A sequence $ a_1, a_2, a_3, \ldots$ is defined recursively by $ a_1 \equal{} 1$ and $ a_{2^k\plus{}j} \equal{} \minus{}a_j$ $ (j \equal{} 1, 2, \ldots, 2^k).$ Prove that this sequence is not periodic.

Prove that if $\frac{AC}{BC}=\frac{AB + BC}{AC}$ in a triangle $ABC$ , then $\angle B = 2 \angle A$ .

Let $f(x) = x^2 + 6x + c$ for all real number s$x$, where $c$ is some real number. For what values of $c$ does $f(f(x))$ have exactly $3$ distinct real roots?

Let $ABC$ be an acute-angled triangle \( ABC \) with \( AC > BC \) and incenter \( I \). Let \( \omega \) be the mixtilinear circle at vertex \( C \), i.e. the circle internally tangent to the circumcircle of \( \triangle ABC \) and also tangent to lines \( AC \) and \( BC \). A circle \( \Gamma \) passes through points \( A \) and \( B \) and is tangent to \( \omega \) at point \( T \), with \( C \notin \Gamma \) and \( I \) being inside \( \triangle ATB \). Prove that: $$\angle CTB + \angle ATI = 180^\circ + \angle BAI - \angle ABI.$$

Let $a,b,c,d$ be positive real numbers. Prove that \[\frac{a}{b+2c+3d}+\frac{b}{c+2d+3a}+\frac{c}{d+2a+3b}+\frac{d}{a+2b+3c} \geq \frac{2}{3}.\]

Let $N\ge2$ be a fixed positive integer. There are $2N$ people, numbered $1,2,...,2N$, participating in a tennis tournament. For any two positive integers $i,j$ with $1\le i<j\le 2N$, player $i$ has a higher skill level than player $j$. Prior to the first round, the players are paired arbitrarily and each pair is assigned a unique court among $N$ courts, numbered $1,2,...,N$. During a round, each player plays against the other person assigned to his court (so that exactly one match takes place per court), and the player with higher skill wins the match (in other words, there are no upsets). Afterwards, for $i=2,3,...,N$, the winner of court $i$ moves to court $i-1$ and the loser of court $i$ stays on court $i$; however, the winner of court 1 stays on court 1 and the loser of court 1 moves to court $N$. Find all positive integers $M$ such that, regardless of the initial pairing, the players $2, 3, \ldots, N+1$ all change courts immediately after the $M$th round. [i]Proposed by Ray Li[/i]

Given a circle $\omega$, a point $A$ lying inside $\omega$, and point $B$ ($B \ne A$). All possible triangles $BXY$ are considered, such that the points $X$ and $Y$ lie on $\omega$ and the chord $XY$ passes through the point $A$. Prove that the centers of the circumcircles of the triangles $BXY$ lie on the same straight line.

Let $\triangle ABC$ and $A'$,$B'$,$C'$ the symmetrics of vertex over opposite sides.The intersection of the circumcircles of $\triangle ABB'$ and $\triangle ACC'$ is $A_1$.$B_1$ and $C_1$ are defined similarly.Prove that lines $AA_1$,$BB_1$ and $CC_1$ are concurent.

[b]Q .[/b]Prove that $$\left(\sum_\text{cyc}(a-x)^4\right)\ +\ 2\left(\sum_\text{sym}x^3y\right)\ +\ 4\left(\sum_\text{cyc}x^2y^2\right)\ +\ 8xyza \geqslant \left(\sum_\text{cyc}(a-x)^2(a^2-x^2)\right)$$where $a=x+y+z$ and $x,y,z \in \mathbb{R}.$ [i]Proposed by srijonrick[/i]

Each of the numbers $x_1, x_2, \dots, x_n$ equals $1$ or $-1$ and \[\sum_{i=1}^n x_i x_{i+1} x_{i+2} x_{i+3} =0.\] where $x_{n+i}=x_i $ for all $i$. Prove that $4\mid n$.

Let $f: Z ^+ \to R$, such that $f (1) = 2018$ and $f (1) + f (2) + ...+ f (n) = n^2f (n)$, for all $n> 1$. Find the value $f (2017)$.

Some positive integers, sum of which is $23$, are written in sequential form. Neither one of the terms nor the sum of some consecutive terms in the sequence is equal to $3$. [b]a) [/b]Is it possible that the sequence contains exactly $11$ terms? [b]b)[/b]Is it possible that the sequence contains exactly $12$ terms?

For a positive integer $n$, there are two countries $A$ and $B$ with $n$ airports each and $n^2-2n+ 2$ airlines operating between the two countries. Each airline operates at least one flight. Exactly one flight by one of the airlines operates between each airport in $A$ and each airport in $B$, and that flight operates in both directions. Also, there are no flights between two airports in the same country. For two different airports $P$ and $Q$, denote by "[i]$(P, Q)$-travel route[/i]" the list of airports $T_0, T_1, \ldots, T_s$ satisfying the following conditions. [list] [*] $T_0=P,\ T_s=Q$ [*] $T_0, T_1, \ldots, T_s$ are all distinct. [*] There exists an airline that operates between the airports $T_i$ and $T_{i+1}$ for all $i = 0, 1, \ldots, s-1$. [/list] Prove that there exist two airports $P, Q$ such that there is no or exactly one [i]$(P, Q)$-travel route[/i]. [hide=Graph Wording]Consider a complete bipartite graph $G(A, B)$ with $\vert A \vert = \vert B \vert = n$. Suppose there are $n^2-2n+2$ colors and each edge is colored by one of these colors. Define $(P, Q)-path$ a path from $P$ to $Q$ such that all of the edges in the path are colored the same. Prove that there exist two vertices $P$ and $Q$ such that there is no or only one $(P, Q)-path$. [/hide]

Prove that for all positive integer $n$, there is a positive integer $m$ that $7^n | 3^m +5^m -1$.

On a table are given $n$ markers, each of which is denoted by an integer. At any time, if some two markers are denoted with the same number, say $k$, we can redenote one of them with $k +1$ and the other one with $k -1$. Prove that after a finite number of moves all the markers will be denoted with different numbers.

Let $ABC$ be a triangle, $O$ is the center of the circle circumscribed around it, $AD$ the diameter of this circle. It is known that the lines $CO$ and $DB$ are parallel. Prove that the triangle $ABC$ is isosceles. (Andrey Mostovy)