Find the minimum value of $xy$, given that $x^2+y^2+z^2=7$, $xz+xy+yz=4$, and $x, y, z$ are real numbers

Determine $$ \left|\prod^{10}_{k=1}(e^{\frac{i \pi}{2^k}}+ 1) \right|$$ 

Let $n>1$ be an integer and $a,b,c$ be three complex numbers such that $a+b+c=0$ and $a^n+b^n+c^n=0$. Prove that two of $a,b,c$ have the same magnitude. [i]Evan O'Dorney.[/i]

Let $z_1,z_2,z_3,\dots,z_{12}$ be the 12 zeroes of the polynomial $z^{12}-2^{36}$. For each $j$, let $w_j$ be one of $z_j$ or $i z_j$. Then the maximum possible value of the real part of $\displaystyle\sum_{j=1}^{12} w_j$ can be written as $m+\sqrt{n}$ where $m$ and $n$ are positive integers. Find $m+n$.

Find all pairs (x; y) of real numbers satisfying the system of equations $x^3 + 1 - xy^2 - y^2 = 0$; $y^3 - 1 - x^2y + x^2 = 0$. Darij

Initially, on a board there a positive integer. If board contains the number $x,$ then we may additionally write the numbers $2x+1$ and $\frac{x}{x+2}.$ At some point 2008 is written on the board. Prove, that this number was there from the beginning.

Let $f(x)$ be a real-valued function satisfying $af(x)+bf(-x)=px^2+qx+r$. $a$ and $b$ are distinct real numbers and $p,q,r$ are non-zero real numbers. Then $f(x)=0$ will have real solutions when (A)$\left(\frac{a+b}{a-b}\right)\leq\frac{q^2}{4pr}$ (B)$\left(\frac{a+b}{a-b}\right)\leq\frac{4pr}{q^2}$ (C)$\left(\frac{a+b}{a-b}\right)\geq\frac{q^2}{4pr}$ (D)$\left(\frac{a+b}{a-b}\right)\geq\frac{4pr}{q^2}$

$n+m-1$ unit squares are arranged in $L$-shape whose one side contains $n$ squares and the other side contains $m$ squares. Ayse and Betul plays a turn based game with following rules: Ayse plays first. At each move, the player captures desired number of adjacent squares in same side of $L$. The one who captures the last square loses the game. If four games are played for pairs $(n,m)=(2003,2003)$, $(2002,2003)$, $(2003,3)$, $(2001,2003)$; how many times can Ayse guarantee to win? $ \textbf{(A)}\ 0 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 2 \qquad\textbf{(D)}\ 3 \qquad\textbf{(E)}\ 4 $

Find all functions $f : \mathbb{N} \rightarrow \mathbb{R}$ such that for all triples $a,b,c$ of positive integers the following holds : $$f(ac)+f(bc)-f(c)f(ab) \ge 1$$ Proposed by [i]Mojtaba Zare[/i]

$n(n\geq6)$ people attend a party. It is known that (1) Anyone knows at least $\left [\frac{n}{2} \right]$ people. (2) For any $\left[\frac{n}{2}\right]$ people, either two people among them know each other, or among people else, there are two people know each other. Prove that there are three people in the $n$ people, they know one another.

It is known that for all $x$ such that $|x| < 1$, the following inequality holds $$ax^2+bx+c\le \frac{1}{\sqrt{1-x^2}}$$Find the greatest value of $a + 2c$.

Let $P$ be a point inside the triangle $ABC$ such that $AB = BC$, $\angle ABC = 80^o$, $\angle PAC = 40^o$ and $\angle ACP = 30^o$. Find $\angle BPC$. (G Galperin)

Let be a circumcircle of radius $ 1 $ of a triangle whose centered representation in the complex plane is given by the affixes of $ a,b,c, $ and for which the equation $ a+b\cos x +c\sin x=0 $ has a real root in $ \left( 0,\frac{\pi }{2} \right) . $ prove that the area of the triangle is a real number from the interval $ \left( 1,\frac{1+\sqrt 2}{2} \right] . $ [i]Gheorghe Iurea[/i]

I start with a sequence of letters $A_1 A_2 \cdots A_{2021} A_1 A_2 \cdots A_{2021} A_1 A_2 \cdots A_{2021}$. I go through $i = 1, 2, 3, \cdots, 6062$ in order, and for each $i$, I can choose to swap letters $i$ and $i+1$. Let $N$ be the number of distinct strings I can end up with. What is the remainder when $N$ is divided by $2017$?

What is the largest positive integer which is equal to the sum of its digits? [i]Proposed by Evan Chen[/i]

Define a search algorithm called $\texttt{powSearch}$. Throughout, assume $A$ is a 1-indexed sorted array of distinct integers. To search for an integer $b$ in this array, we search the indices $2^0,2^1,\ldots$ until we either reach the end of the array or $A[2^k] > b$. If at any point we get $A[2^k] = b$ we stop and return $2^k$. Once we have $A[2^k] > b > A[2^{k-1}]$, we throw away the first $2^{k-1}$ elements of $A$, and recursively search in the same fashion. For example, for an integer which is at position $3$ we will search the locations $1, 2, 4, 3$. Define $g(x)$ to be a function which returns how many (not necessarily distinct) indices we look at when calling $\texttt{powSearch}$ with an integer $b$ at position $x$ in $A$. For example, $g(3) = 4$. If $A$ has length $64$, find \[g(1) + g(2) + \ldots + g(64).\]

Let $ABCD$ be a circumscribed quadrilateral and let $P$ be the orthogonal projection of its in center on $AC$. Prove that $\angle {APB}=\angle {APD}$

The function $ x^2 \plus{} px \plus{} q$ with $ p$ and $ q$ greater than zero has its minimum value when: $ \textbf{(A)}\ x \equal{} \minus{} p \qquad\textbf{(B)}\ x \equal{} \frac {p}{2} \qquad\textbf{(C)}\ x \equal{} \minus{} 2p \qquad\textbf{(D)}\ x \equal{} \frac {p^2}{4q} \qquad\textbf{(E)}\ x \equal{} \frac { \minus{} p}{2}$

In a triangle $ ABC,$ let $ D$ and $ E$ be the intersections of the bisectors of $ \angle ABC$ and $ \angle ACB$ with the sides $ AC,AB,$ respectively. Determine the angles $ \angle A,\angle B, \angle C$ if $ \angle BDE \equal{} 24 ^{\circ},$ $ \angle CED \equal{} 18 ^{\circ}.$

For a positive integer $n$, $n$ vertices which have $10000$ written on them exist on a plane. For $3$ vertices that are collinear and are written positive numbers on them, denote procedure $P$ as subtracting $1$ from the outer vertices and adding $2023$ to the inner vertical. Show that procedure $P$ cannot be repeated infinitely.

You are given a $5\times 6$ checkerboard with squares alternately shaded black and white. The bottom- left square is white. Each square has side length $1$ unit. You can normally travel on this board at a speed of $2$ units per second, but while you travel through the interior (not the boundary) of a black square, you are slowed down to $1$ unit per second. What is the shortest time it takes to travel from the bottom-left corner to the top-right corner of the board?

Let $AD,BE,CF$ be the altitudes of an acute triangle $ABC$ with $AB>AC$. Line $EF$ meets $BC$ at $P$, and line through $D$ parallel to $EF$ meets $AC$ and $AB$ at $Q$ and $R$, respectively. Let $N$ be any poin on side $BC$ such that $\widehat{NQP}+\widehat{NRP}<180^{0}$. Prove that $BN>CN$.

Let $H$ be the orthocenter of the triangle $ABC$ and let $D$, $E$, $F$ be the feet of heights by $A$, $B$, $C$. Let $\omega_D$, $\omega_E$, $\omega_F$ be the incircles of $FEH$, $DHF$, $HED$ and let $I_D$, $I_E$, $I_F$ be their centers. Show that $I_DD$, $I_EE$ and $I_FF$ compete.

An integer $ m > 1$ is given. The infinite sequence $ (x_n)_{n\ge 0}$ is defined by $ x_i\equal{}2^i$ for $ i<m$ and $ x_i\equal{}x_{i\minus{}1}\plus{}x_{i\minus{}2}\plus{}\cdots \plus{}x_{i\minus{}m}$ for $ i\ge m$. Find the greatest natural number $ k$ such that there exist $ k$ successive terms of this sequence which are divisible by $ m$.

Let $n$ be a natural number greater than 2. $l$ is a line on a plane. There are $n$ distinct points $P_1$, $P_2$, …, $P_n$ on $l$. Let the product of distances between $P_i$ and the other $n-1$ points be $d_i$ ($i = 1, 2,$ …, $n$). There exists a point $Q$, which does not lie on $l$, on the plane. Let the distance from $Q$ to $P_i$ be $C_i$ ($i = 1, 2,$ …, $n$). Find $S_n = \sum_{i = 1}^{n} (-1)^{n-i} \frac{c_i^2}{d_i}$.