Question bank APMO 2013

Let $ABC$ be an acute triangle with altitudes $AD,BE,CF$, and let $O$ be its circumcenter. Show that the segments $OA,OF,OB,OD,OC,OE$ dissect $ABC$ into three pairs of triangles with equal areas.

Newly drawn configuration:

Determine all positive integers $n$ for which $\frac{n^2+1}{\lfloor\sqrt n\rfloor^2+2}$ is an integer. Here $\lfloor r\rfloor$ is the greatest integer not exceeding $r$.

For real numbers $a_1,\ldots,a_k,b_1,\ldots,b_k$, define
$$X_n=\sum_{i=1}^k\lfloor a_i n+b_i\rfloor\qquad(n=1,2,\ldots).$$
If $(X_n)$ is an arithmetic progression, prove that $\sum_{i=1}^k a_i$ is an integer. Here $\lfloor r\rfloor$ is the greatest integer not exceeding $r$.

Let $a,b$ be positive integers and $A,B$ finite sets of integers such that:

(i) $A$ and $B$ are disjoint;

(ii) if $i\in A\cup B$, then $i+a\in A$ or $i-b\in B$.

Prove $a|A|=b|B|$, where $|X|$ is the number of elements of $X$.

Let $ABCD$ be a quadrilateral inscribed in a circle $\omega$, and let $P$ lie on the extension of $AC$ such that $PB,PD$ are tangent to $\omega$. The tangent at $C$ meets $PD$ at $Q$ and line $AD$ at $R$. Let $E$ be the second intersection of $AQ$ with $\omega$. Prove that $B,E,R$ are collinear.

Newly drawn configuration: