You can determine all 4-ples $(a,b, c,d)$ of real numbers, which solve the following equation system $\begin{cases} ab + ac = 3b + 3c \\ bc + bd = 5c + 5d \\ ac + cd = 7a + 7d \\ ad + bd = 9a + 9b \end{cases} $

Georg plays the following game. He chooses two positive integers $n$ and $k$. On an $n \times n$ - board where all the tiles are white, Georg paints k of the tiles black. Then he counts the number of black tiles in each row, forms the square of each of these n numbers and adds up the squares. He calls the result $S$. In the same way he counts the number of white tiles in each row, forms the square of each of these n numbers and adds up those squares. He calls the result $H$. Georg would like to achieve $S - H = 49$. Determine all possible values of n and k for which this is possible. Example: If Georg chooses $n = 5$ and$ k = 14$, he could for example paint the board as shown. [img]https://cdn.artofproblemsolving.com/attachments/f/2/d3c778f603f0a43c9aa877a4564734eab50058.png[/img] Then $S = 1^2 + 2^2 + 3^2 + 3^2 + 5^2 = 1 + 4 + 9 + 9 + 25 = 48$, $H = 4^2 + 3^2 + 2^2 + 2^2 + 0^2 = 16 + 9 + 4 + 4 + 0 = 33$, so in this case $S - H = 48 - 33 = 15$.

Solve the system of equations $$\begin{cases} (x+y)(x^2-y^2)=32 \\ (x-y)(x^2+y^2)=20 \end{cases}$$

At a strange party, each person knew exactly $22$ others. For any pair of people $X$ and $Y$ who knew each other, there was no other person at the party that they both knew. For any pair of people $X$ and $Y$ who did not know one another, there were exactly $6$ other people that they both knew. How many people were at the party?

Over each side of a cyclic quadrilateral erect a rectangle whose height is equal to the length of the opposite side. Prove that the centers of these rectangles form another rectangle.

For a positive integer, we define it's [i]set of exponents[/i] the unordered list of all the exponents of the primes, in it`s decomposition. For example, $18=2\cdot 3^{2}$ has it`s set of exponents $1,2$ and $300=2^{2}\cdot 3\cdot 5^{2}$ has it`s set of exponents $1,2,2$. There are given two arithmetical progressions $\big(a_{n}\big)_{n}$ and $\big(b_{n}\big)_{n}$, such that for any positive integer $n$, $a_{n}$ and $b_{n}$ have the same set of exponents. Prove that the progressions are proportional (that is, there is $k$ such that $a_{n}=kb_{n}$ for any $n$). [i]Proposed by A. Golovanov[/i]

Consider, on a plane, the triangle $ ABC, $ vectors $ \vec x,\vec y,\vec z, $ real variable $ \lambda >0 $ and $ M,N,P $ such that $$ \left\{\begin{matrix} \overrightarrow{AM}=\lambda\cdot\vec x\\\overrightarrow{AN}=\lambda\cdot\vec y \\\overrightarrow{AP}=\lambda\cdot\vec z \end{matrix}\right. . $$ Find the locus of the center of mass of $ MNP. $ [i]Dan Brânzei and Gheorghe Iurea[/i]

Find all functions ${ f : N \rightarrow N}$ (where ${N}$ is the set of the natural numbers and is assumed to contain ${0}$), such that ${f(x^2) - f(y^2) = f(x + y)f(x - y)}$ for all ${x, y \in N}$ with ${x \ge y}$.

Let $\phi(n)$ denote the number of positive integers less than or equal to $n$ that are relatively prime to $n$, and let $d(n)$ denote the number of positive integer divisors of $n$. For example, $\phi(6) = 2$ and $d(6) = 4$. Find the sum of all odd integers $n \le 5000$ such that $n \mid \phi(n) d(n)$. [i]Alex Zhu[/i]

Evaluate the integrals as follows. (1) $\int \frac{x^2}{2-x}\ dx$ (2) $\int \sqrt[3]{x^5+x^3}\ dx$ (3) $\int_0^1 (1-x)\cos \pi x\ dx$

Let $N$ be a positive integer. Let $R$ denote the smallest positive number that is the sum of $m$ terms $\sum^m_{i=1}{\pm \sqrt{a_i}}$, where each $a_i, i=1,\cdots, m$ is an integer not larger than $N$. Prove that \[R\le C\cdot N^{-m+\frac{3}{2}}\] for some positive real number $C$. [i]Proposed by Navid[/i] [i](Clarification: note that the constant is allowed to depend on $m$ but should be independent of $N$, i.e. the equation $R(m,N)\le C(m)\cdot N^{-m+\frac{3}{2}}$ should hold for all positive integers $N$)[/i]

Suppose $n$ and $r$ are nonnegative integers such that no number of the form $n^2+r-k(k+1) \text{ }(k\in\mathbb{N})$ equals to $-1$ or a positive composite number. Show that $4n^2+4r+1$ is $1$, $9$, or prime.

Find all triplets $ (x,y,z) $ of positive integers such that \[ x^y + y^x = z^y \]\[ x^y + 2012 = y^{z+1} \]

Let $G$ be a simple graph with $11$ vertices labeled as $v_{1} , v_{2} , ... , v_{11}$ such that the degree of $v_1$ equals to $2$ and the degree of other vertices are equal to $3$.If for any set $A$ of these vertices which $|A| \le 4$ , the number of vertices which are adjacent to at least one verex in $A$ and are not in $A$ themselves is at least equal to $|A|$ , then find the maximum possible number for the diameter of $G$. (The distance between two vertices of graph is the number of edges of the shortest path between them and the diameter of a graph , is the largest distance between arbitrary pairs in $V(G)$. ) [i]Proposed by Alireza Haqi[/i]

Find the greatest possible sum of integers $a$ and $b$ such that $\frac{2021!}{20^a\cdot 21^b}$ is a positive integer. [i]Proposed by Aidan Duncan[/i]

An equilateral triangle with side length 9 is divided into 81 congruent triangles with segments which are parallel to the sides of the triangle. Prove that it cannot be divided into more than 18 parallelograms with sides 1 and 2. [I]Proposed by O.Vanyushina[/i]

Let $a$, $b$, $c$ and $d$ be non-zero pairwise different real numbers such that $$ \frac{a}{b} + \frac{b}{c} + \frac{c}{d} + \frac{d}{a} = 4 \text{ and } ac = bd. $$ Show that $$ \frac{a}{c} + \frac{b}{d} + \frac{c}{a} + \frac{d}{b} \leq -12 $$ and that $-12$ is the maximum.

Consider points $C_1, C_2$ on the side $AB$ of a triangle $ABC$, points $A_1, A_2$ on the side $BC$ and points $B_1 , B_2$ on the side $CA$ such that these points divide the corresponding sides to three equal parts. It is known that all the points $A_1, A_2, B_1, B_2 , C_1$ and $C_2$ are concyclic. Prove that triangle $ABC$ is equilateral.

Let $D$ be a point in the interior of $ABC$. Let $BD$ and $AC$ intersect at $E$ while $CD$ and $AB$ intersect at $F$. Let $EF$ intersect $BC$ at $G$. Let $H$ be an arbitrary point on $AD$. Let $HF$ and $BD$ intersect at $I$. Let $HE$ and $CD$ intersect at $J$ . prove that $G$,$I$ and $J$ are collinear

A [i]magic square[/i] is a table [img]https://cdn.artofproblemsolving.com/attachments/7/9/3b1e2b2f5d2d4c486f57c4ad68b66f7d7e56dd.png[/img] in which all the natural numbers from $1$ to $16$ appear and such that: $\bullet$ all rows have the same sum $s$. $\bullet$ all columns have the same sum $s$. $\bullet$ both diagonals have the same sum $s$ . It is known that $a_{22} = 1$ and $a_{24} = 2$. Calculate $a_{44}$.

Three unit circles $\omega_1$, $\omega_2$, and $\omega_3$ in the plane have the property that each circle passes through the centers of the other two. A square $S$ surrounds three circles in such a way that each of its four sides is tangent to at least one of $\omega_1$, $\omega_2$, and $\omega_3$. Find the side length of the square $S$.

Given $n \ (n \geq 3)$ points in space such that every three of them form a triangle with one angle greater than or equal to $120^\circ$, prove that these points can be denoted by $A_1,A_2, \ldots,A_n$ in such a way that for each $i, j, k, 1 \leq i < j < k \leq n$, angle $A_iA_jA_k$ is greater than or equal to $120^\circ . $

The sequence of numbers $a_1, a_2, a_3, ...$ is defined recursively by $a_1 = 1, a_{n + 1} = \lfloor \sqrt{a_1+a_2+...+a_n} \rfloor $ for $n \ge 1$. Find all numbers that appear more than twice at this sequence.

In a room there are nine men. Among every three of them there are two mutually acquainted. Prove that some four of them are mutually acquainted.

Integers $1 \le n \le 200$ are written on a blackboard just one by one. We surrounded just $100$ integers with circle. We call a square of the sum of surrounded integers minus the sum of not surrounded integers $score$ of this situation. Calculate the average score in all ways.