MathDB

1

Part of 2007 Iran MO (3rd Round)

Problems(4)

Roots of a polynomial form a rhombus

Source: Iranian National Olympiad (3rd Round) 2007

8/27/2007
Let a,b a,b be two complex numbers. Prove that roots of z^{4}\plus{}az^{2}\plus{}b form a rhombus with origin as center, if and only if a2b \frac{a^{2}}{b} is a non-positive real number.
algebrapolynomialgeometryrhombusalgebra proposed
Collinear points

Source: Iranian National Olympiad (3rd Round) 2007

8/27/2007
Let ABC ABC, l l and P P be arbitrary triangle, line and point. A′,B′,C′ A',B',C' are reflections of A,B,C A,B,C in point P P. A′′ A'' is a point on B′C′ B'C' such that AA′′∥l AA''\parallel l. B′′,C′′ B'',C'' are defined similarly. Prove that A′′,B′′,C′′ A'',B'',C'' are collinear.
geometrygeometric transformationreflectionparallelogramgeometry proposed
Another reduced residue system

Source: Iranian National Olympiad (3rd Round) 2007

8/29/2007
Let n n be a natural number, such that (n,2(2^{1386}\minus{}1))\equal{}1. Let {a1,a2,…,aφ(n)} \{a_{1},a_{2},\dots,a_{\varphi(n)}\} be a reduced residue system for n n. Prove that: n|a_{1}^{1386}\plus{}a_{2}^{1386}\plus{}\dots\plus{}a_{\varphi(n)}^{1386}
modular arithmeticnumber theory proposednumber theory
A cutting problem

Source: Iranian National Olympiad (3rd Round) 2007

9/10/2007
Consider two polygons P P and Q Q. We want to cut P P into some smaller polygons and put them together in such a way to obtain Q Q. We can translate the pieces but we can not rotate them or reflect them. We call P,Q P,Q equivalent if and only if we can obtain Q Q from P P(which is obviously an equivalence relation). http://i3.tinypic.com/4lrb43k.png a) Let P,Q P,Q be two rectangles with the same area(their sides are not necessarily parallel). Prove that P P and Q Q are equivalent. b) Prove that if two triangles are not translation of each other, they are not equivalent. c) Find a necessary and sufficient condition for polygons P,Q P,Q to be equivalent.
rotationgeometrygeometric transformationreflectionrectanglegeometry proposed