MathDB

3

Part of 2007 Iran MO (3rd Round)

Problems(4)

Largest T

Source: Iranian National Olympiad (3rd Round) 2007

8/27/2007
Find the largest real T T such that for each non-negative real numbers a,b,c,d,e a,b,c,d,e such that a\plus{}b\equal{}c\plus{}d\plus{}e: \sqrt{a^{2}\plus{}b^{2}\plus{}c^{2}\plus{}d^{2}\plus{}e^{2}}\geq T(\sqrt a\plus{}\sqrt b\plus{}\sqrt c\plus{}\sqrt d\plus{}\sqrt e)^{2}
inequalitiesinequalities proposed
AT is perpendicular to BC

Source: Iranian National Olympiad (3rd Round) 2007

8/27/2007
Let I I be incenter of triangle ABC ABC, M M be midpoint of side BC BC, and T T be the intersection point of IM IM with incircle, in such a way that I I is between M M and T T. Prove that \angle BIM\minus{}\angle CIM\equal{}\frac{3}2(\angle B\minus{}\angle C), if and only if AT⊥BC AT\perp BC.
geometryincentercircumcircletrigonometrygeometry proposed
n does not divide 2^{n-1}+1

Source: Iranian National Olympiad (3rd Round) 2007

8/29/2007
Let n n be a natural number, and n \equal{} 2^{2007}k\plus{}1, such that k k is an odd number. Prove that n\not|2^{n\minus{}1}\plus{}1
modular arithmeticnumber theory proposednumber theory
Sets containing circles in plane

Source: Iranian National Olympiad (3rd Round) 2007

9/10/2007
We call a set A A a good set if it has the following properties: 1. A A consists circles in plane. 2. No two element of A A intersect. Let A,B A,B be two good sets. We say A,B A,B are equivalent if we can reach from A A to B B by moving circles in A A, making them bigger or smaller in such a way that during these operations each circle does not intersect with other circles. Let an a_{n} be the number of inequivalent good subsets with n n elements. For example a_{1}\equal{} 1,a_{2}\equal{} 2,a_{3}\equal{} 4,a_{4}\equal{} 9. http://i5.tinypic.com/4r0x81v.png If there exist a,b a,b such that Aan≤an≤Bbn Aa^{n}\leq a_{n}\leq Bb^{n}, we say growth ratio of an a_{n} is larger than a a and is smaller than b b. a) Prove that growth ratio of an a_{n} is larger than 2 and is smaller than 4. b) Find better bounds for upper and lower growth ratio of an a_{n}.
ratiogeometry proposedgeometry