MathDB

7

Part of 2004 IMO Shortlist

Problems(3)

variable point on the line BC

Source: IMO Shortlist 2004 geometry problem G7

6/14/2005
For a given triangle ABC ABC, let X X be a variable point on the line BC BC such that C C lies between B B and X X and the incircles of the triangles ABX ABX and ACX ACX intersect at two distinct points P P and Q. Q. Prove that the line PQ PQ passes through a point independent of X X.
geometryinradiusincenterIMO ShortlistTriangle
triangle with vertices at three of the given points

Source: IMO Shortlist 2004, number theory problem 7

6/14/2005
Let pp be an odd prime and nn a positive integer. In the coordinate plane, eight distinct points with integer coordinates lie on a circle with diameter of length pnp^{n}. Prove that there exists a triangle with vertices at three of the given points such that the squares of its side lengths are integers divisible by pn+1p^{n+1}.
Proposed by Alexander Ivanov, Bulgaria
analytic geometryTrianglecoordinate geometryDivisibilityIMO Shortlist
Most challenging inequality on the ISL 2004

Source: IMO ShortList 2004, algebra problem 7

6/15/2005
Let a1,a2,…,an{a_1,a_2,\dots,a_n} be positive real numbers, n>1{n>1}. Denote by gng_n their geometric mean, and by A1,A2,…,AnA_1,A_2,\dots,A_n the sequence of arithmetic means defined by Ak=a1+a2+⋯+akk,k=1,2,…,n. A_k=\frac{a_1+a_2+\cdots+a_k}{k},\qquad k=1,2,\dots,n. Let GnG_n be the geometric mean of A1,A2,…,AnA_1,A_2,\dots,A_n. Prove the inequality n \root n\of{\frac{G_n}{A_n}}+ \frac{g_n}{G_n}\le n+1 and establish the cases of equality.
Proposed by Finbarr Holland, Ireland
inequalitiesalgebraIMO Shortlistmeann-variable inequality