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 | A magician with an assistant is going to perform the following trick. A spectator writes a sequence of $N$ digits on a board. The magician's assistant covers two adjacent digits with a black circle. Then the magician enters. His task is to guess both covered digits (and the order in which they are located). For what sm... | 101 | 85 | 3 |
math | ## Task A-2.2.
Determine all pairs $(a, n)$ of natural numbers for which
$$
3 a^{2}+2^{n}=a^{4}
$$ | (,n)=(2,2) | 40 | 8 |
math | 1. A sequence of ants walk from $(0,0)$ to $(1,0)$ in the plane. The $n$th ant walks along $n$ semicircles of radius $\frac{1}{n}$ with diameters lying along the line from $(0,0)$ to $(1,0)$. Let $L_{n}$ be the length of the path walked by the $n$th ant. Compute $\lim L_{n}$. | \pi | 95 | 2 |
math | Biinkov Add:
A thick issue of the newspaper costs 30 rubles, and a thin one is cheaper. A discount of the same percentage is applied to all newspapers for pensioners, so they buy a thin issue of the same newspaper for 15 rubles. It is known that in any case, the newspaper costs a whole number of rubles. How much does ... | 25 | 97 | 2 |
math | 56.
Suppose the brother Alice met said: "I am lying today or I am Trulala." Could it be determined in this case which of the brothers it was? | Trulala | 38 | 3 |
math | $p > 3$ is a prime. Find all integers $a$, $b$, such that $a^2 + 3ab + 2p(a+b) + p^2 = 0$. | (a, b) = (-p, 0) | 44 | 13 |
math | 1. The second term of an infinite decreasing geometric progression is 3. Find the smallest possible value of the sum $A$ of this progression, given that $A>0$. | 12 | 37 | 2 |
math | 3.265. $\sin (2 x-\pi) \cos (x-3 \pi)+\sin \left(2 x-\frac{9 \pi}{2}\right) \cos \left(x+\frac{\pi}{2}\right)$. | \sin3x | 56 | 4 |
math | Task 3.
## Maximum 10 points
In the Country of Wonders, a pre-election campaign is being held for the position of the best tea lover, in which the Mad Hatter, March Hare, and Dormouse are participating. According to a survey, $20 \%$ of the residents plan to vote for the Mad Hatter, $25 \%$ for the March Hare, and $3... | 70 | 171 | 2 |
math | Consider a regular triangular-based right pyramid. The length of the edges meeting at the apex is $a$, and the angle between any two of these edges is $\alpha$. Calculate the volume of the pyramid. If $a$ remains constant, what should $\alpha$ be for the volume to be maximized? ${ }^{1} \sqrt{1}$[^0]
[^0]: ${ }^{1... | \alpha=90 | 232 | 5 |
math | 1. Given the real number pair $(x, y)$ satisfies the equation $(x-2)^{2}+y^{2}=3$, let the minimum and maximum values of $\frac{y}{x}$ be $m$ and $n$ respectively. Then $m+n=$ | 0 | 59 | 1 |
math | 8. Let $x_{1}, x_{2}$ be the roots of the equation $x^{2}-x-3=0$. Find $\left(x_{1}^{5}-20\right) \cdot\left(3 x_{2}^{4}-2 x_{2}-35\right)$. | -1063 | 68 | 5 |
math | 2. (10 points) Calculate: $1+2+4+5+7+8+10+11+13+14+16+17+19+20=$ | 147 | 46 | 3 |
math | 1. $[\mathbf{3}]$ If $a$ and $b$ are positive integers such that $a^{2}-b^{4}=2009$, find $a+b$. | 47 | 42 | 2 |
math | 11. Given real numbers $x_{1}, x_{2}, x_{3}$ satisfy $x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+x_{1} x_{2}+x_{2} x_{3}=2$. Then the maximum value of $\left|x_{2}\right|$ is $\qquad$. | 2 | 82 | 1 |
math | Problem 4. The sum of all sides of two squares is $96 \mathrm{~cm}$. If the side of one square is three times larger than the side of the other square, then calculate the lengths of the sides of the two squares. | 6\mathrm{~},18\mathrm{~} | 53 | 13 |
math | 26th Putnam 1965 Problem A1 How many positive integers divide at least one of 10 40 and 20 30 ? Solution | 2301 | 37 | 4 |
math | 7.54 It is known that $\beta=10^{\frac{1}{1-\lg \alpha}}$ and $\gamma=10^{\frac{1}{1-\operatorname{lg} \beta}}$. Find the dependence of $\alpha$ on $\gamma$. | \alpha=10^{\frac{1}{1-\lg\gamma}} | 60 | 17 |
math | Fomin C.B.
Kolya and Vasya received 20 grades each in January, and Kolya received as many fives as Vasya received fours, as many fours as Vasya received threes, as many threes as Vasya received twos, and as many twos as Vasya received fives. Moreover, their average grade for January is the same. How many twos did Koly... | 5 | 96 | 1 |
math | 799. Solve the equation in integers
$$
x^{2}-y^{2}=1997
$$ | (999;998),(999;-998),(-999;-998),(-999;998) | 26 | 35 |
math | Find the number of trailing zeros at the end of the base-$10$ representation of the integer $525^{25^2}
\cdot 252^{52^5}$ . | 1250 | 43 | 4 |
math | ## Task 1 - 291221
Determine all pairs $(x ; y)$ of real numbers $x, y$ that satisfy the following system of equations (1), (2), (3):
$$
\begin{aligned}
x + xy + xy^2 & = -21 \\
y + xy + x^2 y & = 14 \\
x + y & = -1
\end{aligned}
$$ | -3,2 | 94 | 4 |
math | 4. In triangle $A B C$, medians $B D$ and $C E$ intersect at point $K$. The circle constructed on segment $C K$ as a diameter passes through vertex $B$ and is tangent to line $D E$. It is known that $D K=3$. Find the height $A H$ of triangle $A B C$, the radius of the circle, and the area of triangle $A B C$. | AH=18,R=6,S_{ABC}=54\sqrt{3} | 93 | 18 |
math | [ Systems of linear equations $]$ [ Methods for solving problems with parameters ]
For what values of $m$ do the equations $m x-1000=1001$ and $1001 x=m-1000 x$ have a common root? | \2001 | 59 | 5 |
math | Example 4 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 value of the length of the chord intercepted by the line $y=2 x$ on this family of curves.
(1995, National High School Mathematics Competition) | 8 \sqrt{5} | 91 | 6 |
math | 6. Let $a_{1}, a_{2}, \cdots, a_{2014}$ be a permutation of the positive integers $1,2, \cdots$, 2014. Define
$$
S_{k}=a_{1}+a_{2}+\cdots+a_{k}(k=1,2, \cdots, 2014) .
$$
Then the maximum number of odd numbers in $S_{1}, S_{2}, \cdots, S_{2014}$ is . $\qquad$ | 1511 | 122 | 4 |
math | Let $n \geqslant 1$ and $P \in \mathbb{R}_{n}[X]$ such that $\forall k \in \llbracket 0, n \rrbracket, P(k)=\frac{k}{k+1}$ Calculate $P(n+1)$
Theorem 1 (Viète's Formulas). Let $P=X^{2}-s X+p$ and $\alpha, \beta$ be its roots. Then, $s=\alpha+\beta$ and $p=\alpha \beta$.
Let $P=X^{3}-s X^{2}+t X-p$ and $\alpha, \beta,... | P(n+1)=\frac{n+1+(-1)^n}{n+2} | 227 | 20 |
math | 5. How many solutions in natural numbers does the equation $(a+1)(b+1)(c+1)=2 a b c$ have? | 27 | 31 | 2 |
math | [ $\left.\begin{array}{c}\text { Equations in integers } \\ \text { [Examples and counterexamples. Constructions }\end{array}\right]$
Do there exist natural numbers $a, b, c, d$ such that $a / b + c / d = 1, \quad a / d + c / b = 2008$? | =2009\cdot(2008\cdot2009-1),b=2008\cdot2010\cdot(2008\cdot2009-1),=2008\cdot2009-1,=2008\cdot2010 | 82 | 74 |
math | A finite sequence of numbers $(a_1,\cdots,a_n)$ is said to be alternating if $$a_1>a_2~,~a_2<a_3~,~a_3>a_4~,~a_4<a_5~,~\cdots$$ $$\text{or ~}~~a_1<a_2~,~a_2>a_3~,~a_3<a_4~,~a_4>a_5~,~\cdots$$ How many alternating sequences of length $5$ , with distinct numbers $a_1,\cdots,a_5$ can be formed such that $a_i\in\{1,2,\cdo... | 32 \times \binom{20}{5} | 162 | 13 |
math | Let $ABCD$ be a square with side length $100$. Denote by $M$ the midpoint of $AB$. Point $P$ is selected inside the square so that $MP = 50$ and $PC = 100$. Compute $AP^2$.
[i]Based on a proposal by Amogh Gaitonde[/i] | 2000 | 77 | 4 |
math | ## Task A-4.1.
In a room, there are seven people. Four of them know exactly one person, and the remaining three people know exactly two people. All acquaintances are mutual.
What is the probability that two randomly selected people do not know each other? | \frac{16}{21} | 56 | 9 |
math | 7. Randomly throw three dice. The probability that the sum of the numbers on two of the dice is 7 is $\qquad$ . | \frac{5}{12} | 30 | 8 |
math | 3. The base of a right quadrilateral prism is a rhombus whose diagonals differ by $10 \mathrm{~cm}$. If the larger diagonal is increased by $2 \mathrm{~cm}$, and the smaller one is decreased by $1 \mathrm{~cm}$, the area of the rhombus remains the same.
Calculate the area of the prism if its height is twice the side o... | 1520\mathrm{~}^{2} | 92 | 12 |
math | 12.1.1. In a box, there are 6 identical pairs of black gloves and 4 identical pairs of beige gloves. Find the probability that two randomly drawn gloves form a pair. | \frac{47}{95} | 41 | 9 |
math | ## Task A-4.4.
Determine the set of all values that the function $f: \mathbb{R} \rightarrow \mathbb{R}$
$$
f(x)=\frac{2020 x}{x^{2}+x+1}
$$
achieves. | [-2020,\frac{2020}{3}] | 64 | 15 |
math | ## Problem Statement
Find the derivative.
$$
y=\ln \frac{1+\sqrt{-3+4 x-x^{2}}}{2-x}+\frac{2}{2-x} \sqrt{-3+4 x-x^{2}}
$$ | \frac{4-x}{(2-x)^{2}\cdot\sqrt{-3+4x-x^{2}}} | 52 | 25 |
math | 13.289. There are two alloys consisting of zinc, copper, and tin. It is known that the first alloy contains $40 \%$ tin, and the second - $26 \%$ copper. The percentage of zinc in the first and second alloys is the same. By melting 150 kg of the first alloy and 250 kg of the second, a new alloy was obtained, in which t... | 170 | 112 | 3 |
math | ## Task 6
Divide and justify the results using multiplication. 40 by $4 ; \quad 24$ by $3 ; \quad 40$ by 5 | 10,8,8 | 40 | 6 |
math | Lajcsi and Pali are discussing how often in lottery draws three numbers contain the same digit. After a brief calculation, Lajcsi says, "Out of 100 draws, on average, nearly 7 will have at least three numbers containing the digit 8." To this, Pali replies, "I once made a mistake in a similar calculation and I think you... | 5.3 | 168 | 3 |
math | 10. Given the ellipse $C: \frac{y^{2}}{a^{2}}+\frac{x^{2}}{b^{2}}=1(a>b>0)$ with an eccentricity of $\frac{1}{2}$, the upper and lower foci are $F_{1}, F_{2}$, and the right vertex is $D$. A line perpendicular to $D F_{2}$ through $F_{1}$ intersects the ellipse $C$ at points $A, B$, and $|B D|-\left|A F_{1}\right|=\fra... | \frac{16}{5} | 229 | 8 |
math | 9. In the Cartesian coordinate system, $F_{1}, F_{2}$ are the two foci of the hyperbola $\Gamma: \frac{x^{2}}{3}-y^{2}=1$. A point $P$ on $\Gamma$ satisfies $\overrightarrow{P F_{1}} \cdot \overrightarrow{P F_{2}}=1$. Find the sum of the distances from point $P$ to the two asymptotes of $\Gamma$. | \frac{3\sqrt{2}}{2} | 98 | 12 |
math | 7. Let $a, b$ be real numbers, and the function $f(x)=a x+b$ satisfies: for any $x \in[0,1]$, we have $|f(x)| \leqslant 1$. Then the maximum value of $a b$ is $\qquad$ | \frac{1}{4} | 66 | 7 |
math | Find the constant numbers $ u,\ v,\ s,\ t\ (s<t)$ such that $ \int_{\minus{}1}^1 f(x)\ dx\equal{}uf(s)\plus{}vf(t)$ holds for any polynomials $ f(x)\equal{}ax^3\plus{}bx^2\plus{}cx\plus{}d$ with the degree of 3 or less than. | u = 1 | 84 | 5 |
math | The set $X$ has $1983$ members. There exists a family of subsets $\{S_1, S_2, \ldots , S_k \}$ such that:
[b](i)[/b] the union of any three of these subsets is the entire set $X$, while
[b](ii)[/b] the union of any two of them contains at most $1979$ members. What is the largest possible value of $k ?$ | 31 | 101 | 2 |
math | 9.188. $\sqrt{x+3}+\sqrt{x-2}-\sqrt{2 x+4}>0$. | x\in(\frac{-1+\sqrt{34}}{2};\infty) | 28 | 20 |
math | ## 5. Integers
Let $x_{1}, x_{2}, x_{3}, x_{4}, \ldots, x_{1013}$ be consecutive integers, in increasing order. If
$$
-x_{1}+x_{2}-x_{3}+x_{4}-\cdots-x_{1011}+x_{1012}-x_{1013}=1013
$$
determine the absolute value of the number $x_{1013}$.
Result: $\quad 507$ | 507 | 124 | 3 |
math | Example 1 Given the function $f(x)=\log _{2}\left(x^{2}+1\right)(x \geqslant 0), g(x)=\sqrt{x-a}(a \in \mathbf{R})$.
(1) Try to find the inverse function $f^{-1}(x)$ of the function $f(x)$;
(2) The function $h(x)=f^{-1}(x)+g(x)$, find the domain of the function $h(x)$, and determine the monotonicity of $h(x)$;
(3) If t... | (-\infty,-4]\cup[\log_{2}5,+\infty) | 166 | 19 |
math | 8. Let two strictly increasing sequences of positive integers $\left\{a_{n}\right\}$ and $\left\{b_{n}\right\}$ satisfy $a_{10}=b_{10}<2017$, for any positive integer $n$, there is $a_{n+2}=a_{n+1}+a_{n}, b_{n+1}=2 b_{n}$.
Then all possible values of $a_{1}+b_{1}$ are $\qquad$ | 13, 20 | 109 | 6 |
math | 4. Let's introduce the notations: $B H=2 a, H C=a, B F=y, F C=x$. Since angle $B F H$ is a right angle, by the theorem of relations in a right-angled triangle for the two legs $B H, H C$, we have:
$$
\left\{\begin{array}{l}
a^{2}=x(y+x) \\
4 a^{2}=y(x+y)
\end{array} \quad \Rightarrow \frac{y}{x}=4 \Rightarrow y=4 x\ri... | \frac{1}{10};\frac{1}{2}(\arcsin\frac{4}{5}+\frac{4}{5}) | 522 | 33 |
math | In chess, the winner receives 1 point and the loser 0 points. In the event of a draw (remis), each player receives $1 / 2$ point.
Fourteen chess players, no two of whom were the same age, participated in a tournament where each played against every other. After the tournament, a ranking list was created. In the case of... | 40 | 149 | 2 |
math | 11.039. In a cube, the centers of the bases are connected to the centers of the side faces. Calculate the surface area of the resulting octahedron if the edge of the cube is $a$. | ^{2}\sqrt{3} | 47 | 7 |
math | 12. Given $F_{1}, F_{2}$ are the left and right foci of the ellipse $C: \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1(a>b>0)$, point $P\left(\frac{2 \sqrt{6}}{3}, 1\right)$ lies on the ellipse $C$, and the orthocenter of $\triangle F_{1} P F_{2}$ is $H\left(\frac{2 \sqrt{6}}{3},-\frac{5}{3}\right)$.
(1) Find the equation o... | 2(x-1) | 244 | 5 |
math | 16. (3 points) Wang Lei and her sister walk from home to the gym to play badminton. It is known that her sister walks 20 meters more per minute than Wang Lei. After 25 minutes, her sister arrives at the gym. At this point, her sister realizes she forgot the racket and immediately returns home along the same route to ge... | 1500 | 113 | 4 |
math | How many ordered pairs of integers $(m,n)$ are there such that $m$ and $n$ are the legs of a right triangle with an area equal to a prime number not exceeding $80$? | 87 | 43 | 2 |
math | 8. (FRG 3) For all rational $x$ satisfying $0 \leq x<1, f$ is defined by
$$ f(x)= \begin{cases}f(2 x) / 4, & \text { for } 0 \leq x<1 / 2 \\ 3 / 4+f(2 x-1) / 4, & \text { for } 1 / 2 \leq x<1\end{cases} $$
Given that $x=0 . b_{1} b_{2} b_{3} \ldots$ is the binary representation of $x$, find $f(x)$. | f\left(0 . b_{1} b_{2} \ldots\right)=0 . b_{1} b_{1} b_{2} b_{2} \ldots | 143 | 41 |
math | Determine the number of ways to serve $n$ foods in the cafeteria, knowing that apples are taken in groups of 3, yogurts come in pairs, and one is allowed at most 2 pieces of bread and one bowl of cereal due to a change in provider. | n+1 | 57 | 3 |
math | Solve the equation
$$
x^{2}+\left(\frac{5 x}{x+5}\right)^{2}=1
$$ | x_{1,2}=\frac{\sqrt{26}-5\\sqrt{10\sqrt{26}-49}}{2} | 31 | 32 |
math | 1. Let $n$ be the sum of all ten-digit numbers that have each of the digits $0,1, \ldots, 9$ in their decimal representation. Determine the remainder when $n$ is divided by seventy-seven. | 28 | 50 | 2 |
math | 8. Let $p$ be a prime number, and $A$ be a set of positive integers, satisfying the following conditions:
(1) The set of prime factors of the elements in $A$ contains $p-1$ elements;
(2) For any non-empty subset of $A$, the product of its elements is not a $p$-th power of an integer.
Find the maximum number of elements... | (p-1)^2 | 95 | 5 |
math | 【Question 30】
There are two piles of stones, one with 11 stones and the other with 13 stones, denoted as $(11,13)$. Two players, A and B, take turns to remove stones, following these rules: one can take any number of stones from one pile or the same number of stones from both piles, but must take at least one stone. Th... | (3,5)or(8,13) | 117 | 12 |
math | 5. Let $f(x)=x^{3}+3\left(x^{2}+x+\sin \pi x\right)$, then $f(1-\pi)+f(\pi-3)=$ | -2 | 45 | 2 |
math | 1. Divide the sequence of positive integers $1,2, \cdots$ from left to right into segments such that the first segment has $1 \times 2$ numbers, the second segment has $2 \times 3$ numbers, $\cdots$, the $n$-th segment has $n \times(n+1)$ numbers, $\cdots$. Then 2014 is in the $\qquad$ segment. | 18 | 92 | 2 |
math | An equilateral triangle has sides of length $x+5, y+11$, and 14 . What is the value of $x+y$ ?
## | 12 | 34 | 2 |
math | ## Problem Statement
Find the derivative.
$$
y=\frac{1}{12} \cdot \ln \frac{x^{4}-x^{2}+1}{\left(x^{2}+1\right)^{2}}-\frac{1}{2 \sqrt{3}} \operatorname{arctg} \frac{\sqrt{3}}{2 x^{2}-1}
$$ | \frac{x^{3}}{(x^{4}-x^{2}+1)\cdot(x^{2}+1)} | 85 | 26 |
math | \section*{Task 1 - 091211}
At an evening event, each of the gentlemen present danced with exactly three ladies, and with each exactly once. When all participants were still sitting together in a cozy circle after the dance and reviewing the evening, it was noted that each of the ladies present had danced with exactly ... | =4,=6 | 117 | 5 |
math | 4. (5 points) The last three digits of the expression $1 \times 1+11 \times 11+111 \times 111+\cdots+111 \cdots 111$ (2010 1’s) $\times 111 \cdots 111$ (2010 1’s) are $\qquad$ . | 690 | 90 | 3 |
math | ## Task B-4.7.
In the set of complex numbers, solve the equation $z^{3}+|z|=0$. | z_{1}=0,z_{2}=\frac{1}{2}+i\frac{\sqrt{3}}{2},z_{3}=-1,z_{4}=\frac{1}{2}-i\frac{\sqrt{3}}{2} | 29 | 56 |
math | Let $A$ and $B$ be points on circle $\Gamma$ such that $AB=\sqrt{10}.$ Point $C$ is outside $\Gamma$ such that $\triangle ABC$ is equilateral. Let $D$ be a point on $\Gamma$ and suppose the line through $C$ and $D$ intersects $AB$ and $\Gamma$ again at points $E$ and $F \neq D.$ It is given that points $C, D, E, F$ are... | \frac{38\pi}{15} | 138 | 11 |
math | 6. Given $0 \leqslant y \leqslant x \leqslant \frac{\pi}{2}$, and
$$
4 \cos ^{2} y+4 \cos x \cdot \sin y-4 \cos ^{2} x \leqslant 1 \text {. }
$$
then the range of $x+y$ is | \left[0, \frac{\pi}{6}\right] \cup\left[\frac{5 \pi}{6}, \pi\right] | 83 | 32 |
math | The fraction $\tfrac1{2015}$ has a unique "(restricted) partial fraction decomposition'' of the form \[\dfrac1{2015}=\dfrac a5+\dfrac b{13}+\dfrac c{31},\] where $a$, $b$, and $c$ are integers with $0\leq a<5$ and $0\leq b<13$. Find $a+b$. | 14 | 99 | 2 |
math | 1. Given arrays $\left(a_{1}, a_{2}, \cdots, a_{n}\right)$ and $\left(b_{1}, b_{2}, \cdots, b_{n}\right)$ are both permutations of $1,2, \cdots, n$. Then
$$
a_{1} b_{1}+a_{2} b_{2}+\cdots+a_{n} b_{n}
$$
the maximum value is | \frac{n(n+1)(2 n+1)}{6} | 98 | 15 |
math | 2. Having walked $4 / 9$ of the length of the bridge, the traveler noticed that a car was catching up to him, but it had not yet entered the bridge. Then he turned back and met the car at the beginning of the bridge. If he had continued his movement, the car would have caught up with him at the end of the bridge. Find ... | 9 | 90 | 1 |
math | Example 14. In a batch of 12 parts, 8 are standard. Find the probability that among 5 randomly selected parts, 3 will be standard. | \frac{14}{33} | 36 | 9 |
math | 4. Given prime numbers $p$ and $q$ such that $p^{2}+3 p q+q^{2}$ is a perfect square of some integer. Then the maximum possible value of $p+q$ is $\qquad$ (Bulgaria) | 10 | 57 | 2 |
math | 3.153. $\sin ^{2} \frac{\pi}{8}+\cos ^{2} \frac{3 \pi}{8}+\sin ^{2} \frac{5 \pi}{8}+\cos ^{2} \frac{7 \pi}{8}$. | 2 | 65 | 1 |
math | Example 3 Simplify $\frac{1+\sin \theta+\cos \theta}{1+\sin \theta-\cos \theta}+\frac{1-\cos \theta+\sin \theta}{1+\cos \theta+\sin \theta}$. | 2\csc\theta | 52 | 6 |
math | 1. Find all integers $n$ for which $n^{4}-3 n^{2}+9$ is a prime number. | n\in{-2,-1,1,2} | 28 | 12 |
math | For any real number $x$, we have
$$
|x+a|-|x+1| \leqslant 2 a \text{. }
$$
Then the minimum value of the real number $a$ is $\qquad$ | \frac{1}{3} | 51 | 7 |
math | 1. [5] If five fair coins are flipped simultaneously, what is the probability that at least three of them show heads? | \frac{1}{2} | 26 | 7 |
math | Example 8. The probability density function of a random variable $X$ is given by
$$
p(x)=a x^{2} e^{-k x} \quad(k>0, \quad 0 \leq x<+\infty)
$$
Find the value of the coefficient $a$. Find the distribution function $F(x)$ of the variable $X$. | =\frac{k^3}{2},\quadF(x)=1-\frac{k^2x^2+2kx+2}{2}e^{-kx} | 78 | 36 |
math | 6. Given that $\alpha$ is an acute angle, vectors
$$
a=(\cos \alpha, \sin \alpha), b=(1,-1)
$$
satisfy $a \cdot b=\frac{2 \sqrt{2}}{3}$. Then $\sin \left(\alpha+\frac{5 \pi}{12}\right)=$ $\qquad$ | \frac{2 + \sqrt{15}}{6} | 81 | 14 |
math | Suppose that $a$ is a positive integer with $a>1$. Determine a closed form expression, in terms of $a$, equal to
$$
1+\frac{3}{a}+\frac{5}{a^{2}}+\frac{7}{a^{3}}+\cdots
$$
(The infinite sum includes exactly the fractions of the form $\frac{2 k-1}{a^{k-1}}$ for each positive integer $k$.) | \frac{^{2}+}{(-1)^{2}} | 98 | 14 |
math | 10. A number lies between 2013 and 2156, and when divided by $5$, $11$, and $13$, the remainders are the same. The largest possible remainder is $\qquad$ _. | 4 | 52 | 1 |
math | Let set $A=\{1,2,\ldots,n\} ,$ and $X,Y$ be two subsets (not necessarily distinct) of $A.$ Define that $\textup{max} X$ and $\textup{min} Y$ represent the greatest element of $X$ and the least element of $Y,$ respectively. Determine the number of two-tuples $(X,Y)$ which satisfies $\textup{max} X>\textup{min} Y.$ | 2^{2n} - (n+1)2^n | 98 | 13 |
math | Ni.(16 points) The vertex of the parabola is $O$ and the focus is $F$. When the moving point $P$ moves on the parabola, find the maximum value of the distance ratio $\left|\frac{P O}{P F}\right|$. | \frac{2 \sqrt{3}}{3} | 60 | 12 |
math | 6. In $\triangle A B C$, $\angle B=\frac{\pi}{3}, A C=\sqrt{3}$, point $D$ is on side $A B$, $B D=1$, and $D A=D C$. Then $\angle D C A=$ $\qquad$ | \frac{\pi}{6} | 62 | 7 |
math | Task B-3.2. Determine all real numbers $x$ for which
$$
\frac{1}{\log _{x} 2 \cdot \log _{x} 4}+\frac{1}{\log _{x} 4 \cdot \log _{x} 8}+\ldots+\frac{1}{\log _{x} 2^{2013} \cdot \log _{x} 2^{2014}} \leq 2013 \log _{2} x
$$ | x\in\langle1,2^{2014}] | 122 | 14 |
math | 5. Determine the smallest natural number $N$, among the divisors of which are all numbers of the form $x+y$, where $x$ and $y$ are natural solutions to the equation $6 x y-y^{2}-5 x^{2}=7$. | 55 | 55 | 2 |
math | Example 7 Given $a \neq b$, and
$$
a^{2}-4 a+1=0, b^{2}-4 b+1=0 \text {. }
$$
Find the value of $\frac{1}{a+1}+\frac{1}{b+1}$. | 1 | 65 | 1 |
math | 5. In $\triangle A B C$, $A B=A C=5, B C=6$, and its orthocenter $H$ satisfies $\overrightarrow{A H}=m \overrightarrow{A B}+n \overrightarrow{B C}$. Then $m+n=$ $\qquad$. | \frac{21}{32} | 65 | 9 |
math | 4. Find the function $f: \mathbf{Z}_{+} \rightarrow \mathbf{Z}_{+}$, such that for all $m, n \in \mathbf{Z}_{+}$, we have
$$
(n!+f(m)!) \mid(f(n)!+f(m!)) \text {. }
$$ | f(n)=n | 73 | 4 |
math | A $5 \mathrm{~m} \times 5 \mathrm{~m}$ flat square roof receives $6 \mathrm{~mm}$ of rainfall. All of this water (and no other water) drains into an empty rain barrel. The rain barrel is in the shape of a cylinder with a diameter of $0.5 \mathrm{~m}$ and a height of $1 \mathrm{~m}$. Rounded to the nearest tenth of a pe... | 76.4 | 108 | 4 |
math | Example 9 If the equation $\left(x^{2}-1\right)\left(x^{2}-4\right)=k$ has 4 non-zero real roots, and the 4 points corresponding to them on the number line are equally spaced, then $k=$ $\qquad$ | \frac{7}{4} | 59 | 7 |
math | 4. [3] Suppose that $a, b, c, d$ are real numbers satisfying $a \geq b \geq c \geq d \geq 0, a^{2}+d^{2}=1, b^{2}+c^{2}=1$, and $a c+b d=1 / 3$. Find the value of $a b-c d$. | \frac{2\sqrt{2}}{3} | 84 | 12 |
math | 18. A moving circle is externally tangent to $(x+2)^{2}+y^{2}=4$ and tangent to the line $x=2$. Then, the equation of the locus of the center of the moving circle is | y^{2}+12 x-12 = 0 | 50 | 14 |
math | $$
f(n)=\left\{\begin{array}{ll}
1, & n=1 \text { when; } \\
2, & 1<n \leqslant 3 \text { when; } \\
3, & 3<n \leqslant 6 \text { when; } \\
\cdots \cdots . & \\
m, & \frac{m(m-1)}{2}<n \leqslant \frac{m(m+1)}{2} \text { when; } \\
\cdots \cdots . &
\end{array}\right.
$$
If $S_{n}=\sum_{k=1}^{n} f(k)=2001$, find the v... | 165 | 163 | 3 |
math | ## Task A-3.2.
Determine all pairs of natural numbers $(a, b)$ such that
$$
\begin{gathered}
a^{3}-3 b=15 \\
b^{2}-a=13
\end{gathered}
$$
Note: The only solution to the given system is $(a, b)=(3,4)$. If a student writes that the pair $(a, b)=(3,4)$ is a solution to the given system but does not write any other conc... | (,b)=(3,4) | 122 | 8 |
math | Exercise 10. Count the number of rearrangements $a_{1}, a_{2}, \ldots, a_{2023}$ of the sequence $1,2, \ldots, 2023$ such that $a_{k}>k$ for exactly one value of $k$. | 2^{2023}-2024 | 67 | 11 |
math | In $\triangle A B C, B C=A C-1$ and $A C=A B-1$. If $\cos (\angle B A C)=\frac{3}{5}$, determine the perimeter of $\triangle A B C$. | 42 | 50 | 2 |
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