Let $P_{1}(x)=x^{2}-2$ and $P_{j}(x)=P_{1}(P_{j-1}(x))$ for j$=2,\ldots$ Prove that for any positive integer n the roots of the equation $P_{n}(x)=x$ are all real and distinct.

Let $ABCD$ be a convex quadrilateral with perpendicular diagonals. . Assume that $ABCD$ has been inscribed in the circle with center $O$. Prove that $AOC$ separates $ABCD$ into two quadrilaterals of equal area

Find all sets $X$ consisting of at least two positive integers such that for every two elements $m,n\in X$, where $n>m$, there exists an element $k\in X$ such that $n=mk^2$.

Find all $f : N \longmapsto N$ that there exist $k \in N$ and a prime $p$ that: $\forall n \geq k \ f(n+p)=f(n)$ and also if $m \mid n$ then $f(m+1) \mid f(n)+1$

If $A$ and $B$ are numbers such that the polynomial $x^{2017} + Ax + B$ is divisible by $(x + 1)^2$, what is the value of $B$?

In the plane, $ n \geq 3 $ segments are placed in such a way that every $ 3 $ of them have a common point. Prove that there is a common point for all the segments.

Given a regular 2007-gon. Find the minimal number $k$ such that: Among every $k$ vertexes of the polygon, there always exists 4 vertexes forming a convex quadrilateral such that 3 sides of the quadrilateral are also sides of the polygon.

Suppose two equally strong tennis players play against each other until one player wins three games in a row. The results of each game are independent, and each player will win with probability $\frac{1}{2}$. What is the expected value of the number of games they will play?

Let $ABC$ be a triangle with incenter $I$ and incircle $\omega$. It is given that there exist points $X$ and $Y$ on the circumference of $\omega$ such that $\angle BXC=\angle BYC=90^\circ$. Suppose further that $X$, $I$, and $Y$ are collinear. If $AB=80$ and $AC=97$, compute the length of $BC$.

Prove that among any ten consecutive positive integers at least one is relatively prime to the product of the others.

Let $ a_{1},a_{2},\cdots,a_{n}$ be positive real numbers satisfying $ a_{1} \plus{} a_{2} \plus{} \cdots \plus{} a_{n} \equal{} 1$. Prove that \[\left(a_{1}a_{2} \plus{} a_{2}a_{3} \plus{} \cdots \plus{} a_{n}a_{1}\right)\left(\frac {a_{1}}{a_{2}^2 \plus{} a_{2}} \plus{} \frac {a_{2}}{a_{3}^2 \plus{} a_{3}} \plus{} \cdots \plus{} \frac {a_{n}}{a_{1}^2 \plus{} a_{1}}\right)\ge\frac {n}{n \plus{} 1}\]

Prove that for any fixed integer $a$ equation $$(m!+a)^2=n!+a^2$$ has finitely many solutions in positive integers $m,n$

Let $n\ge 2$ be an integer. Show that there exists a subset $A\in \{1,2,\ldots ,n\}$ such that: i) The number of elements of $A$ is at most $2\lfloor\sqrt{n}\rfloor+1$ ii) $\{ |x-y| \mid x,y\in A, x\not= y\} = \{ 1,2,\ldots n-1 \}$ [i]Radu Todor[/i]

Given are $100$ positive integers whose sum equals their product. Determine the minimum number of $1$s that may occur among the $100$ numbers.

If $ y\equal{}x^2\plus{}px\plus{}q$, then if the least possible value of $ y$ is zero $ q$ is equal to: $ \textbf{(A)}\ 0 \qquad \textbf{(B)}\ \frac{p^2}{4} \qquad \textbf{(C)}\ \frac{p}{2} \qquad \textbf{(D)}\ \minus{}\frac{p}{2} \qquad \textbf{(E)}\ \frac{p^2}{4}\minus{}q$

In rectangular coordinate system, if $m(x^2+y^2+2y+1)=(x-2y+3)^2$ refers to an ellipse, then the range value of $m$ is $\text{(A)}(0,1)\qquad\text{(B)}(1,+\infty)\qquad\text{(C)}(0,5)\qquad\text{(D)}(5,+\infty)$

A man has three daughters. The product of their ages is $168$, and he remembers that the sum of their ages is the number of trees in his yard. He counts the trees but cannot determine any of their ages. What are all possible ages of his oldest daughter?

Let $ABCD$ be a cyclic quadrilateral with $\angle BAD < \angle ADC$. Let $M$ be the midpoint of the arc $CD$ not containing $A$. Suppose there is a point $P$ inside $ABCD$ such that $\angle ADB = \angle CPD$ and $\angle ADP = \angle PCB$. Prove that lines $AD, PM$, and $BC$ are concurrent.

We put pebbles on some unit squares of a $2013 \times 2013$ chessboard such that every unit square contains at most one pebble. Determine the minimum number of pebbles on the chessboard, if each $19\times 19$ square formed by unit squares contains at least $21$ pebbles.

Let $x, y, z$ be non-negative real numbers whose sum is $ 1$. Prove that: $$\sqrt[3]{x - y + z^3} + \sqrt[3]{y - z + x^3} + \sqrt[3]{z - x + y^3} \le 1$$

Consider a cyclic quadrilateral $ABCD$ and define points $X = AB \cap CD$, $Y = AD \cap BC$, and suppose that there exists a circle with center $Z$ inscribed in $ABCD$. Show that the $Z$ belongs to the circle with diameter $XY$ , which is orthogonal to circumcircle of $ABCD$.

For some positive integer $n$, $n$ is considered a $\textbf{unique}$ number if for any other positive integer $m\neq n$, $\{\dfrac{2^n}{n^2}\}\neq\{\dfrac{2^m}{m^2}\}$ holds. Prove that there is an infinite list consisting of composite unique numbers whose elements are pairwise coprime.

Determine whether there exist two infinite point sequences $ A_1,A_2,\ldots$ and $ B_1,B_2,\ldots$ in the plane, such that for all $i,j,k$ with $ 1\le i < j < k$, (i) $ B_k$ is on the line that passes through $ A_i$ and $ A_j$ if and only if $ k=i+j$. (ii) $ A_k$ is on the line that passes through $ B_i$ and $ B_j$ if and only if $ k=i+j$. [i](Proposed by Gerhard Woeginger, Austria)[/i]

For every edge of a tetrahedron, we consider a plane through its midpoint that is perpendicular to the opposite edge. Prove that these six planes intersect in a point symmetric to the circumcenter of the tetrahedron with respect to its centroid.

[u]Set 1 [/u] [b]1.1[/b] Compute the number of real numbers x such that the sequence $x$, $x^2$, $x^3$,$ x^4$, $x^5$, $...$ eventually repeats. (To be clear, we say a sequence “eventually repeats” if there is some block of consecutive digits that repeats past some point—for instance, the sequence $1$, $2$, $3$, $4$, $5$, $6$, $5$, $6$, $5$, $6$, $...$ is eventually repeating with repeating block $5$, $6$.) [b]1.2[/b] Let $T$ be the answer to the previous problem. Nicole has a broken calculator which, when told to multiply $a$ by $b$, starts by multiplying $a$ by $b$, but then multiplies that product by b again, and then adds $b$ to the result. Nicole inputs the computation “$k \times k$” into the calculator for some real number $k$ and gets an answer of $10T$. If she instead used a working calculator, what answer should she have gotten? [b]1.3[/b] Let $T$ be the answer to the previous problem. Find the positive difference between the largest and smallest perfect squares that can be written as $x^2 + y^2$ for integers $x, y$ satisfying $\sqrt{T} \le x \le T$ and $\sqrt{T} \le y \le T$. PS. You should use hide for answers.