Let $S$ be a string of $99$ characters, $66$ of which are $A$ and $33$ are $B$. We call $S$ good if, for each $n$ such that $1\le n \le 99$, the sub-string made from the first $n$ characters of $S$ has an odd number of distinct permutations. How many good strings are there? Which strings are good?

Let $ABC$ be a triangle, let $H$ be the orthocentre and $L,M,N$ the midpoints of the sides $AB, BC, CA$ respectively. Prove that \[HL^{2} + HM^{2} + HN^{2} < AL^{2} + BM^{2} + CN^{2}\] if and only if $ABC$ is acute-angled.

Find all functions $f : \mathbb{Z} \rightarrow \mathbb{Z}$ such that for all integers $m,n$, \[f(m - n + f(n)) = f(m) + f(n).\]

The circles $\gamma_1$ and $\gamma_2$ intersect at the points $Q$ and $R$ and internally touch a circle $\gamma$ at $A_1$ and $A_2$ respectively. Let $P$ be an arbitrary point on $\gamma$. Segments $PA_1$ and $PA_2$ meet $\gamma_1$ and $\gamma_2$ again at $B_1$ and $B_2$ respectively. a) Prove that the tangent to $\gamma_{1}$ at $B_{1}$ and the tangent to $\gamma_{2}$ at $B_{2}$ are parallel. b) Prove that $B_{1}B_{2}$ is the common tangent to $\gamma_{1}$ and $\gamma_{2}$ iff $P$ lies on $QR$.

Click for solution Let the centres of the circles be $O_1$, $O_2$ and $O$ respectively. Then: (a). Connect $O_1B_1$, $O_2B_2$ and $PO$. Then $O_1B_1 || PO || O_2B_2$, which means the tangents at $B_1$ and $B_2$ are also paralel. (b). Since P lies on QR, so $PB_1 \cdot PA_1 = PB_2 \cdot PA_2$. Thus $A_1, A_2, B_2, B_1$ lies on the same circle. Therefore $\angle{PB_2B_1}+\angle{O_2B_2A_2}=\angle{B_1A_1A_2}+\angle{O_2A_2B_2}=$ $=\angle{O_2A_2B_2}+\angle{OA_2A_1}+\angle{OA_1B_1}=\angle{PB_1B_2}+\angle{O_1B_1A_1}$ Thus $\angle{B_1B_2O_2}=\angle{B_2B_1O_1}$. But by the conclusion of the part (a) we have that those two angles have the sum of $180^o$. So there must be $\angle{B_1B_2O_2}=\angle{B_2B_1O_1}=90^o$, which yields $B_1B_2$ is the common tangent of $\gamma_1$ and $\gamma_2$.

Let $n$ be a positive integer, and let $A_{n}$ be the the set of all positive integers $a\le n$ such that $n|a^{n}+1$. a) Find all $n$ such that $A_{n}\neq \emptyset$ b) Find all $n$ such that $|{A_{n}}|$ is even and non-zero. c) Is there $n$ such that $|{A_{n}}| = 130$?

Let $P(x)$ be a polynomial with complex coefficients such that $P(0)\neq 0$. Prove that there exists a multiple of $P(x)$ with real positive coefficients if and only if $P(x)$ has no real positive root.