MathDB

3

Part of 2008 IMO Shortlist

Problems(4)

IMO ShortList 2008, Algebra problem 3

Source: IMO ShortList 2008, Algebra problem 3, German TST 5, P2, 2009

7/9/2009
Let S⊆R S\subseteq\mathbb{R} be a set of real numbers. We say that a pair (f,g) (f, g) of functions from S S into S S is a Spanish Couple on S S, if they satisfy the following conditions: (i) Both functions are strictly increasing, i.e. f(x)<f(y) f(x) < f(y) and g(x)<g(y) g(x) < g(y) for all x x, y∈S y\in S with x<y x < y; (ii) The inequality f(g(g(x)))<g(f(x)) f\left(g\left(g\left(x\right)\right)\right) < g\left(f\left(x\right)\right) holds for all x∈S x\in S. Decide whether there exists a Spanish Couple [*] on the set S \equal{} \mathbb{N} of positive integers; [*] on the set S \equal{} \{a \minus{} \frac {1}{b}: a, b\in\mathbb{N}\} Proposed by Hans Zantema, Netherlands
functionalgebraFunctional inequalityIMO Shortlist
IMO ShortList 2008, Combinatorics problem 3

Source: IMO ShortList 2008, Combinatorics problem 3, German TST 6, P1, 2009

7/9/2009
In the coordinate plane consider the set S S of all points with integer coordinates. For a positive integer k k, two distinct points AA, B∈S B\in S will be called k k-friends if there is a point C∈S C\in S such that the area of the triangle ABC ABC is equal to k k. A set T⊂S T\subset S will be called k k-clique if every two points in T T are k k-friends. Find the least positive integer k k for which there exits a k k-clique with more than 200 elements.
Proposed by Jorge Tipe, Peru
geometryIMO ShortlistcombinatoricsExtremal combinatoricspoint setpigenhole principlebezout s identity
IMO Shortlist 2008, Geometry problem 3

Source: IMO Shortlist 2008, Geometry problem 3

7/9/2009
Let ABCD ABCD be a convex quadrilateral and let P P and Q Q be points in ABCD ABCD such that PQDA PQDA and QPBC QPBC are cyclic quadrilaterals. Suppose that there exists a point E E on the line segment PQ PQ such that \angle PAE \equal{} \angle QDE and \angle PBE \equal{} \angle QCE. Show that the quadrilateral ABCD ABCD is cyclic. Proposed by John Cuya, Peru
geometrycircumcirclehomothetytrigonometryquadrilateralIMO ShortlistInversion
IMO ShortList 2008, Number Theory problem 3

Source: IMO ShortList 2008, Number Theory problem 3

7/9/2009
Let a0 a_0, a1 a_1, a2 a_2, … \ldots be a sequence of positive integers such that the greatest common divisor of any two consecutive terms is greater than the preceding term; in symbols, \gcd (a_i, a_{i \plus{} 1}) > a_{i \minus{} 1}. Prove that an≥2n a_n\ge 2^n for all n≥0 n\ge 0. Proposed by Morteza Saghafian, Iran
greatest common divisornumber theorySequenceIMO Shortlist