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 | 1. Find all integer solutions of the system of equations
$\left\{\begin{array}{c}x+y+z=2, \\ x^{3}+y^{3}+z^{3}=-10 .\end{array}\right.$ | (3,3,-4),(3,-4,3),(-4,3,3) | 54 | 20 |
math | 34. [?] The Game. Eric and Greg are watching their new favorite TV show, The Price is Right. Bob Barker recently raised the intellectual level of his program, and he begins the latest installment with bidding on following question: How many Carmichael numbers are there less than 100,000 ?
Each team is to list one nonne... | 16 | 183 | 2 |
math | Example 3 Solve the equation
$$
x^{2}-2 x-3=12 \cdot\left[\frac{x-1}{2}\right] \text {. }
$$
(1990, Sichuan Province Junior High School Mathematics Competition) | x=1+2 \sqrt{7} \text { or } x=1+2 \sqrt{10} | 56 | 26 |
math | 2A. The positive real numbers satisfy the equation
$$
\log \left(1+a^{2}\right)-\log a-2 \log 2=1-\log \left(100+b^{2}\right)+\log b
$$
Calculate the sum $a+b$. | 11 | 63 | 2 |
math | 2. Solve the system of equations
$$
\left\{\begin{array}{l}
\left(1+4^{2 x-y}\right) 5^{1-2 x+y}=1+2^{2 x-y+1}, \\
y^{3}+4 x+1+\ln \left(y^{2}+2 x\right)=0 .
\end{array}\right.
$$
(1999, Vietnam Mathematical Olympiad) | x=0, y=-1 | 98 | 7 |
math | Find all functions $f:\mathbb{Q}^{+} \to \mathbb{Q}^{+}$ such that for all $x\in \mathbb{Q}^+$: [list] [*] $f(x+1)=f(x)+1$, [*] $f(x^2)=f(x)^2$. [/list] | f(x) = x | 74 | 6 |
math | Four. (20 points) Given the sequence $\left\{a_{n}\right\}$, $S_{n}$ represents the sum of its first $n$ terms. If it satisfies the relation $S_{n}+a_{n}=n^{2}+3 n-1$, find the general formula for the sequence $\left\{a_{n}\right\}$, i.e., the expression for $a_{n}$. | a_{n}=2 n-\frac{1}{2^{n}} | 94 | 15 |
math | ## Problem Statement
Find the distance from point $M_{0}$ to the plane passing through three points $M_{1}, M_{2}, M_{3}$.
$M_{1}(-1 ; -5 ; 2)$
$M_{2}(-6 ; 0 ; -3)$
$M_{3}(3 ; 6 ; -3)$
$M_{0}(10 ; -8 ; -7)$ | 2\sqrt{38} | 93 | 7 |
math | 6. Let $t=\left(\frac{1}{2}\right)^{x}+\left(\frac{2}{3}\right)^{x}+\left(\frac{5}{6}\right)^{x}$, then the sum of all real solutions of the equation $(t-1)(t-2)(t-3)=0$ with respect to $x$ is $\qquad$ . | 4 | 85 | 1 |
math | G1.1 Given that $a$ is an integer. If 50 ! is divisible by $2^{a}$, find the largest possible value of $a$. | 47 | 36 | 2 |
math | 4. Let
$$
n=9+99+999+\cdots+\underbrace{999 \ldots 999}_{99 \text { digits }}
$$
Determine the number and the sum of the digits of the number $n$. | 99 | 59 | 2 |
math | A certain frog that was placed on a vertex of a convex polygon chose to jump to another vertex, either clockwise skipping one vertex, either counterclockwise skipping two vertexes, and repeated the procedure.
If the number of jumps that the frog made is equal to the number of sides of the polygon, the frog has passed t... | \{ n \in \mathbb{N} \mid 6 \nmid n \text{ or } 30 \mid n \} | 103 | 33 |
math | Point $M$ is the midpoint of chord $A B$. Chord $C D$ intersects $A B$ at point $M$. A semicircle is constructed on segment $C D$ as its diameter. Point $E$ lies on this semicircle, and $M E$ is perpendicular to $C D$. Find the angle $A E B$. | 90 | 76 | 2 |
math | 3. If $x^{2}+y^{2}+2 x-4 y+5=0$, what is $x^{2000}+2000 y$? | 4001 | 42 | 4 |
math | In the plane a point $O$ is and a sequence of points $P_1, P_2, P_3, \ldots$ are given. The distances $OP_1, OP_2, OP_3, \ldots$ are $r_1, r_2, r_3, \ldots$ Let $\alpha$ satisfies $0 < \alpha < 1.$ Suppose that for every $n$ the distance from the point $P_n$ to any other point of the sequence is $\geq r^{\alpha}_n.$ De... | \beta = \frac{1}{2(1 - \alpha)} | 166 | 16 |
math | 29. $\int x^{2 / 3} d x$.
Translate the above text into English, please retain the original text's line breaks and format, and output the translation result directly.
29. $\int x^{2 / 3} d x$. | \frac{3}{5}x^{5/3}+C | 56 | 15 |
math | 375. A discrete random variable $X$ is given by the distribution law:
$$
\begin{array}{lrrrr}
X & -1 & -2 & 1 & 2 \\
p & 0.3 & 0.1 & 0.2 & 0.4
\end{array}
$$
Find the distribution law of the random variable $Y=X^{2}$. | \begin{pmatrix}Y&1&4\\p&0.5&0.5\end{pmatrix} | 87 | 27 |
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 $AQ$... | 247 | 140 | 3 |
math | Let $x$ and $y$ be distinct real numbers such that
\[ \sqrt{x^2+1}+\sqrt{y^2+1}=2021x+2021y. \]
Find, with proof, the value of
\[ \left(x+\sqrt{x^2+1}\right)\left(y+\sqrt{y^2+1}\right). \] | \frac{1011}{1010} | 85 | 13 |
math | Let the rest energy of a particle be $E$. Let the work done to increase the speed of this particle from rest to $v$ be $W$. If $ W = \frac {13}{40} E $, then $ v = kc $, where $ k $ is a constant. Find $10000k$ and round to the nearest whole number.
[i](Proposed by Ahaan Rungta)[/i] | 6561 | 95 | 4 |
math | Four squirrels ate a total of 2020 nuts, each at least 103 nuts. The first squirrel ate more nuts than any of the other three squirrels. The second and third squirrels ate a total of 1277 nuts.
How many nuts did the first squirrel eat?
(L. Hozová) | 640 | 71 | 3 |
math | Each integer in $\{1, 2, 3, . . . , 2020\}$ is coloured in such a way that, for all positive integers $a$ and $b$
such that $a + b \leq 2020$, the numbers $a$, $b$ and $a + b$ are not coloured with three different colours.
Determine the maximum number of colours that can be used.
[i]Massimiliano Foschi, Italy[/i] | 11 | 106 | 2 |
math | # 1. CONDITION
Consider quadratic trinomials of the form $x^{2}+p x+q$ with integer coefficients, where $p+q=30$. How many such trinomials have integer roots? | Two\quadratic\polynomials:\x^{2}-34x+64\\x^{2}+30x | 50 | 27 |
math | Problem 3. A team of lumberjacks was cutting trees on a large and a small plot, with the area of the small plot being 3 times less than that of the large plot. In the part of the team that worked on the large plot, there were 8 more lumberjacks than in the part that worked on the small plot. When the tree harvesting on... | 14 | 115 | 2 |
math | A full score of 20 points) Let $x, y$ be non-zero real numbers, and satisfy $\frac{x \sin \frac{\pi}{5}+y \cos \frac{\pi}{5}}{x \cos \frac{\pi}{5}-y \sin \frac{\pi}{5}}=\tan \frac{9 \pi}{20}$.
(1) Find the value of $\frac{y}{x}$; (2) In $\triangle A B C$, if $\tan C=\frac{y}{x}$, find the maximum value of $\sin 2 A+2 \... | \frac{3}{2} | 133 | 7 |
math | Find all prime positive integers $p, q$ such that $2 p^{3}-q^{2}=2(p+q)^{2}$. | (3,2) | 31 | 5 |
math | 10. Solve the following inequality.
$$
\log _{1 / 2} x-\sqrt{2-\log _{4} x}+1 \leq 0
$$ | \frac{1}{\sqrt{2}}\leqx\leq16 | 41 | 18 |
math | 8.192. $\frac{1+\sin 2 x}{1-\sin 2 x}+2 \cdot \frac{1+\tan x}{1-\tan x}-3=0$. | x_{1}=\pik;x_{2}=\operatorname{arctg}2+\pin,wherekn\inZ | 44 | 28 |
math | Let $f^1(x)=x^3-3x$. Let $f^n(x)=f(f^{n-1}(x))$. Let $\mathcal{R}$ be the set of roots of $\tfrac{f^{2022}(x)}{x}$. If
\[\sum_{r\in\mathcal{R}}\frac{1}{r^2}=\frac{a^b-c}{d}\]
for positive integers $a,b,c,d$, where $b$ is as large as possible and $c$ and $d$ are relatively prime, find $a+b+c+d$. | 4060 | 132 | 4 |
math | 6. (IND 2) ${ }^{\mathrm{IMO}(F 2}$ Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that $f\left(x^{2}+f(y)\right)=y+f(x)^{2} \quad$ for all $x, y$ in $\mathbb{R}$. | f(x)=x | 80 | 4 |
math | Example 10. Let the sequence be $u_{1}=3$, $s_{n+1}=\frac{2 u_{n}+3}{u_{n}+4}$, find its general term. | u_{n}=\frac{3 \cdot 5^{n-1}+3}{3 \cdot 5^{n-1}-1} | 47 | 32 |
math | 1 Let $w=-\frac{1}{2}+\frac{\sqrt{3}}{2} \cdot i\left(i^{2}=-1\right)$, then the number of different values that the algebraic expression $w^{m}+w^{n}+w^{l}(m, n, l$ are pairwise distinct integers) can take is $\qquad$ . | 10 | 82 | 2 |
math | We have two gears. By connecting them, the ratio of the rotational speeds of the driving and driven gears (the transmission ratio) is $7: 9$ or $9: 7$. By replacing one of the gears with a gear that has 3 fewer teeth and the other with a gear that has 3 more teeth, the transmission ratio becomes $1: 3$ or $3: 1$. - Whi... | 1, | 109 | 2 |
math | Problem 4. Find all solutions to the puzzle
$$
\mathrm{G}+\mathrm{OH}+\mathrm{OH}+\mathrm{OH}+\mathrm{OH}+\mathrm{OH}+\mathrm{OH}+\mathrm{OH}+\mathrm{OH}=\mathrm{Mb} .
$$
(Same letters encode the same digits, different letters encode different digits.)
[5 points] (D. E. Schnol) | 0+12+12+12+12+12+12+12+12=96 | 92 | 28 |
math | 4. Problem: Determine all prime numbers $p$ such that $p^{2}-6$ and $p^{2}+6$ are both prime numbers. | 5 | 34 | 1 |
math | 5. Bacamo istovremeno četiri simetrična novčića (za simetričan novčić vjerojatnosti da će pasti pismo ili glava su jednake). Kolika je vjerojatnost da su pri jednom bacanju ta četiri novčića pala dva pisma i dvije glave?
## Zadaci za 10 bodova:
Translates to:
5. We toss four symmetrical coins simultaneously (for a s... | \frac{3}{8} | 154 | 7 |
math | 1. A n s w e r: $\frac{1}{4}$.
F i r s t s o l u t i o n. Let the vertices of the square be denoted as $A, B, C, D$, such that $L$ lies on $A B$, $M$ lies on $B C$, $N$ lies on $C D$, and $K$ lies on $A D$. Denote the intersection point of $K M$ and $L N$ as $O$. Let the lengths of segments $A L$ and $A K$ be $x$ and ... | \frac{1}{4} | 312 | 7 |
math | 7. The number of non-negative integer solutions $(x, y, z)$ to the equation $x+2 y+3 z=2014$ is $\qquad$. | 339024 | 38 | 6 |
math | 6. It is known that the sum of three fractions is $\frac{10}{11}$, and they have the same denominator, with the ratio of their numerators being $2: 3: 4$. Therefore, the largest of these three fractions is $\qquad$ . | \frac{40}{99} | 60 | 9 |
math | In how many ways can $n$ identical socks be arranged in a drawer with $k \geqslant 1$ compartments. | \binom{n+k-1}{k-1} | 28 | 12 |
math | ## Task $4 / 62$
What remainder does the number $2^{n}$ leave when divided by 3? | The2^{n}leavesremainderof1whendivided3ifniseven,remainderof2ifnisodd | 26 | 26 |
math | Let's determine those two-digit numbers $\overline{a b}$ in the decimal system for which the greatest common divisor of $\overline{a b}$ and $\overline{b a}$ is $a^{2}-b^{2}$. | (10,1),(21,12),(54,45) | 51 | 18 |
math | 2. For the number $a$, the equality $a+\frac{1}{a}=1$ is satisfied. Calculate the value of
$$
a^{5}+\frac{1}{a^{5}}
$$ | 1 | 45 | 1 |
math | 13.252. The duty maintenance worker descended on a downward-moving metro escalator. His entire journey from the upper platform to the lower one lasted $24 \mathrm{s}$. Then he climbed up and at the same pace descended again, but this time on a stationary escalator. It is known that the descent lasted 42 s. How many sec... | 56 | 96 | 2 |
math | Example 17 Real numbers $x_{1}, x_{2}, \cdots, x_{2001}$ satisfy $\sum_{k=1}^{2000}\left|x_{k}-x_{k+1}\right|=2001$, let $y_{k}=\frac{1}{k}\left(x_{1}+x_{2}+\cdots+\right.$ $\left.x_{k}\right), k=1,2, \cdots, 2001$. Find the maximum possible value of $\sum_{k=1}^{2000} \mid y_{k}-y_{k+1}$ . (2001 Shanghai Competition P... | 2000 | 152 | 4 |
math | Of a rhombus $ABCD$ we know the circumradius $R$ of $\Delta ABC$ and $r$ of $\Delta BCD$. Construct the rhombus. | ABCD | 38 | 3 |
math | One, (40 points) Find the smallest real number $\lambda$, such that there exists a sequence $\left\{a_{n}\right\}$ with all terms greater than 1, for which for any positive integer $n$ we have $\prod_{i=1}^{n+1} a_{i}<a_{n}^{\lambda}$. | 4 | 76 | 1 |
math | In a tennis tournament, each player advances to the next round only in case of victory. If it is not possible for an even number of players to always advance to the next round, the tournament organizers decide which rounds certain players should play. For example, a seeded player can, at the discretion of the organizer... | 2010 | 125 | 4 |
math | 7.222. $3 \cdot 16^{x}+2 \cdot 81^{x}=5 \cdot 36^{x}$. | 0;\frac{1}{2} | 36 | 8 |
math | 2. Given points
$$
A\left(x-\frac{3-\sqrt{5}}{2} \cdot y, 1\right) 、 B\left(x-\frac{3+\sqrt{5}}{2} \cdot y,-1\right)
$$
satisfy $\overrightarrow{O A} \cdot \overrightarrow{O B}=0$. Then the range of $S=x^{2}-x y+y^{2}-1$ is $\qquad$ . | \left[-\frac{2}{5},+\infty\right) | 105 | 16 |
math | Example 1 Find $\sum_{k=1}^{n} k^{2} \mathrm{C}_{n}^{k}$.
(23rd Putnam Mathematical Competition)
Analysis: In this problem, there is a variable $k^{2}$ before $\mathrm{C}_{n}^{k}$, therefore, it is necessary to use identity (III) twice to transform to $n$, then the binomial theorem can be applied to solve it. | n(n+1) 2^{n-2} | 95 | 12 |
math | 4. (10 points) Professor Wang arrived at the station at 8:00 AM. When he boarded the train, the hour and minute hands of the clock on the platform were exactly symmetrical left and right. The train departed at 8:35 AM and arrived at the terminal station at 2:15 PM. When Professor Wang got off the train, the hour and mi... | 360 | 131 | 3 |
math | 158*. Using the digits from 1 to 9 once each, form the smallest nine-digit number that is divisible by 11. | 123475869 | 30 | 9 |
math | 4. (7 points) On the board, 45 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 45 minutes? | 990 | 69 | 3 |
math | ## Task $5 / 87$
Determine all three-digit (proper) natural numbers $z \in N$ in the decimal system that are represented by exactly $n$ digits 1 in the number system with base $n \in N$. | 781 | 52 | 3 |
math | Find all natural numbers $n$, such that there exist relatively prime integers $x$ and $y$ and an integer $k > 1$ satisfying the equation $3^n =x^k + y^k$.
[i]A. Kovaldji, V. Senderov[/i] | n = 2 | 62 | 5 |
math | At a familiar factory, they cut out metal disks with a diameter of 1 m. It is known that a disk with a diameter of exactly 1 m weighs exactly 100 kg. During manufacturing, there is a measurement error, and therefore the standard deviation of the radius is 10 mm. Engineer Sidorov believes that a stack of 100 disks will ... | 4 | 99 | 1 |
math | 10.205. A circle with a radius of 5 cm is inscribed in some angle. The length of the chord connecting the points of tangency is 8 cm. Two tangents, parallel to the chord, are drawn to the circle. Find the sides of the resulting trapezoid. | 20 | 65 | 2 |
math | Let $(x_n)$, $n=1,2, \ldots $ be a sequence defined by $x_1=2008$ and
\begin{align*} x_1 +x_2 + \ldots + x_{n-1} = \left( n^2-1 \right) x_n \qquad ~ ~ ~ \forall n \geq 2 \end{align*}
Let the sequence $a_n=x_n + \frac{1}{n} S_n$, $n=1,2,3, \ldots $ where $S_n$ $=$ $x_1+x_2 +\ldots +x_n$. Determine the values of $n$ for ... | n = 251, 1004, 4016 | 171 | 19 |
math | 6.080. $\left\{\begin{array}{l}u^{2}+u v=15, \\ v^{2}+u v=10\end{array}\right.$ | (-3,-2),(3,2) | 45 | 9 |
math | 714. Find the limits:
1) $\lim _{\substack{x \rightarrow 3 \\ y \rightarrow 0}} \frac{\tan(x y)}{y}$
2) $\lim _{\substack{x \rightarrow 0 \\ y \rightarrow 0}} \frac{x}{x+y}$. | 3 | 67 | 1 |
math | In $\triangle ABC$, the sides have integer lengths and $AB=AC$. Circle $\omega$ has its center at the incenter of $\triangle ABC$. An excircle of $\triangle ABC$ is a circle in the exterior of $\triangle ABC$ that is tangent to one side of the triangle and tangent to the extensions of the other two sides. Suppose that ... | 20 | 118 | 2 |
math | Find all one-to-one mappings $f:\mathbb{N}\to\mathbb{N}$ such that for all positive integers $n$ the following relation holds:
\[ f(f(n)) \leq \frac {n+f(n)} 2 . \] | f(n) = n | 55 | 6 |
math | 17. $[\mathbf{9}]$ Solve the equation
$$
\sqrt{x+\sqrt{4 x+\sqrt{16 x+\sqrt{\ldots+\sqrt{4^{2008} x+3}}}}} \sqrt{x}=1 .
$$
Express your answer as a reduced fraction with the numerator and denominator written in their prime factorization. | \frac{1}{2^{4016}} | 78 | 12 |
math | Problem 2.4. Consider an alphabet of $n$ letters, with which we will form words. We will say that a word contains a palindrome if a segment of that word, of more than one letter, reads the same forwards as backwards. For example, the word OLIMPIADAS contains the palindrome ADA. Given an integer $k$ greater than 2, dete... | n^{2}(n-1)^{k-2} | 109 | 13 |
math | 12. Find the domain of each of the following functions:
1) $y=\sqrt{1-x^{2}}$
2) $u=\frac{x-1}{x^{2}-5 x+6}+\sqrt[3]{2 x+1}$
3) $v=\arccos \frac{1-2 x}{3}$
4) $p=\frac{x}{\sin x}$
5) $q=\log _{2}\left(x^{2}-9\right)$. | \begin{pmatrix}1)[-1;1]\\2)(-\infty,2)\cup(2,3)\cup(3,+\infty)\\3)[-1;2]\\4)(-\infty,0)\cup(0,\pi)\cup(\pi,2\pi)\cup\cdots\\5)(-\infty,-3 | 108 | 75 |
math | 10.86 Try to divide the seven digits $3,4,5,6,7,8,9$ into two groups, and arrange them into a three-digit number and a four-digit number, respectively, such that the product of these two numbers is maximized. How should the groups be divided and arranged, and prove your conclusion.
(China Beijing Junior High School Gra... | 964\cdot8753 | 90 | 9 |
math | $14 \cdot 34$ Calculate the value of the sum $\sum_{n=0}^{502}\left[\frac{305 n}{503}\right]$.
(1st China Northeast Three Provinces Mathematics Invitational Competition, 1986) | 76304 | 62 | 5 |
math | 21 *pure. Let $M=\{1,2, \cdots, 2 n+1\}, A$ be a subset of $M$, find the maximum value of $|A|$.
translated as:
21 *pure. Let $M=\{1,2, \cdots, 2 n+1\}, A$ be a subset of $M$, find the maximum value of $|A|$. | n+1 | 91 | 3 |
math | 3.309. $1-\cos (\pi-8 \alpha)-\cos (\pi+4 \alpha)$. | 4\cos4\alpha\cos(2\alpha+\frac{\pi}{6})\cos(2\alpha-\frac{\pi}{6}) | 27 | 32 |
math | What is the smallest four-digit positive integer that is divisible by both 5 and 9 and has only even digits? | 2880 | 24 | 4 |
math | Example 8 Given that $x, y, z$ are real numbers, and $x+y+z=$ $5, xy+yz+zx=3$. Try to find the maximum and minimum values of $z$. (10th Canadian High School Mathematics Competition) | \frac{13}{3} \text{ and } -1 | 55 | 15 |
math | 4.021. An arithmetic progression has the following property: for any $n$ the sum of its first $n$ terms is equal to $5 n^{2}$. Find the common difference of this progression and its first three terms. | 10;5,15,25 | 51 | 10 |
math | Determine if there exists a positive integer $n$ such that $n$ has exactly $2000$ prime divisors and $2^{n}+1$ is divisible by $n$. | n = 3^s p_1 p_2 \ldots p_{1999} | 42 | 23 |
math |
Problem 11.1 Let $f(x)=\frac{x^{2}+4 x+3}{x^{2}+7 x+14}$.
a) Find the greatest value of $f(x)$;
b) Find the greatest value of the function $\left(\frac{x^{2}-5 x+10}{x^{2}+5 x+20}\right)^{f(x)}$.
| 9 | 90 | 1 |
math | 3. From the edges and the diagonals on the faces of a cube, select $k$ lines such that any two of these lines are skew lines. Then the maximum value of $k$ is . $\qquad$ | 4 | 46 | 1 |
math | 1391. Find for each series the partial sum of the first $n$ terms $\left(S_{n}\right)$; show, using the definition, the convergence (divergence) of the series; find the sum of the series ( $S$ ):
1) $a+a q+a q^{2}+\ldots+a q^{n-1}+\ldots$
$$
\text { 2) } \frac{1}{1 \cdot 2}+\frac{1}{2 \cdot 3}+\frac{1}{3 \cdot 4}+\ldo... | \frac{}{1-q} | 141 | 7 |
math | $2 \cdot 32$ Let $A$ be a set of integers, where the smallest element is 1 and the largest element is 100. Except for 1, each element is equal to the sum of two elements in $A$ (which can be twice a single element). Find the minimum number of elements in set $A$. | 9 | 74 | 1 |
math | [ Divisibility of numbers. General properties ]
The year of the current mathematics festival is divisible by its number: $2006: 17=118$.
a) Name the first number of the math festival for which this was also true.
b) Name the last number of the math festival for which this will also be true.
# | 1989 | 74 | 4 |
math | 14.51. In a drawer, there are red and black socks. If two socks are randomly pulled out of the drawer, the probability that both are red is $1 / 2$.
a) What is the smallest number of socks that can be in the drawer?
b) What is the smallest number of socks that can be in the drawer if it is known that the number of bl... | 4 | 86 | 1 |
math | 9. Let the set $A=\left\{\left.\frac{a_{1}}{9}+\frac{a_{2}}{9^{2}}+\frac{a_{3}}{9^{3}}+\frac{a_{4}}{9^{4}} \right\rvert\, a_{i} \in\{0,1,2, \cdots, 8\}, i=1\right.$, $2,3,4\}$, arrange the numbers in $A$ in descending order, and find the 1997th number. | \frac{6}{9}+\frac{2}{9^{2}}+\frac{3}{9^{3}}+\frac{1}{9^{4}} | 124 | 34 |
math | For which values of $a$ will the expression
$$
97 a^{2}+84 a-55
$$
be a multiple of $a$? | 1,5,11,55,-1,-5,-11,-55 | 37 | 19 |
math | 3.9 Try to find all numbers of the form $p^{p}+1$ that are prime and have no more than 19 digits ($p$ is a natural number).
(28th Moscow Mathematical Olympiad, 1965) | 2,5,257 | 54 | 7 |
math | \section*{Problem 1 - 211031}
Determine all triples \((a, b, c)\) of natural numbers with the following properties!
(1) It holds that \(0 < a \leq b \leq c\).
(2) In a cuboid with length \(a \mathrm{~cm}\), width \(b \mathrm{~cm}\), and height \(c \mathrm{~cm}\), the sum of all edge lengths is equal to the volume in... | (1,5,24),(1,6,14),(1,8,9),(2,3,10),(2,4,6) | 112 | 34 |
math | 37th Putnam 1976 Problem B2 G is a group generated by the two elements g, h, which satisfy g 4 = 1, g 2 ≠ 1, h 7 = 1, h ≠ 1, ghg -1 h = 1. The only subgroup containing g and h is G itself. Write down all elements of G which are squares. Solution | 1,^2,,^2,^3,^4,^5,^6 | 85 | 19 |
math | In a trapezoid $ABCD$, the internal bisector of angle $A$ intersects the base $BC$(or its extension) at the point $E$. Inscribed in the triangle $ABE$ is a circle touching the side $AB$ at $M$ and side $BE$ at the point $P$. Find the angle $DAE$ in degrees, if $AB:MP=2$. | 60^\circ | 86 | 4 |
math | For a sequence $a_{1}, a_{2}, a_{3}, \ldots$ of real numbers it is known that
$$
a_{n}=a_{n-1}+a_{n+2} \quad \text { for } n=2,3,4, \ldots
$$
What is the largest number of its consecutive elements that can all be positive?
Answer: 5. | 5 | 88 | 1 |
math | 4. Determine all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that for all $x, y \in \mathbb{R}$,
$$
f(x f(x)+f(y))=(f(x))^{2}+y
$$
(Bulgaria) | f(x)\equivxf(x)\equiv-x | 65 | 9 |
math | Let $N$ be the number of (positive) divisors of $2010^{2010}$ ending in the digit $2$. What is the remainder when $N$ is divided by 2010? | 503 | 49 | 3 |
math | 1A. 2B. Determine $\log _{a} b, \log _{a b} b, \log _{a b^{2}} b$ and $\log _{a b^{3}} b$, if
$$
\log _{a} b-\log _{a b} b=\log _{a b^{2}} b-\log _{a b^{3}} b
$$ | \log_{}b=-\frac{2}{3},\log_{}b=-2,\log_{^{2}}b=2,\log_{^{3}}b=\frac{2}{3} | 89 | 43 |
math | 42 Let $k$ be a natural number. Try to determine the smallest natural number $n$ such that: in any $n$ integers, there must be two numbers whose sum or difference is divisible by $2 k+1$.
| k+2 | 50 | 3 |
math | 12.143 $a$ is a real number greater than zero. It is known that there exists a unique real number $k$, such that the quadratic equation in $x$, $x^{2}+\left(k^{2}+a k\right) x+1999+k^{2}+a k=0$, has two roots that are both prime numbers. Find the value of $a$.
(China Junior High School Mathematics League, 1999) | 2\sqrt{502} | 103 | 8 |
math | 4. (25th IMO Problem) Find a pair of positive integers $a, b$ satisfying:
(1) $ab(a+b)$ is not divisible by 7.
(2) $(a+b)^{7}-a^{7}-b^{7}$ is divisible by $7^{7}$. | (18,1) | 64 | 6 |
math | 10. (15 points) Given the set $D=\left\{\left(x_{1}, x_{2}\right)\left|x_{1}\right\rangle\right.$ $\left.0, x_{2}>0, x_{1}+x_{2}=k\right\}$, where $k$ is a positive constant. Find the range of $k$ such that the inequality $\left(\frac{1}{x_{1}}-x_{1}\right)\left(\frac{1}{x_{2}}-x_{2}\right) \geqslant$ $\left(\frac{k}{2... | 0<k\leqslant2\sqrt{\sqrt{5}-2} | 165 | 17 |
math | Let $p$ and $q$ be positive integers such that $\frac{5}{8}<\frac{p}{q}<\frac{7}{8}$. What is the smallest value of $p$ for which $p+q=2005$? | 772 | 57 | 3 |
math | The triangle $ABC$ is $| BC | = a$ and $| AC | = b$. On the ray starting from vertex $C$ and passing the midpoint of side $AB$ , choose any point $D$ other than vertex $C$. Let $K$ and $L$ be the projections of $D$ on the lines $AC$ and $BC$, respectively, $K$ and $L$. Find the ratio $| DK | : | DL |$. | \frac{a}{b} | 98 | 7 |
math | 13.235. Two cars and a motorcycle participated in a race over the same distance. The second car took 1 minute longer to complete the entire distance than the first car. The first car moved 4 times faster than the motorcycle. What part of the distance did the second car cover in one minute, if it covered $1 / 6$ of the ... | \frac{2}{3} | 99 | 7 |
math | 10. Among the students in the second year of junior high school, 32 students participated in the math competition, 27 students participated in the English competition, and 22 students participated in the Chinese competition. Among them, 12 students participated in both math and English, 14 students participated in both... | 30 | 111 | 2 |
math | 1. In a tennis tournament, 512 schoolchildren are participating. 1 point is awarded for a win, and 0 points for a loss. Before each round, pairs are formed by lottery among participants with the same number of points (those who do not find a pair are awarded a point without playing). The tournament ends as soon as a so... | 84 | 90 | 2 |
math | 4. Find all functions $f: \mathbf{R}_{+} \rightarrow \mathbf{R}_{+}$, such that for all $x, y \in \mathbf{R}_{+}$,
$$
f(x+f(y))=f(x+y)+f(y),
$$
where $\mathbf{R}_{+}$ is the set of positive real numbers. | f(x)=2x | 80 | 5 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.