MathDB

1

Part of 2007 Iran Team Selection Test

Problems(4)

A computational geometry, midpoint of a side

Source: Iran TST 2007, Day 1

5/7/2007
In triangle ABCABC, MM is midpoint of ACAC, and DD is a point on BCBC such that DB=DMDB=DM. We know that 2BC2−AC2=AB.AC2BC^{2}-AC^{2}=AB.AC. Prove that BD.DC=AC2.AB2(AB+AC)BD.DC=\frac{AC^{2}.AB}{2(AB+AC)}
geometrytrigonometrycircumcircleangle bisectorgeometry proposed
An isosceles right-angled billiards table

Source: Iran TST 2007, Day 2

5/7/2007
In an isosceles right-angled triangle shaped billiards table , a ball starts moving from one of the vertices adjacent to hypotenuse. When it reaches to one side then it will reflect its path. Prove that if we reach to a vertex then it is not the vertex at initial position By Sam Nariman
geometrygeometric transformationreflectionanalytic geometrygeometry proposed
Irreducible sequence

Source: Iran TST 2007, Day 3

5/20/2007
Does there exist a a sequence a0,a1,a2,…a_{0},a_{1},a_{2},\dots in N\mathbb N, such that for each i≠j,(ai,aj)=1i\neq j, (a_{i},a_{j})=1, and for each nn, the polynomial ∑i=0naixi\sum_{i=0}^{n}a_{i}x^{i} is irreducible in Z[x]\mathbb Z[x]? By Omid Hatami
algebrapolynomialnumber theory proposednumber theory
Polynomial with an inequality condition

Source: Iran TST 2007, Day 4

5/28/2007
Find all polynomials of degree 3, such that for each x,y≥0x,y\geq 0: p(x+y)≥p(x)+p(y)p(x+y)\geq p(x)+p(y)
algebrapolynomialinequalitiesalgebra proposed