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Romanian TST II 2011 Problem 4

May 31, 2011 2 comments

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

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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

May 31, 2011 Leave a comment

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

May 31, 2011 Leave a comment

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

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Categories: Geometry, IMO, Olympiad Tags: ,

Romanian TST 2011 Problem 4

April 29, 2011 Leave a comment

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)}.

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Romanian TST 2011 Problem 2

April 21, 2011 9 comments

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.

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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

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Categories: Algebra, Olympiad Tags: ,
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