Anzo's Mathematics

My Problem Proposals to JOM

The idea of hosting a Junior Olympiad of Mathematics (JOM) for the junior students in the IMO camp ignited my thoughts of creating problems for the junior students. In particular, I proposed my first problem to the Olympiad after being inspired by IMO 2012, Problem 4: a problem that caused turmoils both among the students during the contests, and among the leaders and coordinators during the coordination sessions. Nevertheless, some beauty of the problem still exists in this IMO problem, which I extracted it and turned it into a problem that made itself into N6 on the JOM shortlist.

Problem 5 of the same Olympiad also inspired me to create another geometry problem, also for the first edition of the Junior Olympiad. Unwilling to succumb to the fact that I didn't have enough time to attempt this problem on the contest (blame problem 4), I later attempted it again and found an elegant solution using poles and polars. That led me to the unexpected discovery of a geometric identity that eventually became problem G2 on the JOM shortlist.

Below are the 10 problems I proposed for the first three editions of JOM. Have fun solving!

JOM 2015, A6. Let $(a_{n})_{n\ge 0}$ and $(b_{n})_{n\ge 0}$ be two sequences with arbitrary real values $a_0, a_1, b_0, b_1$. For $n\ge 1$, let $a_{n+1}, b_{n+1}$ be defined in this way: $$a_{n+1}=\dfrac{b_{n-1}+b_{n}}{2}, b_{n+1}=\dfrac{a_{n-1}+a_{n}}{2}$$ Prove that for any constant $c>0$ there exists a positive integer $N$ s.t. for all $n>N$, $|a_{n}-b_{n}|< c$.

JOM 2014, C6. Let $n$ be a positive integer. At the beginning, two frogs, $A$ and $B$ are at point 0 of a number line. At each second, they jump to the right and stop after arriving at $n$, following the constraints below:
(i) Each of them must jump either 1 or 2 step(s) every second.
(ii) Both of them must arrive at $n$ at the same time.
(iii) Frog $A$ must never be behind $B$ during the jump.
Let $f(n)$ be the number of arrangements of the sequence of jumps for the two frogs. Prove that $f(n)\ge 2^{n-1}$.

JOM 2014, Problem 5 (G5). Given $\triangle ABC$ with circumcircle $\Gamma$ and circumcentre $O$. Let $X$ to be a point on $\Gamma$. Let $XC_1$, $XB_1$ to be feet of perpendiculars from $X$ to lines $AB$ and $AC$. Let $\omega_C$ to be circle with centre the midpoint of $AB$ and passing through $C_1$ . Define $\omega_B$ similarly.
Prove that $\omega_B$ and $\omega_C$ has a common point on $XO$.

JOM 2014, N11. Prove that for all nonzero integers $c$ there exists composite positive integers $a, b$ such that:
(i) $a - b = c$.
(ii) $\gcd(a, b) = \gcd(a, c) = \gcd(b, c) = 1$.

JOM 2013, A5. Find all polynomials $P\in \mathbb{R}[x]$ such that $P(x)+P(y)+P(z)=0\Rightarrow x+y+z=0$.

JOM 2013, Problem 4 (C3). Let $n$ be a positive integer. A pseudo-Gangnam Style is a dance competition between players $A$ and $B$. At time $0$, both players face to the north. For every $k\ge 1$, at time $2k-1$, player $A$ can either choose to stay stationary, or turn $90^{\circ}$ clockwise, and player $B$ is forced to follow him; at time $2k$, player $B$ can either choose to stay stationary, or turn $90^{\circ}$ clockwise, and player $A$ is forced to follow him.
After time $n$, the music stops and the competition is over. If the final position of both players is north or east, $A$ wins. If the final position of both players is south or west, $B$ wins. Determine who has a winning strategy when:
(a) $n=2013^{2012}$;
(b) $n=2013^{2013}$.

JOM 2013, G2. Let $\omega_1$ and $\omega_2$ be two circles, with centres $O_1$ and $O_2$ respectively, intersecting at $X$ and $Y$. Let a line tangent to both $\omega_1$ and $\omega_2$ at $A$ and $B$, respectively. Let $E,F$ be points on $O_1O_2$ such that $XE$ is tangent to $\omega_1$ and $YF$ is tangent to $\omega_2$. Let $AE\cap \omega_1=A, C$ and $BF\cap\omega_2=B,D$.
Show that line $BO_2$ is tangent to the circumcircle of $\triangle ACD$ and line $AO_2$ is tangent to $\triangle BCD$.

JOM 2013, G5. Let $ABCD$ be a convex quadrilateral, with $AD,BC$ intersecting at $F$. Choose lines $\ell_1$, $\ell_2$ such that $\ell_1$ passes through $F$ and is tangent to circle $FAB$, and $\ell_2$ passes through $F$ and is tangent to circle $FCD$. Let $\ell_1, ell_2$ intersect $AC, BD$ at $W, X, Y, Z$. Prove that $A,B,C,D$ are concyclic if and only if $W,X, Y, Z$ are concyclic.

JOM 2013, N5. Let $p$ be an odd prime such that $2^p+1|p^p+1$. Prove that any prime divisor of $2^p+1$ other than $3$ is greater than $6p$.

JOM 2013, N6. Find all functions $f:\mathbb{Z}\to\mathbb{Z}$ such that for all prime $p$ the following condition holds: $$p|ab+bc+ca\Leftrightarrow p|f(a)f(b)+f(b)f(c)+f(c)f(a).$$