Question bank APMO 2015

Let $ABC$ be a triangle and $D$ a point on side $BC$. A line through $D$ intersects side $AB$ at $X$ and ray $AC$ at $Y$. The circumcircle of $BXD$ meets the circumcircle $\omega$ of $ABC$ again at $Z\ne B$. Lines $ZD,ZY$ meet $\omega$ again at $V,W$, respectively. Prove $AB=VW$.

Proposed by Warut Suksompong, Thailand.

Newly drawn configuration:



Editorial convention: all named circles and intersections are well-defined, and each second circle intersection is distinct from the first.

Let $S=\{2,3,4,\ldots\}$. Does there exist a function $f:S\to S$ such that
$$f(a)f(b)=f(a^2b^2)\quad\text{for all distinct }a,b\in S?$$

Proposed by Angelo Di Pasquale, Australia.

A sequence of real numbers $a_0,a_1,\ldots$ is good if:

(i) $a_0$ is a positive integer;

(ii) for every nonnegative integer $i$, either $a_{i+1}=2a_i+1$ or $a_{i+1}=a_i/(a_i+2)$;

(iii) $a_k=2014$ for some positive integer $k$.

Find the smallest positive integer $n$ for which a good sequence can have $a_n=2014$.

Proposed by Wang Wei Hua, Hong Kong.

Let $n$ be a positive integer. There are $2n$ distinct lines in the plane, no two parallel, of which $n$ are blue and $n$ red. Let $B$ be the set of points on at least one blue line and $R$ the set of points on at least one red line. Prove that a circle exists meeting $B$ in exactly $2n-1$ points and $R$ also in exactly $2n-1$ points.

Proposed by Pakawut Jiradilok and Warut Suksompong, Thailand.

Example arrangement (new):

Determine all sequences $a_0,a_1,a_2,\ldots$ of positive integers with $a_0\ge2015$ such that for every integer $n\ge1$:

(i) $a_n$ divides $a_{n+2}$;

(ii) $|s_{n+1}-(n+1)a_n|=1$, where
$$s_{n+1}=a_{n+1}-a_n+a_{n-1}-\cdots+(-1)^{n+1}a_0.$$

Proposed by Pakawut Jiradilok and Warut Suksompong, Thailand.