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 | Example. Compute the integral
$$
\int_{L} \sin ^{2} z d z
$$
where $L$ is the line segment from the point $z_{1}=0$ to the point $z_{1}=i$. | \frac{i}{4}(2-\operatorname{sh}2) | 53 | 15 |
math | 4. If the real numbers $x, y, z$ satisfy
$$
\sqrt[5]{x-y}+\sqrt[5]{y-z}=3 \text {, }
$$
and $x-z=33$,
then the value of the algebraic expression $x-2 y+z$ is | \pm 31 | 66 | 5 |
math | Let $q$ be a positive rational number. Two ants are initially at the same point $X$ in the plane. In the $n$-th minute $(n = 1,2,...)$ each of them chooses whether to walk due north, east, south or west and then walks the distance of $q^n$ metres. After a whole number of minutes, they are at the same point in the plane... | q = 1 | 120 | 5 |
math | Find all functions $f:\mathbb{R} \to \mathbb{R}$ such that $f(x)\geq 0\ \forall \ x\in \mathbb{R}$, $f'(x)$ exists $\forall \ x\in \mathbb{R}$ and $f'(x)\geq 0\ \forall \ x\in \mathbb{R}$ and $f(n)=0\ \forall \ n\in \mathbb{Z}$ | f(x) = 0 \quad \forall x \in \mathbb{R} | 103 | 20 |
math | 4. The reciprocal value of the difference between two numbers is $\frac{3}{4}$. If the subtrahend is equal to $\frac{5}{18}$, what is the minuend? | \frac{29}{18}=1\frac{11}{18} | 45 | 19 |
math | Let $T=\text{TNFTPP}$. Fermi and Feynman play the game $\textit{Probabicloneme}$ in which Fermi wins with probability $a/b$, where $a$ and $b$ are relatively prime positive integers such that $a/b<1/2$. The rest of the time Feynman wins (there are no ties or incomplete games). It takes a negligible amount of time for t... | a = 1 | 182 | 5 |
math | 2. Tina wrote down one natural number on each of five pieces of paper, but she didn't want to reveal which numbers she had written. Deleted Žan managed to convince her to tell him all the sums of pairs of numbers. He found out that the sums were 17, 20, 28, $14, 42, 36, 28, 39, 25$ and 31. Which numbers did Tina write ... | 3,11,14,17,25 | 103 | 13 |
math | 4. Given that the hyperbola with asymptotes $y= \pm 2 x$ passes through the intersection of the lines $x+y-3=0$ and $2 x-y+6=0$. Then the length of the real axis of the hyperbola is $\qquad$ | 4\sqrt{3} | 62 | 6 |
math | 7. Given the sequence $\left\{a_{n}\right\}$, where $a_{1}=2, a_{n+1}=\frac{2 a_{n}+6}{a_{n}+1}$, then $a_{n}=$ | \frac{2+3\times(-4)^{n}}{(-4)^{n}-1} | 57 | 23 |
math | 1. In the field of real numbers, solve the system of equations
$$
\begin{aligned}
x^{2}-x y+y^{2} & =7, \\
x^{2} y+x y^{2} & =-2 .
\end{aligned}
$$
(J. Földes) | {x,y}={-1,2},{x,y}={1+\sqrt{2},1-\sqrt{2}},{x,y}={\frac{-9+\sqrt{57}}{6},\frac{-9-\sqrt{57}}{6}} | 66 | 56 |
math | XX OM - II - Task 2
Find all four-digit numbers in which the thousands digit is equal to the hundreds digit, and the tens digit is equal to the units digit, and which are squares of integers. | 7744 | 44 | 4 |
math | 7. Five numbers have the property that, when we add any four of these five numbers, we get the sums 186, 203, 214, 228, and 233. What are these numbers?
The use of a pocket calculator or any manuals is not allowed. | 33,38,52,63,80 | 67 | 14 |
math | 11. The terms of the sequence $\left\{a_{n}\right\}$ are positive, and the sum of its first $n$ terms $S_{n}=\frac{1}{2}\left(a_{n}+\frac{1}{a_{n}}\right)$. Then $a_{n}=$ $\qquad$ | \sqrt{n}-\sqrt{n-1} | 73 | 10 |
math | 9
On the sides of a triangle, three squares are constructed externally. What should the angles of the triangle be so that six vertices of these squares, distinct from the vertices of the triangle, lie on one circle? | \angleA=\angleB=\angleC=60or\angleA=\angleC=45,\angleB=90(or\angleB=45,\angleA=90) | 44 | 42 |
math | 45. Simpler than it seems. Calculate the root
$$
\left(\frac{1 \cdot 2 \cdot 4+2 \cdot 4 \cdot 8+3 \cdot 6 \cdot 12+\ldots}{1 \cdot 3 \cdot 9+2 \cdot 6 \cdot 18+3 \cdot 9 \cdot 27+\ldots}\right)^{\frac{1}{3}}
$$ | \frac{2}{3} | 99 | 7 |
math | Problem 9.6. Find all pairs of natural prime numbers $p$, $q$, that satisfy the equation
$$
3 p^{4}+5 q^{4}+15=13 p^{2} q^{2}
$$ | (2,3) | 52 | 5 |
math | ## Task 2 - 200732
Given are seven line segments with lengths $1 \mathrm{~cm}, 3 \mathrm{~cm}, 5 \mathrm{~cm}, 7 \mathrm{~cm}, 9 \mathrm{~cm}, 11 \mathrm{~cm}$, and $15 \mathrm{~cm}$.
a) Give the number of all different ways to select three of these seven line segments! Ways that differ only in the order of the selec... | 31.4 | 184 | 4 |
math | Proizvolov V.V.
There are 19 weights of $1, 2, 3, \ldots, 19$ grams: nine iron, nine bronze, and one gold. It is known that the total weight of all iron weights is 90 grams more than the total weight of the bronze weights. Find the weight of the gold weight. | 10 | 77 | 2 |
math | 2. (6 points) $\square \square \square \square \square+\bigcirc \bigcirc O=39$ liters
■■■+○○○=27 liters
■ represents $\qquad$ liters, ○ represents $\qquad$ liters. | 6;3 | 57 | 3 |
math | 5. Let $a, b$ be real numbers, and the function $f(x) = ax + b$ satisfies: for any $x \in [0,1]$, we have $|f(x)| \leqslant 1$. Find
\[
S = (a+1)(b+1)
\]
the range of values.
(Yang Xiaoming, problem contributor) | S\in[-2,\frac{9}{4}] | 83 | 12 |
math | Let's determine those two-digit numbers which are 3 less than the sum of the cubes of their digits! | 32 | 22 | 2 |
math | 16. Given that $\vec{a}$ and $\vec{b}$ are non-zero vectors, and $\vec{a}+3 \vec{b}$ is perpendicular to $7 \vec{a}-5 \vec{b}$, $\vec{a}-4 \vec{b}$ is perpendicular to $7 \vec{a}-2 \vec{b}$, find the angle between $\vec{a}$ and $\vec{b}$. | 60 | 95 | 2 |
math | Example 2 Suppose the three roots of the cubic equation $x^{3}+p x+1=0$ correspond to points in the complex plane that form an equilateral triangle. Find the area of this triangle. | \frac{3\sqrt{3}}{4} | 45 | 12 |
math | Exercise 2. Let $x, y$ and $z$ be three real numbers such that $0 \leqslant x \leqslant y \leqslant z$ and $x+y+z=1$. Find the maximum value that the expression
$$
(x-y z)^{2}+(y-z x)^{2}+(z-x y)^{2}
$$
can take. | 1 | 87 | 1 |
math | 1. Solve the equation $3 \cdot 2^{x}+1=y^{2}$ in integers. | (0,-2),(0,2),(3,-5),(3,5),(4,-7),(4,7) | 23 | 25 |
math | Find all pairs of primes $(p, q)$ such that $$p^3 - q^5 = (p + q)^2.$$ | (p, q) = (7, 3) | 28 | 13 |
math | Let $a_1, a_2, \ldots , a_{11}$ be 11 pairwise distinct positive integer with sum less than 2007. Let S be the sequence of $1,2, \ldots ,2007$. Define an [b]operation[/b] to be 22 consecutive applications of the following steps on the sequence $S$: on $i$-th step, choose a number from the sequense $S$ at random, say $... | 0 | 360 | 1 |
math | Calculate $1+3+3^{2}+\ldots+3^{2020}$. | \frac{3^{2021}-1}{2} | 22 | 14 |
math | 5. The sequence is defined recursively:
$x_{0}=0, x_{n+1}=\frac{\left(n^{2}+n+1\right) x_{n}+1}{n^{2}+n+1-x_{n}} . \quad$ Find $x_{8453}$. (12 points) | 8453 | 73 | 4 |
math | Solve the following system of equations:
$$
\frac{x+y+1}{x+y-1}=p, \quad \frac{x-y+1}{x-y-1}=q
$$ | \frac{pq-1}{(p-1)(q-1)},\quad\frac{q-p}{(p-1)(q-1)} | 41 | 33 |
math | We wish to write $n$ distinct real numbers $(n\geq3)$ on the circumference of a circle in such a way that each number is equal to the product of its immediate neighbors to the left and right. Determine all of the values of $n$ such that this is possible. | n = 6 | 60 | 5 |
math | Problem 5. Vladо, Boban, and Kатe have a total of 600 denars. If Vladо spends half of his money, Boban spends two-thirds of his money, and Kатe spends four-fifths of her money, then each of them will have the same amount of money left. How much money did each of them have? | Vladhad120denars,Boban180denars,Katarina300denars | 78 | 24 |
math | 1. In a math test, $N<40$ people participate. The passing score is set at 65. The test results are as follows: the average score of all participants is 66, that of the promoted is 71, and that of the failed is 56. However, due to an error in the formulation of a question, all scores are increased by 5. At this point, t... | 12,24,36 | 160 | 8 |
math | 13. In $\triangle A B C$, $\angle A, \angle B, \angle C$ are opposite to sides $a, b, c$ respectively. Let
$$
\begin{array}{l}
f(x)=\boldsymbol{m} \cdot \boldsymbol{n}, \boldsymbol{m}=(2 \cos x, 1), \\
\boldsymbol{n}=(\cos x, \sqrt{3} \sin 2 x), \\
f(A)=2, b=1, S_{\triangle A B C}=\frac{\sqrt{3}}{2} . \\
\text { Then }... | 2 | 158 | 1 |
math | Problem 4.2. Petya took half of the candies from the box and put them in two pockets. Deciding that he took too many, Petya took out 6 candies from each pocket and put them back into the box. By how many more candies did the box have than Petya's pockets? | 24 | 67 | 2 |
math | Find the sum of all positive integers $B$ such that $(111)_B=(aabbcc)_6$, where $a,b,c$ represent distinct base $6$ digits, $a\neq 0$. | 237 | 47 | 3 |
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 | 8 If the four digits of the four-digit number $\overline{a b c d}$ satisfy $a+b=c+d$, it is called a "good number"; for example, 2011 is a "good number". Then, the number of "good numbers" is $\qquad$ . | 615 | 63 | 3 |
math | Fill in numbers in the boxes below so that the sum of the entries in each three consecutive boxes is $2005$. What is the number that goes into the leftmost box?
[asy]
size(300);
label("999",(2.5,.5));
label("888",(7.5,.5));
draw((0,0)--(9,0));
draw((0,1)--(9,1));
for (int i=0; i<=9; ++i)
{
draw((i,0)--(i,1));
}
[/asy... | 118 | 123 | 3 |
math | 4. Natural numbers $a$, $b$, and $c$, greater than 2022, are such that $a+b$ is divisible by $c-2022$, $a+c$ is divisible by $b-2022$, and $b+c$ is divisible by $a-2022$. What is the greatest value that the number $a+b+c$ can take? (S. Berlov) | 2022\cdot85 | 93 | 8 |
math | ## Task 30/61
Five housewives want to buy rolls. After counting the available rolls, the baker allows himself a joke: "If each of you buys half of the currently available rolls plus half a roll, none will be left!" How many rolls did the baker have, and how many would each of the customers have received according to t... | 31 | 75 | 2 |
math | 12.53. In triangle $ABC$, the angle bisectors $AD$ and $BE$ are drawn. Find the measure of angle $C$, given that $AD \cdot BC = BE \cdot AC$ and $AC \neq BC$. | 60 | 54 | 2 |
math | 2. If the sum of the perimeters of two squares is 8 and the difference of their areas is 3, determine the lengths of the sides of both squares. | x=\frac{7}{4},y=\frac{1}{4} | 35 | 16 |
math | In the following division (where identical letters represent identical digits), replace the letters with digits so that the equality is correct:
| abcd : ef $=$ fga |
| :--- |
| hb |
| dic |
| be |
| dhd |
| dhd | | 7981:23=347 | 55 | 11 |
math | 5. In $\triangle A B C$, $\angle A=60^{\circ}, A C=16, S_{\triangle A B C}=220 \sqrt{3}, B C=$ | 49 | 44 | 2 |
math | A positive integer is called sparkly if it has exactly 9 digits, and for any n between 1 and 9 (inclusive), the nth digit is a positive multiple of n. How many positive integers are sparkly? | 216 | 46 | 3 |
math | 3. Some numbers read the same from left to right as they do from right to left (for example, 2772, 515), such numbers are called "palindromic numbers." Now there is a two-digit number, when it is multiplied by 91, 93, 95, 97, the products are all palindromic numbers. This two-digit number is $\qquad$. | 55 | 92 | 2 |
math | Determine all positive integers $n$ such that all positive integers less than or equal to $n$ and relatively prime to $n$ are pairwise coprime. | \{1, 2, 3, 4, 6, 8, 12, 18, 24, 30\} | 34 | 36 |
math | Exercise 1. Find all quadruplets ( $p, q, r, \mathfrak{n}$ ) of strictly positive integers satisfying the following three conditions:
- $p$ and $q$ are prime,
- $p+q$ is not divisible by 3,
$-p+q=r(p-q)^{n}$. | (2,3,5,2n)_{n\in\mathbb{N}^{*}},(3,2,5,n)_{n\in\mathbb{N}^{*}},(3,5,2,2),(5,3,1,3),(5,3,2,2),(5,3,4,1) | 71 | 77 |
math | Suppose tetrahedron $PABC$ has volume $420$ and satisfies $AB = 13$, $BC = 14$, and $CA = 15$. The minimum possible surface area of $PABC$ can be written as $m+n\sqrt{k}$, where $m,n,k$ are positive integers and $k$ is not divisible by the square of any prime. Compute $m+n+k$.
[i]Ray Li[/i] | 346 | 100 | 3 |
math | 3. (3 points) In a certain country, there are 120 cities. The Ministry of Aviation requires that every two cities be connected by a two-way flight operated by exactly one airline, and that it should be possible to travel from any city to any other using the flights of each airline (possibly with layovers). What is the ... | 60 | 86 | 2 |
math | Let $n$ be a positive integer. On a blackboard, Bobo writes a list of $n$ non-negative integers. He then performs a sequence of moves, each of which is as follows:
-for each $i = 1, . . . , n$, he computes the number $a_i$ of integers currently on the board that are at most $i$,
-he erases all integers on the board,
... | 2n | 264 | 4 |
math | 6. In the Cartesian coordinate system $x O y$, the point set $K=\{(x, y) \mid x, y=-1,0,1\}$. If three points are randomly selected from $K$, then the probability that the distance between each pair of these three points is no more than 2 is $\qquad$ . | \frac{5}{14} | 72 | 8 |
math | 10. (20 points) Given the ellipse $C: \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1(a>$ $b>0)$, $F$ is the right focus of the ellipse $C$. A line $l$ is drawn through the intersection of the right directrix $x=2a$ and the $x$-axis, intersecting the ellipse $C$ at points $A$ and $B$. The arithmetic mean of $\frac{1}{|A F|}$... | 192\sqrt{3} | 151 | 8 |
math | 32. There are 2012 students in a secondary school. Every student writes a new year card. The cards are mixed up and randomly distributed to students. Suppose each student gets one and only one card. Find the expected number of students who get back their own cards. | 1 | 58 | 1 |
math | Example 1 Given three points $A(2,1), B(3,2), C(1,4)$, find
$$S_{\triangle A B C}$$ | 2 | 38 | 1 |
math | 4. Let $F_{1}$ and $F_{2}$ be the left and right foci of the hyperbola $C: x^{2}-\frac{y^{2}}{24}=1$, respectively, and let $P$ be a point on the hyperbola $C$ in the first quadrant. If $\frac{\left|P F_{1}\right|}{\left|P F_{2}\right|}=\frac{4}{3}$, then the radius of the incircle of $\triangle P F_{1} F_{2}$ is . $\q... | 2 | 127 | 1 |
math | 1. Find all real solutions of the following nonlinear system:
$$
\begin{aligned}
x+4 y+6 z & =16 \\
x+6 y+12 z & =24 \\
x^{2}+4 y^{2}+36 z^{2} & =76
\end{aligned}
$$ | (6,1,1)(-\frac{2}{3},\frac{13}{3},-\frac{1}{9}) | 72 | 29 |
math | 3. The maximum value of the function $y=\sin 2x-2(\sin x+\cos x)$ is $\qquad$ . | 1+2 \sqrt{2} | 30 | 8 |
math | Example 2 Let $a, b$ be positive integers, $n$ be a given positive odd number, and $a+b=n$. Find the maximum value of $a b$.
Analysis: Since under the given conditions it is impossible to achieve $a=$ $b$, the solution cannot be obtained using the arithmetic mean inequality. | \frac{n^2-1}{4} | 68 | 10 |
math | 6. Points $A_{1}, \ldots, A_{12}$ are the vertices of a regular 12-gon. How many different 11-segment open broken lines without self-intersections with vertices at these points exist? Broken lines that can be transformed into each other by rotation are considered the same. | 1024 | 67 | 4 |
math | 3. Let the function be
$$
f(x)=x^{3}+a x^{2}+b x+c \quad (x \in \mathbf{R}),
$$
where $a, b, c$ are distinct non-zero integers, and
$$
f(a)=a^{3}, f(b)=b^{3} .
$$
Then $a+b+c=$ $\qquad$ | 18 | 85 | 2 |
math | Let $f:\mathbb{N}\mapsto\mathbb{R}$ be the function \[f(n)=\sum_{k=1}^\infty\dfrac{1}{\operatorname{lcm}(k,n)^2}.\] It is well-known that $f(1)=\tfrac{\pi^2}6$. What is the smallest positive integer $m$ such that $m\cdot f(10)$ is the square of a rational multiple of $\pi$? | 42 | 106 | 2 |
math | 2. (5 points) At the World Meteorologists Conference, each participant in turn announced the average monthly temperature in their hometown. At this moment, all the others recorded the product of the temperatures in their and the speaker's cities. In total, 62 positive and 48 negative numbers were recorded. What is the ... | 3 | 78 | 1 |
math | Find all integer solution pairs $(x, y)$ for $y^{3}=8 x^{6}+2 x^{3} y-y^{2}$ when $x \in$ $[0,10]$. | (x, y)=(0,0),(0,-1),(1,2) | 46 | 16 |
math | 11.1. Solve the equation $\arccos \frac{x+1}{2}=2 \operatorname{arctg} x$. | \sqrt{2}-1 | 31 | 6 |
math | 12.227. The base of the pyramid is an isosceles trapezoid, where the lateral side is equal to $a$, and the acute angle is equal to $\alpha$. All lateral faces form the same angle $\beta$ with the base of the pyramid. Find the total surface area of the pyramid. | \frac{2a^2\sin\alpha\cos^2\frac{\beta}{2}}{\cos\beta} | 69 | 27 |
math | 2, ** Divide a circle into $n(\geqslant 2)$ sectors $S_{1}, S_{2}, \cdots, S_{n}$. Now, color these sectors using $m(\geqslant 2)$ colors, with each sector being colored with exactly one color, and the requirement that adjacent sectors must have different colors. How many different coloring methods are there? | (-1)^{n}+(-1)^{n}(-1) | 83 | 16 |
math | 6. Given the ellipse $C: \frac{x^{2}}{9}+\frac{y^{2}}{8}=1$ with left and right foci $F_{1}$ and $F_{2}$, and left and right vertices $A$ and $B$, the line $l: x=m y+1$ passing through the right focus $F_{2}$ intersects the ellipse $C$ at points $M\left(x_{1}, y_{1}\right)$ and $N\left(x_{2}, y_{2}\right)\left(y_{1}>0,... | \frac{\sqrt{3}}{12} | 158 | 11 |
math | 3. (25 points) On a circle, there are $n$ different positive integers $a_{1}$, $a_{2}, \cdots, a_{n}$ placed in a clockwise direction. If for any number $b$ among the ten positive integers $1, 2, \cdots, 10$, there exists a positive integer $i$ such that $a_{i}=b$ or $a_{i}+a_{i+1}=b$, with the convention that $a_{n+1}... | 6 | 129 | 1 |
math | Let $ x,y$ are real numbers such that $x^2+2cosy=1$. Find the ranges of $x-cosy$. | [-1, 1 + \sqrt{3}] | 32 | 12 |
math | At the base of the pyramid $S A B C D$ lies a trapezoid $A B C D$ with bases $B C$ and $A D$, and $B C=2 A D$. Points $K$ and $L$ are taken on the edges $S A$ and $S B$, respectively, such that $2 S K=K A$ and $3 S L=L B$. In what ratio does the plane $K L C$ divide the edge $S D$? | 2:1 | 106 | 3 |
math | 9. For any two distinct real numbers $x, y$, define $D(x, y)$ as the unique integer $d$ satisfying
$$
2^{d} \leqslant|x-y|<2^{d+1}
$$
The "scale" of $x \in \mathscr{F}$ in $\mathscr{F}$ is defined as the value of $D(x, y)$, where $y \in \mathscr{F}, x \neq y$.
Let $k$ be a given positive integer, and assume that each... | 2^{k} | 168 | 4 |
math | Example 2 Solve the system of inequalities: $\left\{\begin{array}{l}x^{2}>x+2, \\ 4 x^{2} \leqslant 4 x+15 .\end{array}\right.$ | [-\frac{3}{2},-1)\cup(2,\frac{5}{2}] | 53 | 21 |
math | 162. Find a perfect number of the form $p q$, where $p$ and $q$ are prime numbers. | 6 | 27 | 1 |
math | Task B-4.2. How many terms in the expansion of the binomial $\left(2 \sqrt{x}-\frac{1}{x^{2}}\right)^{100}, x>0$, have a positive exponent of $x$, and how many of those have a positive even integer exponent of $x$? | 20termshavepositiveexponentofx,5ofthosehavepositiveevenintegerexponentofx | 70 | 22 |
math | 11. Let the complex numbers $z_{1}, z_{2}$ satisfy $\left|z_{1}\right|=\left|z_{1}+z_{2}\right|=3,\left|z_{1}-z_{2}\right|=3 \sqrt{3}$, then $\log _{3}\left|\left(z_{1} \overline{z_{2}}\right)^{2000}+\left(\overline{z_{1}} z_{2}\right)^{2000}\right|=$ $\qquad$ . | 4000 | 122 | 4 |
math | In a society of 30 people, any two people are either friends or enemies, and everyone has exactly six enemies. How many ways are there to choose three people from the society so that any two of them are either all friends or all enemies? | 1990 | 51 | 4 |
math | 4. In the field of real numbers, solve the system of inequalities
$$
\begin{aligned}
& \sin x + \cos y \geq \sqrt{2} \\
& \sin y + \cos z \geq \sqrt{2} \\
& \sin z + \cos x \geq \sqrt{2}
\end{aligned}
$$ | \frac{\pi}{4}+2k_{1}\pi,\quad\frac{\pi}{4}+2k_{2}\pi,\quad\frac{\pi}{4}+2k_{3}\pi\quad(k_{1},k_{2},k_{3}\in\mathbb{Z}) | 78 | 67 |
math | Find the greatest $n$ such that $(z+1)^n = z^n + 1$ has all its non-zero roots in the unitary circumference, e.g. $(\alpha+1)^n = \alpha^n + 1, \alpha \neq 0$ implies $|\alpha| = 1.$ | n = 7 | 68 | 5 |
math | 10) (20 points) Let positive real numbers $x, y, z$ satisfy $xyz=1$. Try to find the maximum value of $f(x, y, z) = (1-yz+z)(1-zx+x)(1-xy+y)$ and the values of $x, y, z$ at that time. | 1 | 72 | 1 |
math | 74. A curious number. Find such a positive number that $\frac{1}{5}$ of it, multiplied by its $\frac{1}{7}$, equals this number. | 35 | 38 | 2 |
math | The lateral side of an isosceles trapezoid is equal to $a$, the midline is equal to $b$, and one angle at the larger base is $30^{\circ}$. Find the radius of the circle circumscribed around the trapezoid. | \sqrt{b^2+\frac{^2}{4}} | 60 | 14 |
math | Example 1 There are 2012 lamps, numbered $1, 2, \cdots, 2012$, arranged in a row in a corridor, and initially, each lamp is on. A mischievous student performed the following 2012 operations: for $1 \leqslant k \leqslant 2012$, during the $k$-th operation, the student toggled the switch of all lamps whose numbers are mu... | 1968 | 118 | 4 |
math | 5. For a given rational number, represent it as a reduced fraction. Then find the product of its numerator and denominator. How many rational numbers between 0 and 1 have a product of their numerator and denominator equal to 20! ? | 128 | 50 | 3 |
math | Find the sum of all integers $n$ for which $n - 3$ and $n^2 + 4$ are both perfect cubes. | 13 | 31 | 2 |
math | 【Example 3】Between 1 and $10^{6}$, how many integers have the sum of their digits equal to 9? | 2002 | 30 | 4 |
math | ## Task B-4.5.
For real numbers $x_{1}, x_{2}, \ldots, x_{2016}$, the following equalities hold:
$$
\frac{x_{1}}{x_{1}+1}=\frac{x_{2}}{x_{2}+2}=\cdots=\frac{x_{2016}}{x_{2016}+2016}, \quad x_{1}+x_{2}+x_{3}+\cdots+x_{2016}=2017
$$
Calculate $x_{1008}$. | 1 | 137 | 1 |
math | 13.311. Two trains, a passenger train and an express train, set off towards each other simultaneously from two points that are 2400 km apart. Each train travels at a constant speed, and at some point in time, they meet. If both trains had traveled at the speed of the express train, they would have met 3 hours earlier t... | 60 | 116 | 2 |
math | 2. (4p) a) The cost of a blouse represents $30 \%$ of the cost of a dress, and $7.5 \%$ of the cost of a coat. What percent does the cost of the dress represent of the cost of the coat?
(3p) b) At a store, the price of a product increased by $25 \%$, then it was reduced by p\%, returning to the initial price. What per... | 20 | 106 | 2 |
math | Find all functions $f: \mathbb{N} \mapsto \mathbb{N}$ so that for any positive integer $n$ and finite sequence of positive integers $a_0, \dots, a_n$, whenever the polynomial $a_0+a_1x+\dots+a_nx^n$ has at least one integer root, so does \[f(a_0)+f(a_1)x+\dots+f(a_n)x^n.\]
[i]Proposed by Sutanay Bhattacharya[/i] | f(n) = n | 109 | 6 |
math | 7.040. $\lg \left(x^{2}+1\right)=2 \lg ^{-1}\left(x^{2}+1\right)-1$.
7.040. $\lg \left(x^{2}+1\right)=2 \lg ^{-1}\left(x^{2}+1\right)-1$. | -3;3 | 77 | 4 |
math | Several gnomes, loading their luggage onto a pony, set off on a long journey. They were spotted by trolls, who counted 36 legs and 15 heads in the caravan. How many gnomes and how many ponies were there?
# | 12 | 54 | 2 |
math | Example 1 Let the sequence of positive integers $\left\{a_{n}\right\}$ satisfy $a_{n+3}=a_{n+2}\left(a_{n+1}+2 a_{n}\right), n=1,2, \cdots$ and $a_{6}=2288$. Find $a_{1}, a_{2}, a_{3}$.
(1988 Sichuan Province Competition Problem) | a_{1}=5,a_{2}=1,a_{3}=2 | 97 | 15 |
math | 10.327. Determine the area of a triangle if two of its sides are 1 and $\sqrt{15}$ cm, and the median to the third side is 2 cm. | \frac{\sqrt{15}}{2}\mathrm{~}^{2} | 42 | 18 |
math | 14. Given the ellipse $\frac{x^{2}}{9}+\frac{y^{2}}{5}=1$ with its right focus at $F$, and $B$ as a moving point on the ellipse, $\triangle F A B$ is an equilateral triangle, and $F, A, B$ are in counterclockwise order. Find the locus of point $A$. | |z-2|+|z-2\sqrt{3}i|=6 | 83 | 18 |
math | [ Counting in two ways ] [ Different tasks on cutting ]
Inside a square, 100 points are marked. The square is divided into triangles in such a way that the vertices of the triangles are only the 100 marked points and the vertices of the square, and for each triangle in the partition, each marked point either lies out... | 202 | 95 | 3 |
math | Problem 10. A cylinder of volume 9 is inscribed in a cone. The plane of the upper base of this cylinder cuts off a frustum of volume 63 from the original cone. Find the volume of the original cone. | 64 | 50 | 2 |
math | 4. A group of 17 middle school students went to several places for a summer social survey, with a budget for accommodation not exceeding $x$ yuan per person per day. One day, they arrived at a place with two hostels, $A$ and $B$. $A$ has 8 first-class beds and 11 second-class beds; $B$ has 10 first-class beds, 4 second... | 10 | 165 | 2 |
math | 4. Let $T$ be a set of ordered triples $(x, y, z)$, where $x, y, z$ are integers, and $0 \leqslant x, y, z \leqslant 9$. Two players, A and B, play the following game: A selects a triple $(x, y, z)$ from $T$, and B has to guess A's chosen triple using several "moves". One "move" consists of: B giving A a triple $(a, b,... | 3 | 167 | 1 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.