task_type stringclasses 1
value | problem stringlengths 23 3.94k | answer stringlengths 1 231 | problem_tokens int64 8 1.39k | answer_tokens int64 1 200 |
|---|---|---|---|---|
math | Problem 6.5. Masha told Sasha that she had thought of two different natural numbers greater than 11. Then she told Sasha their sum.
- Sasha, after thinking, said: "I don't know the numbers you thought of."
- To which Masha replied: "At least one of them is even."
- Then Sasha said: "Now I definitely know the numbers y... | 1216 | 98 | 4 |
math | Problem 5. Solve the equation: $2\left(x^{4}+3 x^{2}+6\right)\left(y^{4}-5 y^{2}+12\right)=69$. | (0;\\sqrt{\frac{5}{2}}) | 46 | 13 |
math | Problem 3. If we divide the numbers 701 and 592 by the same natural number, we get remainders of 8 and 7, respectively. By which number did we divide the given numbers? | 9 | 47 | 1 |
math | Find all ordered pairs $(x,y)$ of real numbers that satisfy the following system of equations:
$$\begin{cases}
y(x+y)^2=2\\
8y(x^3-y^3) = 13.
\end{cases}$$ | (x, y) = \left( \frac{3}{2}, \frac{1}{2} \right) | 54 | 25 |
math | 437. What sequence do the differences between the differences («second differences») of the cubes of consecutive natural numbers form? | 6n+6 | 27 | 4 |
math | 1. (5 points) Find the value of $n$ for which the following equality holds:
$$
\frac{1}{1+\sqrt{2}}+\frac{1}{\sqrt{2}+\sqrt{3}}+\frac{1}{\sqrt{3}+\sqrt{4}}+\ldots+\frac{1}{\sqrt{n}+\sqrt{n+1}}=2018
$$ | 4076360 | 88 | 7 |
math | 10.201. Given a triangle $A B C$ such that $A B=15 \text{~cm}, B C=12$ cm and $A C=18 \text{~cm}$. Calculate in what ratio the incenter of the triangle divides the angle bisector of angle $C$. | 2:1 | 70 | 3 |
math | ## Task 21/82
We are looking for all four-digit numbers (in decimal notation) with the digit sequence $\{a ; b ; c ; d\}$ where $a ; b ; c ; d \in N, 1 \leq a \leq 9,0 \leq b ; c ; d \leq 9$, whose sixfold has the digit sequence $\{a ; a ; c ; b ; d\}$. | 1982 | 98 | 4 |
math | Problem 81. Let \( x, y, z \) be positive real numbers satisfying \( 2xyz = 3x^2 + 4y^2 + 5z^2 \). Find the minimum of the expression \( P = 3x + 2y + z \). | 36 | 63 | 2 |
math | 6. Let $a, b, c, d \in\{-1,0,1\}$. If the ordered array $(a, b, c, d)$ satisfies that $a+b$, $c+d$, $a+c$, and $b+d$ are all distinct, then $(a, b, c, d)$ is called a "good array". The probability that an ordered array $(a, b, c, d)$ is a good array among all possible ordered arrays $(a, b, c, d)$ is $\qquad$ | \frac{16}{81} | 114 | 9 |
math | $4 \cdot 36$ Try to find the $n$ $n$-th roots of 1, and find the sum of their $n$-th powers.
untranslated part:
试求 1 的 $n$ 个 $n$ 次方根,并求它们 $n$ 次幕的和.
translated part:
Try to find the $n$ $n$-th roots of 1, and find the sum of their $n$-th powers. | n | 106 | 1 |
math | 250. $y=\ln \left(x^{3}-1\right)$.
250. $y=\ln \left(x^{3}-1\right)$.
The above text has been translated into English, retaining the original text's line breaks and format. However, since the original text is a mathematical expression, the translation is identical to the original as mathematical expressions are gene... | \frac{3x^{2}}{x^{3}-1} | 89 | 15 |
math | Let $n$ be a positive integer. $f(n)$ denotes the number of $n$-digit numbers $\overline{a_1a_2\cdots a_n}$(wave numbers) satisfying the following conditions :
(i) for each $a_i \in\{1,2,3,4\}$, $a_i \not= a_{i+1}$, $i=1,2,\cdots$;
(ii) for $n\ge 3$, $(a_i-a_{i+1})(a_{i+1}-a_{i+2})$ is negative, $i=1,2,\cdots$.
(1) Fin... | 10 | 175 | 4 |
math | 1. Let $A B C$ be a right triangle, with hypotenuse $A C$, and let $H$ be the foot of the altitude from $B$ to $A C$. Given that the lengths $A B, B C$, and $B H$ form the sides of a new right triangle, determine the possible values of $\frac{A H}{C H}$. | \frac{\sqrt{5}+1}{2} | 81 | 12 |
math | ## Problem 1
Let the sequence $\left(x_{n}\right)_{n \geq 2}$ be such that $x_{2}=1, x_{n+1}=\frac{n^{2}}{n-1} \cdot x_{n}, n \geq 2$.
If $S_{n}=\sum_{k=2}^{n}\left(1-\frac{1}{k}\right) \cdot \frac{1}{x_{k}}$, calculate $\lim _{n \rightarrow \infty} S_{n}$. | e-2 | 122 | 3 |
math | 11. $m, n$ are two natural numbers, satisfying $26019 \times m-649 \times n=118$, then $m=$ $\qquad$ , $n=$ $\qquad$ _. | =2,n=80 | 52 | 6 |
math | 394. The random variable $X$ is given by the probability density function $f(x)=(1 / 2) \sin x$ in the interval $(0, \pi)$; outside this interval, $f(x)=0$. Find the variance of the function $Y=\varphi(X)=X^{2}$, using the density function $g(y)$. | (\pi^{4}-16\pi^{2}+80)/4 | 77 | 17 |
math | 10.1. The first term of the sequence is 934. Each subsequent term is equal to the sum of the digits of the previous term, multiplied by 13. Find the 2013-th term of the sequence. | 130 | 52 | 3 |
math | 7. If the four lines
$$
x=1, y=-1, y=3, y=k x-3
$$
enclose a convex quadrilateral with an area of 12, then the value of $k$ is $\qquad$. | 1 \text{ or } -2 | 55 | 8 |
math | 10. (12th IMO Problem) Let real numbers $x_{1}, x_{2}, \cdots, x_{1997}$ satisfy the following conditions:
(1) $-\frac{1}{\sqrt{3}} \leqslant x_{i} \leqslant \sqrt{3}$, where $i=1,2, \cdots, 1997$.
(2) $x_{1}+x_{2}+\cdots+x_{1997}=-318 \sqrt{3}$.
Find: $x_{1}^{12}+x_{2}^{12}+\cdots+x_{1997}^{12}$'s maximum value. | 189548 | 163 | 6 |
math | Example 4 Find all positive integer triples $(x, y, z)$ such that $y$ is a prime number, and neither 3 nor $y$ is a divisor of $z$, and satisfying $x^{3}-y^{3}=z^{2}$.
---
The translation maintains the original text's format and line breaks. | (x,y,z)=(8,7,13) | 69 | 11 |
math | Example 3.29. Find $\frac{\partial^{2} z}{\partial y \partial x}$ if $Z=f\left(x^{2}+y^{2}, x^{2} y^{2}\right)$. | \frac{\partial^{2}f}{\partialu^{2}}4x^{2}+8x^{2}y^{2}\frac{\partial^{2}f}{\partialu\partialv}+\frac{\partial^{2}f}{\partialv^{2}}4x^{2}y^{4}+\frac{\partialf}{\partialv}4xy | 50 | 81 |
math | [b]p1.[/b] One throws randomly $120$ squares of the size $1\times 1$ in a $20\times 25$ rectangle. Prove that one can still place in the rectangle a circle of the diameter equal to $1$ in such a way that it does not have common points with any of the squares.
[b]p2.[/b] How many digits has the number $2^{70}$ (produc... | 22 | 254 | 2 |
math | ## Task B-4.1.
Solve the equation
$$
\binom{x+1}{x-2}+2\binom{x-1}{3}=7(x-1)
$$ | 5 | 43 | 1 |
math | ## Task B-4.4.
The line $t_{1}$ touches the left, and the line $t_{2}$, which is parallel to it, touches the right branch of the hyperbola $x^{2}-y^{2}=$ 1. If the lines $t_{1}$ and $t_{2}$ intersect the $x$-axis at an angle of $60^{\circ}$, calculate their mutual distance. | \sqrt{2} | 94 | 5 |
math | Point $M$ is the midpoint of chord $A B$. Chord $C D$ intersects $A B$ at point $M$. A semicircle is constructed on segment $C D$ as its diameter. Point $E$ lies on this semicircle, and $M E$ is perpendicular to $C D$. Find the angle $A E B$. | 90 | 76 | 2 |
math | 4.91 Solve the equation $\cos ^{n} x-\sin ^{n} x=1$. Here $n$ is any given natural number. | {\begin{pmatrix}k\pi&(n\text{iseven}),\\2k\pi\text{or}2k\pi-\frac{\pi}{2}&(n\text{isodd}),\end{pmatrix}.} | 34 | 53 |
math | + Find all positive integers $n$ such that there exist $k \in \mathbf{N}^{*}, k \geqslant 2$ and positive rational numbers $a_{1}, a_{2}, \cdots, a_{k}$, satisfying
$$
a_{1}+a_{2}+\cdots+a_{k}=a_{1} \cdots a_{k}=n .
$$ | 4orn\geqslant6 | 90 | 8 |
math | 9. The sequence $\left\{a_{n}\right\}$ satisfies: $a_{1}=a_{2}=a_{3}=1$. Let $b_{n}=a_{n}+a_{n+1}+a_{n+2}\left(n \in \mathbf{N}^{*}\right)$. If the sequence $\left\{b_{n}\right\}$ is a geometric sequence with a common ratio of 3, find the value of $a_{100}$. | \frac{3^{100}+10}{13} | 110 | 16 |
math | ## Task A-2.1.
Determine all real numbers $x$ for which
$$
\sqrt{|x-1|+|x+4|}=|x|
$$ | -\sqrt{5},3 | 39 | 6 |
math | At point $O$ on the shore of the Dongjiang Lake (the lake shore can be considered a straight line), a rescue boat is parked. Due to the sudden breakage of the rope, the boat is blown away, with its direction forming a $15^{\circ}$ angle with the shore, at a speed of $2.5 \mathrm{~km} / \mathrm{h}$. At the same time, a ... | 2\sqrt{2}\mathrm{~}/\mathrm{} | 184 | 13 |
math | \begin{aligned} & \text { Example } 5 \max _{a, b, c \in \mathbf{R}_{+}} \min \left\{\frac{1}{a}, \frac{1}{b^{2}}, \frac{1}{c^{3}}, a+b^{2}+c^{3}\right\} \\ = & \end{aligned} | \sqrt{3} | 83 | 5 |
math | 8.2. The price of a ticket to the stadium was 25 rubles. After the ticket prices were reduced, the number of spectators at the stadium increased by $50 \%$, and the revenue from ticket sales increased by $14 \%$. What is the new price of a ticket to the stadium after the price reduction? | 19 | 69 | 2 |
math | 1. There is a quadratic equation, whose two roots are two-digit numbers formed by the digits 1, 9, 8, and 4. Let the difference between these two roots be $\mathrm{x}$, which makes $\sqrt{1984 \mathrm{x}}$ an integer. Try to find this equation and its two roots. | y_1=49, y_2=18 | 72 | 13 |
math | 10. For what values of the parameter $a$ does the equation $x^{4}-40 x^{2}+144=a\left(x^{2}+4 x-12\right)$ have exactly three distinct solutions? | 48 | 52 | 2 |
math | 8. (15 points) Fill in the 9 cells of the 3x3 grid with 9 different natural numbers, such that: in each row, the sum of the two left numbers equals the rightmost number; in each column, the sum of the two top numbers equals the bottom number. The smallest number in the bottom-right corner is . $\qquad$ | 12 | 77 | 2 |
math | For each pair of positive integers $(x, y)$ a nonnegative integer $x\Delta y$ is defined.
It is known that for all positive integers $a$ and $b$ the following equalities hold:
i. $(a + b)\Delta b = a\Delta b + 1$.
ii. $(a\Delta b) \cdot (b\Delta a) = 0$.
Find the values of the expressions $2016\Delta 121$ and $2016\D... | f(2016, 121) = 16 | 118 | 17 |
math | 2. In a certain year, there are 5 Saturdays and 4 Sundays in October. Question: What day of the week was October 1st of that year? | Thursday | 35 | 1 |
math | 2. Triangle $A B C$ is isosceles at $C$, and let $\Gamma$ be its circumcircle. Let $M$ be the midpoint of the arc $B C$ of $\Gamma$ that does not contain $A$, and let $N$ be the point where the line parallel to $A B$ through $M$ intersects $\Gamma$ again. It is known that $A N$ is parallel to $B C$. What are the measur... | \angleA=\angleB=72,\angleC=36 | 108 | 15 |
math | a) Let $ x_1, x_2, ..., x_n $ ($ n> 2 $) be negative real numbers and $ x_1 + x_2 + ... + x_n = m. $
Determine the maximum value of the sum
$ S = x_1x_2 + x_1x_3 + \dots + x_1x_n + x_2x_3 + x_2x_4 + \dots + x_2x_n + \dots + x_ {n-1} x_n. $
b) Let $ x_1, x_2, ..., x_n $ ($ n> 2 $) be nonnegative natural numbers and $ ... | \frac{(n-1)m^2}{n} | 237 | 12 |
math | ## Problem Statement
Based on the definition of the derivative, find $f^{\prime}(0)$:
$f(x)=\left\{\begin{array}{c}\frac{\ln \left(1+2 x^{2}+x^{3}\right)}{x}, x \neq 0 ; \\ 0, x=0\end{array}\right.$ | 2 | 79 | 1 |
math | ## Task Condition
Find the derivative.
$$
y=e^{a x}\left(\frac{1}{2 a}+\frac{a \cdot \cos 2 b x+2 b \cdot \sin 2 b x}{2\left(a^{2}+4 b^{2}\right)}\right)
$$ | e^{}\cdot\cos^{2} | 68 | 9 |
math | 4. Let $S=\{1,2, \cdots, 1990\}$. If the sum of the elements of a 31-element subset of $S$ is divisible by 5, it is called a good subset of $S$. Find the number of good subsets of $S$.
(IMO - 31 Preliminary Question) | \frac{1}{5}C_{1990}^{31} | 77 | 18 |
math | The numbers $36,27,42,32,28,31,23,17$ are grouped in pairs so that the sum of each pair is the same. Which number is paired with 32 ? | 27 | 51 | 2 |
math | Example 2 A glasses workshop in a factory has received a batch of tasks, requiring the processing of 6000 type $A$ parts and 2000 type $B$ parts. This workshop has 214 workers, each of whom can process 3 type $B$ parts in the time it takes to process 5 type $A$ parts. These people are divided into two groups, both work... | 137 | 117 | 3 |
math | ## Problem Statement
Calculate the limit of the numerical sequence:
$\lim _{n \rightarrow \infty} \sqrt{n^{3}+8}\left(\sqrt{n^{3}+2}-\sqrt{n^{3}-1}\right)$ | \frac{3}{2} | 52 | 7 |
math | 7. (10 points) There is a magical tree with 58 fruits on it. On the first day, 1 fruit will fall from the tree. Starting from the second day, the number of fruits that fall each day is 1 more than the previous day. However, if the number of fruits on the tree is less than the number that should fall on a certain day, t... | 12 | 123 | 2 |
math | 488. Find the integrals:
1) $\int \frac{d x}{x^{2}+4 x+8}$
2) $\int \frac{7-8 x}{2 x^{3}-3 x+1} d x$;
3) $\int \frac{3 x-2}{x^{2}+6 x+9} d x$;
4) $\int \frac{6 x^{3}-7 x^{2}+3 x-1}{2 x-3 x^{2}} d x$. | \begin{aligned}1)&\quad\frac{1}{2}\operatorname{arctg}\frac{x+2}{2}+C,\\2)&\quad\ln|\frac{x-1}{x-0.5}|-2\ln|x^{2}-1.5x+0.5|+C,\\3)&\quad3\ | 116 | 79 |
math | Task 2. Determine all triples $(x, y, z)$ of non-negative real numbers that satisfy the system of equations
$$
\begin{aligned}
& x^{2}-y=(z-1)^{2}, \\
& y^{2}-z=(x-1)^{2}, \\
& z^{2}-x=(y-1)^{2} .
\end{aligned}
$$ | (1,1,1) | 84 | 7 |
math | $2 \cdot 63$ Insert "+" or "-" between $1^{2}, 2^{2}, 3^{2}, \cdots, 1989^{2}$, what is the smallest non-negative number that can be obtained from the resulting sum? | 1 | 57 | 1 |
math | I5.3 It is known that $b^{16}-1$ has four distinct prime factors, determine the largest one, denoted by $c$ | 257 | 33 | 3 |
math | 1. The lieutenant is engaged in drill training with the new recruits. Upon arriving at the parade ground, he saw that all the recruits were lined up in several rows, with the number of soldiers in each row being the same and 5 more than the number of rows. After the training session, the lieutenant decided to line up t... | 24 | 131 | 2 |
math | 39. Find all three-digit numbers $x$, in the notation of which the digits do not repeat, such that the difference between this number and the number written with the same digits but in reverse order is also a three-digit number consisting of the same digits as the number $x$. | 954459 | 58 | 6 |
math | 2.047. $\frac{\frac{a-b}{2a-b}-\frac{a^{2}+b^{2}+a}{2a^{2}+ab-b^{2}}}{\left(4b^{4}+4ab^{2}+a^{2}\right):\left(2b^{2}+a\right)} \cdot\left(b^{2}+b+ab+a\right)$. | \frac{b+1}{b-2a} | 95 | 12 |
math | Let $n\geq 2$ be a given integer. Initially, we write $n$ sets on the blackboard and do a sequence of moves as follows: choose two sets $A$ and $B$ on the blackboard such that none of them is a subset of the other, and replace $A$ and $B$ by $A\cap B$ and $A\cup B$. This is called a $\textit{move}$.
Find the maximum n... | \frac{n(n-1)}{2} | 112 | 10 |
math | 5. (5 points) $A$, $B$, and $C$ are three fractions, where both the numerators and denominators are natural numbers. The ratio of the numerators is $3: 2: 1$, and the ratio of the denominators is $2: 3: 4$. The sum of the three fractions is $\frac{29}{60}$. Then, $A-B-C=$ $\qquad$ . | \frac{7}{60} | 94 | 8 |
math | 3. If the function $f(x)=\cos n x \cdot \sin \frac{4}{n} x(n \in \mathbf{Z})$ has a period of $3 \pi$, then the set of values for $n$ is | {\2,\6} | 54 | 5 |
math | Example 25. Solve the equation
$$
\sqrt{4-6 x-x^{2}}=x+4
$$ | -1 | 28 | 2 |
math | 16. Solve the following recurrence relations:
(1) $\left\{\begin{array}{l}a_{n}=2 a_{n-1}+2^{n} \quad(n \geqslant 1), \\ a_{0}=3 .\end{array}\right.$
(2) $\left\{\begin{array}{l}a_{n}=n a_{n-1}+(-1)^{n} \quad(n \geqslant 1), \\ a_{0}=3 .\end{array}\right.$
(3) $\left\{\begin{array}{l}a_{n}=2 a_{n-1}-1 \quad(n \geqslant... | 2^{n}+1 | 292 | 6 |
math | 2. (5 points) Two different natural numbers end with 7 zeros and have exactly 72 divisors. Find their sum.
# | 70000000 | 29 | 8 |
math | ## Problem Statement
Find the point $M^{\prime}$ symmetric to the point $M$ with respect to the line.
$M(3 ; 3 ; 3)$
$\frac{x-1}{-1}=\frac{y-1.5}{0}=\frac{z-3}{1}$ | M^{\}(1;0;1) | 67 | 10 |
math | Let's round each addend in the following expressions to $n=1,2,3, \ldots$ decimal places, and find the largest value of $n$ for which the rounded values of $K_{1}$ and $K_{2}$ are still equal. Also, examine the question for the case where the addends are rounded to the nearest unit, ten, hundred, and thousand.
$$
K_{1... | 4 | 127 | 1 |
math | 3: Given the function $f(x)=\sqrt{1-x}(x \leqslant 1)$. Then the coordinates of the intersection point of the function $f(x)$ and its inverse function $f^{-1}(x)$ are $\qquad$ | (1,0),(0,1),\left(\frac{-1+\sqrt{5}}{2}, \frac{-1+\sqrt{5}}{2}\right) | 55 | 37 |
math | Let $m$ be a positive integer and let $A$, respectively $B$, be two alphabets with $m$, respectively $2m$ letters. Let also $n$ be an even integer which is at least $2m$. Let $a_n$ be the number of words of length $n$, formed with letters from $A$, in which appear all the letters from $A$, each an even number of times... | 2^{n-m} | 140 | 5 |
math | 21. In a square with side length 5, draw lines parallel to the sides through the 5 equal division points on each side, then the total number of rectangles formed that are not squares is $\qquad$ . | 170 | 46 | 3 |
math | Jana had to calculate the product of two six-digit numbers for her homework. When copying from the board, she omitted one digit from one of the numbers, and instead of a six-digit number, she wrote only 85522. When she got home, she realized her mistake. However, she remembered that the number she had copied incorrectl... | 13 | 108 | 2 |
math | 3.463 Find $\operatorname{ctg} \frac{x}{2}$, if it is known that $\sin x-\cos x=\frac{1+2 \sqrt{2}}{3}$. | \cot\frac{x}{2}=\frac{\sqrt{2}}{2},\cot\frac{x}{2}=3-2\sqrt{2} | 46 | 34 |
math | Example 8 The sequence $a_{n}$ is defined as follows: $a_{1}=0, a_{2}=1$
$$
a_{n}=\frac{1}{2} n a_{n-1}+\frac{1}{2} n(n-1) a_{n-2}+(-1)^{n}\left(1-\frac{n}{2}\right)(n \geqslant 3)
$$
Try to find the simplest expression for $f_{n}=a_{n}+2 C_{n}^{1} a_{n-1}+3 C_{n}^{n} a_{n-2}+\cdots+(n-1) C_{n}^{n-2} a_{2}+n C_{n}^{n-... | f_{n}=2n!-(n+1) | 178 | 12 |
math | 4. If the set $S=\{1,2,3, \cdots, 16\}$ is arbitrarily divided into $n$ subsets, then there must exist a subset in which there are elements $a, b, c$ (which can be the same), satisfying $a+b=c$. Find the maximum value of $n$.
[Note] If the subsets $A_{1}, A_{2}, \cdots, A_{n}$ of set $S$ satisfy the following conditio... | 3 | 200 | 1 |
math | Three. (20 points) In the Cartesian coordinate system $x O y$, a line segment $A B$ of fixed length $m$ has its endpoints $A$ and $B$ sliding on the $x$-axis and $y$-axis, respectively. Let point $M$ satisfy
$$
A M=\lambda A B(\lambda>0, \lambda \neq 1) .
$$
Do there exist two distinct fixed points $E$ and $F$ such th... | E(-\sqrt{1-2 \lambda} m, 0), F(\sqrt{1-2 \lambda} m, 0) | 148 | 31 |
math | The clock hand points to 12. Jack writes a sequence consisting of param 1 symbols, each symbol being plus or minus. After that he gives this sequence to a robot. The robot reads it from right to left. If he sees a plus he turns the clock hand $120^{\circ}$ clockwise and if he sees a minus he turns it $120^{\circ}$ coun... | 682 | 295 | 3 |
math | 12. The smallest positive period of the function $f(x)=\cos ^{3} x-\sin ^{2} x$ is | 2\pi | 30 | 3 |
math | Let $P(x)=ax^3+(b-a)x^2-(c+b)x+c$ and $Q(x)=x^4+(b-1)x^3+(a-b)x^2-(c+a)x+c$ be polynomials of $x$ with $a,b,c$ non-zero real numbers and $b>0$.If $P(x)$ has three distinct real roots $x_0,x_1,x_2$ which are also roots of $Q(x)$ then:
A)Prove that $abc>28$,
B)If $a,b,c$ are non-zero integers with $b>0$,find all their po... | (a, b, c) = (2, 4, 4) | 138 | 18 |
math | 5.4. A smooth sphere with a radius of 1 cm was dipped in red paint and launched between two perfectly smooth concentric spheres with radii of 4 cm and 6 cm, respectively (this sphere ended up outside the smaller sphere but inside the larger one). Upon touching both spheres, the sphere leaves a red trail. During its mov... | 83.25 | 134 | 5 |
math | 15. Kiran is designing a game which involves a bag of twenty-one marbles. Some of the marbles are blue, the rest are red. To play the game, two marbles are drawn out. The game is won if at least one red marble is drawn. To ensure the probability of the game being won is exactly one-half, Kiran uses $B$ blue marbles and... | 261 | 104 | 3 |
math | 10. (3 points) The expressway from Lishan Town to the provincial capital is 189 kilometers long, passing through the county town. The county town is 54 kilometers away from Lishan Town. At 8:30 AM, a bus departs from Lishan Town to the county town, arriving at 9:15. After a 15-minute stop, it continues to the provincia... | 72 | 166 | 2 |
math | Sure, here is the translated text:
```
II. (20 points) Find all positive integers $n$ such that
$$
\left[\frac{n}{2}\right]+\left[\frac{n}{3}\right]+\left[\frac{n}{4}\right]+\left[\frac{n}{5}\right]=69 \text {, }
$$
where $[x]$ denotes the greatest integer not exceeding the real number $x$.
``` | 55 | 94 | 2 |
math | ## Condition of the problem
Find the derivative $y_{x}^{\prime}$.
$\left\{\begin{array}{l}x=(\arcsin t)^{2} \\ y=\frac{t}{\sqrt{1-t^{2}}}\end{array}\right.$ | \frac{1}{2(1-^{2})\arcsin} | 62 | 17 |
math | 8.1. Usually, we write the date in the format of day, month, and year (for example, 17.12.2021). In the USA, however, it is customary to write the month number, day number, and year in sequence (for example, 12.17.2021). How many days in a year cannot be determined unequivocally by its writing? | 132 | 90 | 3 |
math | Task 6. (15 points) On the plane xOy, indicate all points through which no curve passes, given by the equation
$$
\mathrm{a} x^{2}+(1-6 a) x+2-a-2 y+a y^{2}=0
$$ | y^{2}+(x-3)^{2}withoutthepoints(0;1)(4;3) | 62 | 25 |
math | 6.24 Find four numbers, the first three of which form a geometric progression, and the last three form an arithmetic progression. The sum of the extreme numbers is 21, and the sum of the middle numbers is 18. | 3;6;12;18or18.75;11.25;6.75;2.25 | 50 | 31 |
math | 11.233 a) The lengths of the edges $AB, AC, AD$ and $BC$ of an orthocentric tetrahedron are 5, 7, 8, and 6 cm, respectively. Find the lengths of the remaining two edges. b) Is the tetrahedron $ABCD$ orthocentric if $AB=8$ cm, $BC=12$ cm, and $DC=6$ cm? | 5\sqrt{3} | 98 | 6 |
math | Consider real numbers $A$, $B$, \dots, $Z$ such that \[
EVIL = \frac{5}{31}, \;
LOVE = \frac{6}{29}, \text{ and }
IMO = \frac{7}{3}.
\] If $OMO = \tfrac mn$ for relatively prime positive integers $m$ and $n$, find the value of $m+n$.
[i]Proposed by Evan Chen[/i] | 579 | 103 | 3 |
math | 174. Find the derivative of the function $y=x^{2}-3 x+5$. | 2x-3 | 21 | 4 |
math | Example 2 The system of equations in $x, y, z$
$$
\left\{\begin{array}{l}
3 x+2 y+z=a, \\
x y+2 y z+3 z x=6
\end{array}\right.
$$
has real solutions $(x, y, z)$. Find the minimum value of the positive real number $a$. | \sqrt{23} | 81 | 6 |
math | 9. In the sequence $\left\{a_{n}\right\}$, $a_{4}=1, a_{11}=9$, and the sum of any three consecutive terms is 15. Then $a_{2016}=$ | 5 | 54 | 1 |
math | 6. Let point $A(2,0)$, and $B$ be a point on the elliptical arc
$$
\frac{x^{2}}{4}+\frac{y^{2}}{3}=1(x>0, y>0)
$$
Draw a perpendicular from point $B$ to the $y$-axis, and let the foot of the perpendicular be $C$. Then the maximum value of the area of quadrilateral $O A B C$ is $\qquad$ | \frac{9}{4} | 105 | 7 |
math | 7. There are 15 players participating in a Go tournament, where each pair of players needs to play one match. Winning a match earns 2 points, a draw earns 1 point each, and losing a match earns 0 points. If a player's score is no less than 20 points, they will receive a prize. Therefore, the maximum number of players w... | 9 | 87 | 1 |
math | 13.134. According to the program, two machines on the assembly line should process the same number of parts in $a$ hours. The first machine completed the task. The second machine turned out to be not quite serviceable, worked with interruptions, as a result of which it processed $n$ fewer parts than the first in the sa... | \frac{+\sqrt{b^{2}n^{2}+240n}}{2b}\text{}\frac{-+\sqrt{b^{2}n^{2}+240n}}{2b} | 104 | 49 |
math | Ionin Yu.I.
The sum of $n$ positive numbers $x_{1}, x_{2}, x_{3}, \ldots, x_{n}$ is 1.
Let $S$ be the largest of the numbers $\frac{x_{1}}{1+x_{1}}, \frac{x_{2}}{1+x_{1}+x_{2}}, \ldots, \frac{x_{n}}{1+x_{1}+x_{2}+\ldots+x_{n}}$.
Find the smallest possible value of $S$. For which values of $x_{1}, x_{2}, \ldots, x_{n... | 1-\frac{1}{\sqrt[n]{2}} | 141 | 12 |
math | Find the largest constant $K$ such that for all positive real numbers $a, b$, and $c$, we have
$$
\sqrt{\frac{a b}{c}}+\sqrt{\frac{b c}{a}}+\sqrt{\frac{a c}{b}} \geqslant K \sqrt{a+b+c}
$$ | \sqrt{3} | 72 | 5 |
math | 【Example 3】 How many positive integers can 2160 be divided by? How many positive even integers can it be divided by? | 40 | 30 | 2 |
math | 8. Let the complex numbers be $z_{1}=-3-\sqrt{3} \mathrm{i}, z_{2}=\sqrt{3}+\mathrm{i}, z=\sqrt{3} \sin \theta+\mathrm{i}(\sqrt{3} \cos \theta+2)$, then the minimum value of $\left|z-z_{1}\right|+\left|z-z_{2}\right|$ is $\qquad$. | 2(\sqrt{3}+1) | 93 | 9 |
math | 24th ASU 1990 Problem 6 Find three non-zero reals such that all quadratics with those numbers as coefficients have two distinct rational roots. | 1,2,-3 | 35 | 5 |
math | Task 3. (15 points) The function $f(x)$ satisfies the condition: for any real numbers $a$ and $b$, the equality $f\left(\frac{a+2 b}{3}\right)=\frac{f(a)+2 f(b)}{3}$ holds. Find the value of the function $f(2021)$, if $f(1)=5, f(4)=2$. | -2015 | 92 | 5 |
math | 4. For each pair of numbers $x$, let us denote by $s(x, y)$ the smallest of the numbers $x, 1-y, y-x$. What is the largest value that the number $s(x, y)$ can take? | \frac{1}{3} | 52 | 7 |
math | Example 1 Let $S$ be a subset of the set $\{1,2,3, \cdots, 50\}$, and the sum of any two elements in $S$ cannot be divisible by 7. What is the maximum number of elements in $S$? | 23 | 61 | 2 |
math | For example, if $7 x, y, z$ are all positive integers, how many solutions does the equation $x+y+z=15$ have?
(1985 Shanghai Competition Question) | 91 | 42 | 2 |
math | Let $\tau$ be an arbitrary time interval. A freely falling body falls for $\frac{\tau}{2}$, and the distances traveled in the subsequent whole $\tau$ time intervals are in what ratio to each other? | 1:2:3:\ldots | 45 | 8 |
math | 1.63. The entire arc of a circle with radius $R$ is divided into four large and four small segments, alternating with each other. The large segment is twice as long as the small one. Determine the area of the octagon whose vertices are the points of division of the circle's arc. | R^{2}(\sqrt{3}+1) | 63 | 12 |
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