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 | We inscribe a straight circular cone in a sphere such that the center of the sphere divides the height of the cone according to the golden ratio. What is the ratio of the volumes of the two bodies to each other?
将上面的文本翻译成英文，请保留源文本的换行和格式，直接输出翻译结果。 | 4:1 | 66 | 3 |
math | Example 4 (1) Find the number of positive integer solutions to the equation $x_{1}+x_{2}+x_{3}+\cdots+x_{m}=n(m, n \in \mathrm{N}, m<n)$.
(2) Find the number of integer solutions to the equation $x_{1}+x_{2}+x_{3}+\cdots+x_{m}=n(m, n \in \mathrm{N}, n \geqslant(m-2) r+$ $1, r \in \mathbb{Z})$ satisfying $x_{1} \geqslan... | C_{n+(r-1)(2-)}^{-1} | 173 | 14 |
math | The sequence $ \{ a_n \} _ { n \ge 0 } $ is defined by $ a_0 = 2 , a_1 = 4 $ and
\[ a_{n+1} = \frac{a_n a_{n-1}}{2} + a_n + a_{n-1} \]
for all positive integers $ n $. Determine all prime numbers $ p $ for which there exists a positive integer $ m $ such that $ p $ divides the number $ a_m - 1 $. | p > 2 | 111 | 5 |
math | Exercise 2. For any strictly positive integer $n$, we define $a_{n}$ as the last digit of the sum of the digits of the number 20052005...2005, (we write "2005" $n$ times in a row). For example, $a_{1}=7$ and $a_{2}=4$.
a) What are the strictly positive integers $n$ such that $a_{n}=0$?
b) Calculate $a_{1}+a_{2}+\cdot... | 9025 | 128 | 4 |
math | 8. If $p$ is a prime number, and $p+3$ divides $5p$, then the last digit of $p^{2009}$ is $\qquad$ . | 2 | 41 | 1 |
math | Consider the polynomial $P(n) = n^3 -n^2 -5n+ 2$. Determine all integers $n$ for which $P(n)^2$ is a square of a prime.
[hide="Remark."]I'm not sure if the statement of this problem is correct, because if $P(n)^2$ be a square of a prime, then $P(n)$ should be that prime, and I don't think the problem means that.[/hide... | n = -1, -3, 0, 3, 1 | 99 | 17 |
math | (Moscow 2000, problem C2) Let $f(x)=x^{2}+12 x+30$. Solve the equation $f(f(f(f(f(x))))))=0$. | -6\6^{1/32} | 45 | 10 |
math | 2.
(a) Determine the natural number $n$ for which $401 \cdot 401^{3} \cdot 401^{5} \cdot \ldots \cdot 401^{401}=401^{n^{2}}$.
(b) For $n$ determined in point (a), find the remainder of the division of the number $a=1 \cdot 2 \cdot 3 \cdot \ldots \cdot 100-31$ by $n$.
(c) For $n$ determined in point (a), show that $... | 201 | 150 | 3 |
math | 4.109. Find the solution to the Cauchy problem: $y^{\prime \prime}-3 y^{\prime}+2 y=0$, $y(0)=1, y^{\prime}(0)=0$. | y(x)=2e^{x}-e^{2x} | 52 | 13 |
math | 11. Let $f(x)=2^{x}+x^{2}$, write a sequence such that the sum of its first $n$ terms has the form of $f(n)$, $a_{n}=$ $\qquad$ | a_{n}={\begin{pmatrix}3,n=1;\\2^{n-1}+2n-1,n\geqslant20\end{pmatrix}.} | 51 | 43 |
math | For what value of the parameter $p$ will the sum of the squares of the roots of the equation
$$
x^{2}+(3 p-2) x-7 p-1=0
$$
be the smallest? What is this smallest value? | \frac{53}{9} | 55 | 8 |
math | 8.1. In a cinema, five friends took seats numbered 1 to 5 (the leftmost seat is number 1). During the movie, Anya left to get popcorn. When she returned, she found that Varya had moved two seats to the right, Galia had moved one seat to the left, and Diana and Elia had swapped places, leaving the edge seat for Anya. Wh... | 2 | 94 | 1 |
math | Find all solutions of $2^n + 7 = x^2$ in which n and x are both integers . Prove that there are no other solutions. | (n, x) = (1, 3) | 33 | 13 |
math | 3 Let $n$ be a natural number, $f(n)$ be the sum of the digits of $n^{2}+1$ (in decimal), $f_{1}(n)=f(n), f_{k+1}(n)$ $=f\left(f_{k}(n)\right)$, then the value of $f_{100}(1990)$ is $\qquad$. | 11 | 86 | 2 |
math | 15. Find all positive integers $n$, such that $n+36$ is a perfect square, and apart from 2 or 3, $n$ has no other prime factors. | 64,108,288,864,1728,10368 | 41 | 25 |
math | Example 3 If $a \neq 1, b \neq 1, a>0, b>0$, and satisfy the relation $\log _{a} 2=\log _{\frac{a}{2}} 4=\log _{b} 3$, find the values of $a$ and $b$. | =\frac{1}{2},b=\frac{1}{3} | 71 | 15 |
math | Example 1 (to $1^{\circ}$). Investigate the function $z=$ $=x^{3}+y^{3}-3 x y$ for extremum. | z_{\}=z(1,1)=-1 | 38 | 12 |
math | Find all integes $a,b,c,d$ that form an arithmetic progression satisfying $d-c+1$ is prime number and $a+b^2+c^3=d^2b$ | (a, b, c, d) = (n, n+1, n+2, n+3) | 39 | 26 |
math | Problem 8. For what values of the parameter $a$ does the equation
$$
5^{x^{2}+2 a x+a^{2}}=a x^{2}+2 a^{2} x+a^{3}+a^{2}-6 a+6
$$
have exactly one solution? | 1 | 67 | 1 |
math | Problem 3. A pedestrian path with a rectangular shape has a length of $3030 \mathrm{~m}$ and a width of $180 \mathrm{~cm}$. The path needs to be covered with square tiles whose area is $9 \mathrm{~dm}^2$. How many such tiles are needed? | 60600 | 71 | 5 |
math | 2. Determine the pairs of complex numbers $(u, v)$ with $|u|=|v|=1$ such that
$$
|1-u|+\left|v^{2}+1\right|=|1-v|+\left|u^{2}+1\right|=\sqrt{2}
$$
Marius Damian, Brăila | (u,v)\in{(i,i),(-i,i),(i,-i),(-i,-i)} | 74 | 21 |
math | Find all four-digit numbers $\overline{abcd}$ such that they are multiples of $3$ and that $\overline{ab}-\overline{cd}=11$.
($\overline{abcd}$ is a four-digit number; $\overline{ab}$ is a two digit-number as $\overline{cd}$ is). | 1302, 1605, 1908, 2211, 2514, 2817, 3120, 3423, 3726, 4029, 4332, 4635, 4938, 5241, 5544, 5847, 6150, 6453, 6756, 7059, 7362, 7665, 7968, 8271, 8574, 8877, 9180, 9483, 9786, 10089, 10392, 10695, 10998 | 71 | 200 |
math | Problem 3.6. Petya can draw only 4 things: the sun, a ball, a tomato, and a banana. But he does it extremely realistically! Today he drew several things, among which there were exactly 15 yellow, 18 round, and 13 edible. What is the maximum number of balls he could have drawn?
Petya believes that all tomatoes are roun... | 18 | 107 | 2 |
math | 9. (16 points) Given the ellipse $C: \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1(a>b>0)$, points $F_{1}, F_{2}$ are its left and right foci, respectively, and $A$ is the right vertex. A line $l$ passing through $F_{1}$ intersects the ellipse at points $P, Q$, and $\overrightarrow{A P} \cdot \overrightarrow{A Q}=\frac{1}{... | 1-\frac{\sqrt{2}}{2} | 142 | 11 |
math | 1. Find all pairs of positive integers $(n . k)$ so that $(n+1)^{k}-1=n$ !. | (1,1),(2,1),(4,2) | 28 | 13 |
math | 4. for which natural numbers $n$ exists a polynomial $P(x)$ with integer coefficients such that $P(d)=(n / d)^{2}$ holds for all positive divisors $d$ of $n$ ?
## 1st solution | nis,n=1orn=6 | 51 | 7 |
math | 5. The equation of the largest circle above the parabola $y=a x^{2}(a>0)$, which is tangent to the parabola at its vertex, is $\qquad$ | x^{2}+\left(y-\frac{1}{2 a}\right)^{2}=\frac{1}{4 a^{2}} | 42 | 30 |
math | 9. Seawater contains $5 \%$ salt. How many kilograms of fresh water need to be added to 40 kg of seawater to make the salt content $2 \%$? | 60 | 40 | 2 |
math | (BMO 2009)
Find all functions $f: \mathbb{N}^{*} \rightarrow \mathbb{N}^{*}$ such that
$$
f\left(f^{2}(m)+2 f^{2}(n)\right)=m^{2}+2 n^{2}
$$
for all positive integers $m, n$. | f(n)=n | 78 | 4 |
math | 3. In $\triangle A B C$, $B C=a, C A=b, A B=c$. If $2 a^{2}+b^{2}+c^{2}=4$, then the maximum value of the area of $\triangle A B C$ is $\qquad$ (Provided by Li Hongchang) | \frac{\sqrt{5}}{5} | 68 | 10 |
math | 8. Let the sequence $\left\{a_{n}\right\}$ satisfy:
$$
\begin{array}{l}
a_{1}=\frac{1}{4}, a_{n+1}=a_{n}+a_{n}^{2}\left(n \in \mathbf{Z}_{+}\right) . \\
\text {Let } T_{2020}=\frac{1}{a_{1}+1}+\frac{1}{a_{2}+1}+\cdots+\frac{1}{a_{2020}+1} .
\end{array}
$$
If the value of $T_{2020}$ lies in the interval $(k, k+1)$, the... | 3 | 170 | 1 |
math | Problem 11.2. Let $\alpha$ and $\beta$ be the real roots of the equation $x^{2}-x-2021=0$, with $\alpha>\beta$. Denote
$$
A=\alpha^{2}-2 \beta^{2}+2 \alpha \beta+3 \beta+7
$$
Find the greatest integer not exceeding $A$. | -6055 | 83 | 5 |
math | Four, (15 points) Can 2010 be written as the sum of squares of $k$ distinct prime numbers? If so, try to find the maximum value of $k$; if not, please briefly explain the reason. | 7 | 51 | 1 |
math | 1. Write 2004 numbers on the blackboard: $1,2, \cdots, 2004$, in each step, erase some numbers from the blackboard and write down the remainder of their sum divided by 167, after several steps, two numbers remain on the blackboard, one of which is 999, what is the second number? | 3 | 82 | 1 |
math | Thales' Theorem and the Proportional Segments Theorem [The ratio of the areas of triangles with a common base or common height]
In triangle $ABC$, a point $K$ is taken on side $AB$, such that $AK: BK=1: 2$, and a point $L$ is taken on side $BC$, such that $CL: BL=2: 1$. $Q$ is the point of intersection of lines $AL$ a... | \frac{7}{4} | 119 | 7 |
math | Circle $\omega_1$ of radius $1$ and circle $\omega_2$ of radius $2$ are concentric. Godzilla inscribes square $CASH$ in $\omega_1$ and regular pentagon $MONEY$ in $\omega_2$. It then writes down all 20 (not necessarily distinct) distances between a vertex of $CASH$ and a vertex of $MONEY$ and multiplies them all togeth... | 2^{20} + 1 | 102 | 8 |
math | \[
\left.\begin{array}{l}
\text { Systems of algebraic nonlinear equations } \\
\text { [Symmetric systems. Involutory transformations ]}
\end{array}\right]
\]
Solve the system of equations: \( x_{1} x_{2}=x_{2} x_{3}=\ldots=x_{n-1} x_{n}=x_{n} x_{1}=1 \). | x_{1}=x_{2}=\ldots=x_{n}=\1foroddn,x_{1}=x_{3}=\ldots=x_{n-1}=x_{2}=x_{4}=\ldots=x_{n}=1/(\neq0)forevenn | 92 | 62 |
math | 2. Use the four digits 1, 2, 3, 4 to form a four-digit number $\overline{a b c d}$, with the following requirements:
(1) $a$, $b$, $c$, $d$ are all different; (2) $b$ is greater than both $a$ and $d$, and $c$ is also greater than both $a$ and $d$. How many such four-digit numbers are there? $\qquad$ | 4 | 103 | 1 |
math | (7) Set $A=\left\{x \left\lvert\, x=\left[\frac{5 k}{6}\right]\right., k \in \mathbf{Z}, 100 \leqslant k \leqslant 999\right\}$, where $[x]$ denotes the greatest integer not greater than $x$. Then the number of elements in set $A$ is $\qquad$ . | 750 | 97 | 3 |
math | 1. The system of equations in $x, y, z$
$$
\left\{\begin{array}{l}
x y+y z+z x=1, \\
5 x+8 y+9 z=12
\end{array}\right.
$$
all real solutions $(x, y, z)$ are $\qquad$ | \left(1, \frac{1}{2}, \frac{1}{3}\right) | 72 | 21 |
math | Example 3 Try to find a positive integer $k$ other than 1, such that $k$ and $k^{4}$ can both be expressed as the sum of squares of two consecutive integers, and prove that such a $k$ is unique.
Translate the above text into English, please retain the original text's line breaks and format, and output the translation ... | 13 | 78 | 2 |
math | 276. Solve the equation:
$$
\left(x^{2}-16\right)(x-3)^{2}+9 x^{2}=0
$$ | -1\\sqrt{7} | 37 | 7 |
math | ## Task B-2.5.
Ana is visiting her grandmother and wants to bring her a few pieces of fruit in a basket. She has 6 bananas, 5 apples, and 4 peaches available.
a) In how many ways can she choose the fruit for the basket if the basket must not be empty?
b) In how many ways can she choose the fruit for the basket if sh... | 209 | 118 | 3 |
math | 8・165 Given a sequence of natural numbers $\left\{x_{n}\right\}$ that satisfies
$$x_{1}=a, x_{2}=b, x_{n+2}=x_{n}+x_{n+1}, n=1,2,3, \cdots$$
If one of the terms in the sequence is 1000, what is the smallest possible value of $a+b$? | 10 | 95 | 2 |
math | ## SUBJECT I
If the natural number $x$ satisfies the equation $\left\lfloor(4026: 61: 33-x): 2013^{0}+3\right\rfloor \cdot 6-24=0$ and $y=\left(2+2^{2}+\ldots+2^{2013}\right):\left(4^{1007}-2\right) \cdot 2013$, find $x^{y}$ and $y^{x}$. | x^{y}=1,y^{x}=2013 | 118 | 13 |
math | Folklore
$a$ and $b$ are the given sides of a triangle.
How to choose the third side $c$ so that the points of tangency of the incircle and the excircle with this side divide it into three equal segments?
For which $a$ and $b$ does such a side exist
(We consider the excircle that touches side $c$ and the extensions of ... | 3|-b|,whenb<<2bor<b<2a | 92 | 14 |
math | 1. Simplify
$$
\sqrt{1+2 \sin \alpha \cdot \cos \alpha}+\sqrt{1-2 \sin \alpha \cdot \cos \alpha}
$$
$\left(0^{\circ}<\alpha \leqslant 90^{\circ}\right)$ The result is $\qquad$ . | 2 \cos \alpha \text{ or } 2 \sin \alpha | 75 | 16 |
math | If $a>1$ and $b>2$ are positive integers, show that $a^{b}+1 \geq b(a+1)$, and determine when equality holds. | a^b + 1 \geq b(a + 1) | 40 | 16 |
math | Problem 2. One dog, one cat, and one rabbit have a mass of 17 kg, two dogs, one cat, and one rabbit 29 kg, and two dogs, three cats, and one rabbit 35 kg. What are the masses of one dog, one cat, and one rabbit? | 12,3,2 | 66 | 6 |
math | 6 Let $f(x)$ be an odd function defined on $\mathbf{R}$, and when $x \geqslant 0$, $f(x)=x^{2}$. If for any $x \in[a, a+2]$, the inequality $f(x+a) \geqslant 2 f(x)$ always holds, then the range of the real number $a$ is $\qquad$. | [\sqrt{2},+\infty) | 89 | 9 |
math | ## Task Condition
Write the decomposition of vector $x$ in terms of vectors $p, q, r$:
$x=\{15 ;-20 ;-1\}$
$p=\{0 ; 2 ; 1\}$
$q=\{0 ; 1 ;-1\}$
$r=\{5 ;-3 ; 2\}$ | -6p+q+3r | 73 | 8 |
math | 8. (2007 Shanghai College Entrance Examination) The smallest positive period of the function $y=\sin \left(x+\frac{\pi}{3}\right) \sin \left(x+\frac{\pi}{2}\right)$ is $\qquad$. | \pi | 54 | 2 |
math | A piece is placed in the lower left-corner cell of the $15 \times 15$ board. It can move to the cells that are adjacent to the sides or the corners of its current cell. It must also alternate between horizontal and diagonal moves $($the first move must be diagonal$).$ What is the maximum number of moves it can make wit... | (n-1)^2 | 81 | 7 |
math | The first $510$ positive integers are written on a blackboard: $1, 2, 3, ..., 510$. An [i]operation [/i] consists of of erasing two numbers whose sum is a prime number. What is the maximum number of operations in a row what can be done? Show how it is accomplished and explain why it can be done in no more operations. | 255 | 85 | 3 |
math | ## Task 2.
Calculate the sum
$$
\frac{1^{2}+2^{2}}{1 \cdot 2}+\frac{2^{2}+3^{2}}{2 \cdot 3}+\frac{3^{2}+4^{2}}{3 \cdot 4}+\cdots+\frac{99^{2}+100^{2}}{99 \cdot 100}
$$ | \frac{99\cdot201}{100} | 97 | 15 |
math | 2. (19th Iranian Mathematical Olympiad (2nd Round) Problem) Find all functions $f: \mathbf{R} \backslash\{0\} \rightarrow \mathbf{R}$, such that for all $x, y \in \mathbf{R} \backslash\{0\}$, we have $x f\left(x+\frac{1}{y}\right)+y f(y)+\frac{y}{x}=y f\left(y+\frac{1}{x}\right)+x f(x)+\frac{x}{y}$. | f(x)=A+\frac{B}{x}+x | 124 | 13 |
math | II. (50 points) Find all real polynomials $f$ and $g$ such that for all $x \in \mathbf{R}$, we have: $\left(x^{2}+x+1\right) f\left(x^{2}-x+1\right)=$ $\left(x^{2}-x+1\right) g\left(x^{2}+x+1\right)$. | f(x)=kx,(x)=kx | 91 | 10 |
math | 5. Find all prime numbers whose decimal representation has the form 101010 ... 101 (ones and zeros alternate). | 101 | 30 | 3 |
math | ## Task 27/81
What values does the function $y=\sin \left(x^{8}-x^{6}-x^{4}+x^{2}\right)$ take when $x$ is an integer? | 0 | 48 | 1 |
math | 3. Is it possible to represent the number 2017 as the sum of three natural numbers such that no two of them are coprime? | 2017=12+10+1995 | 32 | 15 |
math | Problem 2. The she-rabbit bought seven drums of different sizes and seven pairs of sticks of different lengths for her seven bunnies. If a bunny sees that both its drum is larger and its sticks are longer than those of one of its brothers, it starts to drum loudly. What is the maximum number of bunnies that can start d... | 6 | 85 | 1 |
math | 11. (8 points) This year, Xiaochun is 18 years younger than his brother. In 3 years, Xiaochun's age will be half of his brother's age. How old is Xiaochun this year? $\qquad$ years old. | 15 | 58 | 2 |
math | 13. A student participates in military training and must shoot 10 times. In the 6th, 7th, 8th, and 9th shots, he scored 9.0, 8.4, 8.1, and 9.3 points, respectively. The average score of his first 9 shots is higher than the average score of his first 5 shots. If he wants the average score of 10 shots to exceed 8.8 point... | 9.9 | 138 | 3 |
math | ## Task B-4.7.
Determine all complex numbers $z$ such that $\operatorname{Re} z>0$ and $z^{8}+(1-4 i) z^{4}-4 i=0$. | {\sqrt{2}(\cos\frac{\pi}{8}+i\sin\frac{\pi}{8}),\sqrt{2}(\cos\frac{13\pi}{8}+i\sin\frac{13\pi}{8}),\cos\frac{\pi}{4}+i\sin\frac{\pi}{4},\cos\frac{7\pi}{4}+i} | 49 | 90 |
math | $f(n)$ is the least number that there exist a $f(n)-$mino that contains every $n-$mino.
Prove that $10000\leq f(1384)\leq960000$.
Find some bound for $f(n)$ | 10000 \leq f(1384) \leq 960000 | 62 | 27 |
math | 7. If the parabola $y=x^{2}-a x+3(a \in \mathbf{R})$ is always above the line $y=\frac{9}{4}$ over the interval $[a, a+1]$, then the range of values for $a$ is $\qquad$. | (-\sqrt{3},+\infty) | 67 | 10 |
math | 4. The integer values of $n$ that satisfy the equation $\left(n^{2}-5 n+5\right)^{n+1}=1$ are $\qquad$ .
Translate the text above into English, please retain the original text's line breaks and format, and output the translation result directly. | 1,4,3,-1 | 64 | 7 |
math | Problem 4.5. Mom and Dad have two children: Kolya and Tanya. Dad is 4 years older than Mom. Kolya is also 4 years older than Tanya and half as old as Dad. How old is each of them if the total age of all family members is 130 years? | Tanyais19old,Kolyais23old,momis42old,dadis46old | 69 | 24 |
math | Example 4. Solve the equation $y^{\prime}-y^{2}=4 x^{2}+4 x y+2$. | 2\operatorname{tg}(2x+C)-2x | 29 | 13 |
math | # 6.1. Condition:
Petya thought of a natural number and wrote down the sums of each pair of its digits on the board. After that, he erased some of the sums, and the numbers $2,0,2,2$ remained on the board. What is the smallest number Petya could have thought of? | 2000 | 71 | 4 |
math | $8, Class 3 (1) 21 students made a total of 69 paper airplanes. Each girl made 2, and each boy made 5. So, there are ( ) boys and ( ) girls.
$ | 9 | 49 | 1 |
math | Find all functions $f:[0,1] \to \mathbb{R}$ such that the inequality \[(x-y)^2\leq|f(x) -f(y)|\leq|x-y|\] is satisfied for all $x,y\in [0,1]$ | f(x) = \pm x + C | 60 | 10 |
math | Let $p$ be a prime number. Find all possible values of the remainder when $p^{2}-1$ is divided by 12 . | 3,8,0 | 31 | 5 |
math | Example. Find the derivative of the function $y=y(x)$, given implicitly by the equation
$$
\ln \sqrt{x^{2}+y^{2}}=\operatorname{arctg} \frac{y}{x}
$$ | y^{\}=\frac{x_{0}+y_{0}}{x_{0}-y_{0}} | 51 | 24 |
math | 1. The age of a person this year (1984) is equal to the sum of the digits of their birth year. How old is the person? | 20 | 34 | 2 |
math | Example 4 Given that $\alpha$ is an acute angle, $\beta$ is an obtuse angle, and $\sec (\alpha-2 \beta) 、 \sec \alpha 、 \sec (\alpha+2 \beta)$ form an arithmetic sequence, find the value of $\frac{\cos \alpha}{\cos \beta}$. | -\sqrt{2} | 71 | 5 |
math | 6.073. $\left\{\begin{array}{l}y^{2}-x y=-12, \\ x^{2}-x y=28\end{array}\right.$ | (-7,-3),(7,3) | 43 | 9 |
math | 10. The function $y=f(x)$ defined on $\mathbf{R}$ has the following properties:
（1）For any $x \in \mathbf{R}$, $f\left(x^{3}\right)=f^{3}(x)$;
（2）For any $x_{1} 、 x_{2} \in \mathbf{R}, x_{1} \neq x_{2}$, $f\left(x_{1}\right) \neq f\left(x_{2}\right)$. Then the value of $f(0)+f(1)+f(-1)$ is $\qquad$. | 0 | 137 | 1 |
math | Solve the following equation:
$$
\left(\frac{3}{4}\right)^{\lg x}+\left(\frac{4}{3}\right)^{\lg x}=\frac{25}{12}
$$
(Correction of problem 495.) | x_1=\frac{1}{10},\,x_2=10 | 59 | 19 |
math | 6. A triangle with vertices at $(1003,0),(1004,3)$, and $(1005,1)$ in the $x y$-plane is revolved all the way around the $y$-axis. Find the volume of the solid thus obtained. | 5020\pi | 63 | 6 |
math | ## Problem Statement
Calculate the lengths of the arcs of the curves given by equations in a rectangular coordinate system.
$$
y=\sqrt{1-x^{2}}+\arccos x, 0 \leq x \leq \frac{8}{9}
$$ | \frac{4\sqrt{2}}{3} | 56 | 12 |
math | If $a$, $6$, and $b$, in that order, form an arithmetic sequence, compute $a+b$. | 12 | 25 | 2 |
math | 10. If $\frac{\sin x}{\sin y}=3$ and $\frac{\cos x}{\cos y}=\frac{1}{2}$, find $\frac{\sin 2 x}{\sin 2 y}+\frac{\cos 2 x}{\cos 2 y}$.
(1 mark)若 $\frac{\sin x}{\sin y}=3$ 及 $\frac{\cos x}{\cos y}=\frac{1}{2}$, 求 $\frac{\sin 2 x}{\sin 2 y}+\frac{\cos 2 x}{\cos 2 y}$ 。
( 1 分) | \frac{49}{58} | 140 | 9 |
math | ## Task Condition
Find the derivative.
$$
y=\frac{5^{x}(\sin 3 x \cdot \ln 5-3 \cos 3 x)}{9+\ln ^{2} 5}
$$ | 5^{x}\cdot\sin3x | 49 | 9 |
math | 14. (51st Czech and Slovak Mathematical Olympiad (Open Question) Problem) Solve the equation $\left(x_{5}\right)^{2}+\left(y^{4}\right)_{5}=2 x y^{2}+51$, where $n_{5}$ denotes the closest multiple of 5 to the integer $n$, and $x, y$ are integers. | (52,6),(52,-6),(8,2),(8,-2),(3,3),(3,-3) | 82 | 27 |
math | 1. Use the numbers $2,3, \cdots, 2019$ to form 1009 fractions, with each number appearing only once in either the numerator or the denominator. From these 1009 fractions, select the largest one. Find the minimum value of this number. | \frac{1010}{2019} | 65 | 13 |
math | Example 10 (2003 Vietnam Mathematical Olympiad Problem) Let $F$ be the set of all functions $f$ satisfying the following conditions: $f: \mathbf{R} \rightarrow \mathbf{R}$ and for any positive real number $x$:
$$
f(3 x) \geqslant f[f(2 x)]+x .
$$
Find the maximum real number $\alpha$ such that for all $f \in F$, the f... | \frac{1}{2} | 148 | 7 |
math | Question 8: Let set $A=\left\{(x, y) \mid y^{2}-x-1=0\right\}$, set $B=\left\{(x, y) \mid 4 x^{2}+2 x-2 y+5=\right.$ $0\}$, and set $\mathrm{C}=\{(\mathrm{x}, \mathrm{y}) \mid \mathrm{y}=\mathrm{kx}+\mathrm{b}\}$. Try to find all non-negative integers $\mathrm{k}$ and $\mathrm{b}$, such that $(\mathrm{A} \cup \mathrm{B... | (k,b)=(1,2) | 148 | 7 |
math | Example 2. Compute the integral
$$
\int_{C}\left(z^{2}+z \bar{z}\right) d z
$$
where $C-$ is the arc of the circle $\{z \mid=1(0 \leqslant \arg z \leqslant \pi)$. | -\frac{8}{3} | 70 | 7 |
math | 3. Find the sum of the first 10 elements that are found both in the arithmetic progression $\{5,8,11,13, \ldots\}$ and in the geometric progression $\{20,40,80,160, \ldots\}$. (10 points) | 6990500 | 68 | 7 |
math | 19. $(\operatorname{tg} x)^{\sin ^{2} x-\frac{3}{2} \sin x+\frac{1}{2}}=1$.
19. $(\tan x)^{\sin ^{2} x-\frac{3}{2} \sin x+\frac{1}{2}}=1$. | \pik+(-1)^k\frac{} | 75 | 11 |
math | Find positive reals $a, b, c$ which maximizes the value of $a+ 2b+ 3c$ subject to the constraint that $9a^2 + 4b^2 + c^2 = 91$ | \frac{1}{3} + 2 \cdot \frac{3}{2} + 3 \cdot 9 = \frac{1}{3} + 3 + 27 = 30.333 | 53 | 49 |
math | A palindrome is a positive integer whose digits are the same when read forwards or backwards. For example, 25352 is a five-digit palindrome. What is the largest five-digit palindrome that is a multiple of 15 ? | 59895 | 48 | 5 |
math | Problem 9.2. In four classes of a school, there are more than 70 children, all of whom came to the grade meeting (no other children were present at the meeting).
Each girl who came was asked: "How many people from your class, including you, came to the meeting?"
Each boy who came was asked: "How many boys from your c... | 33 | 158 | 2 |
math | ## Task A-3.5.
Determine the smallest natural number $n$ such that in every set consisting of $n$ integers, there exist three distinct elements $a, b$, and $c$ such that $a b + b c + c a$ is divisible by 3. | 6 | 61 | 1 |
math | 510. Find the approximate value of the function $y=\sqrt{3 x^{2}+1}$ at $x=1.02$. | 2.03 | 33 | 4 |
math | 8. Let $\{a, b, c, d\}$ be a subset of $\{1,2, \cdots, 17\}$. If 17 divides $(a-b+c-d)$, then $\{a, b, c, d\}$ is called a "good subset". Then, the number of good subsets is $\qquad$ | 476 | 77 | 3 |
math | [
Find the height of a right triangle dropped to the hypotenuse, given that the base of this height divides the hypotenuse into segments of 1 and 4.
# | 2 | 38 | 1 |
math | ## Task 1 - 190511
(A historical problem, 2000 years B.C.)
In a cage, rabbits and pheasants are locked up. These animals together have 40 heads and 104 feet.
State the number of all rabbits and the number of all pheasants that are in the cage! | k=12,f=28 | 76 | 8 |
math | ## Task B-2.1.
For which $n \in \mathbb{N}$ does the following hold
$$
1 i^{1}+2 i^{2}+3 i^{3}+4 i^{4}+\cdots+n i^{n}=48+49 i, \text { if } i=\sqrt{-1} ?
$$ | 97 | 78 | 2 |
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