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 | 22. For each positive integer $n$, define $A_{n}=\frac{20^{n}+11^{n}}{n!}$, where $n!=1 \times 2 \times \cdots \times n$. Find the value of $n$ that maximizes $A_{n}$. | 19 | 69 | 2 |
math | ## Task Condition
Find the derivative.
$$
y=\frac{1}{\sqrt{2}} \ln \left(\sqrt{2} \tan x+\sqrt{1+2 \tan^{2} x}\right)
$$ | \frac{1}{\cos^{2}x\sqrt{1+2\tan^{2}x}} | 49 | 24 |
math | What is the greatest common divisor of the numbers $9 m+7 n$ and $3 m+2 n$, if the numbers $m$ and $n$ have no common divisors other than one?
# | 3 | 44 | 1 |
math | 5. (20 points) Professor K., wishing to be known as a wit, plans to tell no fewer than two but no more than three different jokes at each of his lectures. At the same time, the sets of jokes told at different lectures should not coincide. How many lectures in total will Professor K. be able to give if he knows 8 jokes? | 84 | 75 | 2 |
math | Find all natural numbers $k$ for which
$$
1^{k}+9^{k}+10^{k}=5^{k}+6^{k}+11^{k}
$$
holds. | 0,2,4 | 46 | 5 |
math | 2. List all three-digit numbers divisible by 9 whose digits are prime numbers. | 225,252,522,333 | 17 | 15 |
math | \section*{Problem 2}
Find the smallest positive integer which can be represented as \(36^{\mathrm{m}}-5^{\mathrm{n}}\).
\section*{Answer}
11
| 11 | 45 | 2 |
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 | [Coordinate method in space]
Find the angle between the line passing through points $A(-3 ; 0 ; 1)$ and $B(2 ; 1 ;-1)$, and the line passing through points $C(-2 ; 2 ; 0)$ and $D(1 ; 3 ; 2)$. | \arccos\frac{2\sqrt{105}}{35} | 68 | 19 |
math | 9. Point $P$ is a moving point on circle $C:(x+2)^{2}+y^{2}=4$, and it is not at the origin. The fixed point $A$ has coordinates $(2,0)$. The perpendicular bisector of line segment $AP$ intersects line $CP$ at point $Q$. Let $M(-1,0), N(1,0)$, then the product of the slopes of lines $MQ$ and $NQ$ is $\qquad$. | 3 | 107 | 1 |
math | Example 2.5. $I=\int_{-1}^{1} x|x| d x$.
Translate the above text into English, please keep the original text's line breaks and format, and output the translation result directly.
However, since the text provided is already in a form that is commonly used in English for mathematical expressions, the translation is e... | 0 | 101 | 1 |
math | What is the maximum number of terms in a geometric progression with common ratio greater than 1 whose entries all come from the set of integers between 100 and 1000 inclusive? | 6 | 40 | 1 |
math | 2. (1 mark) A clock has an hour hand of length 3 and a minute hand of length 4. From 1:00 am to $1: 00 \mathrm{pm}$ of the same day, find the number of occurrences when the distance between the tips of the two hands is an integer.
(1 分) 一時鐘的時針長為 3 , 分針長為 4 。問在同一天上午一時至下午一時的一段時間内, 時針與分針的端點的距離為整數多少次？ | 132 | 124 | 3 |
math | For each positive integer $n$, define $s(n)$ to be the sum of the digits of $n$. For example, $s(2014)=2+0+1+4=7$. Determine all positive integers $n$ with $1000 \leq n \leq 9999$ for which $\frac{n}{s(n)}=112$. | 1008,1344,1680,2688 | 84 | 19 |
math | 2. Solve the equation $(\cos 2 x-2 \cos 4 x)^{2}=9+\cos ^{2} 5 x$. | \frac{\pi}{2}+k\pi,k\in\mathrm{Z} | 33 | 19 |
math | 7. Given $\alpha, \beta \in[0, \pi]$, then the maximum value of $(\sin \alpha+\sin (\alpha+\beta)) \cdot \sin \beta$ is | \frac{8\sqrt{3}}{9} | 42 | 12 |
math | 1. 182 For 1 to 1000000000, find the sum of the digits of each number; then for the resulting 1 billion numbers, find the sum of the digits of each number, $\cdots$, until obtaining 1 billion single-digit numbers. Question: Among the resulting numbers, are there more 1s or 2s? | 1 | 82 | 1 |
math | ## Task B-1.2.
In a sequence of six natural numbers, the third and each subsequent number is equal to the sum of the two preceding ones. Determine all such sequences of numbers, if the fifth number in the sequence is equal to 25. | \begin{pmatrix}11,1,12,13,25,38\\8,3,11,14,25,39\\5,5,10,15,25,40\\2,7,9,16,25,41\end{pmatrix} | 54 | 75 |
math | 11.048. Find the volume of an oblique triangular prism, the base of which is an equilateral triangle with a side equal to $a$, if the lateral edge of the prism is equal to the side of the base and is inclined to the base plane at an angle of $60^{\circ}$. | \frac{3^{3}}{8} | 68 | 10 |
math | 35. [20] A random permutation of $\{1,2, \ldots, 100\}$ is given. It is then sorted to obtain the sequence $(1,2, \ldots, 100)$ as follows: at each step, two of the numbers which are not in their correct positions are selected at random, and the two numbers are swapped. If $s$ is the expected number of steps (i.e. swap... | 2427 | 170 | 4 |
math | ## Task B-4.4.
A sequence of real numbers $\left(a_{n}\right)$ is defined recursively by
$$
a_{1}=1, \quad a_{2}=\frac{3}{5}, \quad a_{n}=\frac{a_{n-2} \cdot a_{n-1}}{2 a_{n-2}-a_{n-1}} \quad \text { for } n \geqslant 3
$$
Determine $a_{2020}$. | \frac{1}{1347} | 113 | 10 |
math | Example 13 (2004-2005 Hungarian Mathematical Olympiad) Find the largest integer $k$ such that $k$ satisfies the following condition: for all integers $x, y$, if $x y+1$ is divisible by $k$ then $x+y$ is also divisible by $k$. | 24 | 69 | 2 |
math | 8.3. Determine the real numbers $a_{1}, a_{2}, a_{3}, a_{4}$, which satisfy the equality
$$
\sqrt{a_{1}}+\sqrt{a_{2}-1}+\sqrt{a_{3}-2}+\sqrt{a_{4}-3}=\frac{1}{2} \cdot\left(a_{1}+a_{2}+a_{3}+a_{4}\right)-1
$$
Find the real numbers $a_{1}, a_{2}, a_{3}, a_{4}$ which satisfy the equality
$$
\sqrt{a_{1}}+\sqrt{a_{2}-1}... | a_{1}=1,a_{2}=2,a_{3}=3,a_{4}=4 | 200 | 20 |
math | For how many positive integers $n \le 500$ is $n!$ divisible by $2^{n-2}$?
[i]Proposed by Eugene Chen[/i] | 44 | 39 | 2 |
math | 5. Given a natural number $x=5^{n}-1$, where $n-$ is a natural number. It is known that $x$ has exactly three distinct prime divisors, one of which is 11. Find $x$. | 3124 | 51 | 4 |
math | 15. Let $0<\theta<\pi$, find the maximum value of $\sin \frac{\theta}{2}(1+\cos \theta)$. | \frac{4\sqrt{3}}{9} | 34 | 12 |
math | 6. Let $k$ and $m$ be two real constants, and the function $f: \mathbf{R} \rightarrow \mathbf{R}$ satisfies for any $x, y \in \mathbf{R}$,
$$
f(x f(y))=k x y+x+m \text{. }
$$
Then the necessary and sufficient conditions for $k$ and $m$, and all such expressions for $f(x)$ are $\qquad$ | m \neq 0, k=\frac{1}{m^{2}}, f(x)=\frac{x}{m}+m | 99 | 28 |
math | 11. The solution set of the equation $16 \sin \pi x \cdot \cos \pi x=16 x+\frac{1}{x}$ is $\qquad$ . | {\frac{1}{4},-\frac{1}{4}} | 41 | 14 |
math | 2. (2004 College Entrance Examination - Zhejiang Paper) In $\triangle A B C$, the sides opposite to angles $A, B, C$ are $a, b, c$ respectively, and $\cos A=\frac{1}{3}$.
(1) Find the value of $\sin ^{2} \frac{B+C}{2}+\cos 2 A$;
(2) If $a=\sqrt{3}$, find the maximum value of $b c$.
保留了原文的换行和格式。 | \frac{9}{4} | 116 | 7 |
math | 891. Solve the equation in natural numbers
$$
3^{x}-2^{y}=1
$$ | (1;1),(2;3) | 24 | 9 |
math | 8. (5 points) Anya and Kolya were collecting apples. It turned out that Anya collected as many apples as the percentage of the total number of apples collected by Kolya, and Kolya collected an odd number of apples. How many apples did Anya and Kolya collect together? | 25,300,525,1900,9900 | 65 | 20 |
math | Three. (50 points) The sequence $\left\{x_{n}\right\}$ satisfies
$$
x_{1}=3, x_{n+1}=\left[\sqrt{2} x_{n}\right]\left(n \in \mathbf{N}_{+}\right) \text {. }
$$
Find all $n$ such that $x_{n} 、 x_{n+1} 、 x_{n+2}$ form an arithmetic sequence, where $[x]$ denotes the greatest integer not exceeding the real number $x$.
保留... | 1 \text{ or } 3 | 252 | 8 |
math | Let $T_1$ be an isosceles triangle with sides of length 8, 11, and 11. Let $T_2$ be an isosceles triangle with sides of length $b$, 1, and 1. Suppose that the radius of the incircle of $T_1$ divided by the radius of the circumcircle of $T_1$ is equal to the radius of the incircle of $T_2$ divided by the radius of the... | \frac{14}{11} | 126 | 9 |
math | 6. For $0<x<1$, if the complex number
$$
z=\sqrt{x}+\mathrm{i} \sqrt{\sin x}
$$
corresponds to a point, then the number of such points inside the unit circle is $n=$ | 1 | 54 | 1 |
math | 9. (16 points) In $\triangle A B C$, $\overrightarrow{A B} \cdot \overrightarrow{A C}+2 \overrightarrow{B A} \cdot \overrightarrow{B C}=3 \overrightarrow{C A} \cdot \overrightarrow{C B}$.
Find the maximum value of $\sin C$. | \frac{\sqrt{7}}{3} | 75 | 10 |
math | GS. 4 Let $x, y$ and $z$ be real numbers that satisfy $x+\frac{1}{y}=4, y+\frac{1}{z}=1$ and $z+\frac{1}{x}=\frac{7}{3}$. Find the value of $x y z$. (Reference 2010 FG2.2) | 1 | 79 | 1 |
math | 5.1. (12 points) The decreasing sequence $a, b, c$ is a geometric progression, and the sequence $19 a, \frac{124 b}{13}, \frac{c}{13}$ is an arithmetic progression. Find the common ratio of the geometric progression. | 247 | 65 | 3 |
math | 3. On the left half of the board, the number 21 is written, and on the right half, the number 8 is written. It is allowed to take an arbitrary number \(a\) from the left half of the board and an arbitrary number \(b\) from the right half of the board, compute the numbers \(ab\), \(a^3 + b^3\), and write \(ab\) on the l... | No | 138 | 1 |
math | 14. A non-zero natural number $\mathrm{n}$, is both the sum of 2010 natural numbers with the same digit sum, the sum of 2012 natural numbers with the same digit sum, and the sum of 2013 natural numbers with the same digit sum. What is the smallest value of $m$? | 6036 | 74 | 4 |
math | Let $ m, n \geq 1$ be two coprime integers and let also $ s$ an arbitrary integer. Determine the number of subsets $ A$ of $ \{1, 2, ..., m \plus{} n \minus{} 1\}$ such that $ |A| \equal{} m$ and $ \sum_{x \in A} x \equiv s \pmod{n}$. | \frac{1}{n} \binom{m+n-1}{m} | 89 | 18 |
math | In Miroslav's kingdom, the cobbler Matěj used to go not only to sing but also to eat and drink well. For one gold piece, he got a whole goose and one jug of wine. Then, however, they increased the prices by $20 \%$, and for a gold piece, he got only half a jug of wine and a whole goose. It is said that after the full m... | 0.96 | 119 | 4 |
math | $25.$ Let $C$ be the answer to Problem $27.$ What is the $C$-th smallest positive integer with exactly four positive factors?
$26.$ Let $A$ be the answer to Problem $25.$ Determine the absolute value of the difference between the two positive integer roots of the quadratic equation $x^2-Ax+48=0$
$27.$ Let $B$ be the... | 8 | 112 | 3 |
math | ## Problem Statement
Find the distance from point $M_{0}$ to the plane passing through three points $M_{1}, M_{2}, M_{3}$.
$M_{1}(2; -1; -2)$
$M_{2}(1; 2; 1)$
$M_{3}(5; 0; -6)$
$M_{0}(14; -3; 7)$ | 4\sqrt{14} | 91 | 7 |
math | I2.1 Determine the positive real root, $\alpha$, of $\sqrt{(x+\sqrt{x})}-\sqrt{(x-\sqrt{x})}=\sqrt{x}$. | \frac{4}{3} | 37 | 7 |
math | N3. Find the smallest number $n$ such that there exist polynomials $f_{1}, f_{2}, \ldots, f_{n}$ with rational coefficients satisfying
$$
x^{2}+7=f_{1}(x)^{2}+f_{2}(x)^{2}+\cdots+f_{n}(x)^{2} .
$$ | 5 | 79 | 1 |
math | 1. For a regular octagon $A_{1} A_{2} A_{3} A_{4} A_{5} A_{6} A_{7} A_{8}$ with side length 1, if any two points $A_{i} A_{j}$ are chosen, then the maximum value of $\overrightarrow{A_{i} A_{j}} \cdot \overrightarrow{A_{1} A_{2}}$ is . $\qquad$ | \sqrt{2}+1 | 100 | 7 |
math | 2. Positive numbers $a, b, c$ satisfy $\left\{\begin{array}{l}a+b+c=10, \\ a^{2}+b^{2}=c^{2} .\end{array}\right.$ Then the maximum value of $ab$ is $\qquad$ | 50(3-2 \sqrt{2}) | 64 | 11 |
math | 78. Arrange the polynomial $f(x)=x^{4}+2 x^{3}-3 x^{2}-4 x+1$ in powers of $x+1$. | f(x)=(x+1)^{4}-2(x+1)^{3}-3(x+1)^{2}+4(x+1)+1 | 38 | 33 |
math | 9. (2003 Taiwan Training Problem) Find all functions $f: \mathbf{N} \rightarrow \mathbf{N}$, for all $m, n \in \mathbf{N}$ satisfying $f\left(m^{2}+n^{2}\right)=$ $f^{2}(m)+f^{2}(n)$ and $f(1)>0$. | f(n)=n | 84 | 4 |
math | Let $[x]$ be the integer part of a number $x$, and $\{x\}=x-[x]$. Solve the equation
$$
[x] \cdot \{x\} = 1991 x .
$$ | x=-\frac{1}{1992} | 50 | 12 |
math | 2. (3 points) Find all solutions of the inequality $\cos 5+2 x+x^{2}<0$, lying in the interval $\left[-2 ;-\frac{37}{125}\right]$. | x\in(-1-\sqrt{1-\cos5};-\frac{37}{125}] | 47 | 23 |
math | ## Task 2 - 221232
Determine for all 30-tuples $\left(a_{1}, a_{2}, \ldots, a_{30}\right)$ of (not necessarily distinct) positive integers $a_{i}(i=1, \ldots, 30)$, which satisfy
$$
\sum_{i=1}^{30} a_{i}=1983
$$
the greatest value that the greatest common divisor $d$ of the numbers $a_{i}$ can take. | 3 | 118 | 1 |
math | Example 1 For $a, b, c \in \mathbf{R}^{+}$, compare $a^{3}+b^{3}+c^{3}$ with $a^{2} b+b^{2} c+c^{2} a$. | ^{3}+b^{3}+^{3}>^{2}b+b^{2}+^{2} | 56 | 24 |
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 | Ribamko A.V.
In the lower left corner of a $n \times n$ chessboard, there is a knight. It is known that the minimum number of moves it takes for the knight to reach the upper right corner is equal to the minimum number of moves it takes to reach the lower right corner. Find $n$.
# | 7 | 70 | 1 |
math | Problem 1. In two tubs, there are $4 \frac{1}{6}$ and $3 \frac{1}{2}$ liters of water respectively. How much water needs to be transferred from the first tub to the second tub so that after the transfer, both tubs have the same amount of water? Explain your answer! | \frac{1}{3} | 70 | 7 |
math | Find all function $f:\mathbb{Z}\to\mathbb{Z}$ satisfying
$\text{(i)}$ $f(f(m)+n)+2m=f(n)+f(3m)$ for every $m,n\in\mathbb{Z}$,
$\text{(ii)}$ there exists a $d\in\mathbb{Z}$ such that $f(d)-f(0)=2$, and
$\text{(iii)}$ $f(1)-f(0)$ is even. | f(2n) = 2n + 2u \text{ and } f(2n + 1) = 2n + 2v \quad \forall n \in \mathbb{Z} | 109 | 47 |
math | One, (40 points) Integers $a, b, c, d$ satisfy $ad - bc = 1$. Find the minimum value of $a^2 + b^2 + c^2 + d^2 + ab + cd - ac - bd - bc$, and determine all quadruples $(a, b, c, d)$ that achieve this minimum value. | 2 | 79 | 1 |
math | 1. Determine the value of the parameter $k$ so that for the solutions $x_{1}$ and $x_{2}$ of the equation
$$
2 x^{2}+3 x+3 k+1=0
$$
the following holds
$$
2 x_{1}-x_{2}=3
$$ | -1 | 69 | 2 |
math | 1. Let positive numbers $x, y, z$ satisfy
$$
\frac{1}{x^{3}}=\frac{8}{y^{3}}=\frac{27}{z^{3}}=\frac{k}{(x+y+z)^{3}} \text {. }
$$
Then $k=$ $\qquad$ | 216 | 70 | 3 |
math | Example 5 Given $x y z=1, x+y+z=2, x^{2}+$ $y^{2}+z^{2}=16$. Then $\frac{1}{x y+2 z}+\frac{1}{y z+2 x}+\frac{1}{z x+2 y}=$ | -\frac{4}{13} | 70 | 8 |
math | 3. Let $S=\frac{1}{\sqrt{1}+\sqrt{2}}+\frac{1}{\sqrt{3}+\sqrt{4}}+\cdots+\frac{1}{\sqrt{9999}+\sqrt{10000}}$, find $[S]$ | 49 | 66 | 2 |
math | 7. The solution set of the inequality $(x-2) \sqrt{x^{2}-2 x-3} \geqslant 0$ is $\qquad$ . | x \geqslant 3 \text{ or } x = -1 | 38 | 17 |
math | let $x,y,z$ be positive reals , such that $x+y+z=1399$ find the
$$\max( [x]y + [y]z + [z]x ) $$
( $[a]$ is the biggest integer not exceeding $a$) | \max( \lfloor x \rfloor y + \lfloor y \rfloor z + \lfloor z \rfloor x ) = 652400 | 63 | 38 |
math | In the universe of Pi Zone, points are labeled with $2 \times 2$ arrays of positive reals. One can teleport from point $M$ to point $M'$ if $M$ can be obtained from $M'$ by multiplying either a row or column by some positive real. For example, one can teleport from $\left( \begin{array}{cc} 1 & 2 \\ 3 & 4 \end{array} \... | 17 | 257 | 2 |
math | 3. Solve the system of equations $\left\{\begin{array}{l}x^{2} y+x y^{2}+3 x+3 y+24=0, \\ x^{3} y-x y^{3}+3 x^{2}-3 y^{2}-48=0 .\end{array}\right.$ | (-3,-1) | 73 | 5 |
math | Problem 2. (Option 1). Given an acute triangle $\mathrm{ABC}(\mathrm{AB}=\mathrm{BC})$ and $\mathrm{BC}=12$. $A N \perp B C$. On the side $\mathrm{BC}$, a point $M$ (M lies between B and $\mathrm{N}$) is marked such that $\mathrm{AN}=\mathrm{MN}$ and $\angle \mathrm{BAM}=\angle \mathrm{NAC}$. Find $\mathrm{BN}$. | 6\sqrt{3} | 112 | 6 |
math | Between the ports of Mumraj and Chaos, two ships are plying the same route. They spend a negligible amount of time in the ports, immediately turn around, and continue their journey. One morning, at the same moment, a blue ship sets sail from the port of Mumraj and a green ship from the port of Chaos. For the first time... | 35\mathrm{~} | 164 | 7 |
math | Find the smallest positive integer $k$ such that there is exactly one prime number of the form $kx + 60$ for the integers $0 \le x \le 10$. | 17 | 41 | 2 |
math | Determine all triples of natural numbers $(a,b, c)$ with $b> 1$ such that $2^c + 2^{2016} = a^b$. | (a, b, c) = (3 \cdot 2^{1008}, 2, 2019) | 40 | 30 |
math | 1. Given that $n$ is a positive integer. Find the smallest positive integer $k$ such that for any real numbers $a_{1}, a_{2}, \cdots, a_{d}$, and
$$
a_{1}+a_{2}+\cdots+a_{d}=n\left(0 \leqslant a_{i} \leqslant 1, i=1,2, \cdots, d\right) \text {, }
$$
it is always possible to partition these numbers into $k$ groups (some... | 2n-1 | 143 | 4 |
math | Practice problem: Given the sequence $\left\{\frac{1}{n(n+1)}\right\}$, find $S_{n}$ | \frac{n}{n+1} | 30 | 8 |
math | When askes: "What time is it?", father said to a son: "Quarter of time that passed and half of the remaining time gives the exact time". What time was it? | 9:36 | 38 | 4 |
math | The second problem of the second round of the Rákosi Mátyás competition read as follows: "On a railway line, a passenger train departs from location $A$ to location $C$. When the train passes through $B$, a freight train departs from $B$ towards $A$. When the freight train arrives at $A$, a fast train departs from $A$ ... | \begin{gathered}40\leqqc_{1}\leqq50\\40\geqqc_{2}\geqq33\frac{1}{3}\\85\frac{5}{7}\leqqc_{3}\leqq150\end{gathered} | 278 | 66 |
math | Determine all polynomials $P(x)$ with real coefficients and which satisfy the following properties:
i) $P(0) = 1$
ii) for any real numbers $x$ and $y,$
\[|y^2-P(x)|\le 2|x|\quad\text{if and only if}\quad |x^2-P(y)|\le 2|y|.\] | P(x) = x^2 + 1 | 84 | 11 |
math | The function $f: N \to N_0$ is such that $f (2) = 0, f (3)> 0, f (6042) = 2014$ and $f (m + n)- f (m) - f (n) \in\{0,1\}$ for all $m,n \in N$. Determine $f (2014)$. $N_0=\{0,1,2,...\}$ | 671 | 107 | 3 |
math | 3. We remind you that the sum of the angles in a triangle equals 180 degrees. In triangle $A B C$, angle $A$ is a right angle. Let $B M$ be the median of the triangle, and $D$ be the midpoint of $B M$. It turns out that $\angle A B D = \angle A C D$. What are these angles? | 30 | 82 | 2 |
math | 515. Find all natural numbers that decrease by 14 times when the last digit is erased. | 14,28 | 22 | 5 |
math | Example 1 Calculate $(x+y+z)(xy+yz+zx)$.
Analysis: Since both factors in the original expression are cyclic symmetric expressions in $x, y, z$, by property 5, their product is also a cyclic symmetric expression in $x, y, z$. Therefore, we only need to multiply the first letter of the first factor by the second factor, ... | x^{2} y+z x^{2}+y^{2} z+x y^{2}+z^{2} x+y z^{2}+3 x y z | 91 | 37 |
math | Find the maximum possible value of the inradius of a triangle whose vertices lie in the interior, or on the boundary, of a unit square. | \frac{\sqrt{5}-1}{4} | 29 | 11 |
math | Let's determine $m$ such that the expression
$$
m x^{2}+(m-1) x+m-1
$$
is negative for all values of $x$.
---
Determine $m$ so that the expression
$$
m x^{2}+(m-1) x+m-1
$$
is negative for all values of $x$. | <-\frac{1}{3} | 79 | 8 |
math | Given the positive real numbers $a_{1}<a_{2}<\cdots<a_{n}$, consider the function \[f(x)=\frac{a_{1}}{x+a_{1}}+\frac{a_{2}}{x+a_{2}}+\cdots+\frac{a_{n}}{x+a_{n}}\] Determine the sum of the lengths of the disjoint intervals formed by all the values of $x$ such that $f(x)>1$. | \sum_{i=1}^n a_i | 101 | 11 |
math | 15.15 A paper punch can be placed at any point in the plane, and when it operates, it can punch out points at an irrational distance from it. What is the minimum number of paper punches needed to punch out all points in the plane?
(51st Putnam Mathematical Competition, 1990) | 3 | 68 | 1 |
math | 8. In the arithmetic sequence $\left\{a_{n}\right\}$, for any positive integer $n$, it holds that $a_{n}+2 a_{n+1}+3 a_{n+2}=6 n+22$, then $a_{2017}=$ $\qquad$ | \frac{6058}{3} | 69 | 10 |
math | ## Task Condition
Find the derivative.
$y=2 x-\ln \left(1+\sqrt{1-e^{4 x}}\right)-e^{-2 x} \cdot \arcsin \left(e^{2 x}\right)$ | 2e^{-2x}\cdot\arcsin(e^{2x}) | 51 | 16 |
math | 364. Find all natural numbers $a$ for which the number $a^{2}-10 a+21$ is prime. | 8,2 | 30 | 3 |
math | 2. [4 points] Find all integer parameter triples $a, b$, and $c$, for each of which the system of equations
$$
\left\{\begin{array}{l}
a x+2 y+c z=c \\
3 x+b y+4 z=4 b
\end{array}\right.
$$
has no solutions. | (-6,-1,-8),(-3,-2,-4),(3,2,4) | 74 | 20 |
math | For which positive real numbers $a, b$ does the inequality
$$
x_{1} \cdot x_{2}+x_{2} \cdot x_{3}+\cdots+x_{n-1} \cdot x_{n}+x_{n} \cdot x_{1} \geq x_{1}^{a} \cdot x_{2}^{b} \cdot x_{3}^{a}+x_{2}^{a} \cdot x_{3}^{b} \cdot x_{4}^{a}+\cdots+x_{n}^{a} \cdot x_{1}^{b} \cdot x_{2}^{a}
$$
hold for all integers $n>2$ and po... | b=1, a=\frac{1}{2} | 178 | 12 |
math | 3. Find the smallest positive integer $m$ such that $5 m$ is a fifth power of a positive integer, $6 m$ is a sixth power of a positive integer, and $7 m$ is a seventh power of a positive integer.
(2013, Irish Mathematical Olympiad) | 2^{35} \times 3^{35} \times 5^{84} \times 7^{90} | 63 | 29 |
math | 2. Solve the equation $x+\frac{x}{\sqrt{x^{2}-1}}=\frac{35}{12}$. | \frac{5}{4},\frac{5}{3} | 29 | 14 |
math | ## Task A-2.1.
Determine all ordered triples $(x, y, z)$ of real numbers for which
$$
x^{2}+y^{2}=5, \quad x z+y=7, \quad y z-x=1
$$ | (2,1,3)(-\frac{11}{5},\frac{2}{5},-3) | 56 | 25 |
math | 22. Suppose that $x_{1}, x_{2}$ and $x_{3}$ are the three roots of $(11-x)^{3}+(13-x)^{3}=(24-2 x)^{3}$. Find the value of $x_{1}+x_{2}+x_{3}$. | 36 | 72 | 2 |
math | XLVI OM - II - Problem 6
A square with side length $ n $ is divided into $ n^2 $ unit squares. Determine all natural numbers $ n $ for which such a square can be cut along the lines of this division into squares, each of which has a side length of 2 or 3. | nisdivisible2or3 | 67 | 6 |
math | 52. Find the equations of the line passing through the point with coordinates $(3,-2,4)$ and perpendicular to the plane given by the equation $5 x-3 y+3 z=10$.
53*. Prove that if the coordinates of any point are substituted into the left side of the normal equation of a plane, written in the form:
$$
x \cos \alpha + ... | \frac{x-3}{5}=\frac{y+2}{-3}=\frac{z-4}{3} | 143 | 27 |
math | The polynomial $1976(x+x^2+ \cdots +x^n)$ is decomposed into a sum of polynomials of the form $a_1x + a_2x^2 + \cdots + a_nx^n$, where $a_1, a_2, \ldots , a_n$ are distinct positive integers not greater than $n$. Find all values of $n$ for which such a decomposition is possible. | 7, 103, 1975 | 95 | 12 |
math | 3. Let positive integers $a, b$ be such that $15a + 16b$ and $16a - 15b$ are both squares of positive integers. Find the smallest value that the smaller of these two squares can take. ${ }^{[3]}$
(1996, China Mathematical Olympiad) | 481^2 | 73 | 5 |
math | 4. (7 points) On the board, 48 ones are written. Every minute, Karlson erases two arbitrary numbers and writes their sum on the board, and then eats a number of candies equal to the product of the two erased numbers. What is the maximum number of candies he could have eaten in 48 minutes? | 1128 | 69 | 4 |
math | 7. (IMO-10 Problem) Let $[x]$ denote the greatest integer not exceeding $x$. Find the value of $\sum_{k=0}^{\infty}\left[\frac{n+2^{k}}{2^{k+1}}\right]$, where $n$ is any natural number. | n | 68 | 1 |
math | $A$, $B$, $C$, and $D$ are points on a circle, and segments $\overline{AC}$ and $\overline{BD}$ intersect at $P$, such that $AP=8$, $PC=1$, and $BD=6$. Find $BP$, given that $BP<DP$. | 2 | 69 | 1 |
math | Find all pairs of real numbers $(x, y)$, that satisfy the system of equations: $$\left\{\begin{matrix} 6(1-x)^2=\frac{1}{y},\\6(1-y)^2=\frac{1}{x}.\end{matrix}\right.$$ | \left(\frac{3}{2}, \frac{2}{3}\right) | 63 | 19 |
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