VN TST 2011

A grasshopper starts at $(1,1)$ in the coordinate plane. It may jump from $A$ to $B$ when $A,B$ have positive integer coordinates and triangle $OAB$ has area $\tfrac12$, where $O$ is the origin.
1. Determine all points $(m,n)$ it can reach in finitely many jumps.
2. Prove that each such point can be reached in fewer than $|m-n|$ jumps.

Editorial notice: Part 2 above faithfully reproduces a false strict bound in the supplied source. For example, $(2,1)$ requires one jump, not fewer than one. The valid corrected bound is **at most** $|m-n|$ jumps; $(1,1)$ requires zero. The solution proves the corrected statement.

Let $(O)$ be a circle and $A$ a point outside it. Draw tangents $AB,AC$, with points of contact $B,C$. Let $P$ lie on the ray opposite to $BA$ and $Q$ on the ray opposite to $CA$, so that $PQ$ is tangent to $(O)$. The line through $P$ parallel to $AC$ meets line $BC$ at $E$; the line through $Q$ parallel to $AB$ meets line $BC$ at $F$.
1. Prove that $QE$ and $PF$ pass through fixed points $M$ and $N$, respectively.
2. Prove that $PM\cdot QN$ is constant.

Editorial correction: The source accidentally calls $A,B$ the points of contact; these must be $B,C$.

Let $n\ge3$ be an integer and let $x_1,\ldots,x_n$ be real numbers such that
$$\sum_{i=1}^n x_i=0,\qquad \sum_{i=1}^n x_i^2=n(n-1),\qquad x_1\ge x_2\ge\cdots\ge x_n.$$
Find the maximum and minimum of $f=x_1+x_2$.

The sequence $(a_n)$ is defined by $a_0=1$, $a_1=3$ and
$$a_{n+2}=1+\left\lfloor\frac{a_{n+1}^2}{a_n}\right\rfloor\quad(n\ge0).$$
Prove that $a_na_{n+2}-a_{n+1}^2=2^n$ for every integer $n\ge0$.

Find all positive integers $n$ such that
$$2^{n+2}(2^n-1)-8\cdot3^n+1$$
is a perfect square.

There are $n$ students around a round table. Each has some candies, and the total number of candies is divisible by $n$. The following transfer is allowed: if a student has more candies than the student immediately to their right, one candy is transferred from the former to the latter.
Prove that, by a suitable finite sequence of such transfers, all students can be made to have equal numbers of candies.

Editorial clarification: “one candy” is explicit in the source's solution equations but accidentally omitted from its statement.