TH-TMO 2011

Given a natural number $n$ $\geq 3$ . If $p,q$ are primes, such that, $p \mid n!$ and $q \mid (n-1)!-1$ . Prove that, $p<q$

There are $900$ students in an International School. There are $59$ international boys and $59$ international girls. The Students are partitioned into $30$ classrooms (each classrooms have equal number of student) and in each of the classrooms, the student will labelled number from $1$ to $30$ . The Partition must satisfy at least one follow condition: [list=i] [] Any Two international boys in same classroom can't have consecutive numbers. [] For every classroom, the student who is labelled $1$ must be a boy. [/list] Prove that there are $2$ classrooms, each of which has $2$ international boys with their labels difference equal.

For any $0\leq x_1,x_2,\ldots,x_{2011} \leq 1$ , Find the maximum value of \begin{align*} \sum_{k=1}^{2011}(x_k-m)^2 \end{align*} where $m$ is the arithmetic mean of $x_1,x_2,\ldots,x_{2011}$ .

Let $a,b,c,d \in \mathbb{R}^+$ and suppose that all roots of the equation \begin{align*} x^5-ax^4+bx^3-cx^2+dx=1 \end{align*} are real. Prove \begin{align*} \frac{1}{a} + \frac{1}{b} + \frac{1}{c} + \frac{1}{d} \le \frac{3}{5} \end{align*}

Prove that, for all $n \in \mathbb{N}$ \begin{align*} \frac{1}{1}+\frac{1}{3}+\frac{1}{5}+\ldots+\frac{1}{2n+1} \not\in \mathbb{Z} \end{align*}

Does there exists a function $f : \mathbb{N} \longrightarrow \mathbb{N}$ \begin{align*} f \left( m+ f(n) \right) = f(m) +f(n) + f(n+1) \end{align*} for all $m,n \in \mathbb{N}$ ?

$7662$ chairs are placed in a circle around the city of Chiang Mai. They are also marked with a label for either $1$ st, $2$ nd, or $3$ rd grade students, so that there are $2554$ chairs labeled with each label. The following situations happen, in order [list=i] [] $2554$ students each from the $1$ st, $2$ nd, and $3$ rd grades are given a ball as follows: $1$ st grade students receive footballs, $2$ nd grade students receive basketballs, and $3$ rd grade students receive volleyballs. [] The students go sit in chairs labeled for their grade [*] The students simultaneously send their balls to the student to their left, and this happens some positive number of times. [/list] A labelling of the chairs is called lin-ping if it is possible for all $1$ st, $2$ nd, and $3$ rd grade students to now hold volleyballs, footballs, and basketballs respectively. Compute the number of lin-ping labellings