The sequence $(x_n)$ is defined by
$$x_0=2,\qquad x_{n+1}=\left[\sqrt{2x_n(x_n+1)}\right]\quad(n\in\mathbb N^*).$$
Prove that
$$x_{2n}=2^n+[2^n\sqrt2],\qquad x_{2n+1}=2^{n+1}+[2^n\sqrt2]$$
for every $n\in\mathbb N^*$.
An alphabet contains $29$ characters. A finite sequence of characters is a word, and $X_n$ is the set of all words of length $n$. Consider functions $f:X_n\to X_2$ that assign to each word a two-character subsequence obtained by deleting any $n-2$ characters, without changing the order of those retained. Write $V(f)$ for the image. Find the minimum possible $|V(f)|$ when (a) $n=3$; (b) $n=4$.
Does there exist a strictly increasing sequence of integers $0=a_0\lt a_1\lt a_2\lt\cdots$ such that every natural number is the sum of two terms, not necessarily distinct, and $a_n\gt n^2/16$ for every positive integer $n$?
Let $ABCD$ be a cyclic quadrilateral. Let $M,N$ be the midpoints of $AC,BD$, respectively, and let $E=AB\cap CD$, $F=AD\cap BC$. Prove that
$$\frac{2MN}{EF}=\left|\frac{AC}{BD}-\frac{BD}{AC}\right|.$$
Let $H$ be the orthocenter of triangle $ABC$, and let $P$ be any point on its circumcircle other than $A,B,C$. Let $E,F$ be the perpendicular feet from $P$ to the lines $BC,AB$, respectively. Prove that the line $EF$ bisects the segment $PH$.
Let $AB$ be a diameter of a circle $\omega$, and let $M,C$ lie on $\omega$ in different half-planes relative to $AB$. The perpendicular feet from $M$ to the lines $AB,AC$ are $N,K$, respectively. Prove that $KN$ bisects the segment $CM$.
(Igor Nagel)
A mathematics conference has exactly $n$ married couples, where $n\ge8$. All men sit around one circular table with $n$ seats, and all women around another such table. A healthy person becomes infected immediately if at least two of their left neighbor, right neighbor and spouse are infected; otherwise that person remains healthy. Initially $s$ of the $2n$ participants are infected, and eventually everybody becomes infected. What is the minimum possible value of $s$?
Inside a square $ABCD$ there are $2009$ points, with no three of these $2013$ points (including the four vertices) collinear. Segments joining some of the points, including square vertices, partition the square into triangles. Each segment is an edge; a walk along edges is a path.
Partition the $2013$ points into sets $X,Y$, with $A,C\in X$ and $B,D\in Y$. Prove that either there is an $A$-to-$C$ path using only vertices in $X$, or a $B$-to-$D$ path using only vertices in $Y$.
It is known that
$$\frac a{b+c+d}+\frac b{c+d+a}+\frac c{d+a+b}+\frac d{a+b+c}=1.$$
Find
$$\frac {a^2}{b+c+d}+\frac {b^2}{c+d+a}+\frac {c^2}{d+a+b}+\frac {d^2}{a+b+c}.$$
Let $n$ be a positive integer. Prove that there is only one tuple of $n+2$ real numbers $(\alpha_1,\alpha_2,\alpha_3,\ldots,\alpha_n,\alpha_{n+1})$ such that $\alpha_0=\alpha_{n+1}=0$, $|\alpha_i|\le\pi/6$, and
$$1+\sin\alpha_{i+1}+3\sin\alpha_{i-1}=10\sin\alpha_i-2\sin3\alpha_i$$
for every $i=1,2,\ldots,n$.
Let integers $n\ge k\ge3$ be given. There are $n+1$ points in the plane, no three collinear. Color every segment joining two of them with one of $k$ colors. An angle is bicolored when its vertex is one of the points and its two sides are segments of different colors. Prove that some coloring has strictly more than
$$n\left\lfloor n/k\right\rfloor^2\binom{k}{2}$$
bicolored angles. Here $\lfloor t\rfloor$ denotes the greatest integer not exceeding $t$.
A fixed circle $(O,R)$ and a fixed point $A$ on it are given. Points $B,C$ vary on the circle with $\angle BAC=\alpha$ constant. On the rays $BA,CA$ take $E,F$, respectively, such that $BE=CF=BC$.
a) If $r$ is the radius of the circle $(AEF)$, prove that
$$r\ge R\frac{\sin((\pi-3\alpha)/4)}{\sin((\pi+\alpha)/4)}.$$
b) Let $P$ be the midpoint of the arc $AC$ not containing $B$, and $Q$ the midpoint of the arc $AB$ not containing $C$. The line through $O$ perpendicular to $EF$ meets the lines $AB,AC$ at $M,N$, respectively. Prove that $PM\cap QN$ always belongs to a fixed line.
Let $m$ be real, $f(x)=x^2-mx+m-1$ and $g(x)=x^2-2x-m$. Find the values of $m$ for which the minimum values of $f$ and $g$ are equal. For the largest such $m$, solve $2f(x)\ge g(x-1)$.