task_type stringclasses 1
value | problem stringlengths 23 3.94k | answer stringlengths 1 231 | problem_tokens int64 10 1.39k | answer_tokens int64 1 98 |
|---|---|---|---|---|
math | 5. Find all values of the parameter $a$ for which the equation $\left(2 \sin x+a^{2}+a\right)^{3}-(\cos 2 x+3 a \sin x+11)^{3}=12-2 \sin ^{2} x+(3 a-2) \sin x-a^{2}-a$ has two distinct solutions on the interval $\left[-\frac{\pi}{6} ; \frac{3 \pi}{2}\right]$. Specify these solutions for each found $a$.
Solution: $\lef... | \in[2.5,4),x_{1}=\arcsin(-3),x_{2}=\pi-\arcsin(-3);\in[-5,-2),x_{1}=\arcsin(0.5+2),x_{2}=\pi-\arcsin(0.5+2) | 918 | 71 |
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 | + 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 | Example 12 Let $x, y, z$ be real numbers, not all zero, find the maximum value of $\frac{x y+2 y z}{x^{2}+y^{2}+z^{2}}$. | \frac{\sqrt{5}}{2} | 49 | 10 |
math | Consider non-negative real numbers $a, b, c$ satisfying the condition $a^2 + b^2 + c^2 = 2$ . Find the maximum value of the following expression $$P=\frac{\sqrt{b^2+c^2}}{3-a}+\frac{\sqrt{c^2+a^2}}{3-b}+a+b-2022c$$ | 3 | 84 | 1 |
math | A function $f: R \to R$ satisfies $f (x + 1) = f (x) + 1$ for all $x$. Given $a \in R$, define the sequence $(x_n)$ recursively by $x_0 = a$ and $x_{n+1} = f (x_n)$ for $n \ge 0$. Suppose that, for some positive integer m, the difference $x_m - x_0 = k$ is an integer. Prove that the limit $\lim_{n\to \infty}\frac{x_n... | \frac{k}{m} | 133 | 7 |
math | 10.1. Kolya wrote a ten-digit number on the board, consisting of different digits. Sasha added one digit so that the resulting number would be divisible by 9. Which digit could Sasha have added? | 0or9 | 45 | 3 |
math | 4. In the notebook, $n$ integers are written, ordered in descending order $a_{1}>a_{2}>\ldots>a_{n}$ and having a sum of 840. It is known that the $k$-th number written in order, $a_{k}$, except for the last one when $k=n$, is $(k+1)$ times smaller than the sum of all the other written numbers. Find the maximum number ... | n_{\max}=7;a_{1}=280,a_{2}=210,a_{3}=168,a_{4}=140,a_{5}=120,a_{6}=105,a_{7}=-183 | 114 | 56 |
math | 8. Spring has arrived, and the school has organized a spring outing for the students. However, due to certain reasons, the spring outing is divided into indoor and outdoor activities. The number of people participating in outdoor activities is 480 more than those participating in indoor activities. Now, if 50 people fr... | 870 | 116 | 3 |
math | 9.1. The square root of the number 49 can be extracted using this "formula": $\sqrt{49}=4+\sqrt{9}$. Are there other two-digit numbers whose square roots can be extracted in a similar manner and are integers? List all such two-digit numbers. | 6481 | 61 | 4 |
math | 3. Solve the system $\left\{\begin{array}{l}3 x \geq 2 y+16, \\ x^{4}+2 x^{2} y^{2}+y^{4}+25-26 x^{2}-26 y^{2}=72 x y .\end{array}\right.$ | (6;1) | 75 | 5 |
math | 11. Observe the array: $(1),(3,5),(7,9,11),(13,15,17$,
19), $\cdots \cdots$. Then 2003 is in the group. | 45 | 53 | 2 |
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 | Find all positive integers $n$ for which both numbers \[1\;\;\!\!\!\!\underbrace{77\ldots 7}_{\text{$n$ sevens}}\!\!\!\!\quad\text{and}\quad 3\;\; \!\!\!\!\underbrace{77\ldots 7}_{\text{$n$ sevens}}\] are prime. | n = 1 | 86 | 5 |
math | # Problem 7. (4 points)
$O A B C$ is a rectangle on the Cartesian plane, with sides parallel to the coordinate axes. Point $O$ is the origin, and point $B$ has coordinates $(11; 8)$. Inside the rectangle, a point $X$ with integer coordinates is taken. What is the smallest value that the area of triangle $O B X$ can ta... | \frac{1}{2} | 96 | 7 |
math | Find the period of the repetend of the fraction $\frac{39}{1428}$ by using [i]binary[/i] numbers, i.e. its binary decimal representation.
(Note: When a proper fraction is expressed as a decimal number (of any base), either the decimal number terminates after finite steps, or it is of the form $0.b_1b_2\cdots b_sa_1a_2... | 24 | 181 | 2 |
math | Example 3 (APMO) Find all nonempty finite sets $S$ of positive integers such that if $m, n \in$ $S$, then $\frac{m+n}{(m, n)} \in \mathbf{S}, (m, n$ do not have to be distinct). | {2} | 64 | 3 |
math | 1. Let $a=\operatorname{tg} x, b=\operatorname{tg} \frac{y}{2}$. Then $a+b=\frac{4}{\sqrt{3}}$ and $\frac{1}{a}+\frac{1}{b}=\frac{4}{\sqrt{3}}$. From the second equation, it follows that $a b=1$. By Vieta's theorem, $a$ and $b$ satisfy the equation $t^{2}-\frac{4}{\sqrt{3}} t+1=0$, whose roots are $\sqrt{3}$ and $\frac... | \frac{\pi}{3}+\pin,\frac{\pi}{3}+2\pikor\frac{\pi}{6}+\pin,\frac{2\pi}{3}+2\pik | 262 | 45 |
math | If we divide number $19250$ with one number, we get remainder $11$. If we divide number $20302$ with the same number, we get the reamainder $3$. Which number is that? | 53 | 53 | 2 |
math | G3.3 If $f(n)=a^{n}+b^{n}$, where $n$ is a positive integer and $f(3)=[f(1)]^{3}+f(1)$, find the value of $a \cdot b$. | -\frac{1}{3} | 57 | 7 |
math | 7. In a company, several employees have a total monthly salary of 10000 dollars. A kind manager proposes to triple the salary for those earning up to 500 dollars, and increase the salary by 1000 dollars for the rest, so the total salary will become 24000 dollars. A mean manager proposes to reduce the salary to 500 doll... | 7000 | 112 | 4 |
math | 3. Find the natural numbers of the form $\overline{ab}$ which, when divided by 36, give a remainder that is a perfect square.
(7 points)
## E:14178 from GM 11/2011 | 16,25,36,37,40,45,52,61,72,73,76,81,88,97 | 55 | 41 |
math | 1. Find all functions $f: \mathbf{R} \rightarrow \mathbf{R}$, such that for all real numbers $x, y$, we have
$$
\begin{array}{l}
f(2 x y)+f(f(x+y)) \\
=x f(y)+y f(x)+f(x+y) .
\end{array}
$$ | f(x)=0\text{or}f(x)=x\text{or}f(x)=2-x | 77 | 23 |
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 4 Let the function $y=f(x)$ have the domain $\mathbf{R}$, and when $x>0$, we have $f(x)>1$, and for any $x, y \in \mathbf{R}$, we have $f(x+y)=f(x) f(y)$. Solve the inequality
$$
f(x) \leqslant \frac{1}{f(x+1)} .
$$
Analysis: This problem involves an abstract function, and we can find a prototype function $f(x... | x \leqslant -\frac{1}{2} | 164 | 14 |
math | 1. Let $x$ be a positive integer, and $x<50$. Then the number of $x$ such that $x^{3}+11$ is divisible by 12 is $\qquad$. | 5 | 47 | 1 |
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 | [ Degree of a vertex ] $[\underline{\text { Pigeonhole Principle (restated). }}]$
For what $n>1$ can it happen that in a company of $n+1$ girls and $n$ boys, all girls are acquainted with a different number of boys, while all boys are acquainted with the same number of girls? | 2m+1foranyoddn>1 | 74 | 10 |
math | Bertha has $ 6$ daughters and no sons. Some of her daughters have $ 6$ daughters and the rest have none. Bertha has a total of $ 30$ daughters and granddaughters, and no great-grand daughters. How many of Bertha's daughters and granddaughters have no daughters?
$ \textbf{(A)}\ 22\qquad
\textbf{(B)}\ 23\qquad
\text... | 26 | 135 | 2 |
math | For a non-empty integer set $A$, if it satisfies $a \in A, a-1 \notin A, a+1 \notin A$, then $a$ is called an isolated element of set $A$. Question: For the set
$$
M=\{1,2, \cdots, n\}(n \geqslant 3)
$$
how many $k(k \geqslant 3)$-element subsets of $M$ have no isolated elements? | P_{k}=\sum_{l=1}^{r} \mathrm{C}_{k-l-1}^{l-1} \mathrm{C}_{n-k+1}^{l} | 104 | 42 |
math | There exist two distinct positive integers, both of which are divisors of $10^{10}$, with sum equal to $157$. What are they? | 32, 125 | 36 | 7 |
math | $5 \cdot 4$ natural numbers $a_{1}, a_{2}, \cdots, a_{49}$ have a sum of 999, let $d$ be the greatest common divisor of $a_{1}, a_{2}, \cdots, a_{49}$, what is the maximum value of $d$?
(Kiev Mathematical Olympiad, 1979) | 9 | 87 | 1 |
math | 26. Multiply the month number of a student's birthday by 31 and the day number by 12, then add the two products together, the sum is 376. What is the student's birthday? | April\21 | 47 | 4 |
math | Let $ P(x)$ be a nonzero polynomial such that, for all real numbers $ x$, $ P(x^2 \minus{} 1) \equal{} P(x)P(\minus{}x)$. Determine the maximum possible number of real roots of $ P(x)$. | 4 | 57 | 1 |
math | G7.1 Find $3+6+9+\ldots+45$. | 360 | 18 | 3 |
math | Solve the following system of equations:
$$
\begin{gathered}
x^{2}-y z=-23 \\
y^{2}-z x=-4 \\
z^{2}-x y=34
\end{gathered}
$$ | \5,\6,\8 | 52 | 6 |
math | ## [ equations in integers ] Decompositions and partitions $\quad]$ [ GCD and LCM. Mutual simplicity ]
Ostap Bender organized a giveaway of elephants to the population in the city of Fux. 28 union members and 37 non-members showed up for the giveaway, and Ostap distributed the elephants equally among all union members... | 2072 | 127 | 4 |
math | Problem 5. The numbers $a, b$, and $c$ satisfy the equation $\sqrt{a}=\sqrt{b}+\sqrt{c}$. Find $a$, if $b=52-30 \sqrt{3}$ and $c=a-2$. | 27 | 59 | 2 |
math | ### 3.493 Find the sum $1+\cos 4 \alpha+\cos 8 \alpha+\ldots+\cos 4 n \alpha$. | \frac{\sin2\alpha(n+1)\cdot\cos2n\alpha}{\sin} | 35 | 22 |
math | 8. (1 mark) Given that $0.3010<\log 2<0.3011$ and $0.4771<\log 3<0.4772$. Find the leftmost digit of $12^{37}$.
(1 分) 設 $0.3010<\log 2<0.3011$ 及 $0.4771<\log 3<0.4772$, 求 $12^{37}$ 最左的一位數字。 | 8 | 129 | 1 |
math | 7. Given $a, b \in \mathbf{R}^{+}$ and $\frac{\sin ^{4} x}{a}+\frac{\cos ^{4} x}{b}=\frac{1}{a+b}$, then $\frac{\sin ^{8} x}{a^{3}}+\frac{\cos ^{8} x}{b^{3}}=$ $\qquad$ | \frac{1}{(+b)^{3}} | 86 | 11 |
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 | Problem 10.3. Find all pairs of positive integers $(m, n), m>n$, such that
$$
\left[m^{2}+m n, m n-n^{2}\right]+[m-n, m n]=2^{2005}
$$
where $[a, b]$ denotes the least common multiple of $a$ and $b$.
Ivan Landjev | =2^{1002},n=2^{1001} | 86 | 17 |
math | $\because 、\left(20\right.$ points) For a quadratic equation with real coefficients $a x^{2}+2 b x$ $+ c=0$ having two real roots $x_{1}, x_{2}$. Let $d=\left|x_{1}-x_{2}\right|$. Find the range of $d$ when $a>b>c$ and $a+b+c=0$. | \sqrt{3}<d<2 \sqrt{3} | 89 | 13 |
math |
Problem 3. Determine all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that
$$
f\left(x^{3}+y^{3}+x y\right)=x^{2} f(x)+y^{2} f(y)+f(x y)
$$
for all $x, y \in \mathbb{R}$.
| f(x)=xf(1) | 83 | 7 |
math | ## Task Condition
Find the point $M^{\prime}$ symmetric to the point $M$ with respect to the line.
$M(2 ; 1 ; 0)$
$\frac{x-2}{0}=\frac{y+1.5}{-1}=\frac{z+0.5}{1}$ | M^{\}(2;-2;-3) | 69 | 10 |
math | 3. In a convex quadrilateral $A B C D$, the midpoints of consecutive sides are marked: $M, N, K, L$. Find the area of quadrilateral $M N K L$, if $|A C|=|B D|=2 a,|M K|+|N L|=2 b$.
| b^{2}-^{2} | 69 | 7 |
math | The positive integers $a$ and $b$ are such that the numbers $15a+16b$ and $16a-15b$ are both squares of positive integers. What is the least possible value that can be taken on by the smaller of these two squares? | 231361 | 60 | 6 |
math | ## problem statement
Calculate the volume of the tetrahedron with vertices at points $A_{1}, A_{2}, A_{3}, A_{4}$ and its height dropped from vertex $A_{4}$ to the face $A_{1} A_{2} A_{3}$.
$A_{1}(0 ;-1 ;-1)$
$A_{2}(-2 ; 3 ; 5)$
$A_{3}(1 ;-5 ;-9)$
$A_{4}(-1 ;-6 ; 3)$ | \frac{37}{3\sqrt{5}} | 114 | 12 |
math | 15. Let $x>1, y>1, S=\min \left\{\log _{x} 2, \log _{2} y\right.$ , $\left.\log _{y}\left(8 x^{2}\right)\right\}$. Then the maximum value of $S$ is $\qquad$ . | 2 | 74 | 1 |
math | 9.6. Thirty girls - 13 in red dresses and 17 in blue dresses - were dancing in a circle around a Christmas tree. Later, each of them was asked if their right neighbor was in a blue dress. It turned out that those who answered correctly were only the girls standing between girls in dresses of the same color. How many gi... | 17 | 86 | 2 |
math | 13.008. Seawater contains $5 \%$ salt by mass. How much fresh water needs to be added to 30 kg of seawater to make the salt concentration $1.5 \%$? | 70 | 47 | 2 |
math | 4. If $12 x=4 y+2$, determine the value of the expression $6 y-18 x+7$. | 4 | 29 | 1 |
math | Five. (15 points) Given a function $f: \mathbf{R} \rightarrow \mathbf{R}$, such that for any real numbers $x, y, z$ we have
$$
\begin{array}{l}
\frac{1}{2} f(x y)+\frac{1}{2} f(x z)-f(x) f(y z) \geqslant \frac{1}{4} . \\
\text { Find }[1 \times f(1)]+[2 f(2)]+\cdots+[2011 f(2011)]
\end{array}
$$
where $[a]$ denotes th... | 1011030 | 154 | 7 |
math | 4. In the arithmetic sequence $\left\{a_{n}\right\}$, it is known that $\left|a_{5}\right|=\left|a_{11}\right|, d>0$, the positive integer $n$ that makes the sum of the first $n$ terms $S_{n}$ take the minimum value is $\qquad$ . | 7or8 | 78 | 3 |
math | Example 11. Find a particular solution of the equation
$$
y^{\prime \prime}-y=4 e^{x},
$$
satisfying the initial conditions
$$
y(0)=0, \quad y^{\prime}(0)=1
$$ | 2xe^{x}-\operatorname{sh}x | 58 | 12 |
math | The workers in a factory produce widgets and whoosits. For each product, production time is constant and identical for all workers, but not necessarily equal for the two products. In one hour, $100$ workers can produce $300$ widgets and $200$ whoosits. In two hours, $60$ workers can produce $240$ widgets and $300$ whoo... | 450 | 119 | 3 |
math | 4. On a $100 \times 100$ chessboard, 1975 rooks were placed (each rook occupies one cell, different rooks stand on different cells). What is the maximum number of pairs of rooks that could be attacking each other? Recall that a rook can attack any number of cells along a row or column, but does not attack a rook that i... | 3861 | 99 | 4 |
math | [ Arithmetic progression ]
[ Arithmetic. Mental calculation, etc. ]
What is the sum of the digits of all numbers from one to a billion? | 40500000001 | 29 | 11 |
math | ## Task B-3.2.
If $\log _{2}\left[\log _{3}\left(\log _{4} x\right)\right]=\log _{3}\left[\log _{2}\left(\log _{4} y\right)\right]=\log _{4}\left[\log _{3}\left(\log _{2} z\right)\right]=0$, calculate $\log _{y}\left(\log _{z} x\right)$. | \frac{1}{4} | 108 | 7 |
math | The Nováks baked wedding cakes. They took a quarter of them to their relatives in Moravia, gave a sixth to their colleagues at work, and gave a ninth to their neighbors. If they had three more cakes left, it would be half of the original number. How many cakes did they bake? | 108 | 62 | 3 |
math | Three lines are drawn parallel to each of the three sides of $\triangle ABC$ so that the three lines intersect in the interior of $ABC$. The resulting three smaller triangles have areas $1$, $4$, and $9$. Find the area of $\triangle ABC$.
[asy]
defaultpen(linewidth(0.7)); size(120);
pair relpt(pair P, pair Q, real a, ... | 36 | 279 | 2 |
math | Determine the smallest integer $n$ whose unit digit is 5, such that $\sqrt{n}$ is an integer whose sum of digits is 9. | 2025 | 32 | 4 |
math | 3 Suppose 2005 line segments are connected end-to-end, forming a closed polyline, and no two segments of the polyline lie on the same straight line. Then, what is the maximum number of intersection points where the polyline intersects itself?
| 2007005 | 50 | 7 |
math | Example 3. Find the variance of a random variable $X$ that has a geometric distribution. | \frac{1-p}{p^{2}} | 20 | 10 |
math | ## Problem Statement
Find the coordinates of point $A$, which is equidistant from points $B$ and $C$.
$A(x ; 0 ; 0)$
$B(1 ; 5 ; 9)$
$C(3 ; 7 ; 11)$ | A(18;0;0) | 62 | 9 |
math | Example 1 Calculate
$$
S\left(9 \times 99 \times 9999 \times \cdots \times \underset{2^{\circ} \uparrow}{99 \cdots 9}\right) \text {. }
$$
(1992, USA Mathematical Olympiad)
[Analysis] If you are familiar with Lemma 1 and have a bit of a sense of magnitude, you can get the answer right away. | 9 \times 2^{n} | 100 | 8 |
math | Example 3.7. Is the function $z=$ $=f(x, y)=\sqrt{4-x^{2}-y^{2}}$ bounded above (below)? | 0\leqslantf(x,y)\leqslant2 | 37 | 15 |
math | 6. In the Lemon Kingdom, there are 2020 villages. Some pairs of villages are directly connected by paved roads. The road network is arranged in such a way that there is exactly one way to travel from any village to any other without passing through the same road twice. Agent Orange wants to fly over as many villages as... | 2019 | 122 | 4 |
math | 18. There are 2012 students standing in a row, numbered from left to right as $1, 2, \cdots \cdots 2012$. In the first round, they report numbers from left to right as “1, 2”, and those who report 2 stay; from the second round onwards, each time the remaining students report numbers from left to right as “1, 2, 3”, and... | 1458 | 118 | 4 |
math | Let $z$ be a complex number such that $|z| = 1$ and $|z-1.45|=1.05$. Compute the real part of $z$. | \frac{20}{29} | 41 | 9 |
math | In the acute-angled triangle $ABC$ the angle$ \angle B = 30^o$, point $H$ is the intersection point of its altitudes. Denote by $O_1, O_2$ the centers of circles inscribed in triangles $ABH ,CBH$ respectively. Find the degree of the angle between the lines $AO_2$ and $CO_1$. | 45^\circ | 86 | 4 |
math | 10. Let $a, b \in [0,1]$, find the maximum and minimum values of $S=\frac{a}{1+b}+\frac{b}{1+a}+(1-a)(1-b)$. | 1 | 49 | 1 |
math | 2. If $\sin x+\cos x=\frac{\sqrt{2}}{2}$, then $\sin ^{3} x+\cos ^{3} x=$ $\qquad$ | \frac{5\sqrt{2}}{8} | 40 | 12 |
math | Example 9. Simplify $1-\frac{1}{4} \sin ^{2} 2 \alpha-\sin ^{2} \beta-\cos ^{4} \alpha$ into a product form of trigonometric functions. (84 College Entrance Examination for Liberal Arts)
(84年高考文科试题)
Note: The last line is kept in Chinese as it is a reference to the source of the problem and does not need to be trans... | \sin (\alpha+\beta) \sin (\alpha-\beta) | 120 | 14 |
math | If $\frac{x}{2}-5=9$, what is the value of $\sqrt{7 x}$ ? | 14 | 23 | 2 |
math | 31st IMO 1990 shortlist Problem 26 Find all positive integers n such that every positive integer with n digits, one of which is 7 and the others 1, is prime. Solution | 1,2 | 45 | 3 |
math | Example 1. Solve the inequality
$$
25^{x}>125^{3 x-2}
$$ | x<\frac{6}{7} | 25 | 9 |
math | 6. Let $x_{k} 、 y_{k} \geqslant 0(k=1,2,3)$. Calculate:
$$
\begin{array}{l}
\sqrt{\left(2018-y_{1}-y_{2}-y_{3}\right)^{2}+x_{3}^{2}}+\sqrt{y_{3}^{2}+x_{2}^{2}}+ \\
\sqrt{y_{2}^{2}+x_{1}^{2}}+\sqrt{y_{1}^{2}+\left(x_{1}+x_{2}+x_{3}\right)^{2}}
\end{array}
$$
the minimum value is | 2018 | 155 | 4 |
math | Example 3 A person walks from place A to place B, and there are regular buses running between A and B, with equal intervals for departures from both places. He notices that a bus going to A passes by every 6 minutes, and a bus going to B passes by every 12 minutes. How often do the buses depart from their respective st... | 8 | 87 | 1 |
math | 236. A cube with edge $a$ is standing on a plane. The light source is located at a distance $b(b>a)$ from the plane. Find the minimum value of the area of the shadow cast by the cube on the plane. | (\frac{}{b-})^{2} | 52 | 10 |
math | 4- $124 m, n$ are two distinct positive integers, find the common complex roots of the equations
$$x^{m+1}-x^{n}+1=0$$
and
$$x^{n+1}-x^{m}+1=0$$ | x=\frac{1}{2} \pm \frac{\sqrt{3}}{2} i | 60 | 21 |
math | 1st Irish 1988 Problem 8 The sequence of nonzero reals x 1 , x 2 , x 3 , ... satisfies x n = x n-2 x n-1 /(2x n-2 - x n-1 ) for all n > 2. For which (x 1 , x 2 ) does the sequence contain infinitely many integral terms? | x_1=x_2= | 81 | 7 |
math | 1. Does there exist a positive integer divisible by 2020, in whose representation the digits $0,1, \cdots, 9$ appear the same number of times? | 12123434565679798080 | 40 | 20 |
math | $(MON 1)$ Find the number of five-digit numbers with the following properties: there are two pairs of digits such that digits from each pair are equal and are next to each other, digits from different pairs are different, and the remaining digit (which does not belong to any of the pairs) is different from the other di... | 1944 | 67 | 4 |
math | Let $ (x_1,x_2,\cdots)$ be a sequence of positive numbers such that $ (8x_2 \minus{} 7x_1)x_1^7 \equal{} 8$ and
\[ x_{k \plus{} 1}x_{k \minus{} 1} \minus{} x_k^2 \equal{} \frac {x_{k \minus{} 1}^8 \minus{} x_k^8}{x_k^7x_{k \minus{} 1}^7} \text{ for }k \equal{} 2,3,\ldots
\]
Determine real number $ a$ such that if ... | a = 8^{1/8} | 180 | 10 |
math | 1. Given points $A(1,1)、B(3,2)、C(2,3)$ and line $l: y=k x(k \in \mathbf{R})$. Then the maximum value of the sum of the squares of the distances from points $A、B、C$ to line $l$ is $\qquad$ | 27 | 75 | 2 |
math | 1. Given complex numbers $z_{1}, z_{2}$ satisfy $\left|z_{1}\right|=1,\left|z_{2}\right|=2,3 z_{1}-$ $z_{2}=2+\sqrt{3} \mathrm{i}$. Then $2 z_{1}+z_{2}=$ $\qquad$ . | 3-\sqrt{3} \mathrm{i} \text{ or } -\frac{9}{7}+\frac{13 \sqrt{3}}{7} \mathrm{i} | 76 | 40 |
math | 8.373. $\operatorname{tg} x+\operatorname{tg} \alpha+1=\operatorname{tg} x \operatorname{tg} \alpha$.
8.373. $\tan x+\tan \alpha+1=\tan x \tan \alpha$. | -\alpha+\frac{\pi}{4}(4k-1)for\alpha\neq\frac{\pi}{4},k\inZ | 63 | 31 |
math | 1.- Find two positive integers $a$ and $b$ given their sum and their least common multiple. Apply this in the case where the sum is 3972 and the least common multiple is 985928.
## SOLUTION: | =1964,b=2008 | 53 | 11 |
math | 16(!). Determine \(p\) so that the sum of the absolute values of the roots of the equation \(z^{2}+p z-6=0\) is equal to 5. | 1or-1 | 41 | 4 |
math | $605$ spheres of same radius are divided in two parts. From one part, upright "pyramid" is made with square base. From the other part, upright "pyramid" is made with equilateral triangle base. Both "pyramids" are put together from equal numbers of sphere rows. Find number of spheres in every "pyramid" | n = 10 | 73 | 6 |
math | $A B$ draw a right-angled triangle $A B C$ over the leg $A B$ such that the sum of the hypotenuse $B C$ and the leg $C A$ is equal to twice the length of $A B$.
Let's denote the length of $A B$ as $x$. Therefore, we have:
\[ BC + CA = 2x \]
Given that $A B C$ is a right-angled triangle with the right angle at $A$... | AC=\frac{3}{4},\quadCB=\frac{5}{4} | 508 | 18 |
math | In trapezoid $ABCD$, the sides $AD$ and $BC$ are parallel, and $AB = BC = BD$. The height $BK$ intersects the diagonal $AC$ at $M$. Find $\angle CDM$. | \angle CDM = 90^\circ | 50 | 11 |
math | An eccentric mathematician has a ladder with $ n$ rungs that he always ascends and descends in the following way: When he ascends, each step he takes covers $ a$ rungs of the ladder, and when he descends, each step he takes covers $ b$ rungs of the ladder, where $ a$ and $ b$ are fixed positive integers. By a sequence ... | a + b - \gcd(a, b) | 132 | 11 |
math | 8. Let $A B C$ be an equilateral triangle with side length 8 . Let $X$ be on side $A B$ so that $A X=5$ and $Y$ be on side $A C$ so that $A Y=3$. Let $Z$ be on side $B C$ so that $A Z, B Y, C X$ are concurrent. Let $Z X, Z Y$ intersect the circumcircle of $A X Y$ again at $P, Q$ respectively. Let $X Q$ and $Y P$ inters... | 304 | 134 | 3 |
math | Find all the integer solutions of the equation
$$
9 x^{2} y^{2}+9 x y^{2}+6 x^{2} y+18 x y+x^{2}+2 y^{2}+5 x+7 y+6=0
$$ | (-2,0),(-3,0),(0,-2),(-1,2) | 61 | 19 |
math | 4. (2005 Croatian Mathematical Olympiad) Find all positive integers $n$ such that $2^{4}+2^{7}+2^{n}$ is a perfect square. | 8 | 41 | 1 |
math | For every $A \subset S$, let
$$
S_{\mathrm{A}}=\left\{\begin{array}{ll}
(-)^{\mid \mathrm{A}} \mid \sum_{\mathbf{a} \in \mathrm{A}} a, & A \neq \varnothing, \\
0, & A=\varnothing .
\end{array}\right.
$$
Find $\sum_{\mathrm{A} \subset \mathrm{S}} S_{\mathrm{A}}$. | 0 | 109 | 1 |
math | $$
\begin{array}{l}
\text { Three. (20 points) (1) The quadratic function } \\
f(x)=a x^{2}+b x+c(a, b, c \in \mathbf{R}, a \neq 0)
\end{array}
$$
satisfies
(i) For $x \in \mathbf{R}$, there is $4 x \leqslant f(x) \leqslant \frac{1}{2}(x+2)^{2}$
always holds;
(ii) $f(-4+2 \sqrt{3})=0$.
Find $f(x)$.
(2) Let $f_{1}(x)=... | f_{2009}(0)=\frac{3^{2010}+3}{3^{2010}-1} | 196 | 31 |
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