task_type stringclasses 4
values | problem stringlengths 21 5.23k | answer stringlengths 1 8.29k | problem_tokens int64 11 1.16k | answer_tokens int64 1 2.04k |
|---|---|---|---|---|
math | 5. In the sum $1+3+5+\ldots+k$ of consecutive odd natural numbers, determine the largest addend $k$ such that $1+3+5+\ldots+k=40000$.
## Tasks worth 10 points: | 399 | 58 | 3 |
math | In trapezoid $A B C E$ base $A E$ is equal to $16, C E=8 \sqrt{3}$. The circle passing through points $A, B$ and $C$ intersects line $A E$ again at point $H ; \angle A H B=60^{\circ}$. Find $A C$. | 8 | 77 | 1 |
math | 23. Let $a=\frac{\sqrt{5}+\sqrt{3}}{\sqrt{5}-\sqrt{3}}$ and $b=\frac{\sqrt{5}-\sqrt{3}}{\sqrt{5}+\sqrt{3}}$. Find the value of $a^{4}+b^{4}+(a+b)^{4}$. | 7938 | 77 | 4 |
math | 3. The solution to the equation $\arctan 2^{x}-\arctan 2^{-x}=\frac{\pi}{6}$
is . $\qquad$ | \log_{2}\sqrt{3} | 39 | 9 |
math | $11 \cdot 14 n$ is a non-negative integer. Given $f(0)=0, f(1)=1, f(n)=$ $f\left(\left[\frac{n}{2}\right]\right)+n-2\left[\frac{n}{2}\right]$. Determine $f(n)$. Find the maximum value of $f(n)$ for $0 \leqslant n \leqslant 1991$. (Here $[x]$ denotes the greatest integer not exceeding $x$)
(Japan Mathematical Olympiad, ... | 10 | 127 | 2 |
math | 21.3. Find all two-digit numbers that are equal to three times the product of their digits.
$$
(7-8 \text {th grades })
$$ | 1524 | 35 | 4 |
math | 1. Let $A B C$ be a triangle with $\angle A=60^{\circ}$. Line $\ell$ intersects segments $A B$ and $A C$ and splits triangle $A B C$ into an equilateral triangle and a quadrilateral. Let $X$ and $Y$ be on $\ell$ such that lines $B X$ and $C Y$ are perpendicular to $\ell$. Given that $A B=20$ and $A C=22$, compute $X Y$... | 21 | 109 | 2 |
math | An integer has two prime factors. It has 6 divisors, and the sum of its divisors is 28. Which number is this? | 12 | 31 | 2 |
math | 6. Given a line $l$ in the plane and a point $P$ at a distance of 1 from $l$, 100 points $A_{1}, A_{2}, \cdots, A_{100}$ on $l$ satisfy
$$
\begin{array}{l}
\overrightarrow{P A_{i}} \cdot \overrightarrow{P A_{i+1}}=-\frac{2}{i}(i=1,2, \cdots, 99) . \\
\text { Then } \overrightarrow{P A_{100}} \cdot \overrightarrow{P A_{... | -\frac{51}{50} | 146 | 9 |
math | 9.6. First, Betya writes the numbers $1, 2$ in her notebook, and Nik writes $3, 4$ in his notebook. Then, at the beginning of each minute, Nik and Betya each write down a quadratic polynomial with real coefficients, whose roots are the two numbers in their notebooks, denoted as $f(x)$ and $g(x)$, respectively. If the e... | \frac{14}{5} | 205 | 8 |
math | 31. When Vitya becomes twice as old, Kolya will be 4 years younger than Vera. Last year, Kolya was half as old as Vera, and 3.5 times younger than Vitya. How old is each of them now? | Kolya:5,Vera:9,Vitya:15 | 57 | 15 |
math | ## 36. San Salvador Embankment
Would you come to have dinner with me tonight? I live in one of the eleven houses on San Salvador Embankment; however, to find out which one, you will have to think.
When, from my home, I look at the sea and multiply the number of houses to my left by the number of houses to my right, I... | 4 | 103 | 1 |
math | Let $a_{0}, a_{1}, a_{2}, \ldots$ be a sequence of real numbers such that $a_{0}=0, a_{1}=1$ and for all $n \geqslant 2$ there exists $1 \geqslant k \geqslant n$ such that
$$
a_{n}=\frac{a_{n-1}+\cdots+a_{n-k}}{k}
$$
What is the maximum value of $a_{2018}-a_{2017}$? | \frac{2016}{2017^{2}} | 122 | 15 |
math | 10. Finde die grösste natürliche Zahl $n$, sodass für alle reellen Zahlen $a, b, c, d$ folgendes gilt:
$$
(n+2) \sqrt{a^{2}+b^{2}}+(n+1) \sqrt{a^{2}+c^{2}}+(n+1) \sqrt{a^{2}+d^{2}} \geq n(a+b+c+d)
$$ | 24 | 100 | 2 |
math | 11. If from any $\mathrm{n}$ numbers, one can always select 4 numbers such that their sum is a multiple of 4, then what is the minimum value of $n$? | 7 | 41 | 1 |
math | 6・136 Let $N$ be the set of natural numbers. Let $f: N \rightarrow N$ satisfy the conditions: $f(1)=1$, and for any natural number $n$,
$$
\left\{\begin{array}{l}
3 f(n) f(2 n+1)=f(2 n)(1+3 f(n)), \\
f(2 n)<6 f(n) .
\end{array}\right.
$$
Find all solutions to the equation
$$
f(k)+f(l)=293, k<l
$$ | \begin{pmatrix}{\begin{pmatrix}k=5,\\=47;\end{pmatrix}.\\{\begin{pmatrix}k=13,\\=39;\end{pmatrix}.\\{\begin{pmatrix}k=7,\\=45;\end{pmatrix}.\\{\begin} | 122 | 73 |
math | Example 4. Find the asymptotes of the curve $f(x)=\frac{x^{2}+1}{x}$. | x | 27 | 1 |
math | Kovaldji A.K.
A football is sewn from 32 patches: white hexagons and black pentagons. Each black patch borders only white ones, and each white one borders three black and three white. How many white patches are there? | 20 | 54 | 2 |
math | 22. N4 (BUL) Find all positive integers $a$ and $b$ for which
$$
\left[\frac{a^{2}}{b}\right]+\left[\frac{b^{2}}{a}\right]=\left[\frac{a^{2}+b^{2}}{a b}\right]+a b
$$
where as usual, $[t]$ refers to greatest integer that is less than or equal to $t$. | (n,n^2+1) | 99 | 7 |
math | 13.221. Two riders set out simultaneously from points $A$ and $C$ to point $B$, and despite the fact that $C$ was 20 km farther from $B$ than $A$ was from $B$, they arrived at $B$ simultaneously. Find the distance from $C$ to $B$, if the rider who set out from $C$ traveled each kilometer 1 minute 15 seconds faster than... | 80 | 124 | 2 |
math | 2. Thirty beads (blue and green) were laid out in a circle. For 26 beads, the neighboring one was blue, and for 20 beads, the neighboring one was green. How many blue beads were there? | 18 | 48 | 2 |
math | Positive integers $m$ and $n$ satisfy $m n=5000$. If $m$ is not divisible by 10 and $n$ is not divisible by 10 , what is the value of $m+n$ ? | 633 | 52 | 3 |
math | G3.4 Given that the function $f$ satisfies $f(2+x)=f(2-x)$ for every real number $x$ and that $f(x)=0$ has exactly four distinct real roots. Find the sum of these four distinct real roots. | 8 | 55 | 1 |
math | [Example 3.3.4] Divide 1983 into the sum of 12 positive integers, so that the product of their factorials is minimized. | (165!)^{9}\times(166!)^{3} | 36 | 17 |
math | 1. Andrei, Boris, and Valentin participated in a 1 km race. (We assume that each of them ran at a constant speed). Andrei was 100 m ahead of Boris at the finish line. And Boris was 60 m ahead of Valentin at the finish line. What was the distance between Andrei and Valentin at the moment Andrei finished? | 154\mathrm{} | 80 | 6 |
math | Let $n$ be a fixed positive integer.
- Show that there exist real polynomials $p_1, p_2, p_3, \cdots, p_k \in \mathbb{R}[x_1, \cdots, x_n]$ such that
\[(x_1 + x_2 + \cdots + x_n)^2 + p_1(x_1, \cdots, x_n)^2 + p_2(x_1, \cdots, x_n)^2 + \cdots + p_k(x_1, \cdots, x_n)^2 = n(x_1^2 + x_2^2 + \cdots + x_n^2)\]
- Fi... | k = n-1 | 193 | 6 |
math | 2. Given that all terms of the sequence $\left\{a_{n}\right\}$ are positive, and the sum of the first $n$ terms $S_{n}$ satisfies $6 S_{n}=a_{n}^{2}+3 a_{n}+2$. If $a_{2}$, $a_{4}$, and $a_{9}$ form a geometric sequence, then the general term of the sequence $\left\{a_{n}\right\}$ is $a_{n}=$ $\qquad$ | 3n-2 | 114 | 4 |
math | ## Task Condition
Find the derivative.
$$
y=\sqrt[3]{\operatorname{ctg} 2}-\frac{1}{20} \cdot \frac{\cos ^{2} 10 x}{\sin 20 x}
$$ | \frac{1}{4\sin^{2}10x} | 57 | 15 |
math | Let $S$ be the set of all positive rational numbers $r$ such that when the two numbers $r$ and $55r$ are written as fractions in lowest terms, the sum of the numerator and denominator of one fraction is the same as the sum of the numerator and denominator of the other fraction. The sum of all the elements of $S$ can be... | 719 | 108 | 3 |
math | Example 3. Let $f(x)=\left\{\begin{array}{ll}0, & -1 \leqslant x<0, \\ 1, & 0<x \leqslant 1 .\end{array}\right.$ Find $\int_{-1}^{1} f(x) d x$.
untranslated text is kept in its original format and alignment. | 1 | 84 | 1 |
math | ## Problem Statement
Calculate the limit of the function:
$$
\lim _{x \rightarrow-1} \frac{x^{3}-3 x-2}{x+x^{2}}
$$ | 0 | 40 | 1 |
math | Example 7 (2006 Beijing College Entrance Examination Final Question) In the sequence $\left\{a_{n}\right\}$, if $a_{1}, a_{2}$ are positive integers, and $a_{n}=\left|a_{n-1}-a_{n-2}\right|$, $n=3,4,5, \cdots$, then $\left\{a_{n}\right\}$ is called an "absolute difference sequence".
(1) Give an example of an "absolute ... | 6 | 277 | 1 |
math | ## Task 4 - 160624
A feed mixture for breeding boars consists of oat groats, wheat bran, barley groats, minerals, and water, and is composed as follows: half of the mixture is oat groats, $\frac{1}{10}$ of the mixture is wheat bran, $\frac{1}{4}$ of the mixture is barley groats, $\frac{1}{100}$ of the mixture is miner... | 4.9 | 133 | 3 |
math | A rectangular floor is covered by a certain number of equally large quadratic tiles. The tiles along the edge are red, and the rest are white. There are equally many red and white tiles. How many tiles can there be? | 48, 60 | 45 | 6 |
math | 503. Three workers need to make 80 identical parts. Together, they make 20 parts per hour. The first worker started the job alone. He made 20 parts, spending more than 3 hours on their production. The remaining work was done by the second and third workers together. The entire job took 8 hours. How many hours would it ... | 16 | 89 | 2 |
math | ## Task 1 - 211221
If $a_{1}$ and $d$ are given real numbers, let $\left(a_{n}\right)$ be the arithmetic sequence with $a_{n}=a_{1}+(n-1) d$ for $n=1,2,3, \ldots$. Furthermore, for $n=1,2,3, \ldots$ define:
$$
s_{n}=\sum_{k=1}^{n} a_{k} \quad ; \quad z_{n}=\sum_{k=1}^{n} s_{k}
$$
a) Determine $a_{1}$ and $d$ such th... | a_{1}=\frac{5}{2},=-1 | 234 | 13 |
math | Example 14 (2002 Shanghai Competition Problem) Let $F$ be the set of all ordered $n$-tuples $\left(A_{1}, A_{2}, \cdots, A_{n}\right)$, where $A_{i}$ $(1 \leqslant i \leqslant n)$ are subsets of the set $\{1,2,3, \cdots, 2002\}$. Let $|A|$ denote the number of elements in set $A$. For all elements $\left(A_{1}, A_{2}, ... | 2002(2^{2002n}-2^{2001n}) | 230 | 21 |
math | Find all positive integers $N$ that are perfect squares and their decimal representation consists of $n$ digits equal to 2 and one digit equal to 5, where $n$ takes positive integer values. | 25 \text{ and } 225 | 42 | 11 |
math | Task 3. (15 points) At the research institute, a scientific employee, Ivan Ivanovich, received an object for research containing about 300 oil samples (a container designed for 300 samples, which was almost completely filled). Each sample has certain characteristics in terms of sulfur content - either low-sulfur or hig... | 120 | 167 | 3 |
math | ## Task A-4.2.
Determine all functions $f: \mathbb{N}_{0} \rightarrow \mathbb{N}_{0}$ such that for all $x \in \mathbb{N}_{0}, y \in \mathbb{N}$ the following holds:
$$
(f(x)+1)(f(y)+1)=(x+1)(f(y-1)+1)+f(x+1)
$$ | f(x)=x | 91 | 4 |
math | 9. (16 points) Let the function
$$
f(x)=3 a x^{2}-2 b x+c \text {. }
$$
If $a-b+c=0, f(0)>0, f(1)>0$, find the range of $\frac{a+3 b+7 c}{2 a+b}$. | (\frac{4}{3},\frac{7}{2}) | 72 | 14 |
math | 8. Given that $a_{1}, a_{2}, a_{3}, a_{4}, a_{5}$ is a permutation of $1,2,3,4,5$, and satisfies $\left|a_{i}-a_{i+1}\right| \neq 1(i=1,2,3,4)$. Then the number of permutations $a_{1}, a_{2}, a_{3}, a_{4}, a_{5}$ that meet the condition is $\qquad$. | 14 | 108 | 2 |
math | ## Task A-2.4.
Determine the largest real constant $\lambda$ such that for all positive real numbers $u$, $v, w$ for which $u \sqrt{v w}+v \sqrt{w u}+w \sqrt{u v} \geq 1$, the inequality $u+v+w \geq \lambda$ holds. | \sqrt{3} | 78 | 5 |
math | ## Task Condition
Find the derivative.
$y=\operatorname{arctg}\left(e^{x}-e^{-x}\right)$ | \frac{e^{x}+e^{-x}}{e^{2x}+e^{-2x}-1} | 29 | 26 |
math | 18. The number 2024 may be split into its first two digits and its last two digits to form the numbers 20 and 24. The highest common factor of these numbers, $\operatorname{HCF}(20,24)$ is equal to 4. Similarly, 2025 may be split, and $\operatorname{HCF}(20,25)=5$.
For how many remaining years this century (i.e. after ... | 30 | 128 | 2 |
math | 8. (17th "Hope Cup" Senior High School Grade 1 Training Question) If the function $f(x)$ is defined on $\mathbf{R}$ as an odd function and is increasing, for any $\theta \in \mathbf{R}$, the inequality $f(\cos 2 \theta-5)+f(2 m+4 \sin \theta)>0$ always holds, then the range of $m$ is
A. $m>5$
B. $m>2$
C. $24$ | >5 | 114 | 2 |
math | ## Task Condition
Find the derivative.
$y=-\frac{\operatorname{ch} x}{2 \operatorname{sh}^{2} x}-\frac{1}{2} \ln \left(\operatorname{th} \frac{x}{2}\right)$ | \frac{1}{\operatorname{sh}^{3}x} | 58 | 16 |
math | 5. Given the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1(a>b>0)$ with left and right foci $F_{1}$ and $F_{2}$, respectively, and $P$ as any point on the ellipse not coinciding with the left or right vertices, points $I$ and $G$ are the incenter and centroid of $\triangle P F_{1} F_{2}$, respectively. When $I G$ ... | \frac{1}{3} | 132 | 7 |
math | Let $F$ be the set of real polynomials $f(x)$ satisfying
(1) the degree of $f(x)$ is less than or equal to 3;
(2) for any $x \in [0,1]$, $|f(x)| \leqslant 1$.
Find $\max _{f \in F} f(2)$. | 99 | 80 | 2 |
math | 【Question 6】
It is known that a certain month has 31 days, and the number of Mondays and Fridays are the same. What day of the week is the 10th of this month? | Thursday | 45 | 1 |
math | 4. (8 points) In another 12 days, it will be 2016, Hao Hao sighs: I have only experienced 2 leap years so far, and the year I was born is a multiple of 9. So how old will Hao Hao be in 2016? | 9 | 65 | 1 |
math | 2. (3 points) A student, while solving a calculation problem, was supposed to divide by 20 in the last step, but mistakenly added 20, thus obtaining an incorrect result of 180. What should the correct answer to this calculation problem be? $\qquad$ . | 8 | 62 | 1 |
math | Three, (18 points) Solve the system of equations:
$$
\left\{\begin{array}{l}
2(x+y+z)-5 \sqrt{x+y+z+5}=2, \\
\frac{x}{3}=\frac{y}{4}=\frac{z}{5} .
\end{array}\right.
$$ | x=\frac{11}{4}, y=\frac{11}{3}, z=\frac{55}{12} | 71 | 28 |
math | 6. The probability that the product of the points obtained from rolling a die three times is divisible by 6 is $\qquad$ | \frac{133}{216} | 27 | 11 |
math | Let $a_{0}, a_{1}, a_{2}, \ldots$ be a sequence of real numbers such that $a_{0}=0, a_{1}=1$, and for every $n \geqslant 2$ there exists $1 \leqslant k \leqslant n$ satisfying $$ a_{n}=\frac{a_{n-1}+\cdots+a_{n-k}}{k} $$ Find the maximal possible value of $a_{2018}-a_{2017}$. (Belgium) Answer: The maximal value is ... | \frac{2016}{2017^{2}} | 148 | 15 |
math | 3. For the quadratic trinomials $f_{1}(x)=a x^{2}+b x+c_{1}, f_{2}(x)=a x^{2}+b x+c_{2}$, $\ldots, f_{2020}(x)=a x^{2}+b x+c_{2020}$, it is known that each of them has two roots.
Denote by $x_{i}$ one of the roots of $f_{i}(x)$, where $i=1,2, \ldots, 2020$. Find the value
$$
f_{2}\left(x_{1}\right)+f_{3}\left(x_{2}\r... | 0 | 189 | 1 |
math | 4. Find the limit of the variable quantity $x=\frac{a z+1}{z}$ as $z \rightarrow \infty$. | a | 30 | 1 |
math | Problem 1. Solve in the set of complex numbers the equation $|z-| z+1||=|z+| z-1||$. | bi,b\in\mathbb{R}or,\in[-1,1] | 32 | 18 |
math | 6. Given that the odd number $n$ is a three-digit number, and the sum of the last digits of all its factors (including 1 and $n$) is 33. Then $n$ $=$ . $\qquad$ | 729 | 52 | 3 |
math | ## Task 4 - 060734
The number $\frac{16}{15}$ is to be represented in the form $\frac{16}{15}=\frac{a}{m}+\frac{b}{n}$. Here, $a, b, m, n$ should be natural numbers such that the fractions $\frac{a}{m}$ and $\frac{b}{n}$ are not reducible and are not whole numbers.
Give three examples of such a representation, where
... | \frac{16}{15}=\frac{2}{15}+\frac{14}{15},\frac{16}{15}=\frac{2}{3}+\frac{2}{5},\frac{16}{15}=\frac{8}{15}+\frac{8}{15} | 161 | 74 |
math | 8. Evaluate $\frac{1}{3}+\frac{1}{15}+\frac{1}{35}+\frac{1}{63}+\frac{1}{99}+\frac{1}{143}+\frac{1}{195}$. | \frac{7}{15} | 61 | 8 |
math | 69. Let $f(x)=(x-1)(x-3)(x-5)(x-7)(x-9)$, then $f(0)+f(1)+f(2)+\ldots+f(10)=$ | 0 | 52 | 1 |
math | 10. Given the sequence $\left\{x_{n}\right\}, x_{1}=1$, and $x_{n+1}=\frac{\sqrt{3} x_{n}+1}{\sqrt{3}-x_{n}}, x_{1999}-x_{601}=$ | 0 | 68 | 1 |
math | Example 10 Let real numbers $x_{1}, x_{2}, \cdots, x_{n}$ satisfy
$$
\sum_{i=1}^{n} x_{i}^{2}=1 \quad(n \geqslant 3) \text {. }
$$
Find $\min _{1 \leqslant i<j \leqslant n}\left\{\left|x_{i}-x_{j}\right|\right\}$'s maximum value. | \sqrt{\frac{12}{n\left(n^{2}-1\right)}} | 105 | 19 |
math | ## Problem Statement
Find the derivative.
$$
y=x\left(2 x^{2}+1\right) \sqrt{x^{2}+1}-\ln \left(x+\sqrt{x^{2}+1}\right)
$$ | 8x^{2}\sqrt{x^{2}+1} | 51 | 13 |
math | Problem 3. Mario distributed the numbers 1, 2, ..., 8 at the vertices of a cube. When he calculated the sums obtained from each face, he noticed that all the sums were equal.
a) What is the value of each such sum?
b) Find one arrangement of the numbers 1, 2, ..., 8 at the vertices of the cube that has the desired pro... | 18 | 83 | 2 |
math | If $E, U, L, S, R, T$ represent $1, 2, 3, 4, 5, 6$ (different letters represent different numbers), and satisfy:
(1) $E+U+L=6$;
(2) $S+R+U+T=18$;
(3) $U \times T=15$;
(4) $S \times L=8$.
Then the six-digit number $\overline{E U L S R T}=$ _. $\qquad$ | 132465 | 119 | 6 |
math | 6.1. Find the largest six-digit number, all digits of which are different, and each of the digits, except for the extreme ones, is either the sum or the difference of the adjacent digits. | 972538 | 42 | 6 |
math | 12.66 In the interval $1 \leqslant n \leqslant 10^{6}$, how many integers $n$ are there such that the equation $n=x^{v}$ has non-negative integer solutions $x, y$, and $x \neq n$.
(Shanghai, China High School Mathematics Competition, 1990) | 1111 | 81 | 4 |
math | Putnam 1993 Problem B2 A deck of 2n cards numbered from 1 to 2n is shuffled and n cards are dealt to A and B. A and B alternately discard a card face up, starting with A. The game when the sum of the discards is first divisible by 2n + 1, and the last person to discard wins. What is the probability that A wins if neith... | 0 | 95 | 1 |
math | $22$ football players took part in the football training. They were divided into teams of equal size for each game ($11:11$). It is known that each football player played with each other at least once in opposing teams. What is the smallest possible number of games they played during the training. | 5 | 65 | 1 |
math | 1. Consider the 2022 fractions
$$
\frac{0}{2022}, \frac{1}{2021}, \frac{2}{2020}, \ldots, \frac{2021}{1}
$$
in the form of the ratio of two non-negative integers, whose sum for each fraction is equal to 2022. How many of them take integer values?
(Jaroslav Zhouf) | 8 | 100 | 1 |
math | 2. (10 points) Calculate: $1+2+4+5+7+8+10+11+13+14+16+17+19+20=$ | 147 | 46 | 3 |
math | Given a positive integer $n>1$. In the cells of an $n\times n$ board, marbles are placed one by one. Initially there are no marbles on the board. A marble could be placed in a free cell neighboring (by side) with at least two cells which are still free. Find the greatest possible number of marbles that could be placed ... | n^2 - n | 84 | 6 |
math | ## 4. The Hound and the Fox
The hound spots a fox 123 meters away and rushes after it. The fox runs ahead of the hound, and both move in the same direction. Each hound's leap is 2 meters long, and each fox's leap is 1 meter. While the hound jumps twice, the fox jumps three times. At what distance from its starting poi... | 492 | 104 | 3 |
math | 6. From the first 2005 natural numbers, $k$ of them are arbitrarily chosen. What is the least value of $k$ to ensure that there is at least one pair of numbers such that one of them is divisible by the other? | 1004 | 53 | 4 |
math | Find the $(x, y)$ natural integers such that $x^{2}=y^{2}+7 y+6$ | (6,3) | 26 | 5 |
math | 1. (6 points) $\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+\frac{1}{32}$. | \frac{31}{32} | 44 | 9 |
math | 11.5. A cube of size $n \times n \times n$, where $n$ is a natural number, was cut into 99 smaller cubes, of which exactly one has an edge length different from 1 (each of the others has an edge length of 1). Find the volume of the original cube. | 125 | 69 | 3 |
math | 9. (40 points) What is the maximum number of numbers that can be chosen among the natural numbers from 1 to 3000 such that the difference between any two of them is different from 1, 4, and 5? | 1000 | 53 | 4 |
math | 3. Find the sum: $S_{n}=1^{2}+2^{2}+3^{2}+\cdots+n^{2}$. | \frac{1}{6}n(n+1)(2n+1) | 33 | 17 |
math | 3. Determine all triples $(a, b, c)$ of positive integers for which
$$
2^{a+2 b+1}+4^{a}+16^{b}=4^{c} .
$$ | (,b,)=(2b,b,2b+1) | 46 | 14 |
math | A grasshopper is jumping about in a grid. From the point with coordinates $(a, b)$ it can jump to either $(a + 1, b),(a + 2, b),(a + 1, b + 1),(a, b + 2)$ or $(a, b + 1)$. In how many ways can it reach the line $x + y = 2014?$ Where the grasshopper starts in $(0, 0)$. | \frac{3}{4} \cdot 3^{2014} + \frac{1}{4} | 103 | 25 |
math | # 9. Problem 9
Vasya throws three dice (cubes with numbers from 1 to 6 on the faces) and adds the numbers that come up. Additionally, if all three numbers are different, he can throw all three dice again and add the numbers that come up to the already accumulated sum, and continue doing so until at least two of the th... | 23.625 | 92 | 6 |
math | 9・13 Solve the inequality
$$\sqrt{\frac{\pi}{4}-\operatorname{arctg} \frac{|x|+|y|}{\pi}}+\operatorname{tg}^{2} x+1 \leqslant \sqrt{2}|\operatorname{tg} x|(\sin x+\cos x)$$ | (x, y) \in \left\{\left(\frac{\pi}{4}, \frac{3\pi}{4}\right), \left(\frac{\pi}{4}, -\frac{3\pi}{4}\right)\right\} | 77 | 53 |
math | 1. $2001 \times 2001$ trees are arranged in a square grid in a park. What is the largest number of trees that can be cut down so that no stump is visible from another tree?
(The trees should have a diameter of 0)
## Solution | 1001^2 | 60 | 6 |
math | 13. Xiao Li and Xiao Zhang are running at a constant speed on a circular track. They start at the same time and place, with Xiao Li running clockwise and completing a lap every 72 seconds; Xiao Zhang running counterclockwise and completing a lap every 80 seconds. A quarter-circle arc interval is marked on the track, ce... | 46 | 104 | 2 |
math | Kovaldjei A.K.
A student did not notice the multiplication sign between two three-digit numbers and wrote a single six-digit number. The result turned out to be three times larger.
Find these numbers. | 167334 | 43 | 6 |
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 | |
| $[$ Decimal numeral system $]$ | | |
Find the last digit of the number $7^{7}$.
# | 3 | 28 | 1 |
math | 5. Form an $n$-digit number using the digits $1, 2, 3$, requiring that each of $1, 2, 3$ appears at least once in the $n$-digit number. The total number of such $n$-digit numbers is $\qquad$ . | 3^{n}-3\cdot2^{n}+3 | 65 | 13 |
math | 12. $[\mathbf{8}]$ A sequence of integers $\left\{a_{i}\right\}$ is defined as follows: $a_{i}=i$ for all $1 \leq i \leq 5$, and $a_{i}=a_{1} a_{2} \cdots a_{i-1}-1$ for all $i>5$. Evaluate $a_{1} a_{2} \cdots a_{2011}-\sum_{i=1}^{2011} a_{i}^{2}$. | -1941 | 125 | 5 |
math | Example 3 (2003 Beijing Competition Problem) Given $n$ distinct positive integers $a_{1}, a_{2}, \cdots, a_{n}$, then among the integers of the form $t_{1} a_{1}+t_{2} a_{2}+\cdots+t_{n} a_{n}$ (where $t_{i}$ takes 1 or $-1, i=1,2, \cdots, n$), there exist $\frac{n^{2}+n+2}{2}$ distinct integers that are either all odd... | \frac{n^{2}+n+2}{2} | 127 | 13 |
math | 14. (6 points) If the sum of the "day" numbers in three consecutive days is 18, then these three "day" numbers are 5, 6, 7. If the sum of the "day" numbers in three consecutive days is 33, then these three "day" numbers are $\qquad$ | 10,11,12 | 72 | 8 |
math | 27.13*. (Belgium, 82). On the inner wrapper of each chocolate bar in the "Great Mathematician" series, one of the $n$ outstanding mathematicians is depicted, and the portrait of each of them appears with equal probability, equal to $1 / n$. On average, how many chocolate bars need to be bought to collect the complete s... | n(1+\frac{1}{2}+\ldots+\frac{1}{n-1}+\frac{1}{n}) | 81 | 29 |
math | [ Inequality problems. Case analysis]
A biologist sequentially placed 150 beetles into ten jars. Moreover, in each subsequent jar, he placed more beetles than in the previous one. The number of beetles in the first jar is no less than half the number of beetles in the tenth jar. How many beetles are in the sixth jar?
... | 16 | 75 | 2 |
math | Example $\mathbf{3}$ Given the Fibonacci sequence $\left\{a_{n}\right\}$ satisfies
$$
\left\{\begin{array}{l}
a_{1}=a_{2}=1, \\
a_{n}=a_{n-1}+a_{n-2}(n \geqslant 3) .
\end{array}\right.
$$
Find the general term formula $a_{n}$ of the sequence $\left\{a_{n}\right\}$. | a_{n}=\frac{\sqrt{5}}{5}\left[\left(\frac{1+\sqrt{5}}{2}\right)^{n}-\left(\frac{1-\sqrt{5}}{2}\right)^{n}\right] | 108 | 55 |
math |
1. In the domain of non-negative real numbers solve the system of equations
$$
\begin{aligned}
\lfloor 3 x+5 y+7 z\rfloor & =7 z, \\
\lfloor 3 y+5 z+7 x\rfloor & =7 x, \\
\lfloor 3 z+5 x+7 y\rfloor & =7 y .
\end{aligned}
$$
(Tomáš Bárta)
| (x,y,z)\in{(0,0,0),(\frac{1}{7},0,0),(0,\frac{1}{7},0),(0,0,\frac{1}{7})} | 101 | 44 |
math | 2. Find all prime numbers $p$ such that $17p + 1$ is a perfect square of a natural number. | 19 | 28 | 2 |
math | Given an integer $n \geqslant 3$. Let $a_{1}, a_{2}, \cdots, a_{2 n}, b_{1}, b_{2}, \cdots, b_{2 n}$ be $4 n$ non-negative real numbers, satisfying
$$
a_{1}+a_{2}+\cdots+a_{2 n}=b_{1}+b_{2}+\cdots+b_{2 n}>0 \text {, }
$$
and for any $i=1,2, \cdots, 2 n$, we have $a_{i} a_{i+2} \geqslant b_{i}+b_{i+1}$, where $a_{2 n+1}... | 12forn=3,16forn\geqslant4 | 213 | 17 |
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