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 natural numbers for which the following is true: if we add the sum of its digits to the number, we get 313. | 296,305 | 32 | 7 |
math | \section*{Problem 1 - 121241}
Determine whether among all pairs \((a, b)\) of positive real numbers, there exist such pairs for which
\[
f(a, b)=\frac{a^{4}}{b^{4}}+\frac{b^{4}}{a^{4}}-\frac{a^{2}}{b^{2}}-\frac{b^{2}}{a^{2}}+\frac{a}{b}+\frac{b}{a}
\]
attains a minimum value. If so, then this minimum value should be... | 2 | 129 | 1 |
math | 27. Suppose $a$ and $b$ are the roots of $x^{2}+x \sin \alpha+1=0$ while $c$ and $d$ are the roots of the equation $x^{2}+x \cos \alpha-1=0$. Find the value of $\frac{1}{a^{2}}+\frac{1}{b^{2}}+\frac{1}{c^{2}}+\frac{1}{d^{2}}$. | 1 | 102 | 1 |
math | 9. Find the result of the following binary subtraction operations:
(i) $(1010111)_{2}-(11001)_{2}-(11110)_{2}=$ ?
(ii) $(10110001)_{2}-(1101100)_{2}-(11110)_{2}=$ ? | (100111)_{2} | 87 | 11 |
math | Determine an equation of third degree with integral coefficients having roots $\sin \frac{\pi}{14}, \sin \frac{5 \pi}{14}$ and $\sin \frac{-3 \pi}{14}.$ | 8x^3 - 4x^2 - 4x - 1 = 0 | 48 | 20 |
math | 4. Find all positive integers $n$ such that every positive integer whose decimal representation consists of $n-1$ digits 1 and one digit 7 is a prime number.
(31st IMO Shortlist) | n=1,2 | 45 | 5 |
math | 3. Let $z_{n}=\left(\frac{1-\mathrm{i}}{2}\right)^{n}, n \in \mathbf{N}_{+}$, and let $S_{n}=\sum_{k=1}^{n}\left|z_{k+1}-z_{k}\right|$, then $\lim _{n \rightarrow \infty} S_{n}=$ | 1+\frac{\sqrt{2}}{2} | 87 | 11 |
math | 10. Non-negative real numbers $a_{i}(i=1,2, \cdots, n)$, satisfy: $a_{1}+a_{2}+a_{3}+\cdots+a_{n}=1$, find the minimum value of $\frac{a_{1}}{1+a_{2}+\cdots+a_{n}}+\frac{a_{2}}{1+a_{1}+a_{3}+\cdots+a_{n}}+\cdots+\frac{a_{n}}{1+a_{1}+a_{2}+\cdots+a_{n-1}}$. | \frac{n}{2n-1} | 132 | 9 |
math | 12. (10 points) In a math competition, each team can only score 0 points, 3 points, or 5 points per question. At the end of the competition, the total score of three teams is 32 points. If any team's total score can reach 32 points, how many different combinations of total scores are there for these three teams? | 255 | 79 | 3 |
math | Tokorevev. S.
Among 2000 indistinguishable balls, half are aluminum with a mass of 10 g, and the rest are duralumin with a mass of 9.9 g. It is required to separate the balls into two piles such that the masses of the piles are different, but the number of balls in them is the same. What is the smallest number of weig... | 1 | 97 | 1 |
math | 7.1. A natural number $n$ was multiplied by the sum of the digits of the number $3 n$, and the resulting number was then multiplied by 2. As a result, 2022 was obtained. Find $n$. | 337 | 52 | 3 |
math | 6. Given the sequence $\left\{a_{n}\right\}$ satisfies
$$
a_{n}^{2}=a_{n+1} a_{n}-1\left(n \in \mathbf{Z}_{+}\right) \text {, and } a_{1}=\sqrt{2} \text {. }
$$
Then the natural number closest to $\sqrt{a_{2014}}$ is $\qquad$ | 8 | 96 | 1 |
math | 1. A natural number is called a palindrome if it remains unchanged when its digits are written in reverse order (for example, 626 is a palindrome, while 2015 is not). Represent the number 2015 as the sum of two palindromes. | 2015=1551+464 | 60 | 13 |
math | We are given a convex quadrilateral $ABCD$ in the plane.
([i]i[/i]) If there exists a point $P$ in the plane such that the areas of $\triangle ABP, \triangle BCP, \triangle CDP, \triangle DAP$ are equal, what condition must be satisfied by the quadrilateral $ABCD$?
([i]ii[/i]) Find (with proof) the maximum possible num... | 1 | 109 | 3 |
math | 1. Consider a rectangular parallelepiped $A B C D A^{\prime} B^{\prime} C^{\prime} D^{\prime}$, where $A B C D$ is the bottom face with the letters assigned in a clockwise direction, and $A, B, C$, and $D$ are directly below $A^{\prime}, B^{\prime}, C^{\prime}$, and $D^{\prime}$ respectively. The parallelepiped is divi... | 2015 | 236 | 4 |
math | II. (50 points) Try to find the smallest positive integer $m$, such that the following conditions are satisfied simultaneously:
(1) $\left[\frac{2}{1977} m^{2}\right] \geqslant m+2006$ ( $[x]$ denotes the greatest integer not exceeding $x$);
(2) $99^{m}$ leaves a remainder of 11 when divided by 190. | 2004 | 98 | 4 |
math | 6. Let the side length of rhombus $A_{1} A_{2} A_{3} A_{4}$ be $1, \angle A_{1} A_{2} A_{3}=$ $\frac{\pi}{6}, P$ be a point in the plane of rhombus $A_{1} A_{2} A_{3} A_{4}$. Then the minimum value of $\sum_{1 \leqslant i<j \leqslant 4} \overrightarrow{P A_{i}} \cdot \overrightarrow{P A_{j}}$ is $\qquad$ | -1 | 133 | 2 |
math | 1. Given are fifty natural numbers, of which half do not exceed 50, and the other half are greater than 50 but less than 100. The difference between any two of the given numbers is not 0 or 50. Find the sum of these numbers. | 2525 | 61 | 4 |
math | Example 4. Convert $\cos \alpha+\cos 3 \alpha+\cos 5 \alpha$ $+\cos 7 \alpha$ into a product. | 4 \cos \alpha \cos 2 \alpha \cos 4 \alpha | 34 | 17 |
math | 1. Given non-empty sets
$$
\begin{array}{l}
A=\{x \mid m+1 \leqslant x \leqslant 2 m-1\}, \\
B=\left\{x \mid x^{2}-2 x-15 \leqslant 0\right\},
\end{array}
$$
and $A \subseteq B$.
Then the range of real number $m$ is | [2,3] | 97 | 5 |
math | 6. Given that the three vertices of $\triangle A B C$ are all on the parabola $y^{2}=2 p x(p>0)$, and the centroid of $\triangle A B C$ is exactly the focus of the parabola. If the equation of the line on which side $B C$ lies is $4 x+y$ $-20=0$, then $p=$ $\qquad$ . | 8 | 90 | 1 |
math | 3. For an integer $n \geq 3$, we say that $A=\left(a_{1}, a_{2}, \ldots, a_{n}\right)$ is an $n$-list if every $a_{k}$ is an integer in the range $1 \leq a_{k} \leq n$. For each $k=1, \ldots, n-1$, let $M_{k}$ be the minimal possible non-zero value of $\left|\frac{a_{1}+\ldots+a_{k+1}}{k+1}-\frac{a_{1}+\ldots+a_{k}}{k}... | 4(n-1) | 245 | 5 |
math | Let $a$ and $b$ be positive integers not divisible by $5$. A sequence of integers is constructed as follows: the first term is $5$, and every consequent term is obtained by multiplying its precedent by $a$ and adding $b$. (For example, if $a = 2$ and $b = 4$, the first three terms are $5,14,32$.) What is the maximum po... | 5 \text{ consecutive primes} | 107 | 7 |
math | 1. A number is called non-null if it is whole and positive and contains no zeros. You can nullify a positive whole number by simply removing the zeros. We denote this with square brackets, for example $[2050]=25$ and $[13]=13$. For multiplication, addition, and subtraction, we use square brackets to indicate when we ar... | (4,5),(8,25),(2,55),(5,22),(11,91),(13,77),(25,44) | 195 | 38 |
math | [Arithmetic. Mental arithmetic, etc.] [Theory of algorithms (other).]
There are two hourglasses - one for 7 minutes and one for 11 minutes. An egg needs to be boiled for 15 minutes. How can you measure this time using the available hourglasses?
# | 15 | 63 | 2 |
math | 1. Solve, in the set of complex numbers, the system of equations $z^{19} w^{25}=1$, $z^{5} w^{7}=1, z^{4}+w^{4}=2$. | (1,1),(-1,-1),(i,i),(-i,-i) | 50 | 18 |
math | 4. For any positive real numbers $x, y$,
$$
\frac{(x y+x+y)(x+y+1)}{(x+y)(x+1)(y+1)}
$$
the range of values is | (1,\frac{9}{8}] | 47 | 9 |
math | Find how many committees with a chairman can be chosen from a set of n persons. Hence or otherwise prove that
$${n \choose 1} + 2{n \choose 2} + 3{n \choose 3} + ...... + n{n \choose n} = n2^{n-1}$$ | n \cdot 2^{n-1} | 67 | 11 |
math | Calvin was asked to evaluate $37 + 31 \times a$ for some number $a$. Unfortunately, his paper was tilted 45 degrees, so he mistook multiplication for addition (and vice versa) and evaluated $37 \times 31 + a$ instead. Fortunately, Calvin still arrived at the correct answer while still following the order of operations.... | 37 | 99 | 2 |
math | 7 (1248). The number a is $80 \%$ of the number b, and the number c is $140 \%$ of the number b. Find the numbers a, b, and c, given that c is 72 more than a. | =96,b=120,=168 | 58 | 13 |
math | If $a>1$ and $b>2$ are positive integers, show that $a^{b}+1 \geq b(a+1)$, and determine when equality holds. | a^b + 1 \geq b(a + 1) | 40 | 16 |
math | 3. Determine a six-digit number which, when multiplied by 2, 3, 4, 5, and 6, gives six-digit numbers written with the same digits as the original number. | 142857 | 42 | 6 |
math | 7.2. On a certain island, only knights, who always tell the truth, and liars, who always lie, live. One day, 99 inhabitants of this island stood in a circle, and each of them said: "All ten people following me in a clockwise direction are liars." How many knights could there be among those standing in the circle? | 9 | 76 | 1 |
math | 1. Calculate $2015-5 \cdot(25 \cdot 13+13+3 \cdot 25-10)$. | 0 | 35 | 1 |
math | Let $P(X)$ be a monic polynomial of degree 2017 such that $P(1)=1, P(2)=2, \ldots, P(2017)=$ 2017. What is the value of $P(2018)$? | 2017!+2018 | 64 | 10 |
math | 12. Let $x \in R$, then the minimum value of the function $f(x)=|2 x-1|+|3 x-2|+|4 x-3|+|5 x-4|$ is | 1 | 49 | 1 |
math | \section*{Problem 3B - 111043B}
Dirk explains to Jürgen the usefulness of differential calculus using the solution to the following problem: Let \(A B C D E\) be a planar convex pentagon such that \(A, B, C, E\) are the vertices of a rectangle and \(C, D, E\) are the vertices of an equilateral triangle. The area of th... | 1+\frac{\sqrt{3}}{3} | 211 | 11 |
math | 7. Given the sequence $\left\{a_{n}\right\}$:
$$
a_{n}=2^{n}+3^{n}+6^{n}+1\left(n \in \mathbf{Z}_{+}\right) \text {. }
$$
Does there exist an integer $k \geqslant 2$, such that $k$ is coprime with all numbers in the sequence $\left\{a_{n}\right\}$? If it exists, find the smallest integer $k$; if not, explain the reaso... | 23 | 129 | 2 |
math | 6. Let $\left(1+x+x^{2}\right)^{n}=\sum_{k=0}^{2 n} a_{k} x^{k}$. Then $\sum_{k=1}^{n} a_{2 k}=$ $\qquad$ | \frac{3^{n}-1}{2} | 58 | 11 |
math | 45th Putnam 1984 Problem A3 Let A be the 2n x 2n matrix whose diagonal elements are all x and whose off-diagonal elements a ij = a for i + j even, and b for i + j odd. Find lim x→a det A/(x - a) 2n-2 . Solution | n^2(^2-b^2) | 74 | 9 |
math | 3. Three numbers form a geometric progression. If the second term is increased by 8, the progression turns into an arithmetic one, but if then the third term of the obtained progression is increased by 64, it turns back into a geometric progression. Find these numbers. | a_{1}=\frac{4}{9},\quadb_{1}=-\frac{20}{9},\quadc_{1}=\frac{100}{9},\quada_{2}=4,\quadb_{2}=12,\quadc_{2}=36 | 56 | 64 |
math | 2. (10 points) When the escalator is not operating, it takes a child 90 seconds to walk the entire 60 meters of the escalator. When the escalator is operating, it takes 60 seconds to transport passengers from the bottom to the top. If the child walks on the operating escalator, how many seconds will it take for the chi... | 36 | 96 | 2 |
math | Let $R$ be the set of points $(x, y)$ such that $\lfloor x^2 \rfloor = \lfloor y \rfloor$ and $\lfloor y^2 \rfloor = \lfloor x \rfloor$. Compute the area of region $R$. Recall that $\lfloor z \rfloor$ is the greatest integer that is less than or equal to $z$. | 4 - 2\sqrt{2} | 87 | 9 |
math | 29. Find a function $f(x)$ such that for any real $x$, except 0 and 1, $f(1 / x) + f(1 - x) = x$. | \frac{x^{3}-x^{2}+1}{2x(1-x)} | 42 | 19 |
math | 30. Let $a, b \in \mathbf{R}^{+}$, find the minimum value of $y=\frac{a}{\sin ^{3} \theta}+\frac{b}{\cos ^{3} \theta}, \theta \in\left(0, \frac{\pi}{2}\right)$. | (^{\frac{2}{5}}+b^{\frac{2}{5}})^{\frac{5}{2}} | 74 | 26 |
math | 2 Let $a, b, c$ be positive real numbers, find the minimum value of
$$
\frac{a+3 c}{a+2 b+c}+\frac{4 b}{a+b+2 c}-\frac{8 c}{a+b+3 c}
$$
(Supplied by Li Shenghong) | -17+12\sqrt{2} | 72 | 11 |
math | 4・ 203 There are two coal mines, A and B. Coal from mine A releases 4 calories when burned per gram, and coal from mine B releases 6 calories when burned per gram. The price of coal at the origin is: 20 yuan per ton for mine A, and 24 yuan per ton for mine B. It is known that: the transportation cost of coal from mine ... | 18 | 131 | 2 |
math | 66. In how many ways can three items be chosen from $n$ items? | \frac{n(n-1)(n-2)}{6} | 18 | 14 |
math | Problem 1. Sasho thought of a number and multiplied it by 7 and by 16. He added the obtained products and got the number 230. Which number did Sasho think of? | 10 | 46 | 2 |
math | 83. Even or odd sum of all natural numbers from 1 to 17? | odd | 19 | 1 |
math | Compute the number of non-empty subsets $S$ of $\{-3, -2, -1, 0, 1, 2, 3\}$ with the following property: for any $k \ge 1$ distinct elements $a_1, \dots, a_k \in S$ we have $a_1 + \dots + a_k \neq 0$.
[i]Proposed by Evan Chen[/i] | 24 | 93 | 2 |
math | 12. (12 points) Calculate.
$$
\begin{array}{l}
9.5 \times 101 \\
12.5 \times 8.8 \\
38.4 \times 187-15.4 \times 384+3.3 \times 16 \\
5.29 \times 73+52.9 \times 2.7
\end{array}
$$ | 959.5,110,1320,529 | 98 | 18 |
math | Determine all positive integers$ n$ such that $f_n(x,y,z) = x^{2n} + y^{2n} + z^{2n} - xy - yz - zx$ divides $g_n(x,y, z) = (x - y)^{5n} + (y -z)^{5n} + (z - x)^{5n}$, as polynomials in $x, y, z$ with integer coefficients. | n = 1 | 98 | 5 |
math | Let $p,q$ be positive integers. For any $a,b\in\mathbb{R}$ define the sets $$P(a)=\bigg\{a_n=a \ + \ n \ \cdot \ \frac{1}{p} : n\in\mathbb{N}\bigg\}\text{ and }Q(b)=\bigg\{b_n=b \ + \ n \ \cdot \ \frac{1}{q} : n\in\mathbb{N}\bigg\}.$$
The [i]distance[/i] between $P(a)$ and $Q(b)$ is the minimum value of $|x-y|$ as $x\i... | \frac{1}{2 \cdot \text{lcm}(p, q)} | 185 | 17 |
math | A hollow glass sphere with uniform wall thickness, which is empty inside, has an outer diameter of $16 \mathrm{~cm}$ and floats in water such that $\frac{3}{8}$ of its surface remains dry. What is the wall thickness if the specific gravity of the glass is $s=2.523$? | 0.8\mathrm{~} | 70 | 8 |
math | (21) The center of the ellipse $C$ is the origin $O$, with foci on the $y$-axis, the distance from the foci to the corresponding directrix, and the eccentricity are both $\frac{\sqrt{2}}{2}$. The line $l$ intersects the $y$-axis at point $P(0, m)$, and intersects the ellipse $C$ at two distinct points $A$ and $B$, and ... | (-1,-\frac{1}{2})\cup(\frac{1}{2},1) | 162 | 21 |
math | Example 18 Among the positive integers less than 10000, how many integers $x$ can make $2^{x}-x^{2}$ divisible by 7? | 2857 | 39 | 4 |
math | 11. (20 points) Find the smallest integer \( n (n > 1) \), such that there exist \( n \) integers \( a_{1}, a_{2}, \cdots, a_{n} \) (allowing repetition) satisfying
$$
a_{1}+a_{2}+\cdots+a_{n}=a_{1} a_{2} \cdots a_{n}=2013 .
$$
12. (20 points) Let positive integers \( a, b, c, d \) satisfy
$$
a^{2}=c(d+13), b^{2}=c(d-1... | 5 | 157 | 1 |
math | ## Task Condition
Find the derivative.
$y=\frac{\sqrt{1-x^{2}}}{x}+\arcsin x$ | -\frac{\sqrt{1-x^{2}}}{x^{2}} | 29 | 15 |
math | Example 5 (2005 National High School Mathematics Competition Question) Define the function
$$f(k)=\left\{\begin{array}{l}
0, \text { if } k \text { is a perfect square } \\
{\left[\frac{1}{\{\sqrt{k}\}}\right], \text { if } k \text { is not a perfect square }}
\end{array} \text {, find } \sum_{k=1}^{240} f(k)\right. \t... | 768 | 114 | 3 |
math | 18. Find the largest integer $n$ such that $n$ is a divisor of $a^{5}-a$ for all integers $a$. | 30 | 32 | 2 |
math | 7. If for any real number $x$, the function
$$
f(x)=x^{2}-2 x-|x-1-a|-|x-2|+4
$$
is always a non-negative real number, then the maximum value of the real number $a$ is | 1 | 61 | 1 |
math | 2. On a line, several points were marked, including points $A$ and $B$. All possible segments with endpoints at the marked points are considered. Vasya calculated that point $A$ is inside 50 of these segments, and point $B$ is inside 56 segments. How many points were marked? (The endpoints of a segment are not consider... | 16 | 81 | 2 |
math | Consider all words containing only letters $A$ and $B$. For any positive integer $n$, $p(n)$ denotes the number of all $n$-letter words without four consecutive $A$'s or three consecutive $B$'s. Find the value of the expression
\[\frac{p(2004)-p(2002)-p(1999)}{p(2001)+p(2000)}.\] | 2 | 101 | 1 |
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 | 8. Let the sequence $a_{n}=\left[(\sqrt{2}+1)^{n}+\left(\frac{1}{2}\right)^{n}\right], n \geq 0$, where $[x]$ denotes the greatest integer less than or equal to $x$. Then
$$
\sum_{n=1}^{\infty} \frac{1}{a_{n-1} a_{n+1}}=
$$ | \frac{1}{8} | 99 | 7 |
math | Given a positive integer $n$, find the largest real number $C$ such that: if the sum of the reciprocals of a set of integers greater than 1 (which can be the same) is less than $C$, then it is always possible to divide this set of numbers into no more than $n$ groups, such that the sum of the reciprocals of the numbers... | C_{\max}=\frac{n+1}{2} | 89 | 13 |
math | 3. Find all positive integers $n \geqslant 4$ such that for any distinct and non-zero complex numbers $a, b, c$, if
$$
(a-b)^{n}+(b-c)^{n}+(c-a)^{n}=0,
$$
then $a, b, c$ are the complex coordinates of the vertices of an equilateral triangle. | 4 | 82 | 1 |
math | 5. (5 points) From the numbers $1, 2, 3, 4, \cdots, 30$, if you arbitrarily select 10 consecutive numbers, the number of situations where there are exactly 2 prime numbers is:
cases | 4 | 54 | 1 |
math | Two workers, $A$ and $B$, completed a task assigned to them as follows. First, only $A$ worked for $\frac{2}{3}$ of the time it would take $B$ to complete the entire task alone. Then, $B$ took over from $A$ and finished the work. In this way, the work took 2 hours longer than if they had started working together and co... | A=6,B=3 | 143 | 6 |
math | 5th Irish 1992 Problem A2 How many (x, y, z) satisfy x 2 + y 2 + z 2 = 9, x 4 + y 4 + z 4 = 33, xyz = -4? | 12 | 57 | 2 |
math | 5.93 $n$ primary school students sit around a circle. A teacher walks along the circle in a counterclockwise direction and gives out candies to the students. Starting from a certain student, the teacher skips 1 student and gives a candy to the next student; then skips 2 students and gives a candy to the next student; t... | 2^{} | 120 | 3 |
math | Find all real solutions of the equation: $$x=\frac{2z^2}{1+z^2}$$ $$y=\frac{2x^2}{1+x^2}$$ $$z=\frac{2y^2}{1+y^2}$$ | (0, 0, 0) | 56 | 10 |
math | 1. In the field of real numbers, solve the system of equations
$$
\begin{aligned}
2 x+\lfloor y\rfloor & =2022, \\
3 y+\lfloor 2 x\rfloor & =2023 .
\end{aligned}
$$
(The symbol $\lfloor a\rfloor$ denotes the floor of the real number $a$, i.e., the greatest integer not greater than $a$. For example, $\lfloor 1.9\rfloo... | (1011,\frac{1}{3}) | 137 | 12 |
math | Let $f$ be a continuously differentiable function on $[0, 1]$ and $m \in \mathbb{N}$. Let $A = f(1)$ and let $B=\int \limits_{0}^1 x^{-\frac{1}{m}}f(x)dx$. Calculate $$\lim \limits_{n \to \infty} n\left(\int \limits_{0}^1 f(x)dx-\sum \limits_{k=1}^n \left(\frac{k^m}{n^m}-\frac{(k-1)^m}{n^m}\right)f\left(\frac{(k-1)^m}{... | \frac{m}{2}A - \frac{m-1}{2}B | 163 | 19 |
math | 3. $n$ is a positive integer, $A$ is a set of subsets of the set $\{1,2, \cdots, n\}$, such that no element of $A$ contains another element of $A$. Find the maximum number of elements in $A$.
(1994 Bulgarian Olympiad Problem) | C_{n}^{[\frac{n}{2}]} | 71 | 12 |
math | ## Task 1 - 080711
The greatest common divisor of two natural numbers is 6, and their least common multiple is 210.
Determine all pairs of numbers with the given properties! | (6,210)(30,42) | 47 | 13 |
math | 13.428 A batch of identical parts was processed on three machines of different designs in the following sequence: first, only the first machine worked for as many hours as it would take for the second and third machines to complete the entire job together; then, only the second machine worked for as many hours as it wo... | 4 | 129 | 1 |
math | 3. The function $f$ is defined on the set of non-negative integers $\mathbb{N}_{0}=\mathbb{N} \cup\{0\}$, and takes values in the same set $\mathbb{N}_{0}$. For every $n \in \mathbb{N}_{0}$, it holds that $f(f(n))+f(n)=2 n+3$. Find $f(2012)$. | 2013 | 95 | 4 |
math | 4. Karlo watched a movie that started at 5:50 PM. During the movie, there were two advertisements, one lasting 4 minutes and the other 6 minutes. The movie ended at 7:45 PM. How long would the movie have lasted without the advertisements? | 105 | 60 | 3 |
math | Determine all rational numbers $a$ for which the matrix
$$\begin{pmatrix}
a & -a & -1 & 0 \\
a & -a & 0 & -1 \\
1 & 0 & a & -a\\
0 & 1 & a & -a
\end{pmatrix}$$
is the square of a matrix with all rational entries.
[i]Proposed by Daniël Kroes, University of California, San Diego[/i] | a = 0 | 112 | 5 |
math | Example. Let $m, n$ be positive integers, find the minimum value of $\left|12^{m}-5^{n}\right|$. | 7 | 32 | 1 |
math | 1. If the cold water tap is opened, the bathtub fills up in 5 minutes and 20 seconds. If both the cold water tap and the hot water tap are opened simultaneously, the bathtub fills up to the same level in 2 minutes. How long will it take to fill the bathtub if only the hot water tap is opened? Give your answer in second... | 192 | 83 | 3 |
math | How many prime numbers are there among the four-digit numbers whose digits are $1,2,3$ and $4$ in some order?
$^{1}$ Let's prove and use the fact that to determine the divisibility of a number, it is sufficient to test its divisibility only by prime numbers whose squares are not greater than the number in question. | 4 | 73 | 1 |
math | 8-6. In a kindergarten, 150 children are standing in a circle. Each child is looking at the teacher standing in the center of the circle. Some children are wearing blue jackets, and the rest are wearing red ones. There are 12 children in blue jackets whose left neighbor is in a red jacket. How many children have a left... | 126 | 82 | 3 |
math | ## Task B-1.6.
The sum of the fractions $\frac{12}{23}+\frac{1212}{2323}+\frac{121212}{232323}+\cdots+\frac{1212 \ldots 12}{2323 \ldots 23}$ is 528. The number of digits 1 and 2 in the numerator and the number of digits 2 and 3 in the denominator of these fractions increase by one. How many times does the digit 2 appe... | 2024 | 132 | 4 |
math | 7. Given positive numbers $a, b$ satisfy $2 a+b=1$, then the maximum value of $4 a^{2}+b^{2}+4 \sqrt{a b}$ is $\qquad$ | \frac{1}{2}+\sqrt{2} | 47 | 12 |
math | The non-negative numbers $x,y,z$ satisfy the relation $x + y+ z = 3$. Find the smallest possible numerical value and the largest possible numerical value for the expression
$$E(x,y, z) = \sqrt{x(y + 3)} + \sqrt{y(z + 3)} + \sqrt{z(x + 3)} .$$ | 3 \leq E(x, y, z) \leq 6 | 76 | 18 |
math | 2 Let $x, y, z \in \mathbf{R}^{+}$, and $x+y+z \geqslant 6$. Find
$$M=\sum x^{2}+\sum \frac{x}{y^{2}+z+1}$$
the minimum value, where " $\sum$ " denotes the cyclic sum. | \frac{90}{7} | 75 | 8 |
math | 12.179. Find the sine of the angle at the vertex of an isosceles triangle, given that the perimeter of any inscribed rectangle, two vertices of which lie on the base, has a constant value. | \frac{4}{5} | 48 | 7 |
math | # 3.1. Condition:
Vanya bought balloons, red ones were 7 times more than blue ones. While Vanya was walking home, some of the balloons burst, and among the burst balloons, there were 3 times fewer red ones than blue ones. What is the smallest number of balloons Vanya could have bought? | 24 | 68 | 2 |
math | A student wrote down the following sequence of numbers : the first number is 1, the second number is 2, and after that, each number is obtained by adding together all the previous numbers. Determine the 12th number in the sequence. | 1536 | 51 | 4 |
math | 11. Given $S_{1}=1, S_{2}=1-2, S_{3}=1-2+3, S_{4}=1-2+3-4, S_{5}=1-2+3-4+5, \cdots$, then $S_{1}+S_{2}+S_{3}+\cdots+S_{299}=$ $\qquad$. | 150 | 89 | 3 |
math | Let $ p > 2$ be a prime number. Find the least positive number $ a$ which can be represented as
\[ a \equal{} (X \minus{} 1)f(X) \plus{} (X^{p \minus{} 1} \plus{} X^{p \minus{} 2} \plus{} \cdots \plus{} X \plus{} 1)g(X),
\]
where $ f(X)$ and $ g(X)$ are integer polynomials.
[i]Mircea Becheanu[/i]. | p | 113 | 1 |
math | Example 6. Find the inverse function of the function $y=\sin x, x \in[-\pi$, $-\frac{\pi}{2}$]. | y=-\pi-\arcsin x, x \in[-1,0] | 33 | 18 |
math | Example 9 Find all positive integer tuples $(a, b, c, x, y, z)$ such that
$$\left\{\begin{array}{l}
a+b+c=x y z, \\
x+y+z=a b c,
\end{array}\right.$$
where $a \geqslant b \geqslant c, x \geqslant y \geqslant z$. | (2,2,2,6,1,1),(5,2,1,8,1,1),(3,3,1,7,1,1),(3,2,1,3,2,1),(6,1,1,2,2,2),(8,1,1,5,2,1),(7,1,1,3,3,1) | 88 | 85 |
math | 5-36 Try to find 3 distinct non-zero integers $a, b, c$, such that the algebraic expression $x(x-a)(x-b)(x-c)+1$
can be expressed as the product of two polynomials with integer coefficients. | (1,2,3),(-1,-2,-3),(1,-1,2),(1,-1,-2) | 53 | 26 |
math | 466. Find the corrected sample variance for the given sample distribution $n=10$:
$$
\begin{array}{lccc}
x_{i} & 102 & 104 & 108 \\
n_{i} & 2 & 3 & 5
\end{array}
$$ | 6.93 | 71 | 4 |
math | Solve the following inequality:
$$
\frac{2+7 x-15 x^{2}}{5-x+6 x^{2}}>0
$$ | -\frac{1}{5}<x<\frac{2}{3} | 34 | 16 |
math | ## Task A-1.3. (8 points)
Determine the largest integer $n$ for which the inequality $3\left(n-\frac{5}{3}\right)-2(4 n+1)>6 n+5$ holds. | -2 | 52 | 2 |
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