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 | 3. A certain number of boys and girls went camping during the summer break. They planned an ecological action that they would finish in 29 days if each child worked evenly - working the same part of the job in any given days. The boys worked a bit faster; in the same time, 2 boys do as much work as 3 girls. Fortunately... | 5 | 142 | 1 |
math | 31. $[\mathbf{1 9}]$ Let $S_{7}$ denote all the permutations of $1,2, \ldots, 7$. For any $\pi \in S_{7}$, let $f(\pi)$ be the smallest positive integer $i$ such that $\pi(1), \pi(2), \ldots, \pi(i)$ is a permutation of $1,2, \ldots, i$. Compute $\sum_{\pi \in S_{7}} f(\pi)$. | 29093 | 113 | 5 |
math | On a typical morning Aiden gets out of bed, goes through his morning preparation, rides the bus, and walks from the bus stop to work arriving at work 120 minutes after getting out of bed. One morning Aiden got out of bed late, so he rushed through his morning preparation getting onto the bus in half the usual time, the... | 126 | 198 | 3 |
math | 25. For a real number $x$, let $\lfloor x\rfloor$ denote the greatest integer not exceeding $x$. Consider the function
$$
f(x, y)=\sqrt{M(M+1)}(|x-m|+|y-m|),
$$
where $M=\max (\lfloor x\rfloor,\lfloor y\rfloor)$ and $m=\min (\lfloor x\rfloor,\lfloor y\rfloor)$. The set of all real numbers $(x, y)$ such that $2 \leq x, ... | 2021 | 190 | 4 |
math | 242. Sum of squares of binomial coefficients. Find the sum of the squares of the coefficients in the expansion of $(a+b)^{n}$. | C_{2n}^{n}=\sum_{i=0}^{n}(C_{n}^{i})^{2} | 33 | 28 |
math | 10CT2 ** In the convex quadrilateral $ABCD$, $AC$ and $BD$ intersect at point $P, \angle DBC=60^{\circ}$, $\angle ACB=50^{\circ}, \angle ABD=20^{\circ}, \angle ACD=30^{\circ}$, find $\angle ADB$. | 30 | 79 | 2 |
math | 27. Find the number of positive integers $x$, where $x \neq 9$, such that
$$
\log _{\frac{x}{9}}\left(\frac{x^{2}}{3}\right)<6+\log _{3}\left(\frac{9}{x}\right) .
$$ | 223 | 67 | 3 |
math | # 2. Option 1
Given a rectangular grid. We will call two cells adjacent if they share a side. Let's count the number of cells that have exactly four adjacent cells. It turned out to be 23. How many cells have exactly three adjacent cells? | 48 | 57 | 2 |
math | Task 2. (10 points) Simplify the expression $3+33+333+\ldots+\underbrace{333 . .3}_{2021}$.
---
The translation maintains the original text's line breaks and format as requested. | \frac{10^{2022}-18199}{27} | 57 | 20 |
math | Lattice points are points on a rectangular coordinate plane having both $x$ - and $y$-coordinates being integers. A moving point $P$ is initially located at $(0,0)$. It moves 1 unit along the coordinate lines (in either directions) in a single step.
G10.1 If $P$ moves 1 step then $P$ can reach $a$ different lattice poi... | =4,b=13,=16,=\frac{1}{10} | 196 | 19 |
math | Let $u, v$ be real numbers. The minimum value of $\sqrt{u^2+v^2} +\sqrt{(u-1)^2+v^2}+\sqrt {u^2+ (v-1)^2}+ \sqrt{(u-1)^2+(v-1)^2}$ can be written as $\sqrt{n}$. Find the value of $10n$. | 80 | 85 | 2 |
math | 6.12. (GDR, 78). Solve the system
$$
\left\{\begin{array}{l}
x + xy + y = 2 + 3 \sqrt{2} \\
x^{2} + y^{2} = 6
\end{array}\right.
$$ | x_{1}=2,y_{1}=\sqrt{2}x_{2}=\sqrt{2},y_{2}=2 | 67 | 28 |
math | 7. Represent the number 2017 as the sum of some number of natural numbers so that their product is the largest.
Answer. $2017=3+3+\ldots+3+2+2$ (671 threes and two twos). | 2017=3+3+\ldots+3+2+2 | 59 | 17 |
math | Consider a rectangular tiled room with dimensions $m\times n$, where the tiles are $1\times1$ in size. Compute all ordered pairs $(m,n)$ with $m\leq n$ such that the number of tiles on the perimeter is equal to the number of tiles in the interior (i.e. not on the perimeter). | \{(5,12),(6,8)\} | 70 | 13 |
math | 10,11
The base of the pyramid is a right-angled triangle with an area of $S$. The lateral edges of the pyramid are equal to each other. The dihedral angles at the legs of its base are $\alpha$ and $\beta$. Find the volume of the pyramid. | \frac{1}{6}S\sqrt{2S\tan\alpha\tan\beta} | 62 | 22 |
math | 14. A four-digit number, if a comma is placed between the hundreds and tens place, can be written as two two-digit numbers $(3126 \rightarrow 31,62)$. If the two two-digit numbers have an integer multiple relationship, we call such a four-digit number a "clever number". If you select 4 numbers from $1, 2, 4, 6, 8$ to f... | 12 | 111 | 2 |
math | Example 2. Compute the integral
$$
\int_{C}\left(z^{2}+z \bar{z}\right) d z
$$
where $C-$ is the arc of the circle $\{z \mid=1(0 \leqslant \arg z \leqslant \pi)$. | -\frac{8}{3} | 70 | 7 |
math | ## Task B-2.7.
In the set $A$, there are $m$ consecutive integers whose sum is $2 m$, and in the set $B$, there are $2 m$ consecutive integers whose sum is $m$. The absolute value of the difference between the largest elements of $A$ and $B$ is 99. Determine $m$, and then the sets $A$ and $B$. | =201,A={-98,-97,-96,-95,\ldots,102},B={-200,-199,-198,-197,\ldots,201} | 87 | 52 |
math | Determine the lowest positive integer n such that following statement is true:
If polynomial with integer coefficients gets value 2 for n different integers,
then it can't take value 4 for any integer. | n = 4 | 41 | 5 |
math | 7. (5 points) A natural number $n$ greater than 0 is a multiple of 3, and $3 n$ is a multiple of 5, then the minimum value of $n$ is | 15 | 44 | 2 |
math | Problem 1. Seven students in the class receive one two every two days of study, and nine other students receive one two every three days each. The rest of the students in the class never receive twos. From Monday to Friday, 30 new twos appeared in the journal. How many new twos will appear in the class journal on Satur... | 9 | 73 | 1 |
math | Proizvolov V.V.
Find all natural numbers $a$ and $b$ such that $\left(a+b^{2}\right)\left(b+a^{2}\right)$ is an integer power of two. | =b=1 | 44 | 3 |
math | I4.1 The average of $p, q, r$ is 12 . The average of $p, q, r, t, 2 t$ is 15 . Find $t$.
I4.2 $k$ is a real number such that $k^{4}+\frac{1}{k^{4}}=t+1$, and $s=k^{2}+\frac{1}{k^{2}}$. Find $s$.
I4.3 $M$ and $N$ are the points $(1,2)$ and $(11,7)$ respectively. $P(a, b)$ is a point on $M N$ such that $M P: P N=1: s$. F... | =13,=4,=3,=12 | 191 | 13 |
math | \section*{Problem 1 - 161231}
Give all pairs \((x, y)\) of real numbers for which the following holds:
\[
x^{2}+y=2 \quad(1) \quad \text { and } \quad y^{2}+x=2
\] | (1,1),(-2,-2),(\frac{1+\sqrt{5}}{2},\frac{1-\sqrt{5}}{2}),(\frac{1-\sqrt{5}}{2},\frac{1+\sqrt{5}}{2}) | 70 | 58 |
math | A crazy physicist has discovered a new particle called an omon. He has a machine, which takes two omons of mass $a$ and $b$ and entangles them; this process destroys the omon with mass $a$, preserves the one with mass $b$, and creates a new omon whose mass is $\frac 12 (a+b)$. The physicist can then repeat the process ... | 9 | 156 | 1 |
math | (a) Determine $\mathrm{a}, \mathrm{b}$ and $\mathrm{c}$ such that the equality
$$
(n+2)^{2}=a(n+1)^{2}+b n^{2}+c(n-1)^{2}
$$
is true for any number $n$.
(b) Suppose that $x_{1}, x_{2}, \ldots, x_{7}$ satisfy the system
$$
\left\{\begin{array}{l}
x_{1}+4 x_{2}+9 x_{3}+16 x_{4}+25 x_{5}+36 x_{6}+49 x_{7}=1 \\
4 x_{1}... | 334 | 317 | 3 |
math | Example 6. Suppose that at a constant temperature, the dissolution rate of a solid in a liquid is proportional to the amount of this substance that can still dissolve in the liquid until it becomes saturated (it is assumed that the substances entering the solution do not chemically interact with each other, and the sol... | P(1-e^{-k}) | 90 | 7 |
math | 15. Given the function $f(x)=\log _{a}\left(x+\sqrt{x^{2}-2}\right)(a>0, a \neq 1)$ with its inverse function $f^{-1}(x)$, let $g(n)=\frac{\sqrt{2}}{2} f^{-1}\left(n+\log _{a} \sqrt{2}\right)$. If $g(n)<\frac{3^{n}+3^{-n}}{2}(n \in$ $\left.\mathbf{N}_{+}\right)$, find the range of values for $a$. | 1<<3 | 132 | 3 |
math | 12.107. A truncated cone is described around a sphere, with the area of one base being four times the area of the other base. Find the angle between the slant height of the cone and the plane of its base. | \arccos(\frac{1}{3}) | 50 | 11 |
math | For any real number $x$, $[x]$ represents the greatest integer not exceeding $x$, and $\{x\}$ represents the fractional part of $x$. Then
$$
\begin{array}{l}
\left\{\frac{2014}{2015}\right\}+\left\{\frac{2014^{2}}{2015}\right\}+\cdots+\left\{\frac{2014^{2014}}{2015}\right\} \\
=
\end{array}
$$ | 1007 | 122 | 4 |
math | 10. (20 points) Given points $M(-1,0), N(1,0)$, the perimeter of $\triangle M N Q$ is 6, and the trajectory of the moving point $Q$ is the curve $C$. $P$ is any point on the circle $x^{2}+y^{2}=4$ (not on the x-axis), and $P A, P B$ are tangent to the curve $C$ at points $A, B$ respectively. Find the maximum value of t... | \frac{3}{2} | 129 | 7 |
math | Example 10 Let $x>y>0, xy=1$, find the minimum value of $\frac{3x^3+125y^3}{x-y}$.
Translate the text above into English, please keep the original text's line breaks and format, and output the translation result directly.
Example 10 Let $x>y>0, xy=1$, find the minimum value of $\frac{3x^3+125y^3}{x-y}$. | 25 | 104 | 2 |
math | Professor Carlão decided to create a math question worth a total of 10 points and consisting of three items: $a, b$, and $c$. After formulating the items, he became unsure about the best way to distribute the 10 points among the items so that each one is worth a positive integer number of points.
a) Joana, a teacher f... | 36 | 186 | 2 |
math | The pages of a book are numbered 1 through $n$. When the page numbers of the book were added, one of the page numbers was mistakenly added twice, resulting in an incorrect sum of 1986. What was the number of the page that was added twice? | 33 | 57 | 2 |
math | The numbers $1, 1/2, \ldots, 1/n$ are written on a board. It is allowed to erase any two numbers $a$ and $b$ and replace them with the number $ab + a + b$.
What number will remain after $n-1$ such operations? | n | 66 | 1 |
math | In a 6-car metro train, there are 4 drunk passengers among the travelers. What is the probability that at most two cars have drunk passengers? | \frac{1}{6} | 31 | 7 |
math | Let $G$ be the set of points $(x, y)$ such that $x$ and $y$ are positive integers less than or equal to 20. Say that a ray in the coordinate plane is [i]ocular[/i] if it starts at $(0, 0)$ and passes through at least one point in $G$. Let $A$ be the set of angle measures of acute angles formed by two distinct ocular ... | \frac{1}{722} | 116 | 9 |
math | Example 8. Find the sum: $\cos \alpha+\cos 3 \alpha+\cos 5 \alpha+\cdots$ $+\cos (2 n-1) \alpha$. | \frac{\sin 2 n \alpha}{2 \sin \alpha} | 40 | 16 |
math | Solve the equation $x^{2 y}+(x+1)^{2 y}=(x+2)^{2 y}$ in natural numbers. | 3,1 | 32 | 3 |
math | $9 \cdot 37$ Find the largest real number $\alpha$ such that for any positive integers $m$ and $n$ satisfying $\frac{m}{n}<\sqrt{7}$, we have
$$
\frac{\alpha}{n^{2}} \leqslant 7-\frac{m^{2}}{n^{2}} \text {. }
$$ | 3 | 81 | 1 |
math | Example 5.11. Estimate the error made when replacing the sum of the series
$$
1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\ldots+(-1)^{n-1} \frac{1}{n}+\ldots
$$
with the sum of its first four terms. | 0.2 | 77 | 3 |
math | 12. Given $x, y, z \in \mathbf{Z}$, and $x+y+z=3, x^{3}+y^{3}+z^{3}=3$, then $x^{2}+y^{2}+z^{2}=$ | 3or57 | 60 | 4 |
math | 1. Which whole numbers from 1 to 60000 (inclusive) are there more of, and by how many: those containing only even digits or those containing only odd digits? | 780 | 40 | 3 |
math | 2. Find all pairs of numbers $(a, b)$ for which the function
$$
f(x)=\frac{(a+5 b) x+a+b}{a x+b}
$$
is constant over its entire domain of definition.
(7 points) | \\sqrt{5}b,b\neq0 | 53 | 11 |
math | Solve the equation $\frac{2003 x}{2004}=2003^{\log _{x} 2004}$. | \frac{1}{2003}or2004 | 36 | 15 |
math | 13. Given that $f(x)$ is defined on $(-1,1)$, $f\left(\frac{1}{2}\right)=-1$, and satisfies $x, y \in (-1,1)$, we have $f(x) + f(y) = f\left(\frac{x+y}{1+xy}\right)$.
(1) The sequence $\{x_n\}$ satisfies
$$
x_1 = \frac{1}{2}, \quad x_{n+1} = \frac{2x_n}{1 + x_n^2}.
$$
Let $a_n = f(x_n)$. Find the general term formula ... | 0 | 251 | 1 |
math | 2. Given $f(x)=\frac{1}{x^{2}}+\frac{1}{x^{4}}$, if $f(a-2)<f(2 a+1)$ holds, then the range of values for $a$ is | (-3,-\frac{1}{2})\cup(-\frac{1}{2},\frac{1}{3}) | 52 | 27 |
math | Example 2. In an urn, there are 7 white and 3 black balls. The condition of the experiment is that each drawn ball is returned to the urn. Event $A$: a white ball is drawn in the first trial; event $B$: a white ball is drawn in the second trial. We have $p(A)=p(B)=\frac{7}{10}$ (if the first ball was black, $p(B)$ woul... | \frac{1}{9} | 603 | 7 |
math | Problem 11.3. On the coordinate plane, all points $(x, y)$ such that $x$ and $y$ are integers satisfying the inequalities $0 \leqslant x \leqslant 2$ and $0 \leqslant y \leqslant 26$ are marked. How many lines exist that pass through exactly 3 of the marked points? | 365 | 85 | 3 |
math | For each positive integer $n$, write the sum $\sum_{m=1}^n 1/m$ in the form $p_n/q_n$, where $p_n$ and $q_n$ are relatively prime positive integers. Determine all $n$ such that 5 does not divide $q_n$. | \{1, 2, 3, 4, 20, 21, 22, 23, 24, 100, 101, 102, 103, 104, 120, 121, 122, 123, 124\} | 65 | 85 |
math | Task B-4.5. In a regular $n$-gon, the radius of the circumscribed circle is $2, S$ is its center, $A, B$ are consecutive vertices of the $n$-gon. Determine $n$ and the interior angle of the regular $n$-gon if the dot product $\overrightarrow{S A} \cdot \overrightarrow{S B}=2 \sqrt{2}$. | 8, | 93 | 2 |
math | Test $\mathbf{E}$ Function $f(x, y)$ satisfies for all non-negative integers $x, y$:
(1) $f(0, y)=y+1$;
(2) $f(x+1,0)=f(x, 1)$;
(3) $f(x+1, y+1)=f(x, f(x+1, y))$.
Determine $f(4,1981)$. | 2^{2^{1984}}-3 | 97 | 11 |
math | 5.9. a) Factorize $x^{8}+x^{4}+1$ into two factors.
b) Factorize $x^{8}+x^{4}+1$ into four factors, allowing square roots of natural numbers as coefficients.
## 5.2. Proof of identities | (x^{2}+x+1)(x^{2}-x+1)(x^{2}+\sqrt{3}x+1)(x^{2}-\sqrt{3}x+1) | 64 | 43 |
math | Let $n$ be a positive integer and $A=\{ 1,2,\ldots ,n\}$. A subset of $A$ is said to be connected if it consists of one element or several consecutive elements. Determine the maximum $k$ for which there exist $k$ distinct subsets of $A$ such that the intersection of any two of them is connected. | \left\lfloor \frac{n + 1}{2} \right\rfloor \cdot \left\lceil \frac{n + 1}{2} \right\rceil | 78 | 40 |
math | 12.43 Find all integer pairs $(x, y)$ that satisfy the equation $x^{2}=y^{2}+2 y+13$.
(46th Moscow Mathematical Olympiad, 1983) | (x,y)=(4,-3),(4,1),(-4,1),(-4,-3) | 50 | 21 |
math | 11. (20 points) Given positive real numbers $p, q$. It is known that the sequence of positive real numbers $\left\{a_{n}\right\}$ satisfies:
$$
a_{0}=1, a_{n+2}=p a_{n}-q a_{n+1}(n \in \mathbf{N}) .
$$
Find all possible values of $a_{1}$ (expressed in terms of $p, q$). | \frac{-q+\sqrt{q^{2}+4p}}{2} | 100 | 18 |
math | Example 10. Solve the equation
$$
5^{2 x-1}=7^{3-x}
$$ | \frac{1+3\log_{5}7}{2+\log_{5}7} | 24 | 21 |
math | (22) Let $i_{1}, i_{2}, \cdots, i_{10}$ be a permutation of $1,2, \cdots, 10$, and let
$$
S=\left|i_{1}-i_{2}\right|+\left|i_{3}-i_{4}\right|+\cdots+\left|i_{9}-i_{10}\right| \text{, }
$$
Find all possible values of $S$. | 5,7,9,11,13,15,17,19,21,23,25 | 101 | 29 |
math | Three boxes each contain an equal number of hockey pucks. Each puck is either black or gold. All 40 of the black pucks and exactly $\frac{1}{7}$ of the gold pucks are contained in one of the three boxes. Determine the total number of gold hockey pucks. | 140 | 62 | 3 |
math | 9. Given that the area of square $A B C D$ is 35 square centimeters, $E, F$ are points on sides $A B$, $B C$ respectively, $A F$ and $C E$ intersect at $G$, and the area of $\triangle A B F$ is 5 square centimeters, the area of $\triangle B C E$ is 14 square centimeters. Then, the area of quadrilateral $B E G F$ is $\q... | \frac{128}{27} | 111 | 10 |
math | ## Task 22/90
Determine all four-digit natural numbers $n$ in the decimal system with the following properties:
1. All digits $a_{i}$ and the cross sum $Q$ of $n$ are prime numbers.
2. It holds that $n=Q \cdot P+2$, where $P$ is the cross product of $n$. | 3572 | 79 | 4 |
math | Are there any digit-matching equalities similar to $(30+25)^{2}=3025$ among two-digit, or four-digit natural numbers? | (98+01)^2=9801,(20+25)^2=2025,(30+25)^2=3025 | 35 | 39 |
math | We want to install an electric bell. The ringing location is 30 meters away from the bell. The internal resistance of the bell is 2 ohms. We intend to use 2 series-connected Leclanché cells as the power source, each with an electromotive force of 1.5 Volts and an internal resistance of 1 ohm. What diameter copper wire ... | 0.63 | 144 | 4 |
math | (12) $n$ football teams participate in a round-robin tournament, where each pair of teams plays one match. The winning team gets 2 points, the losing team gets 0 points, and in the case of a draw, both teams get 1 point. Three referees each calculated the total points of all teams, resulting in three different totals: ... | 47 | 151 | 2 |
math | G2.2 Let $[x]$ be the largest integer not greater than $x$.
If $B=[10+\sqrt{10+\sqrt{10+\sqrt{10+\cdots}}}]$, find the value of $B$. | 13 | 54 | 2 |
math | Example 4 For each $n \in \mathbf{N}^{*}$, solve the equation
$$
\sin x \sin 2 x \cdots \sin n x+\cos x \cos 2 x \cdots \cos n x=1 .
$$ | x=2m\piorx=2k\pi+\frac{\pi}{2}forn=1;\,x=2m\piforn=4l-2orn=4l+1;\,x=\piforn=4lorn=4 | 59 | 56 |
math | 7. Given the sequence $\left\{a_{n}\right\}$ satisfies: $a_{n}=\left[(2+\sqrt{5})^{n}+\frac{1}{2^{n}}\right]\left(n \in \mathbf{N}^{*}\right)$, where $[x]$ denotes the greatest integer not exceeding the real number $x$. Let $C$ be a real number, and for any positive integer $n$, we have $\sum_{k=1}^{n} \frac{1}{a_{k} a... | \frac{1}{288} | 144 | 9 |
math | Example 2 Given points $A_{1}(-1,0), A_{2}(1, -1), A_{3}(2,0), A_{4}(3,3), A_{5}(0,5)$ on a plane, find the area of the pentagon $A_{1} A_{2} A_{3} A_{4} A_{5}$. | 14.5 | 81 | 4 |
math | Factorize the following expression:
$$
\left[(a-c)^{2}+(b-d)^{2}\right] \cdot\left(a^{2}+b^{2}\right)-(a d-b c)^{2}
$$ | [^{2}+b^{2}-(+)]^{2} | 50 | 15 |
math | Question 127, Find the last two digits of $\left[(2+\sqrt{3})^{2^{2020}}\right]$.
Translate the text above into English, please keep the original text's line breaks and format, and output the translation result directly. | 53 | 58 | 2 |
math | 2. In $\square A B C D$, $A B<A C<B C$. From point $D$ draw tangents to the circumcircle $\Gamma$ of $\triangle A B C$, the points of tangency are $E$ and $F$. If segment $A D$ intersects $C E$, and $\angle A B F=\angle D C E$, find $\angle A B C$. | 60^{\circ} | 83 | 6 |
math | Find all integers $n \geq 1$ such that $3^{n-1}+5^{n-1}$ divides $3^{n}+5^{n}$. | 1 | 39 | 1 |
math | Let $x_n=2^{2^{n}}+1$ and let $m$ be the least common multiple of $x_2, x_3, \ldots, x_{1971}.$ Find the last digit of $m.$ | 9 | 54 | 1 |
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 | 262. Find a triangular number whose square is also a triangular number. | 6 | 16 | 1 |
math | 3. Find the smallest positive integer $m$ such that $5 m$ is a fifth power of a positive integer, $6 m$ is a sixth power of a positive integer, and $7 m$ is a seventh power of a positive integer.
(2013, Irish Mathematical Olympiad) | 2^{35} \times 3^{35} \times 5^{84} \times 7^{90} | 63 | 29 |
math | 1. Calculate the sum $\sum_{n=0}^{502}\left[\frac{305 n}{503}\right]$. | 76304 | 33 | 5 |
math | ## Zadatak A-3.1.
Izračunaj
$$
\frac{\operatorname{tg} 192^{\circ}+\operatorname{tg} 48^{\circ}}{1+\operatorname{tg} 168^{\circ} \cdot \operatorname{tg} 408^{\circ}}
$$
| \sqrt{3} | 82 | 5 |
math | ## Task 1 - 170711
Matthias was in an international pioneer tent camp during the summer holidays. He reports to his classmates:
"A quarter of all participants and four pioneers came from the Soviet Union, a fifth of all participants and five pioneers from the GDR, a sixth of all participants and six pioneers from Cze... | 45 | 147 | 2 |
math | For a given integer $k \geq 1$, find all $k$-tuples of positive integers $(n_1,n_2,...,n_k)$ with $\text{GCD}(n_1,n_2,...,n_k) = 1$ and $n_2|(n_1+1)^{n_1}-1$, $n_3|(n_2+1)^{n_2}-1$, ... , $n_1|(n_k+1)^{n_k}-1$. | (1, 1, \ldots, 1) | 112 | 13 |
math | 7. There are 15 players participating in a Go tournament, where each pair of players needs to play one match. Winning a match earns 2 points, a draw earns 1 point each, and losing a match earns 0 points. If a player's score is no less than 20 points, they will receive a prize. Therefore, the maximum number of players w... | 9 | 87 | 1 |
math | 6. A point on the coordinate plane whose both horizontal and vertical coordinates are integers is called an integer point. The number of integer points in the region enclosed by the parabola $y=x^{2}+1$ and the line $2 x-y+81=0$ is $\qquad$ . | 988 | 64 | 3 |
math | # Problem 6. (4 points)
Solve the equation $a b c d e f=a+b+c+d+e+f$ in natural numbers. | (1,1,1,1,2,6) | 32 | 13 |
math | 4. A4 (MON) Let $\mathbb{R}$ denote the set of all real numbers and $\mathbb{R}^{+}$ the subset of all positive ones. Let $\alpha$ and $\beta$ be given elements in $\mathbb{R}$, not necessarily distinct. Find all functions $f: \mathbb{R}^{+} \rightarrow \mathbb{R}$ such that
$$ f(x) f(y)=y^{\alpha} f\left(\frac{x}{2}\... | f(x) \equiv 2^{1-\alpha} x^{\alpha} \text{ or } f(x) \equiv 0 | 158 | 29 |
math | 5. The engine of a car traveling at a speed of $v_{0}=72 \mathrm{km} / \mathbf{h}$ operates with a power of $P=50$ kW. Determine the distance from the point of engine shutdown at which the car will stop, if the resistance force is proportional to the car's speed. The mass of the car is m=1500 kg. (15
## points) | 240 | 93 | 3 |
math | 3. Given the quadratic function
$$
y=3 a x^{2}+2 b x-(a+b) \text {, }
$$
when $x=0$ and $x=1$, the value of $y$ is positive. Then, when $0<x<1$, the parabola intersects the $x$-axis at $\qquad$ points. | 2 | 80 | 1 |
math | 9. Given $n(n>1)$ integers (which can be the same) $a_{1}$, $a_{2}, \cdots, a_{n}$ satisfy
$$
a_{1}+a_{2}+\cdots+a_{n}=a_{1} a_{2} \cdots a_{n}=2007 .
$$
Then the minimum value of $n$ is $\qquad$ | 5 | 91 | 1 |
math | Given a game board consisting of $n \times n$ square fields, where $n \geq 2$. Fields that are directly horizontally or vertically adjacent to a field are called its neighbors. At the beginning, $k$ game pieces are distributed on the fields, with multiple or no game pieces possibly being on a single field.
In each mov... | 3n^{2}-4n+1 | 167 | 9 |
math | In the acute-angled triangle $ABC$ the angle$ \angle B = 30^o$, point $H$ is the intersection point of its altitudes. Denote by $O_1, O_2$ the centers of circles inscribed in triangles $ABH ,CBH$ respectively. Find the degree of the angle between the lines $AO_2$ and $CO_1$. | 45^\circ | 86 | 4 |
math | There are 1994 points on a circle, which are painted in several different colors, and the number of points of each color is different. Now, take one point from each color set to form a polygon with vertices of different colors inside the circle. To maximize the number of such polygons, how many different colors should ... | 61 | 86 | 2 |
math | 5 Given a circle $(x-14)^{2}+(y-12)^{2}=36^{2}$, a point $C(4,2)$ inside the circle, and two moving points $A, B$ on the circumference, such that $\angle A C B=90^{\circ}$. The equation of the locus of the midpoint of the hypotenuse $A B$ is $\qquad$ . | (x-9)^{2}+(y-7)^{2}=13\times46 | 92 | 21 |
math | 10.4. Given the function $f(x)=\sqrt{x^{2}-x}$. Find the domain of the function $y=f(f(x))$. | (-\infty,\frac{1-\sqrt{5}}{2}]\cup{0}\cup{1}\cup[\frac{1+\sqrt{5}}{2},+\infty) | 34 | 42 |
math | Determine all triplets of complex numbers $(x, y, z)$ satisfying the following system:
$$
\begin{aligned}
x+y+z & =1 \\
x y z & =1 \\
|x|=|y|=|z| & =1
\end{aligned}
$$ | {x,y,z}={-i,1,i} | 59 | 11 |
math | Example 5. Integrate the equation
$$
\left(1+x^{2}\right) y^{\prime \prime}-2 x y^{\prime}=0
$$ | C_{1}(\frac{x^{3}}{3}+x)+C_{2} | 38 | 20 |
math | 6. (10 points) For a natural number $N$, if at least seven of the nine natural numbers from 1 to 9 are factors of $N$, then $N$ is called a "Seven-Star Number". Among the natural numbers greater than 2000, the smallest "Seven-Star Number" is $\qquad$ | 2016 | 71 | 4 |
math | 4. A group of toddlers in a kindergarten has 90 teeth in total. Any two toddlers together do not have more than 9 teeth. What is the minimum number of toddlers that can be in the group | 23 | 43 | 2 |
math | 59 If three lines: $4 x+y=4, m x+y=0,2 x-3 m y=4$. cannot form a triangle, then the value of $m$ is . $\qquad$ | 4,-\frac{1}{6},-1,\frac{2}{3} | 46 | 18 |
math | 1. A coin was tossed 2021 times. What is the probability that an even number of "heads" will appear? | \frac{1}{2} | 28 | 7 |
math | Example. Solve the equation by the Bubnov-Galerkin method
$$
\varphi(x)=x+\int_{-1}^{1} x t \varphi(t) d t
$$ | \varphi_{3}(x)=3x | 43 | 10 |
math | 5. In the plane, there are 200 points, no three of which are collinear, and each point is labeled with one of the numbers $1, 2, 3$. All pairs of points labeled with different numbers are connected by line segments, and each line segment is labeled with a number 1, 2, or 3, which is different from the numbers at its en... | 199 | 121 | 3 |
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