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 | 1. (5 points) Find the value of the function $f(x)$ at the point $x_{0}=4000$, if $f(0)=1$ and for any $x$ the equality $f(x+2)=f(x)+3 x+2$ holds. | 11998001 | 61 | 8 |
math |
6. Let $\mathbf{R}$ denote the set of all real numbers. Find all functions $f: \mathbf{R} \rightarrow \mathbf{R}$ satisfying the condition
$$
f(x+y)=f(x) f(y) f(x y)
$$
for all $x, y$ in $\mathbf{R}$.
| f(x)=0,f(x)=1,f(x)=-1 | 75 | 13 |
math | Find all real polynomials $\mathrm{P}$ such that $\mathrm{P}(0)=0$ and $\mathrm{P}\left(\mathrm{X}^{2}+1\right)=\mathrm{P}(\mathrm{X})^{2}+1$.
| P(X)=X | 58 | 4 |
math | ## Problem Statement
Find the derivative.
$$
y=\sqrt{1-3 x-2 x^{2}}+\frac{3}{2 \sqrt{2}} \arcsin \frac{4 x+3}{\sqrt{17}}
$$ | -\frac{2x}{\sqrt{1-3x-2x^{2}}} | 54 | 19 |
math | Find all functions $f: \mathbb{R} \longrightarrow \mathbb{R}$ such that $f(0)=0$, and for all $x, y \in \mathbb{R}$,
$$
(x-y)\left(f\left(f(x)^{2}\right)-f\left(f(y)^{2}\right)\right)=(f(x)+f(y))(f(x)-f(y))^{2}
$$ | f(x)=cx | 91 | 4 |
math | 一、Fill in the Blanks (Total 3 questions, 10 points each)
1. In a 1000-meter race, when A reaches the finish line, B is 50 meters from the finish line; when B reaches the finish line, C is 100 meters from the finish line. So when A reaches the finish line, C is $\qquad$ meters from the finish line. | 145 | 88 | 3 |
math | 21st CanMO 1989 Problem 2 Each vertex of a right angle triangle of area 1 is reflected in the opposite side. What is the area of the triangle formed by the three reflected points? Solution | 3 | 46 | 1 |
math | Consider a stripe of $n$ fieds, numbered from left to right with the integers $1$ to $n$ in ascending order. Each of the fields is colored with one of the colors $1$, $2$ or $3$. Even-numbered fields can be colored with any color. Odd-numbered fields are only allowed to be colored with the odd colors $1$ and $3$.
How m... | 2 \cdot 3^k | 101 | 9 |
math | Find all $ n > 1$ such that the inequality \[ \sum_{i\equal{}1}^nx_i^2\ge x_n\sum_{i\equal{}1}^{n\minus{}1}x_i\] holds for all real numbers $ x_1$, $ x_2$, $ \ldots$, $ x_n$. | n \le 5 | 76 | 6 |
math | ## Task 2 - 340822
From the digits $0,1,2,3,4,5,6,7,8,9$, two fractions $\frac{a}{b}$ and $\frac{c}{d}$ are to be formed such that each digit is used exactly once in the digit representations of the four natural numbers $a, b, c, d$. For the fractions formed, $\frac{a}{b}+\frac{c}{d}=1$ should hold. In the first of th... | \frac{485}{970} | 172 | 11 |
math | Example 6. Let $M=\frac{1}{1+\sqrt{2}}+\frac{1}{\sqrt{2}+\sqrt{3}}+$ $\cdots+\frac{1}{\sqrt{1993}+\sqrt{1994}}, N=1-2+3-4+\cdots$ $+1993-1994$. Then the value of $\frac{N}{(M+1)^{2}}$ is ( ). | -\frac{1}{2} | 104 | 7 |
math | Ed and Sue bike at equal and constant rates. Similarly, they jog at equal and constant rates, and they swim at equal and constant rates. Ed covers $74$ kilometers after biking for $2$ hours, jogging for $3$ hours, and swimming for $4$ hours, while Sue covers $91$ kilometers after jogging for $2$ hours, swimming for $... | 314 | 128 | 3 |
math | 【Question 7】
A wire 78 cm long, with a red dot painted every 3 cm. The wire is bent at the red dots to form a rectangle. What is the maximum area of the rectangle formed in square centimeters? | 378 | 51 | 3 |
math | 1. Given $a+\frac{1}{a+1}=b+\frac{1}{b-1}-2$, and $a-$ $b+2 \neq 0$. Then the value of $a b-a+b$ is $\qquad$ . | 2 | 56 | 1 |
math | 23. Determine the largest prime factor of the sum $\sum_{k=1}^{11} k^{5}$. | 263 | 27 | 3 |
math | ## Task 2 - 140812
Determine all ordered pairs $(x, y)$ of natural numbers $x, y$ for which the equation $13 x+5 y=82$ holds! | (4,6) | 48 | 5 |
math | $$
\begin{array}{l}
\text { 1. Let } A=\left\{x \in \mathbf{Z} \left\lvert\, \frac{x^{2}}{2009}+\frac{y^{2}}{2008}=1\right.\right\}, \\
B=\{x \mid x=2 n+1, n \in \mathbf{Z}\},
\end{array}
$$
Set $M$ is a subset of $A$, but not a subset of $B$. Then the number of all such sets $M$ is $\qquad$ | 2^{44}\left(2^{45}-1\right) | 135 | 16 |
math | 4. Given a tetrahedron $ABCD$ satisfying: $AB=2, CD=2\sqrt{5}, AC=BD=3, AD=BC=\sqrt{5}$, then the volume of the tetrahedron is $\qquad$. | \frac{4}{3} | 56 | 7 |
math | 4. How many orderings $\left(a_{1}, \ldots, a_{8}\right)$ of $(1,2, \ldots, 8)$ exist such that $a_{1}-a_{2}+a_{3}-a_{4}+a_{5}-a_{6}+a_{7}-a_{8}=0$ ? | 4608 | 77 | 4 |
math | ii. $n \geqslant 5$ is a positive integer, $a_{1}, a_{2}, \ldots, a_{n}, b_{1}, b_{2}, \ldots, b_{n} \geqslant 0$. It is known that
$$
\sum_{i=1}^{n} a_{i}^{2}=1, \sum_{i=1}^{n} b_{i}=1
$$
Find
$$
\sum_{i=1}^{n} a_{i}^{1+b_{i}}
$$
the maximum value. | \sqrt{n-1} | 132 | 6 |
math | 2. Find all solutions to the equation
$$
x^{4}=y^{2}+2 y+2
$$
where $x, y$ are integers. | (-1,-1)(1,-1) | 36 | 9 |
math | 6. (15 points) From a homogeneous straight rod, a piece of length $s=60 \mathrm{~cm}$ was cut. By how much did the center of gravity of the rod move as a result? | 30\, | 47 | 4 |
math | 3. (2000 National High School Competition Question) Given that $A$ is the left vertex of the hyperbola $x^{2}-y^{2}=1$, and points $B$ and $C$ are on the right branch of the hyperbola, $\triangle A B C$ is an equilateral triangle, then the area of $\triangle A B C$ is | 3\sqrt{3} | 81 | 6 |
math | Let $ABCD$ be a square with side $4$. Find, with proof, the biggest $k$ such that no matter how we place $k$ points into $ABCD$, such that they are on the interior but not on the sides, we always have a square with sidr length $1$, which is inside the square $ABCD$, such that it contains no points in its interior(they ... | 15 | 90 | 4 |
math | Example 7. Solve the integral equation
$$
\varphi(x)=x e^{-x}+\lambda \int_{0}^{\infty} J_{0}(2 \sqrt{x t}) \varphi(t) d t \quad(|\lambda| \neq 1)
$$ | \varphi(x)=e^{-x}(\frac{x}{1+\lambda}+\frac{\lambda}{1-\lambda^{2}}) | 63 | 29 |
math | Find all the real numbers $k$ that have the following property: For any non-zero real numbers $a$ and $b$, it is true that at least one of the following numbers: $$a, b,\frac{5}{a^2}+\frac{6}{b^3}$$is less than or equal to $k$. | 2 | 72 | 1 |
math | ## Problem Statement
Calculate the limit of the function:
$\lim _{x \rightarrow \frac{\pi}{3}} \frac{1-2 \cos x}{\sin (\pi-3 x)}$ | -\frac{\sqrt{3}}{3} | 44 | 10 |
math | 7. There are less than 100 cells in a row, and at the beginning, only the cells at both ends contain a chess piece each. Several students take turns placing chess pieces in the cells. Each person places one piece at a time, and it must be placed in the middle of two adjacent cells that already contain pieces (for examp... | 7 | 189 | 1 |
math | Let $N$ be the number of complex numbers $z$ with the properties that $|z|=1$ and $z^{6!}-z^{5!}$ is a real number. Find the remainder when $N$ is divided by $1000$.
| 440 | 57 | 3 |
math | 8. Given the ellipse $\frac{x^{2}}{4}+\frac{y^{2}}{3}=1$ with left and right foci $F_{1}, F_{2}$, a line $l$ passing through the right focus intersects the ellipse at points $P, Q$. Then the maximum value of the area of the incircle of $\triangle F_{1} P Q$ is $\qquad$ | \frac{9\pi}{16} | 88 | 10 |
math | Given the family of curves
$2(2 \sin \theta-\cos \theta+3) x^{2}-(8 \sin \theta+\cos \theta+1) y=0, \theta$ is a parameter. Find the maximum length of the chord intercepted by the line $y=2 x$ on this family of curves. | 8\sqrt{5} | 72 | 6 |
math | 11.4. Nyusha has 2022 coins, and Barash has 2023. Nyusha and Barash toss all their coins simultaneously and count how many heads each of them gets. The one who gets more heads wins, and in case of a tie, Nyusha wins. $C$ What is the probability that Nyusha wins? | 0.5 | 80 | 3 |
math |
A2. Find the largest possible value of the expression $\left|\sqrt{x^{2}+4 x+8}-\sqrt{x^{2}+8 x+17}\right|$ where $x$ is a real number.
| \sqrt{5} | 50 | 5 |
math | 2. Determine all positive integers that are equal to 300 times the sum of their digits. | 2700 | 21 | 4 |
math | Example 5 Simplify $\frac{\sin x+\sin 3 x+\sin 5 x+\cdots+\sin (2 n-1) x}{\cos x+\cos 3 x+\cos 5 x+\cdots+\cos (2 n-1) x}$. | \tannx | 60 | 3 |
math | 2.287. For what values of $a$ and $b$ does the quadratic trinomial $16 x^{2}+144 x+(a+b)$ represent a perfect square, given that $b-a=-7$? | =165.5;b=158.5 | 53 | 13 |
math | Let $p$ be any (positive) prime number. Give all positive integers $b$ for which the roots of the quadratic equation $x^{2}-b x+b p = 0$ are integers. | (p+1)^2or4p | 43 | 8 |
math | 2. In a chess tournament, everyone played against each other once. The winner won half of the games and drew the other half. It turned out that he scored 13 times fewer points than all the others. (1 point for a win, 0.5 for a draw, 0 for a loss.) How many chess players were there in the tournament? | 21 | 75 | 2 |
math | 6. (8 points) On the board, 27 ones are written. Every minute, Karlson erases two arbitrary numbers and writes their sum on the board, and then eats a number of candies equal to the product of the two erased numbers. What is the maximum number of candies he could have eaten in 27 minutes? | 351 | 69 | 3 |
math | 12. (5 points) From the 200 natural numbers from 1 to 200, how many numbers must be selected to ensure that there are at least 2 numbers whose sum is a multiple of 5? | 82 | 49 | 2 |
math | 8,9
How many parts do the diagonals of an $n$-gon divide it into, if no three diagonals intersect at the same point? | \frac{1}{24}n(n-1)(n-2)(n-3)+\frac{1}{2}n(n-3)+1 | 33 | 34 |
math | ## Problem Statement
Find the second-order derivative $y_{x x}^{\prime \prime}$ of the function given parametrically.
$\left\{\begin{array}{l}x=e^{t} \\ y=\arcsin t\end{array}\right.$ | \frac{^{2}+-1}{e^{2}\cdot\sqrt{(1-^{2})^{3}}} | 58 | 25 |
math | [ Measurement of segment lengths and angles. Adjacent angles.]
How many times in a day do the hour and minute hands coincide? Form a straight angle? Form a right angle? | 22 | 36 | 2 |
math | 6.145. $(x+1)^{2}(x+2)+(x-1)^{2}(x-2)=12$.
Translate the above text into English, please retain the original text's line breaks and format, and output the translation result directly.
6.145. $(x+1)^{2}(x+2)+(x-1)^{2}(x-2)=12$. | 1 | 90 | 1 |
math | 14. Given the functions
$$
\begin{array}{l}
f(x)=\left\{\begin{array}{ll}
-x^{2}+x, & x \leqslant 1 ; \\
\log _{\frac{1}{3}} x, & x>1,
\end{array}\right. \\
g(x)=|x-k|+|x-1| .
\end{array}
$$
If for any $x_{1} 、 x_{2} \in \mathbf{R}$, we have $f\left(x_{1}\right) \leqslant$ $g\left(x_{2}\right)$, then the range of the ... | k \leqslant \frac{3}{4} \text{ or } k \geqslant \frac{5}{4} | 156 | 31 |
math | Let $a, b, c$, and $d$ be real numbers such that $a^2 + b^2 + c^2 + d^2 = 3a + 8b + 24c + 37d = 2018$. Evaluate $3b + 8c + 24d + 37a$. | 1215 | 77 | 4 |
math | # Task 7.2
There are 30 logs, the lengths of which are 3 or 4 meters, and their total length is one hundred meters. How many cuts are needed to saw all these logs into pieces 1 meter long? (Each cut saws exactly one log).
Points 7
# | 70 | 66 | 2 |
math | A cyclic $n$-gon is divided by non-intersecting (inside the $n$-gon) diagonals to $n-2$ triangles. Each of these triangles is similar to at least one of the remaining ones. For what $n$ this is possible? | n = 4 | 57 | 5 |
math | In a tournament, each participant plays a match against every other participant. The winner of a match earns 1 point, the loser 0 points, and if the match is a draw, both players earn half a point. At the end of the tournament, the participants are ranked according to their score (in case of a tie, the order is arbitra... | 25 | 100 | 2 |
math | 6th Chinese 1991 Problem A3 10 points are arranged in the plane, so that given any 5, at least 4 lie on a circle. M is the maximum number of points on a circle. What is the minimum possible value of M? Solution | 9 | 58 | 1 |
math | Example 13. Solve the inequality
$$
\log _{1 / 3}\left(x^{2}-6 x+18\right)-2 \log _{1 / 3}(x-4)<0
$$ | 4<x<+\infty | 50 | 6 |
math | 3. Let $x_{k}=\tan \frac{k \pi}{17}(k=1,2, \cdots, 16)$. Then $\sum_{k=1}^{16} \frac{1}{1+x_{k}^{2}}=$ $\qquad$ | \frac{15}{2} | 64 | 8 |
math | 10,11
In a slanted parallelepiped, the projection of a lateral edge onto the base plane is 5, and the height is 12. A section perpendicular to the lateral edge is a rhombus with an area of 24 and a diagonal of 8. Find the lateral surface area and the volume of the parallelepiped. | 260;312 | 76 | 7 |
math | 3rd ASU 1969 Problem 9 Every city in a certain state is directly connected by air with at most three other cities in the state, but one can get from any city to any other city with at most one change of plane. What is the maximum possible number of cities? Solution | 10 | 62 | 2 |
math | Find the minimal positive integer $m$, so that there exist positive integers $n>k>1$, which satisfy
$11...1=11...1.m$, where the first number has $n$ digits $1$, and the second has $k$ digits $1$. | 101 | 58 | 3 |
math | Find all functions $ f: \mathbb{R} \to \mathbb{R}$ satisfying
\[ f\left(\frac {x \plus{} y}{x \minus{} y}\right) \equal{} \frac {f\left(x\right) \plus{} f\left(y\right)}{f\left(x\right) \minus{} f\left(y\right)}
\]
for all $ x \neq y$. | f(x) = x | 95 | 6 |
math | 5. In $\triangle A B C$, $A B=6, B C=4$, the length of the median on side $A C$ is $\sqrt{10}$, then the value of $\sin ^{6} \frac{A}{2}+\cos ^{6} \frac{A}{2}$ is $\qquad$ | \frac{211}{256} | 74 | 11 |
math | A secret agent is trying to decipher an access code. So far, he has obtained the following information:
- it is a four-digit number,
- it is not divisible by seven,
- the digit in the tens place is the sum of the digit in the units place and the digit in the hundreds place,
- the number formed by the first two digits ... | 4583 | 130 | 4 |
math | 7.191. $\log _{10} x+\log _{\sqrt{10}} x+\log _{\sqrt[3]{10}} x+\ldots+\log _{\sqrt[1]{10}} x=5.5$. | \sqrt[10]{10} | 56 | 9 |
math | 8.377. $2^{\sin ^{2} x}+4 \cdot 2^{\cos ^{2} x}=6$.
8.377. $2^{\sin ^{2} x}+4 \cdot 2^{\cos ^{2} x}=6$. | \frac{\pi}{2}(2k+1),k\inZ | 69 | 16 |
math | Task 2. The sets $A$ and $B$ are subsets of the positive integers. The sum of any two different elements from $A$ is an element of $B$. The quotient of any two different elements of $B$ (where we divide the larger by the smaller) is an element of $A$. Determine the maximum number of elements in $A \cup B$. | 5 | 79 | 1 |
math | A tournament will take place with 100 competitors, all with different skill levels. The most skilled competitor always wins against the least skilled competitor. Each participant plays exactly twice, with two randomly drawn opponents (once against each). A competitor who wins two matches receives a medal. Determine the... | 1 | 70 | 1 |
math | 3. In the set of integers, solve the equation:
$$
y^{4}-x(x+1)(x+2)(x+3)=1
$$ | (0,-1),(0,1),(-3,1),(-3,-1),(-1,1),(-1,-1),(-2,1),(-2,-1) | 34 | 39 |
math | Task 4. (20 points) For the numerical sequence $\left\{x_{n}\right\}$, all terms of which, starting from $n \geq 2$, are distinct, the relation $x_{n}=\frac{x_{n-1}+198 x_{n}+x_{n+1}}{200}$ holds. Find $\sqrt{\frac{x_{2023}-x_{1}}{2022} \cdot \frac{2021}{x_{2023}-x_{2}}}+2022$. | 2023 | 129 | 4 |
math | ## Task 3 - 020623
If you swap the units and tens digits of a two-digit number, you get a new number that is $4 \frac{1}{2}$ times as large as the original number.
a) What is the number?
b) How did you find it?
Show that there is only one such number! | 18 | 75 | 2 |
math | . Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ so that
$$
f(f(x)+x+y)=f(x+y)+y f(y)
$$
for all real numbers $x, y$. | f(x)=0 | 52 | 4 |
math | 4. Let $S$ be a set of $n$ distinct real numbers, and $A_{s}$ be the set of all distinct averages of pairs of elements from $S$. For a given $n \geqslant 2$, what is the minimum number of elements that $A_{s}$ can have?
(1993 Putnam Competition) | 2n-3 | 76 | 4 |
math | 8. (10 points) On the way from Xiaoming's house to the swimming pool, there are 200 trees. On the round trip, Xiaoming used red ribbons to mark some trees. On the way to the swimming pool, he marked the 1st tree, the 6th tree, the 11th tree, ... each time marking a tree after skipping 4 trees; on the way back, he marke... | 140 | 143 | 3 |
math | $$
\begin{array}{l}
\text { 4. If } a=1+\mathrm{i}, b=2+\mathrm{i}, c=3+\mathrm{i}, \\
x=-\frac{1}{2}+\frac{\sqrt{3}}{2} \mathrm{i},
\end{array}
$$
then $\left|a+b x+c x^{2}\right|=$ | \sqrt{3} | 84 | 5 |
math | 905. Solve the equation
$$
x^{3}=3 y^{3}+9 z^{3}
$$
in non-negative integers $x, y$ and $z$. | (0;0;0) | 40 | 7 |
math | 6. Given that $A$ is a subset of $S=\{1,2,3,4,5,6\}$ with at least 2 elements, and $a, b$ are two distinct elements in $A$. When $A$ ranges over $S$ and $a, b$ range over $A$, the total sum of the product $ab$ is $M=$ $\qquad$ | 2800 | 86 | 4 |
math | 4. Solve the numerical riddle: TETA+BETA=GAMMA. (Different letters - different digits.)
# | 4940+5940=10880 | 24 | 15 |
math | Yashchenko I.v.
A number was multiplied by the sum of its digits and the result was 2008. Find this number.
# | 251 | 32 | 3 |
math | A cricket can jump two distances: 9 and 8 meters. It competes in a 100-meter race to the edge of a cliff. How many jumps must the cricket make to reach the end of the race without going past the finish line and falling off the cliff?
# | 12 | 59 | 2 |
math | 10. For the sequence $\left\{a_{n}\right\}$, if there exists a sequence $\left\{b_{n}\right\}$, such that for any $n \in \mathbf{Z}_{+}$, we have $a_{n} \geqslant b_{n}$, then $\left\{b_{n}\right\}$ is called a "weak sequence" of $\left\{a_{n}\right\}$. Given
$$
\begin{array}{l}
a_{n}=n^{3}-n^{2}-2 t n+t^{2}\left(n \in... | \left(-\infty, \frac{1}{2}\right] \cup\left[\frac{3}{2},+\infty\right) | 233 | 33 |
math | ## Task 1 - 210821
Determine all natural numbers $a$ for which $\frac{1}{4}<\frac{a}{a+12}<\frac{1}{3}$ holds! | 5 | 49 | 1 |
math | 9. The solution set of the inequality $|x|^{3}-2 x^{2}-4|x|+3<0$ is $\qquad$ . | (-3,-\frac{\sqrt{5}-1}{2})\cup(\frac{\sqrt{5}-1}{2},3) | 34 | 29 |
math | 8. If $n$ is a natural number less than 50, find all values of $n$ such that the values of the algebraic expressions $4n+5$ and $7n+6$ have a common divisor greater than 1. | 7,18,29,40 | 54 | 10 |
math | Question 110, Given that the set of points in the complex plane corresponding to the complex number $z$ satisfying the condition $\left|z^{2}\right|+\left|z^{2}-1\right|=7$ forms a quadratic curve, find the eccentricity $e$ of this quadratic curve.
---
The above text translated into English, preserving the original t... | \frac{1}{2} | 154 | 7 |
math | $1^{2} / 3 \%$ of the country's students are engaged with math magazines, specifically $2^{1} / 3 \%$ of the boys and $2 / 3 \%$ of the girls. How many boys and how many girls are engaged with the magazines among 75000 students? | u=45000,v=30000 | 68 | 14 |
math | Example 11 Given that $\alpha$ is a root of the equation $x^{2}+x-\frac{1}{4}=0$. Find the value of $\frac{\alpha^{3}-1}{\alpha^{5}+\alpha^{4}-\alpha^{3}-\alpha^{2}}$.
(1995, National Junior High School Mathematics League) | 20 | 79 | 2 |
math | $\overline{x y z}$ is a three-digit number written in the decimal system, its square is a five-digit number: $\overline{x y z^{2}}=\overline{a b c d e}$ and $\overline{z y x^{2}}=\overline{e d c b a}$. How many such three-digit numbers are there and which are they? | 101,102,103,201,202,301,111,112,113,211,212,311,121,122,221 | 80 | 59 |
math | 8. Let $a_{1}, a_{2}, \cdots, a_{105}$ be a permutation of $1,2, \cdots, 105$, satisfying: for any $m \in\{3,5,7\}$, for all $n$ such that $1 \leqslant n<n+m \leqslant 105$, we have $m \mid\left(a_{n+m}-a_{n}\right)$. Then the number of different permutations that meet the requirement is $\qquad$ (answer with a specifi... | 3628800 | 127 | 7 |
math | 2. Given the set $A=\{k+1, k+2, \cdots, k+n\}, k, n$ are positive integers, if the sum of all elements in set $A$ is 2019, then when $n$ takes the maximum value, the set $A=$ | {334,335,336,337,338,339} | 66 | 25 |
math | Example 4.10. Investigate the convergence of the series $\sum_{n=1}^{\infty} \frac{2 n+1}{a^{n}}$, $(a>0)$. | if\>1\the\series\converges,\if\0<<1\or\=1\the\series\diverges | 45 | 31 |
math | 291. Find all integers $n$ for which the fraction $\frac{n-2}{n+1}$ takes integer values. | 2,0,-2,-4 | 28 | 7 |
math | Example 3 Real numbers $x_{1}, x_{2}, x_{3}, x_{4} \in [0,1]$. Find $K_{\max }$, where,
$$
\begin{aligned}
K= & \left|x_{1}-x_{2}\right|\left|x_{1}-x_{3}\right|\left|x_{1}-x_{4}\right|\left|x_{2}-x_{3}\right| \cdot \\
& \left|x_{2}-x_{4}\right|\left|x_{3}-x_{4}\right| .
\end{aligned}
$$ | \frac{\sqrt{5}}{125} | 129 | 12 |
math | 6. 28 Given the value of $\sin \alpha$. Try to find: (a) $\sin \frac{\alpha}{2}$, (b) $\sin \frac{\alpha}{3}$, respectively, how many different values can they have at most? | 4, 3 | 56 | 4 |
math | In a bag are all natural numbers less than or equal to $999$ whose digits sum to $6$. What is the probability of drawing a number from the bag that is divisible by $11$? | \frac{1}{7} | 44 | 7 |
math | 49.2. It is known that the parabola $y=x^{2}+p x+q$ touches the line $y=2 x-3$ at the point $M(2 ; 1)$. Find $p$ and $q$. | x^{2}-2x+1 | 56 | 8 |
math | 3. In $\triangle A B C$, the lengths of the sides opposite to $\angle A$, $\angle B$, and $\angle C$ are $a$, $b$, and $c$ respectively. Point $G$ satisfies
$$
\overrightarrow{G A}+\overrightarrow{G B}+\overrightarrow{G C}=\mathbf{0}, \overrightarrow{G A} \cdot \overrightarrow{G B}=0 \text {. }
$$
If $(\tan A+\tan B) ... | \frac{1}{2} | 131 | 7 |
math | Example 1 Given in the sequence $\left\{a_{n}\right\}$, $a_{1}=1$, and $a_{n+1}=\frac{1}{16}\left(1+4 a_{n}+\sqrt{1+24 a_{n}}\right)$.
Find $a_{n}$. 1$]$
Translate the above text into English, please retain the original text's line breaks and format, and output the translation result directly. | a_{n}=\frac{1}{24}\left[\left(\frac{1}{2}\right)^{n-2}+3\right]^{2}-\frac{1}{24} | 103 | 45 |
math | 6. Let $[x]$ denote the greatest integer not exceeding the real number $x$,
$$
a_{k}=\left[\frac{2014}{k}\right](k=1,2, \cdots, 100) \text {. }
$$
Then, among these 100 integers, the number of distinct integers is | 69 | 77 | 2 |
math | Given is a $n\times n$ grid with all squares on one diagonal being forbidden. You are allowed to start from any square, and move one step horizontally, vertically or diagonally. You are not allowed to visit a forbidden square or previously visited square. Your goal is to visit all non forbidden squares. Find, with proo... | 2\lfloor\frac{n}{2}\rfloor - 1 | 82 | 17 |
math | 13.2 設 $n$ 為整數。求 $n^{a}-n$ 除以 30 的稌值 $b$ 。
Let $n$ be an integer. Determine the remainder $b$ of $n^{a}-n$ divided by 30 . | 0 | 66 | 1 |
math | Find all triples of real numbers $(x, y, z)$ that satisfy
$$
x^{2}+y^{2}+z^{2}+1=x y+y z+z x+|x-2 y+z|
$$ | (y+1, y, y+1) \text{ and } (y-1, y, y-1) \text{ for any } y \in \mathbb{R} | 49 | 41 |
math | \section*{Exercise 4 - 241044}
Determine all pairs \((a ; b)\) of natural numbers for which \(a! + b! = (a + b)!\) holds! Hint: For every natural number \(n \geq 1\), \(n!\) is defined as the product of all natural numbers \(k\) such that \(1 \leq k \leq n\); furthermore, \(0! = 1\) is defined. | (1;1) | 105 | 5 |
math | 2. The fraction $A=\frac{2 x-14}{x-4}$, where $x$ is an integer different from 4.
a) For which values of the number $x$ is the fraction $A$ an integer?
b) For which values of the natural number $x(x>4)$ does the fraction $A$ have the smallest value? | x\in{-2,1,2,3,5,6,7,10} | 78 | 21 |
math | 32. $\left\{\begin{array}{l}\log _{y} \log _{y} x=\log _{x} \log _{x} y \\ \log _{a}^{2} x+\log _{a}^{2} y=8 .\end{array}\right.$ | x_{1}=y_{1}=^{2},\quadx_{2}=y_{2}=^{-2} | 69 | 24 |
math | Find all pairs of integers $(m,n)$ such that an $m\times n$ board can be totally covered with $1\times 3$ and $2 \times 5$ pieces. | 4 \times 4 | 41 | 7 |
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