MathDB

2

Part of 2004 IMO Shortlist

Problems(4)

A circle, many lines and points - ref. of G in AB lies on CF

Source: ISL 2004; Greek TST 2005; Moldova TST 2005; etc.

1/19/2005
Let Γ\Gamma be a circle and let dd be a line such that Γ\Gamma and dd have no common points. Further, let ABAB be a diameter of the circle Γ\Gamma; assume that this diameter ABAB is perpendicular to the line dd, and the point BB is nearer to the line dd than the point AA. Let CC be an arbitrary point on the circle Γ\Gamma, different from the points AA and BB. Let DD be the point of intersection of the lines ACAC and dd. One of the two tangents from the point DD to the circle Γ\Gamma touches this circle Γ\Gamma at a point EE; hereby, we assume that the points BB and EE lie in the same halfplane with respect to the line ACAC. Denote by FF the point of intersection of the lines BEBE and dd. Let the line AFAF intersect the circle Γ\Gamma at a point GG, different from AA. Prove that the reflection of the point GG in the line ABAB lies on the line CFCF.
geometryreflectioncircumcircleIMO Shortlist
Multiplicative function

Source: IMO Shortlist 2004, number theory problem 2

3/23/2005
The function ff from the set N\mathbb{N} of positive integers into itself is defined by the equality f(n)=∑k=1ngcd⁡(k,n),n∈N.f(n)=\sum_{k=1}^{n} \gcd(k,n),\qquad n\in \mathbb{N}. a) Prove that f(mn)=f(m)f(n)f(mn)=f(m)f(n) for every two relatively prime m,n∈N{m,n\in\mathbb{N}}.
b) Prove that for each a∈Na\in\mathbb{N} the equation f(x)=axf(x)=ax has a solution.
c) Find all a∈N{a\in\mathbb{N}} such that the equation f(x)=axf(x)=ax has a unique solution.
functionnumber theorygreatest common divisorequationIMO Shortlist
Can this sequence be bounded?

Source: German pre-TST 2005, problem 4, ISL 2004, algebra problem 2

1/19/2005
Let a0a_0, a1a_1, a2a_2, ... be an infinite sequence of real numbers satisfying the equation an=∣an+1−an+2∣a_n=\left|a_{n+1}-a_{n+2}\right| for all n≥0n\geq 0, where a0a_0 and a1a_1 are two different positive reals.
Can this sequence a0a_0, a1a_1, a2a_2, ... be bounded?
Proposed by Mihai Bălună, Romania
IMO ShortlistalgebraSequencebounded
Colored circle intersections, german pre-tst 2005, problem 6

Source: IMO ShortList 2004, combinatorics problem 2

1/19/2005
Let n{n} and kk be positive integers. There are given n{n} circles in the plane. Every two of them intersect at two distinct points, and all points of intersection they determine are pairwise distinct (i. e. no three circles have a common point). No three circles have a point in common. Each intersection point must be colored with one of nn distinct colors so that each color is used at least once and exactly kk distinct colors occur on each circle. Find all values of n≥2n\geq 2 and kk for which such a coloring is possible.
Proposed by Horst Sewerin, Germany
geometrymodular arithmeticcombinatoricscirclesIntersectionIMO Shortlistdouble counting