MathDB

1

Part of 2014 Moldova Team Selection Test

Problems(3)

Moldova TST, equation in positive integers

Source: Moldova TST 2014, First Day, Problem 1

3/4/2014
Find all pairs of non-negative integers (x,y)(x,y) such that x+y−x−y+2=0.\sqrt{x+y}-\sqrt{x}-\sqrt{y}+2=0.
quadraticsnumber theory proposednumber theory
Sequence inequality

Source: Moldova TST 2014, Second Day, Problem 1

3/30/2014
Consider n≥2n \geq 2 positive numbers 0<x1≤x2≤...≤xn0<x_1 \leq x_2 \leq ... \leq x_n, such that x1+x2+...+xn=1x_1 + x_2 + ... + x_n = 1. Prove that if xn≤23x_n \leq \dfrac{2}{3}, then there exists a positive integer 1≤k≤n1 \leq k \leq n such that 13≤x1+x2+...+xk<23\dfrac{1}{3} \leq x_1+x_2+...+x_k < \dfrac{2}{3}.
inequalitiesalgebra proposedalgebra
Distance of 4 points

Source: Moldova TST 2014, Third Day, Problem 1

3/31/2014
Prove that there do not exist 44 points in the plane such that the distances between any pair of them is an odd integer.
geometry3D geometrytetrahedronnumber theory proposednumber theory