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 | . What is the value of the sum of binomial coefficients
$$
\binom{n}{0}+\binom{n}{1}+\binom{n}{2}+\cdots+\binom{n}{n-1}+\binom{n}{n} ?
$$ | 2^n | 57 | 2 |
math | [Example 4.4.4] There is a tetrahedron $A-BCD$, where the sum of the dihedral angles at each vertex is $180^{\circ}$, and it has three unequal edge lengths of $\sqrt{34}$, $\sqrt{58}$, and $\sqrt{74}$. Find the volume of this tetrahedron. | 35 | 84 | 2 |
math | Let's simplify the following fraction:
$$
\frac{18 x^{4}-9 x^{3}-86 x^{2}+16 x+96}{18 x^{4}-63 x^{3}+22 x^{2}+112 x-96}
$$ | \frac{2x+3}{2x-3} | 66 | 13 |
math | 14. A. From the 2010 positive integers $1,2, \cdots, 2010$, what is the maximum number of integers that can be selected such that the sum of any three selected numbers is divisible by 33? | 61 | 56 | 2 |
math | 4. (6 points) There are three natural numbers $a, b, c$. It is known that $b$ divided by $a$ gives a quotient of 3 and a remainder of 3; $c$ divided by $a$ gives a quotient of 9 and a remainder of 11. Then, the remainder when $c$ is divided by $b$ is $\qquad$ | 2 | 84 | 1 |
math | 8. Problem: Find all real numbers $x$ such that $-1<x \leq 2$ and
$$
\sqrt{2-x}+\sqrt{2+2 x}=\sqrt{\frac{x^{4}+1}{x^{2}+1}}+\frac{x+3}{x+1} .
$$ | 1 | 71 | 1 |
math | Petya and Vasya came up with ten polynomials of the fifth degree. Then Vasya sequentially called out natural numbers (starting from some number), and Petya substituted each called number into one of the polynomials of his choice and wrote down the obtained values on the board from left to right. It turned out that the ... | 50 | 98 | 2 |
math | 3. Given an angle of $13^{0}$. How to obtain an angle of $11^{0}$? | 11 | 26 | 2 |
math | 4. (7 points) A fair coin is tossed $n$ times. Find the probability that no two consecutive tosses result in heads? | \frac{f_{n+2}}{2^n} | 29 | 13 |
math | 2. Let $m, n \in N$, and $m>n$, set $A=\{1$, $2,3, \cdots, m\}, B=\{1,2,3, \cdots, n\}$, and $C \subset A$. Then the number of $C$ that satisfies $B \cap C \neq \varnothing$ is $\qquad$ . | 2^{m-n}\left(2^{n}-1\right) | 86 | 15 |
math | Problem 6.6. On an island, there live knights who always tell the truth, and liars who always lie. One day, 65 inhabitants of the island gathered for a meeting. Each of them, in turn, made the statement: "Among the previously made statements, there are exactly 20 fewer true statements than false ones." How many knights... | 23 | 80 | 2 |
math | 8. In the complex plane, the point corresponding to the complex number $z_{1}$ moves on the line segment connecting 1 and $\mathrm{i}$, and the point corresponding to the complex number $z_{2}$ moves on the circle centered at the origin with a radius of 1. Then the area of the region where the point corresponding to th... | 2 \sqrt{2}+\pi | 94 | 8 |
math | Example 7 Find all real number pairs $(x, y)$ that satisfy the equation $x^{2}+(y-1)^{2}+(x-y)^{2}=$ $\frac{1}{3}$. (Example 5 from [1]) | x=\frac{1}{3}, y=\frac{2}{3} | 54 | 16 |
math | Given a finite set $S \subset \mathbb{R}^3$, define $f(S)$ to be the mininum integer $k$ such that there exist $k$ planes that divide $\mathbb{R}^3$ into a set of regions, where no region contains more than one point in $S$. Suppose that
\[M(n) = \max\{f(S) : |S| = n\} \text{ and } m(n) = \min\{f(S) : |S| = n\}.\]
Eval... | 2189 | 135 | 4 |
math | 3.178. Calculate $(1+\operatorname{tg} \alpha)(1+\operatorname{tg} \beta)$, if $\alpha+\beta=\frac{\pi}{4}$. | 2 | 42 | 1 |
math | 3. Round the difference between the largest and smallest five-digit numbers with distinct digits that are divisible by 15 to the nearest ten. | 88520 | 28 | 5 |
math | In the class, there are $a_{1}$ students who received at least one two during the year, $a_{2}$ students who received at least two twos, ..., $a_{k}$ students who received at least $k$ twos. How many twos are there in total in this class?
(It is assumed that no one has more than $k$ twos.) | a_{1}+a_{2}+\ldots+a_{k} | 80 | 16 |
math | Task B-3.4. How many isosceles triangles have side lengths that are integers, and a perimeter of $30 \mathrm{~cm}$? | 7 | 35 | 1 |
math | Define a sequence recursively by $f_1(x)=|x-1|$ and $f_n(x)=f_{n-1}(|x-n|)$ for integers $n>1$. Find the least value of $n$ such that the sum of the zeros of $f_n$ exceeds $500,000$. | 101 | 70 | 3 |
math | Let $S = \{1, 2, \ldots, 2016\}$, and let $f$ be a randomly chosen bijection from $S$ to itself. Let $n$ be the smallest
positive integer such that $f^{(n)}(1) = 1$, where $f^{(i)}(x) = f(f^{(i-1)}(x))$. What is the expected value of $n$?
| \frac{2017}{2} | 99 | 10 |
math | What is the value of $\frac{6}{3} \times \frac{9}{6} \times \frac{12}{9} \times \frac{15}{12}$ ? | 5 | 43 | 1 |
math | 2. Let $n$ be a positive integer. If $n$ is divisible by 2010 and exactly one of the digits of $n$ is even, find the smallest possible value of $n$.
(1 mark)
Let $n$ be a positive integer. If $n$ is divisible by 2010 and exactly one of the digits of $n$ is even, find the smallest possible value of $n$. | 311550 | 93 | 6 |
math | Given two sequences of positive numbers $\{a_k\}$ and $\{b_k\} \ (k \in \mathbb N)$ such that:
[b](i)[/b] $a_k < b_k,$
[b](ii) [/b] $\cos a_kx + \cos b_kx \geq -\frac 1k $ for all $k \in \mathbb N$ and $x \in \mathbb R,$
prove the existence of $\lim_{k \to \infty} \frac{a_k}{b_k}$ and find this limit. | 0 | 127 | 1 |
math | Find the functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that for all real numbers $x, y$, we have
$$
f(x-f(y))=1-x-y
$$ | x\mapsto\frac{1}{2}-x | 47 | 12 |
math | Example 3. Find the integral $\int \operatorname{sh}^{2} x \operatorname{ch}^{2} x d x$. | \frac{1}{32}\sinh4x-\frac{1}{8}x+C | 32 | 21 |
math | Putnam 1994 Problem B2 For which real α does the curve y = x 4 + 9x 3 + α x 2 + 9x + 4 contain four collinear points? Solution | 30.375 | 47 | 6 |
math | 23. How many ordered pairs of integers $(m, n)$ where $0<m<n<2008$ satisfy the equation $2008^{2}+m^{2}=2007^{2}+n^{2} ?$ | 3 | 55 | 1 |
math | ## Task 12/68
Calculate the sum
$$
\sum_{k=1}^{n} k \cdot\binom{n}{k}
$$ | n\cdot2^{n-1} | 36 | 9 |
math | Example 5.17 Given non-negative real numbers $a, b, c$ satisfy $a b + b c + c a + 6 a b c = 9$. Determine the maximum value of $k$ such that the following inequality always holds.
$$a + b + c + k a b c \geqslant k + 3$$ | 3 | 75 | 1 |
math | In parallelogram $ABCD$, the longer side is twice the shorter side. Let $XYZW$ be the quadrilateral formed by the internal bisectors of the angles of $ABCD$. If the area of $XYZW$ is $10$, find the area of $ABCD$ | 40 | 61 | 2 |
math | 2. How many natural numbers greater than one exist, the product of which with their smallest prime divisor does not exceed 100? | 33 | 28 | 2 |
math | 1. Find all real pairs $(x, y)$ that satisfy:
$$
\left\{\begin{array}{l}
x^{2}+y^{2}+x+y=x y(x+y)-\frac{10}{27}, \\
|x y| \leqslant \frac{25}{9} .
\end{array}\right.
$$ | (x,y)=(-\frac{1}{3},-\frac{1}{3}),(\frac{5}{3},\frac{5}{3}) | 78 | 32 |
math | 10,11 |
Let $x, y, z$ be positive numbers and $x y z(x+y+z)=1$. Find the minimum value of the expression $(x+y)(x+z)$. | 2 | 43 | 1 |
math | 9.102. For what values of $n$ are both roots of the equation $(n-2) x^{2}-$ $-2 n x+n+3=0$ positive? | n\in(-\infty;-3)\cup(2;6] | 42 | 16 |
math | Example 5 Let $a, b, c, x, y, z$ be real numbers, and
$$a^{2}+b^{2}+c^{2}=25, x^{2}+y^{2}+z^{2}=36, a x+b y+c z=30 .$$
Find the value of $\frac{a+b+c}{x+y+z}$. | \frac{5}{6} | 86 | 7 |
math | The sides of a triangle and the measurements of two medians, arranged in ascending order, are as follows: $13,16,17,18$, 22. What is the length of the third median? | 18.44 | 48 | 5 |
math | $A, B, C$, and $D$ are all on a circle, and $ABCD$ is a convex quadrilateral. If $AB = 13$, $BC = 13$, $CD = 37$, and $AD = 47$, what is the area of $ABCD$? | 504 | 68 | 3 |
math | [ Formulas for abbreviated multiplication (other).] Case analysis
Solve the equation $\left(x^{2}-y^{2}\right)^{2}=16 y+1$ in integers.
# | (\1,0),(\4,3),(\4,5) | 42 | 15 |
math | 13.171. In a family, there is a father, a mother, and three daughters; together they are 90 years old. The age difference between the girls is 2 years. The mother's age is 10 years more than the sum of the daughters' ages. The difference in age between the father and the mother is equal to the age of the middle daughte... | 38,31,5,7,9 | 91 | 11 |
math | Find all positive integers that satisfy the following conditions: its first digit is 6, and removing this 6, the resulting integer is $\frac{1}{25}$ of the original integer;
Prove that there is no such integer: its first digit is 6, and removing its first digit results in $\frac{1}{35}$ of the original integer. | 625,6250,62500,\cdots | 76 | 17 |
math | 7. In the complex plane, the points corresponding to the complex numbers $z_{1}, z_{2}, z_{3}$ are $Z_{1}, Z_{2}, Z_{3}$, respectively. If
$$
\begin{array}{l}
\left|z_{1}\right|=\left|z_{2}\right|=\sqrt{2}, \overrightarrow{O Z_{1}} \cdot \overrightarrow{O Z_{2}}=0, \\
\left|z_{1}+z_{2}-z_{3}\right|=1,
\end{array}
$$
t... | [1,3] | 143 | 5 |
math | Let $m \geq 3$ and $n$ be positive integers with $n > m(m-2)$. Find the largest positive integer $d$ such that $d \mid n!$ and $k \nmid d$ for all $k \in \{m, m+1, \ldots, n\}$. | m-1 | 73 | 3 |
math | There are $100$ white points on a circle. Asya and Borya play the following game: they alternate, starting with Asya, coloring a white point in green or blue. Asya wants to obtain as much as possible pairs of adjacent points of distinct colors, while Borya wants these pairs to be as less as possible. What is the maxima... | 50 | 95 | 2 |
math | Fifteen distinct points are designated on $\triangle ABC$: the 3 vertices $A$, $B$, and $C$; $3$ other points on side $\overline{AB}$; $4$ other points on side $\overline{BC}$; and $5$ other points on side $\overline{CA}$. Find the number of triangles with positive area whose vertices are among these $15$ points. | 390 | 89 | 3 |
math | ## 5. The Story of the Vagabonds
At the very moment when Pierrot left the "Commercial" bar, heading for the "Theatrical" bar, Jeanno was leaving the "Theatrical" bar, making his way to the "Commercial" bar. They were walking at a constant (but different) speed. When the vagabonds met, Pierrot proudly noted that he had... | 1000 | 196 | 4 |
math | (Benelux 2010) Find all polynomials $P \in \mathbb{R}[X]$ such that for all real numbers $a, b, c$ we have:
$$
p(a+b-2 c)+p(b+c-2 a)+p(c+a-2 b)=3 p(a-b)+3 p(b-c)+3 p(c-a)
$$ | P()=aX^2+bX | 81 | 9 |
math | 9.1. What is the minimum sum of digits in the decimal representation of the number $f(n)=17 n^{2}-11 n+1$, where $n$ runs through all natural numbers?
# Answer. 2. | 2 | 50 | 1 |
math | Suppose that each of $n$ people knows exactly one piece of information and all $n$ pieces are different. Every time person $A$ phones person $B$, $A$ tells $B$ everything he knows, while tells $A$ nothing. What is the minimum of phone calls between pairs of people needed for everyone to know everything? | 2n - 2 | 74 | 5 |
math | 8. $[7]$ Let $T=\int_{0}^{\ln 2} \frac{2 e^{3 x}+e^{2 x}-1}{e^{3 x}+e^{2 x}-e^{x}+1} d x$. Evaluate $e^{T}$. | \frac{11}{4} | 65 | 8 |
math | 6.96 Let $a$ and $d$ be non-negative numbers, $b$ and $c$ be positive numbers, and $b+c \geqslant a+d$. Find the minimum value of the following expression:
$$\frac{b}{c+d}+\frac{c}{a+b} .$$ | \sqrt{2}-\frac{1}{2} | 68 | 12 |
math | We break a thin wooden stick into 3 pieces. What is the probability that we can form a triangle with these pieces?
[^0]: ${ }^{1 *}$ We are happy to accept solutions to problems marked with a star from university students as well. | \frac{1}{4} | 53 | 7 |
math | Let $n,k$ be positive integers such that $n>k$. There is a square-shaped plot of land, which is divided into $n\times n$ grid so that each cell has the same size. The land needs to be plowed by $k$ tractors; each tractor will begin on the lower-left corner cell and keep moving to the cell sharing a common side until it... | (n-k)^2 | 112 | 4 |
math | 8.1. In the expansion of the function $f(x)=\left(1+x-x^{2}\right)^{20}$ in powers of $x$, find the coefficient of $x^{3 n}$, where $n$ is equal to the sum of all coefficients of the expansion. | 760 | 62 | 3 |
math | 3. Given $x^{2}-x-1=0$. Then, the value of the algebraic expression $x^{3}$ $-2 x+1$ is $\qquad$ . | 2 | 41 | 1 |
math | 6. An isosceles obtuse triangle $P Q T$ with base $P T$ is inscribed in a circle $\Omega$. Chords $A B$ and $C D$, parallel to the line $P T$, intersect the side $Q T$ at points $K$ and $L$ respectively, and $Q K=K L=L T$. Find the radius of the circle $\Omega$ and the area of triangle $P Q T$, if $A B=2 \sqrt{66}, C D... | R=12.5,S=108 | 120 | 11 |
math | Example 5: A and B take turns picking distinct numbers from $0,1, \cdots, 81$, with A starting first, and each person picks one number from the remaining numbers each time. After all 82 numbers are picked, let $A$ and $B$ be the sums of all numbers chosen by A and B, respectively. During the process of picking numbers,... | 41 | 163 | 2 |
math | Let's divide 17 into two parts such that $3/4$ of one part is $a$ greater than $5/6$ of the other part, where $a>0$. What are the limits within which the value of $a$ can vary to ensure that the problem is always solvable? | <12\frac{3}{4} | 65 | 10 |
math | 2B. Solve the system of equations in $\mathbb{R}$: $x+y=z, x^{2}+y^{2}=z, x^{3}+y^{3}=z$. | (0,0,0),(0,1,1),(1,0,1),(1,1,2) | 43 | 25 |
math | ## Task Condition
Are the vectors $a, b$ and $c$ coplanar?
$a=\{3 ; 1 ; 0\}$
$b=\{-5 ;-4 ;-5\}$
$c=\{4 ; 2 ; 4\}$ | -18\neq0 | 56 | 7 |
math | ## Task A-3.2.
Determine all triples of natural numbers ( $p, m, n$ ) such that $p$ is a prime number and
$$
2^{m} p^{2}+1=n^{5}
$$ | =1,n=3,p=11 | 52 | 9 |
math | 7. Let $f(x)$ be an odd function defined on $\mathbf{R}$, and when $x \geqslant 0$, $f(x)=x^{2}$. If for any $x \in$ $[a, a+2]$, the inequality $f(x+a) \geqslant 2 f(x)$ always holds, then the range of the real number $a$ is $\qquad$ | [\sqrt{2},+\infty) | 93 | 9 |
math | The real numbers x and у satisfy the equations
$$\begin{cases} \sqrt{3x}\left(1+\dfrac{1}{x+y}\right)=2 \\ \sqrt{7y}\left(1-\dfrac{1}{x+y}\right)=4\sqrt2 \end{cases}$$
Find the numerical value of the ratio $y/x$. | \frac{y}{x} = 6 | 81 | 10 |
math | [Decimal numeral system]
Find all natural numbers that increase by 9 times when a zero is inserted between the units digit and the tens digit.
# | 45 | 30 | 2 |
math | 2.045. $\left(\frac{\sqrt[4]{x^{3}}-\sqrt[4]{x}}{1-\sqrt{x}}+\frac{1+\sqrt{x}}{\sqrt[4]{x}}\right)^{2} \cdot\left(1+\frac{2}{\sqrt{x}}+\frac{1}{x}\right)^{-1 / 2}$. | \frac{1-\sqrt{x}}{1-x} | 84 | 12 |
math | 13. Let $a_{1}=1, a_{2}=2$ and for all $n \geq 2, a_{n+1}=\frac{2 n}{n+1} a_{n}-\frac{n-1}{n+1} a_{n-1}$. It is known that $a_{n}>2+\frac{2009}{2010}$ for all $n \geq m$, where $m$ is a positive integer. Find the least value of $m$. | 4021 | 114 | 4 |
math | 4.99 Solve the system of equations $\left\{\begin{array}{l}x+y=4, \\ \left(x^{2}+y^{2}\right)\left(x^{3}+y^{3}\right)=280 .\end{array}\right.$
(Kyiv Mathematical Olympiad, 1935) | {\begin{array}{}{x_{1}=3,}\{y_{1}=1;}\end{pmatrix}\quad\text{or}\quad{\begin{pmatrix}x_{2}=1,\\y_{2}=30\end{pmatrix}..}} | 75 | 61 |
math | 4. In all other cases - o points.
## Task 2
Maximum 15 points
Solve the equation $2 \sqrt{2} \sin ^{3}\left(\frac{\pi x}{4}\right)=\cos \left(\frac{\pi}{4}(1-x)\right)$.
How many solutions of this equation satisfy the condition:
$0 \leq x \leq 2020 ?$ | 505 | 92 | 3 |
math | 18. Let $x_{1}, x_{2}, \cdots, x_{n}$ be positive real numbers, satisfying $\frac{1}{x_{1}}+\frac{1}{x_{2}}+\cdots+\frac{1}{x_{n}}=n$, find the minimum value of $x_{1}+\frac{x_{2}^{2}}{2}+$ $\frac{x_{3}^{2}}{3}+\cdots+\frac{x_{n}^{n}}{n}$. (1995 Polish Mathematical Olympiad Problem) | 1+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{n} | 122 | 25 |
math | Example 7. Determine the number of roots of the equation
$$
z^{6}-6 z+10=0
$$
inside the circle $|z|<1$. | 0 | 39 | 1 |
math | \section*{Problem 4 - 051044}
Calculate the difference \(D\) between the sum of the squares of all even natural numbers \(\leq 100\) and the sum of the squares of all odd natural numbers \(<100\)! | 5050 | 60 | 4 |
math | Question 1. Find a prime number that remains prime when 10 or 14 is added to it. | 3 | 24 | 1 |
math | 1. Given the complex number $z$ satisfies $|z|-z=3-\mathrm{i}$. Then $z=$
$\qquad$ . | -\frac{4}{3}+\mathrm{i} | 32 | 11 |
math | In the symmetric trapezoid $ABCD$, the diagonals $AC$ and $BD$ intersect at point $O$. The area of $\triangle AOB$ is $52 \, \text{m}^2$, and the area of $\triangle COD$ is $117 \, \text{m}^2$. Calculate the area of the trapezoid! | 325 | 82 | 3 |
math | 520. A circle of radius $r$ is inscribed in a triangle with perimeter $p$ and area $S$. How are these three quantities related? | \frac{pr}{2} | 34 | 7 |
math | Consider the set of continuous functions $f$, whose $n^{\text{th}}$ derivative exists for all positive integer $n$, satisfying $f(x)=\tfrac{\text{d}^3}{\text{dx}^3}f(x)$, $f(0)+f'(0)+f''(0)=0$, and $f(0)=f'(0)$. For each such function $f$, let $m(f)$ be the smallest nonnegative $x$ satisfying $f(x)=0$. Compute all poss... | 0, \frac{2\pi \sqrt{3}}{3} | 120 | 16 |
math | Find the angle between the line of intersection of the planes $2 x-y-3 z+5=0$ and $x+y-2=0$ and the plane passing through the points $M(-2 ; 0 ; 3), N(0 ; 2 ; 2)$ and $K(3 ;-3 ; 1)$.
# | \arcsin\frac{22}{3\sqrt{102}} | 73 | 18 |
math | 5. (3 points) Find all functions continuous on the entire number line that satisfy the identity $5 f(x+y)=f(x) f(y)$ and the condition $f(1)=10$.
Answer: $f(x)=5 \cdot 2^{x}$. | f(x)=5\cdot2^{x} | 58 | 10 |
math | (1) Let real numbers $x_{1}, x_{2}, \cdots, x_{1997}$ satisfy the following two conditions:
(1) $-\frac{1}{\sqrt{3}} \leqslant x_{i} \leqslant \sqrt{3}(i=1,2, \cdots, 1997)$;
(2) $x_{1}+x_{2}+\cdots+x_{1977}=-318 \sqrt{3}$.
Find: $x_{1}^{12}+x_{2}^{12}+\cdots+x_{1997}^{12}$'s maximum value, and explain the reason. | 189548 | 159 | 6 |
math | 4. It is known that the equation $x^{3}-x-1=0$ has a unique real root $x_{0}$. Come up with at least one equation of the form
$$
a \cdot z^{3}+b \cdot z^{2}+c \cdot z+d=0
$$
where $a, b, c, d$ are integers and $a \neq 0$, one of the roots of which is the number
$$
z=x_{0}^{2}+3 \cdot x_{0}+1
$$ | z^{3}-5z^{2}-10z-11=0 | 122 | 17 |
math | ## Task Condition
Find the derivative of the specified order.
$y=\frac{\ln x}{x^{2}}, y^{IV}=?$ | \frac{-154+120\lnx}{x^{6}} | 30 | 18 |
math | \section*{Exercise 6 - 031016}
In the football Toto, on the betting slip, 12 matches need to be marked for which team is expected to win or if the match will end in a draw. For each match, there are three possibilities:
Team A wins, Team B wins, or a draw.
How many betting slips would someone need to fill out to ens... | 531441 | 103 | 6 |
math | 21. Find the smallest natural number that is a multiple of 36 and in whose representation all 10 digits appear exactly once. | 1023457896 | 29 | 10 |
math | For which natural numbers $n$ is the expression $P=\left(n^{2}-4\right)\left(n^{2}-1\right)\left(n^{2}+3\right)$ divisible by 2880? | 1,2,7,14,17,23,31,34,41,46,47,49,62,71,73,79,82,89,94,97,98,103,113,119 | 49 | 71 |
math | 6.065. $\sqrt{2 x+5}+\sqrt{5 x+6}=\sqrt{12 x+25}$. | 2 | 33 | 1 |
math | Find a positive integer $x$, when divided by 2 the remainder is 1, when divided by 5 the remainder is 2, when divided by 7 the remainder is 3, when divided by 9 the remainder is 4.
Ask for the original number. (Trans: Find a positive integer $x$, when divided by 2 the remainder is 1, when divided by 5 the remainder is... | x=157+630k, \quad k=0,1,2, \cdots | 111 | 24 |
math | 3. Let three positive integers $a$, $x$, $y$ greater than 100 satisfy $y^{2}-1=a^{2}\left(x^{2}-1\right)$.
Find the minimum value of $\frac{a}{x}$. | 2 | 55 | 1 |
math | Determine all triplets $(p, n, k)$ of strictly positive integers such that $p$ is prime and satisfy the following equation:
$$
144 + p^n = k^2
$$ | (5,2,13),(2,8,20),(3,4,15) | 43 | 22 |
math | 5. Given that the length of the major axis of an ellipse is 4, the left vertex is on the parabola $y^{2}=x-1$, and the left directrix is the $y$-axis. Then the maximum value of the eccentricity of such an ellipse is $\qquad$
Translate the above text into English, please retain the original text's line breaks and forma... | \frac{2}{3} | 90 | 7 |
math | Find all 7-divisible six-digit numbers of the form $\overline{A B C C B A}$, where $\overline{A B C}$ is also divisible by 7. ( $A$, $B, C$ represent different digits.) | 168861,259952,861168,952259 | 53 | 27 |
math | 16. Let the positive even function $f(x)$ satisfy $f(1)=2$, and when $x y \neq 0$, $f\left(\sqrt{x^{2}+y^{2}}\right) \leqslant \frac{f(x) f(y)}{f(x)+f(y)}$, then $f(5)$ $\qquad$ $\frac{2}{25}$ (fill in one of $>,<, \geqslant,=, \leqslant$ ). | f(5)\leqslant\frac{2}{25} | 113 | 16 |
math | A random walk is a process in which something moves from point to point, and where the direction of movement at each step is randomly chosen. Suppose that a person conducts a random walk on a line: he starts at $0$ and each minute randomly moves either $1$ unit in the positive direction or $1$ unit in the negative dire... | \frac{15}{8} | 104 | 8 |
math | 150. Find: $i^{28} ; i^{33} ; i^{135}$. | i^{28}=1;i^{33}=i;i^{135}=-i | 26 | 20 |
math | Given a positive integer $n$. Determine the maximum value of $\operatorname{gcd}(a, b)+\operatorname{gcd}(b, c)+\operatorname{gcd}(c, a)$, under the condition that $a, b$ and $c$ are positive integers with $a+b+c=5 n$. | 5 n \text{ if } 3 \mid n, \text{ and } 4 n \text{ if } 3 \nmid n | 68 | 32 |
math | If the price of a product increased from 5.00 to 5.55 reais, what was the percentage increase? | 11 | 28 | 2 |
math | 5. The absolute value of a number $x$ is equal to the distance from 0 to $x$ along a number line and is written as $|x|$. For example, $|8|=8,|-3|=3$, and $|0|=0$. For how many pairs $(a, b)$ of integers is $|a|+|b| \leq 10$ ? | 221 | 85 | 3 |
math | Consider coins with positive real denominations not exceeding 1. Find the smallest $C>0$ such that the following holds: if we have any $100$ such coins with total value $50$, then we can always split them into two stacks of $50$ coins each such that the absolute difference between the total values of the two stacks is ... | \frac{50}{51} | 90 | 9 |
math | 2B. A four-digit number is a perfect square, such that if the same digit is subtracted from its digits, the resulting number is also a perfect square. Determine all such natural numbers. | 3136,4489 | 40 | 9 |
math | 3. $(5-7,8)$ Nезнayka and Ponchik have the same amounts of money, composed of coins worth $1, 3, 5$, and 7 ferthings.
Nезнayka has as many 1-ferthing coins as Ponchik has 3-ferthing coins;
3-ferthing coins - as many as Ponchik has 5-ferthing coins; 5-ferthing coins - as many as Ponchik has 7-ferthing coins; and 7-fer... | 5 | 156 | 1 |
math | 11. The line $l:(2 m+1) x+(m+1) y-7 m-$ $4=0$ is intersected by the circle $C:(x-1)^{2}+(y-2)^{2}=25$ to form the shortest chord length of $\qquad$ . | 4 \sqrt{5} | 68 | 6 |
math | In the $xOy$ system consider the lines $d_1\ :\ 2x-y-2=0,\ d_2\ :\ x+y-4=0,\ d_3\ :\ y=2$ and $d_4\ :\ x-4y+3=0$. Find the vertices of the triangles whom medians are $d_1,d_2,d_3$ and $d_4$ is one of their altitudes.
[i]Lucian Dragomir[/i] | A(1, 0), B(0, 4), C(5, 2) | 107 | 21 |
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