}+y^{2})(x+y+z)(x+y-z) | 62 | 17 |
math | Solve the equation
$$
2 \cos 5 x+2 \cos 4 x+2 \cos 3 x+2 \cos 2 x+2 \cos x+1=0
$$ | \frac{2\pi}{11},\quad=1,2,\ldots,10 | 45 | 22 |
math | 7. Given the parabola $y^{2}=4 x$, with its focus at $F$, a line passing through the focus $F$ and with an inclination angle of $\theta\left(0<\theta<\frac{\pi}{2}\right)$ intersects the parabola at points $A$ and $B$. $A O(O$ being the origin) intersects the directrix at point $B^{\prime}$, and $B O$ intersects the di... | \frac{8}{\sin^{3}\theta} | 130 | 12 |
math | 2. Let $x, y, z$ be non-negative real numbers, and $x+y+z=$ 2. Then the sum of the maximum and minimum values of $x^{2} y^{2}+y^{2} z^{2}+z^{2} x^{2}$ is $\qquad$ . | 1 | 68 | 1 |
math | 11.4. Given the function $f(x)=\left(1-x^{3}\right)^{-1 / 3}$. Find $f(f(f \ldots f(2018) \ldots))$ (the function $f$ is applied 2019 times) | 2018 | 64 | 4 |
math | 3. Solve the system of equations:
$$
\left\{\begin{array}{l}
2 x^{2}+3 y+5=2 \sqrt{2 z+5} \\
2 y^{2}+3 z+5=2 \sqrt{2 x+5} \\
2 z^{2}+3 x+5=2 \sqrt{2 y+5}
\end{array}\right.
$$ | -0.5 | 90 | 4 |
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