VN-TST 2011

A grasshopper rests on the point $(1,1)$ on the plane. Denote by $O,$ the origin of coordinates. From that point, it jumps to a certain lattice point under the condition that, if it jumps from a point $A$ to $B,$ then the area of $\triangle AOB$ is equal to $\frac 12.$ $(a)$ Find all the positive integral poijnts $(m,n)$ which can be covered by the grasshopper after a finite number of steps, starting from $(1,1).$ $(b)$ If a point $(m,n)$ satisfies the above condition, then show that there exists a certain path for the grasshopper to reach $(m,n)$ from $(1,1)$ such that the number of jumps does not exceed $|m-n|.$

$A$ is a point lying outside a circle $(O)$ . The tangents from $A$ drawn to $(O)$ meet the circle at $B,C.$ Let $P,Q$ be points on the rays $AB, AC$ respectively such that $PQ$ is tangent to $(O).$ The parallel lines drawn through $P,Q$ parallel to $CA, BA,$ respectively meet $BC$ at $E,F,$ respectively. $(a)$ Show that the straight lines $EQ$ always pass through a fixed point $M,$ and $FP$ always pass through a fixed point $N.$ $(b)$ Show that $PM\cdot QN$ is constant.

Let $n$ be a positive integer $\geq 3.$ There are $n$ real numbers $x_1,x_2,\cdots x_n$ that satisfy: $$\left\{\begin{aligned}&\ x_1\ge x_2\ge\cdots \ge x_n;& \ x_1+x_2+\cdots+x_n=0;& \ x_1^2+x_2^2+\cdots+x_n^2=n(n-1).\end{aligned}\right.$$ Find the maximum and minimum value of the sum $S=x_1+x_2.$

Let $\langle a_n\rangle_{n\ge 0}$ be a sequence of integers satisfying $a_0=1, a_1=3$ and $a_{n+2}=1+\left\lfloor \frac{a_{n+1}^2}{a_n}\right\rfloor \ \ \forall n\ge0.$ Prove that $a_n\cdot a_{n+2}-a_{n+1}^2=2^n$ for every natural number $n.$

Find all positive integers $n$ such that $A=2^{n+2}(2^n-1)-8\cdot 3^n +1$ is a perfect square.

Let $n$ be an integer greater than $1.$ $n$ pupils are seated around a round table, each having a certain number of candies (it is possible that some pupils don't have a candy) such that the sum of all the candies they possess is a multiple of $n.$ They exchange their candies as follows: For each student's candies at first, there is at least a student who has more candies than the student sitting to his/her right side, in which case, the student on the right side is given a candy by that student. After a round of exchanging, if there is at least a student who has candies greater than the right side student, then he/she will give a candy to the next student sitting to his/her right side. Prove that after the exchange of candies is completed (ie, when it reaches equilibrium), all students have the same number of candies.