### Archive

Posts Tagged ‘TST’

## Romanian TST II 2011 Problem 4

Show that

(a) There are infinitely many positive integers $n$ such that there exists a square equal to the sum of the squares of $n$ consecutive positive integers (for instance, $2$ and $11$ are such, since $5^2=3^2+4^2$, and $77^2=18^2+19^2+...+28^2$).

(b) If $n$ is a positive integer which is not a perfect square, and if $x_0$ is an integer number such that $x_0^2+(x_0+1)^2+...+(x_0+n-1)^2$ is a perfect square, then there are infinitely many positive integers $x$ such that $x^2+(x+1)^2+...+(x+n-1)^2$ is a perfect square.

Romanian TST 2011

Categories: IMO, Number theory, Olympiad Tags: ,

## Romanian TST II Problem 3

May 31, 2011 1 comment

Given a positive integer $n$, determine the maximum number of edges a simple graph of $n$ edges vertices may have, in order that it won’t contain any cycles of even length.

Romanian TST 2011

Categories: Combinatorics, IMO, Olympiad Tags: ,

## Romanian TST II 2011 Problem 2

Let $A_0A_1A_2$ be a triangle. The incircle of the triangle $A_0A_1A_2$ touches the side $A_iA_{i+1}$ at the point $T_{i+2}$ (all indices are considered modulo $3$). Let $X_i$ be the foot of the perpendicular dropped from the point $T_i$ onto the line $T_{i+1}T_{i+2}$. Show that the lines $A_iX_i$ are concurrent at a point situated on the Euler line of the triangle $T_0T_1T_2$.

Romanian TST 2011

Categories: Geometry, IMO, Olympiad Tags: , ,

## Romanian TST II 2011 Problem 1

A square of sidelength $\ell$ is contained in the unit square whose centre is not interior to the former. Show that $\ell \leq 1/2$.

Romanian TST 2011

Categories: Geometry, IMO, Olympiad Tags: ,

## Romanian TST 2011 Problem 4

Suppose $n \geq 2$ and denote $S_n$ the permutations of the set $\{1,2,..,n\}$. For $\sigma \in S_n$ denote by $\ell(\sigma)$ the number of cycles in the disjoint cycle decomposition of the permutation $\sigma$. Calculate the sum $\displaystyle \sum_{\sigma \in S_n} \text{sgn}(\sigma) n^{\ell(\sigma)}$.

## Romanian TST 2011 Problem 2

Denote $S=\{\lfloor n \pi \rfloor : n \in \Bbb{N} \}$.

Prove that:

a) For any positive integer $m \geq 3$ there exists an arithmetic progression of length $m$ in $S$.

b) Prove that $S$ does not contain a infinite length arithmetic progression.

Categories: Algebra, IMO, Problem Solving

## Romanian TST 2011 Problem 1

April 20, 2011 1 comment

Find all functions $f: \Bbb{R} \to \Bbb{R}$ for which we have $2f(x)=f(x+y)+f(x+2y),\ \forall x \in \Bbb{R},\ \forall y \geq 0$.

Romanian TST 2011