MathDB

2

Part of 2007 IMO Shortlist

Problems(3)

f(m + n) >= f(m) + f(f(n)) - 1

Source: IMO Shortlist 2007, A2, AIMO 2008, TST 2, P1, Ukrainian TST 2008 Problem 8

7/13/2008
Consider those functions f:N↦N f: \mathbb{N} \mapsto \mathbb{N} which satisfy the condition f(m \plus{} n) \geq f(m) \plus{} f(f(n)) \minus{} 1 for all m,n∈N. m,n \in \mathbb{N}. Find all possible values of f(2007). f(2007).
Author: Nikolai Nikolov, Bulgaria
algebraFunctional inequalityIMO Shortlist
MTB - CTM does not depend on choice of X

Source: ISL 2007, G2, AIMO 2008, TST 1, P3, Ukrainian TST 2008 Problem 1

6/3/2008
Denote by M M midpoint of side BC BC in an isosceles triangle △ABC \triangle ABC with AC=AB AC = AB. Take a point X X on a smaller arc \overarc{MA} of circumcircle of triangle △ABM \triangle ABM. Denote by T T point inside of angle BMA BMA such that ∠TMX=90 \angle TMX = 90 and TX=BX TX = BX.
Prove that ∠MTB−∠CTM \angle MTB - \angle CTM does not depend on choice of X X.
Author: Farzan Barekat, Canada
geometrycircumcirclereflectioncyclic quadrilateralIMO Shortlistgeometry solvedmixtilinear incircle
Integer a_k such that b - a^n_k is divisible by k

Source: IMO Shortlist 2007, N2, Ukrainian TST 2008 Problem 10

7/13/2008
Let b,n>1b,n > 1 be integers. Suppose that for each k>1k > 1 there exists an integer aka_k such that b−aknb - a^n_k is divisible by kk. Prove that b=Anb = A^n for some integer AA.
Author: Dan Brown, Canada
modular arithmeticnumber theoryDivisibilityIMO ShortlistHi