MathDB

2

Part of 2007 Iran Team Selection Test

Problems(4)

Maximum subset wth no element divinding two other elements

Source: Iran TST 2007, Day 1

5/6/2007
Let AA be the largest subset of {1,…,n}\{1,\dots,n\} such that for each x∈Ax\in A, xx divides at most one other element in AA. Prove that 2n3≤∣A∣≤⌈3n4⌉.\frac{2n}3\leq |A|\leq \left\lceil \frac{3n}4\right\rceil.
ceiling functionpigeonhole principlemodular arithmeticsearchcombinatorics proposedcombinatorics
A polynomial that its image is closed under multiplication

Source: Iran TST 2007, Day 2

5/7/2007
Find all monic polynomials f(x)f(x) in Z[x]\mathbb Z[x] such that f(Z)f(\mathbb Z) is closed under multiplication. By Mohsen Jamali
algebrapolynomialsearchnumber theory proposednumber theory
An iscoceles triangle

Source: Iran TST 2007, Day 4

5/28/2007
Triangle ABCABC is isosceles (AB=ACAB=AC). From AA, we draw a line ℓ\ell parallel to BCBC. P,QP,Q are on perpendicular bisectors of AB,ACAB,AC such that PQ⊥BCPQ\perp BC. M,NM,N are points on ℓ\ell such that angles ∠APM\angle APM and ∠AQN\angle AQN are π2\frac\pi2. Prove that 1AM+1AN≤2AB\frac{1}{AM}+\frac1{AN}\leq\frac2{AB}
analytic geometrygeometrytrigonometryinequalitiescircumcirclegeometry proposed
Angles between lines

Source: Iran TST 2007, Day 3

5/23/2007
Suppose nn lines in plane are such that no two are parallel and no three are concurrent. For each two lines their angle is a real number in [0,π2][0,\frac{\pi}2]. Find the largest value of the sum of the (n2)\binom n2 angles between line. By Aliakbar Daemi
combinatorics proposedcombinatorics