MathDB

4

Part of 2007 Pre-Preparation Course Examination

Problems(3)

Prove that a=0 - Iran 3rd round-Number Theory 2007

Source:

7/28/2010
a,b∈Za,b \in \mathbb Z and for every n∈N0n \in \mathbb{N}_0, the number 2na+b2^na+b is a perfect square. Prove that a=0a=0.
number theory unsolvednumber theory
Constant point

Source: Iran PPCE 2007

3/23/2007
Let (C)(C) and (L)(L) be a circle and a line. P1,…,P2n+1P_{1},\dots,P_{2n+1} are odd number of points on (L)(L). A1A_{1} is an arbitrary point on (C)(C). Ak+1A_{k+1} is the intersection point of AkPkA_{k}P_{k} and (C)(C) (1≤k≤2n+11\leq k\leq 2n+1). Prove that A1A2n+2A_{1}A_{2n+2} passes through a constant point while A1A_{1} varies on (C)(C).
geometry proposedgeometry
Sum

Source: Iran PPCE 2007

3/25/2007
Prove that ∑i=−20072007∣i+1∣(2)∣i∣>∑i=−20072007∣i∣(2)∣i∣\sum_{i=-2007}^{2007}\frac{\sqrt{|i+1|}}{(\sqrt2)^{|i|}}>\sum_{i=-2007}^{2007}\frac{\sqrt{|i|}}{(\sqrt2)^{|i|}}
inequalities proposedinequalities