In the triangle $ABC$, the excircle opposite to the vertex $A$ with centre at $I$ touches the side $BC$ at $D$. (The circle also touches the sides $AB$, $AC$ extended.) Let $M$ be the midpoint of $BC$ and $N$ the midpoint of $AD$. Prove that $I,M,N$ are collinear.
Find, with justification, all positive real numbers $a,b,c$ satisfying the system of equations: $$a\sqrt b=a+c,\qquad b\sqrt c=b+a,\qquad c\sqrt a=c+b.$$
Some blue and red circular disks of identical size are packed together to form a triangle. The top level has one disk and each level has 1 more disk tan the level above it. Each disk not at the bottom level touches two disks below it and its colour is blue if these two disks are of the same colour. Otherwise its colour is red.
Suppose the bottom level has 2048 disks of which 2014 are red, What is the colour of the disk at the top?
For each positive integer $n$, let $$x_n=p_1+\cdots+p_n$$ where $p_1,\ldots,p_n$ are the first few $n$ primes. Prove that for each positive integer $n$, there is an integer $k_n$ such that $x_n\lt k_n^2\lt x_{n+1}$.
Alice and Bob play a number game. Starting with a positive integer $n$, they take turns changing the numvber with Alice going first. Each player may change the current number $k$ to either $k-1$ or $\lceil k/2\rceil$. The person who changes 1 to 0 wins. Determine all $n$ such that Alice has a winnin strategy. (For each real number $x$, its ceiling $\lceil x\rceil$ is the smallest intger $\ge x$.