Romanian TST II 2011 Problem 4

Show that

(a) There are infinitely many positive integers such that there exists a square equal to the sum of the squares of consecutive positive integers (for instance, and are such, since , and ).

(b) If is a positive integer which is not a perfect square, and if is an integer number such that is a perfect square, then there are infinitely many positive integers such that is a perfect square.

Romanian TST 2011

Romanian TST II Problem 3

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

Romanian TST 2011

Romanian TST II 2011 Problem 2

Let be a triangle. The incircle of the triangle touches the side at the point (all indices are considered modulo ). Let be the foot of the perpendicular dropped from the point onto the line . Show that the lines are concurrent at a point situated on the Euler line of the triangle .

Romanian TST 2011

Romanian TST II 2011 Problem 1

A square of sidelength is contained in the unit square whose centre is not interior to the former. Show that .

Romanian TST 2011

Romanian TST 2011 Problem 4

Suppose and denote the permutations of the set . For denote by the number of cycles in the disjoint cycle decomposition of the permutation . Calculate the sum .

Romanian TST 2011 Problem 2

Denote .

Prove that:

a) For any positive integer there exists an arithmetic progression of length in .

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

Romanian TST 2011 Problem 1

Find all functions for which we have .

Romanian TST 2011

Weird sequence

Suppose the sequence is given by and . Prove that the sequence is bounded if and only if it consists only of two values, one of which is .
IMO Shortlist 2004, various Team Selection Tests

Irreductible Polynomial TST 2003

Let be an irreducible polynomial over the ring of integer polynomials, such that is not a perfect square. Prove that if the leading coefficient of is 1 (the coefficient of the term having the highest degree in ) then is also irreducible in the ring of integer polynomials.

Mihai Piticari, Romanian TST 2003

Cover of the unit segment

Suppose that a segment of length is covered by a number of segments. Prove that we can pick some of these segments who are pairwise disjoint and whose sum of lengths is at least .

Sufficient condition for a sequence to be an enumeration of integers

Let be a sequence if integers with infinitely many positive and negative terms. Suppose that for any the numbers have different remainders modulo . Prove that each integer occurs exactly once in the given sequence.
IMO 2005 Problem 2

Positive integers partitioned by translations of the same set

It is possible to partition the set in infinitely many parts each of which having infinitely many elements such that there exists one member of the partition such that all other members of the partiton are translates of the first set with an integer?

Interesting recurrence

Suppose that is a sequence of integers such that for all positive integers . Prove that for all .

IMO 2010 Problem 6

Let be a sequence of positive real numbers, and be a positive integer, such that
.
Prove there exist positive integers and , such that
.
IMO 2010 Problem 6

IMO 2010 Problem 5

Each of the six boxes , , , , , initially contains one coin. The following operations are allowed

Type 1) Choose a non-empty box , , remove one coin from and add two coins to ;

Type 2) Choose a non-empty box , , remove one coin from and swap the contents (maybe empty) of the boxes and .

Determine if there exists a finite sequence of operations of the allowed types, such that the five boxes , , , , become empty, while box contains exactly coins.
IMO 2010 Problem 5

IMO 2010 Problem 4

Let be a point interior to triangle (with ). The lines , and meet again its circumcircle at , , respectively . The tangent line at to meets the line at . Show that from follows .
IMO 2010 Problem 4

Invisible points

We will cal a lattice point invisible if and only if the segment which joins it to the origin contains at least another lattice point. Prove that there are squares of any size whose inside points are all invisible.
Romanian TST 2003

Circles inside a square

Inside a square consider some circles such that the sum of their lengths is twice the perimeter of the square.
a) Find the least number of circles having this property.
b) Prove that there exist infinitely many lines intersecting at least three of the given circles.
SEEMOUS 2010

Sum independent of triangulation

In a circle with center is inscribed a polygon which is triangulated. Show that the sum of the squares of the distances from to the incenters of the formed triangles is independent of the triangulation.
Romanian TST 2007

Variable points in plane

Let be variable points in the plane such that . Find the maximum value of the area of .
Romanian TST 1999