Find all pairs of real numbers $x$ and $y$ which satisfy the following equations: \begin{align*} x^2 + y^2 - 48x - 29y + 714 & = 0 \\ 2xy - 29x - 48y + 756 & = 0 \end{align*}

Do there exist 12 rectangular parallelepipeds $P_1,\,P_2,\ldots,P_{12}$ with edges parallel to coordinate axes $OX,\,OY,\,OZ$ such that $P_i$ and $P_j$ have a common point iff $i\ne j\pm 1$ modulo 12?

For $a,b,c>0$, prove that (i) $\frac{a+b+c}{2}-\frac{ab}{a+b}-\frac{bc}{b+c}-\frac{ca}{c+a}\ge 0$ (ii) $a(1+b)+b(1+c)+c(1+a)\ge 6\sqrt{abc}$

The numbers from $1$ through $100$ are written in some order on a circle. We call a pair of numbers on the circle [i]good [/i] if the two numbers are not neighbors on the circle and if at least one of the two arcs they determine on the circle only contains numbers smaller then both of them. What may be the total number of good pairs on the circle.

Assume $a,b,c$ are positive numbers, such that \[ a(1-b) = b(1-c) = c(1-a) = \dfrac14 \] Prove that $a=b=c$.

An acute angle $XAY$ and a point $P$ inside the angle are given. Construct (using a ruler and a compass) a line that passes through $P$ and intersects the rays $AX$ and $AY$ at $B$ and $C$ such that the area of the triangle $ABC$ equals $AP^2$. [i]Greece[/i]

Given a nonzero real number $a\leq 1/e$, let $z_1, ..., z_n \in C$ be non-real numbers for which $ze^z + a = 0$ holds, and let $c_1, ..., c_n \in C$ be arbitrary. Show that the function $f(x)=Re(\sum_{j=1}^n c_j e^{z_j x})$ ($x \in R$) has a zero in every closed interval of length 1.

For an integer $n$ with $b$ digits, let a [i]subdivisor[/i] of $n$ be a positive number which divides a number obtained by removing the $r$ leftmost digits and the $l$ rightmost digits of $n$ for nonnegative integers $r,l$ with $r+l<b$ (For example, the subdivisors of $143$ are $1$, $2$, $3$, $4$, $7$, $11$, $13$, $14$, $43$, and $143$). For an integer $d$, let $A_d$ be the set of numbers that don't have $d$ as a subdivisor. Find all $d$, such that $A_d$ is finite.

The circles $k_1$ and $k_2$ intersect at points $A$ and $B$, and $k_1$ passes through the center $O$ of the circle $k_2$. The line $p$ intersects $k_1$ at the points $K ,O$ and $k_2$ at the points $L ,M$ so that $L$ lies between $K$ and $O$. The point $P$ is the projection of $L$ on the line $AB$. Prove that $KP$ is parallel to the median of triangle $ABM$ drawn from the vertex $M$.

Let $O$ be the center of the circle circumscribed to $\vartriangle ABC$, and let $ P$ be any point on $BC$ ($P \ne B$ and $P \ne C$). Suppose that the circle circumscribed to $\vartriangle BPO$ intersects $AB$ at $R$ ($R \ne A$ and $R \ne B$) and that the circle circumscribed to $\vartriangle COP$ intersects $CA$ at point $Q$ ($Q \ne C$ and $Q \ne A$). 1) Show that $\vartriangle PQR \sim \vartriangle ABC$ and that$ O$ is orthocenter of $\vartriangle PQR$. 2) Show that the circles circumscribed to the triangles $\vartriangle BPO$, $\vartriangle COP$, and $\vartriangle PQR$ all have the same radius.

Find all primes $ p$ for that there is an integer $ n$ such that there are no integers $ x,y$ with $ x^3 \plus{} y^3 \equiv n \mod p$ (so not all residues are the sum of two cubes). E.g. for $ p \equal{} 7$, one could set $ n \equal{} \pm 3$ since $ x^3,y^3 \equiv 0 , \pm 1 \mod 7$, thus $ x^3 \plus{} y^3 \equiv 0 , \pm 1 , \pm 2 \mod 7$ only.

Find the orthogonal trajectories of the family of conics $(x+2y)^{2} = a(x+y)$. At what angle do the curves of one family cut the curves of the other family at the origin?

Determine all integers $n \geq 2$ for which the number $11111$ in base $n$ is a perfect square.

Circle $\omega$ touches the side $AC$ of the equilateral triangle $ABC$ at point $D$, and its circumcircle at the point $E$ lying on the arc $\overarc{BC}$. Prove that with segments $AD$, $BE$ and $CD$, you can form a triangle, in which the difference of two of its angles is $60^{\circ}$.

An $8 \times 8 \times 8$ cube is painted red on $3$ faces and blue on $3$ faces such that no corner is surrounded by three faces of the same color. The cube is then cut into $512$ unit cubes. How many of these cubes contain both red and blue paint on at least one of their faces? [i]Author: Ray Li[/i] [hide="Clarification"]The problem asks for the number of cubes that contain red paint on at least one face and blue paint on at least one other face, not for the number of cubes that have both colors of paint on at least one face (which can't even happen.)[/hide]

a) Let $n$ be a positive integer $G$ be a a group with $|G|<\frac{4n^2}{n-\varphi(n)}$. Suppose that $Z(G)$ contains at least $\varphi(n)+1$ elements of order $n$. Prove that $G$ is abelian. b) Find a noncommutative group $G$ with $16$ elements such that $Z(G)$ contains two elements of order two. [i]Robert Rogozsan & Filip Munteanu[/i]

Let $ f:[0,1]\longrightarrow\mathbb{R}_+^* $ be a Riemann-integrable function. Calculate $ \lim_{n\to\infty}\left(-n+\sum_{i=1}^ne^{\frac{1}{n}\cdot f\left(\frac{i}{n}\right)}\right) . $

Let $f: \mathbb{N} \to \mathbb{N}$ be a function that satisfies: \[ f(1) = 2008, \] \[ f(4n^2) = 4f(n^2), \] \[ f(4n^2 + 2) = 4f(n^2) + 3, \] \[ f(4n(n+1)) = 4f(n(n+1)) + 1, \] \[ f(4n(n+1) + 3) = 4f(n(n+1)) + 4. \] Determine whether there exists a natural number $m$ such that: \[ 1^2 + 2^2 + \dots + m^2 + f(1^2) + \dots + f(m^2) = 2008m + 251. \]

Let $f: \mathbb{R} \to \mathbb{R}$ be a differentiable function such that $$f(x)=f \left( \frac{x}{2} \right) + \frac{x}{2} f'(x), ~\forall x \in \mathbb{R}.$$ Prove that $f$ is a polynomial function of degree at most one. [hide=Note]The problem was posted quite a few times before: [url]https://artofproblemsolving.com/community/c7h100225p566080[/url] [url]https://artofproblemsolving.com/community/q11h564540p3300032[/url] [url]https://artofproblemsolving.com/community/c7h2605212p22490699[/url] [url]https://artofproblemsolving.com/community/c7h198927p1093788[/url] I'm reposting it just to have a more suitable statement for the [url=https://artofproblemsolving.com/community/c13_contests]Contest Collections[/url]. [/hide]

Find all positive integers $n$ for which there exist non-negative integers $a_1, a_2, \ldots, a_n$ such that \[ \frac{1}{2^{a_1}} + \frac{1}{2^{a_2}} + \cdots + \frac{1}{2^{a_n}} = \frac{1}{3^{a_1}} + \frac{2}{3^{a_2}} + \cdots + \frac{n}{3^{a_n}} = 1. \] [i]Proposed by Dusan Djukic, Serbia[/i]

Lucas writes two distinct positive integers on a whiteboard. He decreases the smaller number by $20$ and increases the larger number by $23,$ only to discover the product of the two original numbers is equal to the product of the two altered numbers. Compute the minimum possible sum of the original two numbers on the board.

Let $X$ be a invertible matrix with columns $X_{1},X_{2}...,X_{n}$. Let $Y$ be a matrix with columns $X_{2},X_{3},...,X_{n},0$. Show that the matrices $A=YX^{-1}$ and $B=X^{-1}Y$ have rank $n-1$ and have only $0$´s for eigenvalues.

Given positive numbers $a_1, \ldots , a_n$, $b_1, \ldots , b_n$, $c_1, \ldots , c_n$. Let $m_k$ be the maximum of the products $a_ib_jc_l$ over the sets $(i, j, l)$ for which $max(i, j, l) = k$. Prove that $$(a_1 + \ldots + a_n) (b_1 +\ldots + b_n) (c_1 +\ldots + c_n) \leq n^2 (m_1 + \ldots + m_n).$$

The centre of the coin with radius $r$ is moved along some polygon with the perimeter $P$, that is circumscribed around the circle with radius $R$ ($R>r$). Find the coin trace area (a sort of polygon ring).

Does $ a^2 \plus{} b^2 \plus{} c^2 \leqslant 2(ab \plus{} bc \plus{} ca)$ hold for every $ a,b,c$ if it is known that $ a^4 \plus{} b^4 \plus{} c^4 \leqslant 2(a^2 b^2 \plus{} b^2 c^2 \plus{} c^2 a^2 )$.