Let $ABC$ be a triangle that is not right-angled at $A$, and let $AM$ be its median. A point $D$ varies on line $AM$. Let $(O_1)$ and $(O_2)$ be circles passing through $D$ and tangent to $BC$ at $B$ and $C$, respectively. Let $P$ and $Q$ be the intersections of line $AB$ with $(O_1)$ and line $AC$ with $(O_2)$, respectively. Prove that:
1. The tangents to $(O_1)$ at $P$ and to $(O_2)$ at $Q$ meet at a point $S$.
2. As $D$ varies on $AM$, the point $S$ lies on a fixed line.
Interpretation needed for the printed construction: $D\ne M$, so the two circles exist; $P$ and $Q$ mean the second intersections other than $B$ and $C$.
For each positive integer $n$, consider
$$T_n=\{11(k+h)+10(n^k+n^h):1\le k,h\le10\}.$$
Find all $n$ for which there do not exist distinct numbers $a,b\in T_n$ such that $110$ divides $a-b$.
A $1\times2$ rectangle is called a simple tile. A $2\times3$ rectangle with two diagonally opposite corner unit squares removed (leaving four unit squares) is called a double tile. Simple and double tiles fit together without gaps or overlaps to form a $2008\times2010$ rectangular board. Find the minimum possible number of simple tiles.
Let $a,b,c$ be positive real numbers satisfying
$$16(a+b+c)\ge\frac1a+\frac1b+\frac1c.$$
Prove that
$$\frac1{(a+b+\sqrt{2(a+c)})^3}+\frac1{(b+c+\sqrt{2(b+a)})^3}+\frac1{(c+a+\sqrt{2(c+b)})^3}\le\frac89.$$
Determine when equality holds.
At a conference there are $n$ countries, each represented by $k$ delegates, where $n>k>1$. The $nk$ delegates are divided into $n$ groups of $k$ people each, with no two people in a group coming from the same country. Prove that one can select $n$ delegates belonging to different groups and coming from different countries.
Let $S_n$ be the sum of the squares of the coefficients in the expansion of $(1+x)^n$, where $n$ is a positive integer and $x$ is any real number. Prove that $S_{2n}+1$ is not divisible by $3$ for any positive integer $n$.
The real sequence $(a_n)$ is defined by
$$a_1=5,\qquad a_n=\sqrt[n]{a_{n-1}^{n-1}+2^{n-1}+2\cdot3^{n-1}}\quad(n\ge2).$$
1. Find the general term of $(a_n)$.
2. Prove that $(a_n)$ is decreasing.
In the plane, let $(O)$ be a circle and let $B,C$ be fixed points on it such that chord $BC$ is not a diameter. Let $A$ vary on $(O)$ with $AB\ne AC$ and $A$ distinct from $B,C$. Let $D,E$ be the intersections of line $BC$ with the internal and external bisectors, respectively, of $\angle BAC$. Let $I$ be the midpoint of $DE$. The line through the orthocenter of triangle $ABC$ perpendicular to $AI$ meets $AD,AE$ at $M,N$, respectively.
1. Prove that line $MN$ always passes through a fixed point.
2. Determine the positions of $A$ for which triangle $AMN$ has greatest area.
Let $n$ be a positive integer. A $3\times3$ grid of square cells is colored using $n$ colors, with each cell receiving one color. Two colorings are considered the same if one is obtained from the other by a rotation about the center of the grid. How many pairwise different colorings are there?
Note: A coloring need not use all $n$ colors.