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 | 2. Determine the largest value of $x$ for which
$$
\left|x^{2}-4 x-39601\right| \geq\left|x^{2}+4 x-39601\right|
$$ | 199 | 54 | 3 |
math | 15. A total of 99 people participated in a mathematics competition, which was divided into three sessions, testing the contestants' abilities in geometry, number theory, and combinatorics. Xiao Ming ranked 16th in the number theory exam, 30th in the combinatorics exam, and 23rd in the geometry exam, and he did not tie with anyone in any of the three exams (the full score for each exam is not necessarily 100 points). The final overall ranking is determined by adding up the scores from the three exams and ranking them from highest to lowest. If we use the $\mathrm{A}$th place to represent the best possible overall ranking Xiao Ming could achieve (the smaller the $\mathrm{A}$, the better the overall ranking), and the $\mathrm{B}$th place to represent the worst possible overall ranking, then $100 A+B=(\quad)$. | 167 | 192 | 3 |
math | B2. How many four-digit numbers are there with the following properties:
- the second digit is the average of the first digit and the third digit,
- the third digit is the average of the second digit and the fourth digit? (A number does not start with the digit 0.) | 30 | 59 | 2 |
math | 16. Given $\vec{a}=\{1,2\}, \vec{b}=\{-3,2\}$, find the real number $k$ such that $k \vec{a}+\vec{b}$ is in the same direction or opposite direction to $\vec{a}-3 \vec{b}$. | -\frac{1}{3} | 71 | 7 |
math | 7. (5 points) It's the New Year, and the students are going to make some handicrafts to give to the elderly in the nursing home. At the beginning, the students in the art group work for one day, then 15 more students join them and they work together for two more days, just completing the task. Assuming each student has the same work efficiency, and one student would need 60 days to complete the task alone. How many students are in the art group? $\qquad$ | 10 | 107 | 2 |
math | 9. (16 points) Let the sequence $\left\{a_{n}\right\}$ satisfy
$$
a_{1}=1, a_{n+1}=2 a_{n}+n\left(1+2^{n}\right)(n=1,2, \cdots) \text {. }
$$
Try to find the general term $a_{n}$ of the sequence. | a_{n}=2^{n-2}\left(n^{2}-n+6\right)-n-1(n \geqslant 2) | 86 | 33 |
math | ## Aufgabe 1/81
Man bestimme alle Paare $(n ; m)$ nichtnegativer ganzer Zahlen, für die gilt:
$$
\sum_{i=1}^{n} i=25^{m}+2
$$
| (2,0) | 58 | 5 |
math | Problem 6. Thirty-three bogatyrs (Russian knights) were hired to guard Lake Luka for 240 coins. The cunning Chernomor can divide the bogatyrs into squads of any size (or record all in one squad), and then distribute the entire salary among the squads. Each squad divides its coins equally, and the remainder goes to Chernomor. What is the maximum number of coins Chernomor can get if:
a) Chernomor distributes the salary among the squads as he pleases
b) Chernomor distributes the salary among the squads equally? [8 points] (I.V.Raskina, A.V.Khachaturyan) | 31 | 139 | 2 |
math | 23. For each positive integer $n \geq 1$, we define the recursive relation given by
$$
a_{n+1}=\frac{1}{1+a_{n}} \text {. }
$$
Suppose that $a_{1}=a_{2012}$. Find the sum of the squares of all possible values of $a_{1}$. | 3 | 80 | 1 |
math | Find the real number $\alpha$ such that the curve $f(x)=e^x$ is tangent to the curve $g(x)=\alpha x^2$. | \frac{e^2}{4} | 34 | 9 |
math | 2. In the plane, there are $h+s$ lines, among which $h$ lines are horizontal, and $s$ lines satisfy:
(1) None of them are horizontal lines;
(2) No two of them are parallel;
(3) Any 3 of these $h+s$ lines do not intersect at the same point.
If these $h+s$ lines exactly divide the plane into 1992 regions, find all pairs of positive integers $(h, s)$. | (995,1),(176,10),(80,21) | 102 | 20 |
math | 6. Let $D$ be a point on side $BC$ of $\triangle ABC$, and points $E, F$ be the centroids of $\triangle ABD$ and $\triangle ACD$, respectively. Connecting $EF$ intersects $AD$ at point $G$. What is the value of $\frac{DG}{GA}$?
(1991-1992 Guangzhou and Four Other Cities Competition Question) | \frac{DG}{GA}=1:2 | 88 | 10 |
math | ## 22. Age Difference
The sums of the digits that make up the birth years of Jean and Jacques are equal to each other, and the age of each of them starts with the same digit. Could you determine the difference in their ages? | 9 | 51 | 1 |
math | 107. An Aeroflot cashier needs to deliver tickets to five groups of tourists. Three of these groups are staying at the hotels "Druzhba", "Rossiya", and "Minsk". The address of the fourth group will be provided by the tourists from "Rossiya", and the address of the fifth group will be provided by the tourists from "Minsk". In how many ways can the cashier choose the order of visiting the hotels to deliver the tickets? | 30 | 97 | 2 |
math | A prison has $2004$ cells, numbered $1$ through $2004$. A jailer, carrying out the terms of a partial amnesty, unlocked every cell. Next he locked every second cell. Then he turned the key in every third cell, locking the opened cells, and unlocking the locked ones. He continued this way, on $n^{\text{th}}$ trip, turning the key in every $n^{\text{th}}$ cell, and he finished his mission after $2004$ trips. How many prisoners were released? | 44 | 120 | 2 |
math | 1. Given that $a$ is a root of the equation $x^{2}-5 x+1=0$. Then the unit digit of $a^{4}+a^{-4}$ is $\qquad$ . | 7 | 46 | 1 |
math | ## Task B-1.1.
The sum of two numbers is 6. If the sum of their cubes is 90, what is the sum of their squares? | 22 | 36 | 2 |
math | Consider all $6$-digit numbers of the form $abccba$ where $b$ is odd. Determine the number of all such $6$-digit numbers that are divisible by $7$. | 70 | 42 | 2 |
math | Solve the inequality
$$
\frac{1}{x}+\frac{1}{x^{3}}+\frac{1}{x^{5}}+\frac{1}{x^{7}}+\frac{1}{x^{9}}+\frac{1}{x^{11}}+\frac{1}{x^{13}} \leq \frac{7}{x^{7}}
$$ | 1orx<0 | 83 | 5 |
math | 7. Given the height of the cylinder $O O_{1}=12$, the base radius $r=5$, there is 1 point $A$ and 1 point $B$ on the circumference of the upper and lower base circles, respectively, and $A B=13$. Then, the distance between the axis of the cylinder $O O_{1}$ and $A B$ is | \frac{5}{2}\sqrt{3} | 83 | 11 |
math | 7. If the parabola $y=\frac{1}{2} x^{2}-m x+m-1$ intersects the $x$-axis at integer points, then the equation of the axis of symmetry of the parabola is $\qquad$. | x=1 | 55 | 3 |
math | 53. Find the matrix inverse of the matrix
$$
A=\left(\begin{array}{rrr}
1 & 2 & 3 \\
0 & -1 & 2 \\
3 & 0 & 7
\end{array}\right)
$$ | (\begin{pmatrix}-7&-14&7\\6&-2&-2\\3&6&-1\end{pmatrix}) | 56 | 34 |
math | Tomonvo A.K.
Petya's bank account contains 500 dollars. The bank allows only two types of transactions: withdrawing 300
dollars or adding 198 dollars.
What is the maximum amount Petya can withdraw from his account if he has no other money? | 498 | 64 | 3 |
math | Question 5 Find all positive integers $k$, such that for any positive numbers $a, b, c$ satisfying $abc=1$, the following inequality holds:
$$\frac{1}{a^{2}}+\frac{1}{b^{2}}+\frac{1}{c^{2}}+3 k \geqslant(k+1)(a+b+c)$$ | k=1 | 79 | 3 |
math | Find all polynomials $P$ such that $P(0)=0$ and $P\left(X^{2}+1\right)=P(X)^{2}+1$ | P(X)=X | 39 | 4 |
math | Solve the equation
\[
\sin 9^\circ \sin 21^\circ \sin(102^\circ + x^\circ) = \sin 30^\circ \sin 42^\circ \sin x^\circ
\]
for $x$ where $0 < x < 90$. | x = 9^\circ | 71 | 7 |
math | What is the last two digits of the number $(11^2 + 15^2 + 19^2 + ... + 2007^2)^2$? | 0 | 46 | 1 |
math | Example 1 Let $x_{1}, x_{2}, \cdots, x_{n} \geqslant 0$ and $\sum_{i=1}^{n} x_{i} \geqslant k$, where $k(k \geqslant 1)$ is a positive constant. Find
$$
f\left(x_{1}, x_{2}, \cdots, x_{n}\right)=\frac{x_{1} \sqrt{\sum_{i=1}^{n} x_{i}}}{\left(\sum_{i=1}^{n-1} x_{i}\right)^{2}+x_{n}}
$$
the maximum value. | \frac{\sqrt{k}}{2 \sqrt{k}-1} | 150 | 14 |
math | 20.3. The collective farm placed a certain amount of money in the savings bank. If the number of hundreds is added to the number formed by the last two digits, then the result is the annual income from this amount, calculated at $2 \%$ per annum. What is the amount of the deposit placed in the savings bank?
$(7-8$ grades$)$ | 4950 | 77 | 4 |
math | 16th ASU 1982 Problem 2 The sequence a n is defined by a 1 = 1, a 2 = 2, a n+2 = a n+1 + a n . The sequence b n is defined by b 1 = 2, b 2 = 1, b n+2 = b n+1 + b n . How many integers belong to both sequences? | 1,2,3 | 89 | 5 |
math | The sum of non-negative numbers $x_{1}, x_{2}, \ldots, x_{10}$ is 1. Find the maximum possible value of the sum $x_{1} x_{2}+$ $x_{2} x_{3}+\ldots+x_{9} x_{10}$ | 0.25 | 67 | 4 |
math | 26th Putnam 1965 Problem B1 X is the unit n-cube, [0, 1] n . Let k n = ∫ X cos 2 ( π(x 1 + x 2 + ... + x n )/(2n) ) dx 1 ... dx n . What is lim n→∞ k n ? | k_n=\frac{1}{2} | 75 | 9 |
math | Example 6 Given 2014 real numbers $x_{1}, x_{2}, \cdots, x_{2014}$ satisfy the system of equations
$$
\sum_{k=1}^{2014} \frac{x_{k}}{n+k}=\frac{1}{2 n+1}(n=1,2, \cdots, 2014) .
$$
Try to calculate the value of $\sum_{k=1}^{2014} \frac{x_{k}}{2 k+1}$. | \frac{1}{4}(1-\frac{1}{4029^{2}}) | 123 | 21 |
math | Find all positive integer solutions $(x, y, u, v)$ for the equation
$$
\left|2^{x} \cdot 3^{y}-2^{u} \cdot 5^{v}\right|=2
$$ | (x,y,u,v)=(1,2,2,1),(4,1,1,2),(1,4,5,1),(2,1,1,1) | 50 | 37 |
math | Let $M$ be a finite sum of numbers, such that among any three of its elements there are two whose sum belongs to $M$. Find the greatest possible number of elements of $M$. | 7 | 41 | 3 |
math | 14. Let $f(x)$ be an odd function defined on $\mathbf{R}$, and for any $x \in \mathbf{R}$, we have
$$
\begin{aligned}
f(x+2) & =f(x)+2, \\
\text { then } \sum_{k=1}^{2014} f(k) & =
\end{aligned}
$$ | 2029105 | 87 | 7 |
math | Find all triples $(a,b,c)$ of positive integers such that if $n$ is not divisible by any prime less than $2014$, then $n+c$ divides $a^n+b^n+n$.
[i]Proposed by Evan Chen[/i] | (1, 1, 2) | 55 | 10 |
math | 8,9
[ Ratio in which the bisector divides the side ]
In an isosceles triangle $A B C$, a rhombus $D E C F$ is inscribed such that vertex $E$ lies on side $B C$, vertex $F$ on side $A C$, and vertex $D$ on side $A B$. Find the length of the side of the rhombus if $A B=B C=12, A C=6$. | 4 | 100 | 1 |
math | Find all relatively distinct integers $m, n, p\in \mathbb{Z}_{\ne 0}$ such that the polynomial
$$F(x) = x(x - m)(x - n)(x - p) + 1$$is reducible in $\mathbb{Z}[x].$ | (3, 1, 2), (-3, -1, -2), (1, -1, 2), (-1, 1, -2) | 65 | 37 |
math | 【Question 8】
The sum of five consecutive even numbers is a multiple of 7. The smallest possible sum of these five numbers is
保留了源文本的换行和格式。 | 70 | 40 | 2 |
math | 3. Let $F_{1}$ and $F_{2}$ be the left and right foci of the ellipse $C$:
$$
\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1(a>b>0)
$$
If there exists a point $P$ on the ellipse $C$ such that $P F_{1} \perp P F_{2}$, then the range of the eccentricity $e$ of the ellipse is $\qquad$ | \left[\frac{\sqrt{2}}{2}, 1\right) | 111 | 17 |
math | I4.1 若 $m$ 和 $n$ 為正整數及 $a=\log _{2}\left[\left(\frac{m^{4} n^{-4}}{m^{-1} n}\right)^{-3} \div\left(\frac{m^{-2} n^{2}}{m n^{-1}}\right)^{5}\right]$, 求 $a$ 的值。
If $m$ and $n$ are positive integers and $a=\log _{2}\left[\left(\frac{m^{4} n^{-4}}{m^{-1} n}\right)^{-3} \div\left(\frac{m^{-2} n^{2}}{m n^{-1}}\right)^{5}\right]$, determine the value of $a$. | 0 | 175 | 1 |
math | 4. Express the sum $8+88+888+8888+\ldots+8 \ldots 8$ in terms of n, if the last term in its notation contains n eights | \frac{8}{81}(10^{n+1}-10-9n) | 46 | 21 |
math | ## Task B-3.1.
Simplify the numerical expression $\frac{6-\log _{14} 2401}{4-\log _{14} 49}$ and write it in the form of a logarithm $\log _{28} n$, where $n$ is a natural number. | \log_{28}56 | 70 | 8 |
math | In $\triangle ABC$ with side lengths $AB = 13,$ $BC = 14,$ and $CA = 15,$ let $M$ be the midpoint of $\overline{BC}.$ Let $P$ be the point on the circumcircle of $\triangle ABC$ such that $M$ is on $\overline{AP}.$ There exists a unique point $Q$ on segment $\overline{AM}$ such that $\angle PBQ = \angle PCQ.$ Then $AQ$ can be written as $\frac{m}{\sqrt{n}},$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$ | 247 | 140 | 3 |
math | Example 5. (1MO-23-1)
The function $f(n)$ is defined for all positive integers $n$, taking non-negative integer values. For all positive integers $m, n$, $f(m+n)-f(m)-f(n)=0$ or 1;
and $f(2)=0, f(3)>0, f(9999)=3333$. Find $f(1982)$. | 660 | 96 | 3 |
math | Example 1 Given that the sum of several positive integers is 1976. Find the maximum value of their product. | 2 \times 3^{658} | 26 | 10 |
math |
2. We consider number sequences $a_{1}, a_{2}, a_{3}, \ldots$ such that $a_{n+1}=\frac{a_{n}+a_{1}}{a_{n}+1}$ holds for all $n \geqslant 1$.
(a) Suppose that $a_{1}=-3$. Compute $a_{2020}$.
(b) Suppose that $a_{1}=2$. Prove that $\frac{4}{3} \leqslant a_{n} \leqslant \frac{3}{2}$ holds for all $n \geqslant 2$.
| -3 | 145 | 2 |
math | Example 1 (9th American Invitational Mathematics Examination AIME Problem) Let $r$ be a real number satisfying the condition:
$$
\left[r+\frac{19}{100}\right]+\left[r+\frac{20}{100}\right]+\left[r+\frac{21}{100}\right]+\cdots+\left[r+\frac{91}{100}\right]=546 \text {. }
$$
Find $[100 r]$. | 743 | 107 | 3 |
math | Let $(a_n)\subset (\frac{1}{2},1)$. Define the sequence $x_0=0,\displaystyle x_{n+1}=\frac{a_{n+1}+x_n}{1+a_{n+1}x_n}$. Is this sequence convergent? If yes find the limit. | 1 | 70 | 3 |
math | ## C2.
There are 2016 costumers who entered a shop on a particular day. Every customer entered the shop exactly once. (i.e. the customer entered the shop, stayed there for some time and then left the shop without returning back.)
Find the maximal $k$ such that the following holds:
There are $k$ customers such that either all of them were in the shop at a specific time instance or no two of them were both in the shop at any time instance.
| 45 | 103 | 2 |
math | 6. Pack 35 egg yolk mooncakes, there are two packaging specifications: large bags with 9 mooncakes per pack, and small bags with $\mathbf{4}$ mooncakes per pack. The requirement is that no mooncakes should be left over, so a total of $\qquad$ packs were made. | 5 | 67 | 1 |
math | 459. Find all values of the greatest common divisor of the numbers $8 a+3$ and $5 a+2$, where $a$ is a natural number. | 1 | 37 | 1 |
math | 17. What odd digit can the sum of two prime numbers end with, if it is not a single-digit number? | 1,3,5,9 | 25 | 7 |
math | 7. If the equation $\cot (\arcsin x)=\sqrt{a^{2}-x^{2}}$ has real solutions for $x$, then the range of real values for $a$ is $\qquad$ . | (-\infty,-1]\cup[1,+\infty) | 48 | 15 |
math | 3. In the coordinate plane, the area enclosed by the curve ||$x|+| x-1||+|y|=2$ is equal to $\qquad$
Translate the above text into English, please retain the original text's line breaks and format, and output the translation result directly. | 3 | 61 | 1 |
math | 2. (5 points) Two different natural numbers end with 9 zeros and have exactly 110 divisors. Find their sum. | 7000000000 | 29 | 10 |
math | Example 15 There are three workers transporting bricks from the brick pile to the scaffolding for wall construction. It takes Worker A 15.6 minutes, Worker B 16.8 minutes, and Worker C 18.2 minutes for a round trip. Now, all three start from the brick pile at the same time. What is the minimum number of minutes it will take for all three to return to the brick pile at the same time? | 218.4 | 94 | 5 |
math | $3 \cdot 26$ If real numbers $a, b, x, y$ satisfy $a x+b y=3, a x^{2}+b y^{2}=7, a x^{3}+$ $b y^{3}=16, a x^{4}+b y^{4}=42$, find the value of $a x^{5}+b y^{5}$. | 20 | 88 | 2 |
math | 183. Solve the equation in integers:
$$
7 x+5 y=62
$$ | 31-5t,7t-31 | 22 | 11 |
math | Fabián has four cards, on each of which he wrote a different positive integer less than 10. He wrote the numbers in different colors, with the following conditions:
- The product of the green and yellow numbers is the green number.
- The blue number is the same as the red number.
- The product of the red and blue numbers is a two-digit number written with the green and yellow digits (in that order).
Determine these four numbers.
(M. Petrová) | 9,9,8,1 | 100 | 7 |
math | 1037. Find the integrals:
1) $\int \frac{d x}{2 x^{2}+9}$
2) $\int \frac{d x}{\sqrt{5-4 x^{2}}}$
3) $\int \frac{d x}{\sqrt{2 x^{2}-3}}$
4) $\int \frac{d x}{4 x^{2}-3}$
5) $\int \cos ^{2} \frac{x}{2} d x$
6) $\int \operatorname{tg}^{2} x d x$ | -\frac{\sqrt{2}}{6}\operatorname{arctg}\frac{x\sqrt{2}}{3}+C | 125 | 29 |
math | 12.28 ** In the Cartesian coordinate system, there are two fixed points $A$ and $B$ on the $x$-axis, different from the origin $O(0,0)$. A moving point $C$ varies on the $y$-axis, and a line $l$ passes through $O(0,0)$ and is perpendicular to the line $A C$. Find the locus of the intersection point of line $l$ and line $B C$. | \frac{(x-\frac{b}{2})^2}{\frac{b^2}{4}}+\frac{y^2}{\frac{}{4}}=1 | 102 | 38 |
math | 4. Let the function $y=f(x)$ be defined on $\mathbf{R}$, and have an inverse function $f^{-1}(x)$. It is known that the inverse function of $y=f(x+1)-2$ is $y=f^{-1}(2 x+1)$, and $f(1)=4$. If $n \in \mathbf{N}_{+}$, then $f(n)=$ $\qquad$ | 3+\left(\frac{1}{2}\right)^{n-1} | 95 | 17 |
math | ## Task 1 - 100911
When asked about his age, Mr. $X$ said:
"The sum of the digits of the number of years I have lived is exactly one third of this number. The square of the sum of the digits of the number of years I have lived is exactly three times the number of years I have lived."
Can Mr. $X$'s statements be true? If so, how old is Mr. $X$? (In full years of life) | 27 | 106 | 2 |
math | Example 8 For positive real numbers $a, b$ that satisfy $a+b=1$, find
$$\left(a+\frac{1}{a}\right)^{2}+\left(b+\frac{1}{b}\right)^{2}$$
the minimum value. | \frac{25}{2} | 59 | 8 |
math | 8.1. Given positive numbers $a, b, c, d, e$. It is known that $a b=2, b c=3, c d=4, d e=15$, $e a=10$. What is the value of $a$? | \frac{4}{3} | 60 | 7 |
math | 2. (2 points) Solve the puzzle $T O K=K O T+K T O$. (Different letters represent different digits, numbers cannot start with zero). | 954=459+495 | 35 | 11 |
math | The positive integer $m$ is a multiple of $101$, and the positive integer $n$ is a multiple of $63$. Their sum is $2018$. Find $m - n$. | 2 | 45 | 1 |
math | If $\frac{99 !}{101 !-99 !}=\frac{1}{n}$, determine the value of $n$.
(If $m$ is a positive integer, then $m$ ! represents the product of the integers from 1 to $m$, inclusive. For example, $5 !=5(4)(3)(2)(1)=120$ and $99 !=99(98)(97) \cdots(3)(2)(1)$. | 10099 | 108 | 5 |
math | 62*. Given two points: $A_{1}\left(x_{1}, y_{1}, z_{1}\right)$ and $A_{2}\left(x_{2}, y_{2}, z_{2}\right)$. Point $P$ defines two vectors: $\overrightarrow{P A}_{1}=\vec{q}_{1}$ and $\overrightarrow{P A}_{2}=\vec{q}_{2}$. What does the set of points $P$ represent, for which the condition $\vec{q}_{1} \vec{q}_{2}=0$ is satisfied? | (x-x_{0})^{2}+(y-y_{0})^{2}+(z-z_{0})^{2}-R^{2}=0 | 125 | 31 |
math | One, (20 points) Given real numbers $a, b, c, d$ satisfy $2a^2 + 3c^2 = 2b^2 + 3d^2 = (ad - bc)^2 = 6$. Find the value of $\left(a^2 + \dot{b}^2\right)\left(c^2 + d^2\right)$. | 6 | 86 | 1 |
math | Solve the following system of equations:
$$
\begin{gathered}
x+y+\sqrt{x+y-2}=14 \\
\frac{x^{2} y^{2}}{3}-\frac{3 x y}{2}=255
\end{gathered}
$$ | x_{1}=y_{2}=5,x_{2}=y_{1}=6,x_{3}=y_{4}=\frac{11}{2}-\frac{1}{2}\sqrt{223},x_{4}=y_{3}=\frac{11}{2}+\frac{1}{2}\sqrt{223} | 61 | 76 |
math | Solve the following system of equations:
$$
\left\{\begin{array}{l}
\frac{7}{2 x-3}-\frac{2}{10 z-3 y}+\frac{3}{3 y-8 z}=8 \\
\frac{2}{2 x-3 y}-\frac{3}{10 z-3 y}+\frac{1}{3 y-8 z}=0 \\
\frac{5}{2 x-3 y}-\frac{4}{10 z-3 y}+\frac{7}{3 y-8 z}=8
\end{array}\right.
$$ | 5,3,1 | 135 | 5 |
math | Find those triples of integers, the product of which is four times their sum, and one of which is twice the sum of the other two. | (,b,)=(\1,\6,\14),(\2,\3,\10),(,-,0) | 29 | 25 |
math | Let $f: \mathbf{R} \rightarrow \mathbf{R}$ be a function for which we know that $f(1)=0$, and for any two real numbers $(x, y)$, $|f(x)-f(y)|=|x-y|$. Determine the function $f$! | f(x)=x-1orf(x)=1-x | 66 | 11 |
math | 15. Given a positive integer $n$ and a positive number $M$. For all arithmetic sequences $a_{1}, a_{2}, a_{3}, \cdots$ satisfying the condition $a_{1}^{2}+a_{n+1}^{2} \leqslant M$, find the maximum value of $S=a_{n+1}+a_{n+2}+\cdots+a_{2 n+1}$. | \frac{(n+1)\sqrt{10M}}{2} | 97 | 16 |
math | 7. Let $M=\{1,2,3,4,5\}$. Then the number of mappings $f: M \rightarrow M$ such that
$$
f(f(x))=f(x)
$$
is $\qquad$ | 196 | 52 | 3 |
math | 9. (5 points) A non-zero natural number $a$ satisfies the following two conditions:
(1) $0.2 a=m \times m$;
(2) $0.5 a=n \times n \times n$.
where $m, n$ are natural numbers, then the minimum value of $a$ is $\qquad$
Translate the above text into English, please retain the original text's line breaks and format, and output the translation result directly. | 2000 | 99 | 4 |
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 | 9. (12 points) Two people, A and B, stand facing each other 30 meters apart, playing "Rock, Paper, Scissors". The winner moves forward 3 meters, the loser moves back 2 meters, and in the case of a tie, both move forward 1 meter. After 15 rounds, A is 17 meters from the starting point, and B is 2 meters from the starting point. How many times did A win? $\qquad$ | 7 | 103 | 1 |
math | 14. [40] Find an explicit, closed form formula for
$$
\sum_{k=1}^{n} \frac{k \cdot(-1)^{k} \cdot\binom{n}{k}}{n+k+1}
$$ | \frac{-1}{\binom{2n+1}{n}} | 55 | 16 |
math | 4. A die is rolled. Find the probability that: a) an even number of points will fall (event $A$); b) a number of points divisible by 3 will fall (event $B$); c) any number of points except 5 will fall (event $C$). | P(A)=\frac{1}{2},P(B)=\frac{1}{3},P(C)=\frac{5}{6} | 62 | 30 |
math | 1. (2 points) Find all numbers $x, y$ for which the equality
$$
4 x^{2}+4 x y \sqrt{7}+8 y^{2}+2 y \sqrt{3}-\frac{2 y}{\sqrt{3}}+\frac{7}{3}=1
$$ | -\frac{2}{\sqrt{3}},\quad\frac{\sqrt{7}}{\sqrt{3}} | 70 | 24 |
math | 58. The son of a mathematics professor. The mathematics professor wrote a polynomial $f(x)$ with integer coefficients on the board and said:
- Today is my son's birthday. If his age $A$ is substituted into this polynomial instead of $x$, then the equation $f(A)=A$ holds. Note also that $f(0)=P$, where $P$ is a prime number greater than $A$.
How old is the professor's son? | 1 | 96 | 1 |
math | 11. (22 points) Let $x, y \in [0,1]$. Find the range of
$$
f(x, y)=\sqrt{\frac{1+x y}{1+x^{2}}}+\sqrt{\frac{1-x y}{1+y^{2}}}
$$ | [1,2] | 63 | 5 |
math | 2. Find the sum of the first twelve terms of an arithmetic progression if its fifth term $a_{5}=1$, and the seventeenth term $a_{17}=18$. | 37.5 | 39 | 4 |
math | 2. Determine the sum of the numbers
$$
1+2 \cdot 2+3 \cdot 2^{2}+4 \cdot 2^{3}+5 \cdot 2^{4}+\ldots+2001 \cdot 2^{2000}
$$ | 2000\cdot2^{2001}+1 | 64 | 15 |
math | 2. The sides of a rectangle were reduced: the length - by $10 \%$, the width - by $20 \%$. As a result, the perimeter of the rectangle decreased by $12 \%$. By what percentage will the perimeter of the rectangle decrease if its length is reduced by $20 \%$ and its width is reduced by $10 \%$? | 18 | 77 | 2 |
math | Let $ a_1,a_2, \cdots ,a_{2015} $ be $2015$-tuples of positive integers (not necessary distinct) and let $ k $ be a positive integers. Denote $\displaystyle f(i)=a_i+\frac{a_1a_2 \cdots a_{2015}}{a_i} $.
a) Prove that if $ k=2015^{2015} $, there exist $ a_1, a_2, \cdots , a_{2015} $ such that $ f(i)= k $ for all $1\le i\le 2015 $.\\
b) Find the maximum $k_0$ so that for $k\le k_0$, there are no $k$ such that there are at least $ 2 $ different $2015$-tuples which fulfill the above condition. | k_0 = 2 | 206 | 7 |
math | Question 8 Find $A^{2}$, where $A$ is the sum of the absolute values of all roots of the equation
$$
x=\sqrt{19}+\frac{91}{\sqrt{19}+\frac{91}{\sqrt{19}+\frac{91}{\sqrt{19}+\frac{91}{\sqrt{19}+\frac{91}{x}}}}}
$$
(9th American Invitational Mathematics Examination) | 383 | 106 | 3 |
math | 5.1. Among the numbers exceeding 2013, find the smallest even number $N$ for which the fraction $\frac{15 N-7}{22 N-5}$ is reducible. | 2144 | 46 | 4 |
math | 7. The function $y=2 \cos \left(x+\frac{\pi}{4}\right) \cos \left(x-\frac{\pi}{4}\right)+\sqrt{3} \sin 2 x$ has a maximum value and a minimum positive period. The sum of these two values is | 2+\pi | 64 | 3 |
math | Simplify the following fraction as much as possible:
$$
\frac{(x+y-z)^{3}}{(x+y)(z-x)(z-y)}
$$
given that $z^{3}=x^{3}+y^{3}$. | 3 | 51 | 1 |
math | 1026. Find the indices of all Fibonacci sequence members that are divisible by 3. | allnaturaldivisible4 | 20 | 5 |
math | ## Task A-4.7.
Determine all natural numbers $n$ such that $\log _{2}\left(3^{n}+7\right)$ is also a natural number. | 2 | 41 | 1 |
math | Solve the following equation:
$$
11 x-x^{2}=24-2 \sqrt{x}
$$ | x_{2}=9,\quadx_{3}=\frac{9-\sqrt{17}}{2} | 24 | 24 |
math | Find the largest positive integer $n$ ($n \ge 3$), so that there is a convex $n$-gon, the tangent of each interior angle is an integer. | n_{\text{max}} = 8 | 38 | 11 |
math | Example 8 Find the minimum value of $\frac{a}{\sin \theta}+\frac{b}{\cos \theta}\left(a, b>0, \theta \in\left(0, \frac{\pi}{2}\right)\right)$. | (\sqrt[3]{^{2}}+\sqrt[3]{b^{2}})^{4} | 56 | 21 |
math | Problem 10.3. Yura has $n$ cards, on which numbers from 1 to $n$ are written. After losing one of them, the sum of the numbers on the remaining cards turned out to be 101. What number is written on the lost card? | 4 | 61 | 1 |
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