MathDB

2

Part of 2010 Tuymaada Olympiad

Problems(4)

Tuymaada 2010, Junior League, Problem 2

Source:

7/18/2010
Let ABCABC be an acute triangle, HH its orthocentre, DD a point on the side [BC][BC], and PP a point such that ADPHADPH is a parallelogram. Show that ∠BPC>∠BAC\angle BPC > \angle BAC.
geometrycircumcirclegeometric transformationreflectionparallelogramcalculustrigonometry
Tuymaada 2010, Junior League, Problem 6

Source:

7/18/2010
We have a number nn for which we can find 5 consecutive numbers, none of which is divisible by nn, but their product is. Show that we can find 4 consecutive numbers, none of which is divisible by nn, but their product is.
modular arithmeticnumber theory unsolvednumber theory
Angle inequality in acute triangle with orthocenter

Source: XVII Tuymaada Mathematical Olympiad (2010), Senior Level

7/31/2011
In acute triangle ABCABC, let HH denote its orthocenter and let DD be a point on side BCBC. Let PP be the point so that ADPHADPH is a parallelogram. Prove that ∠DCP<∠BHP\angle DCP<\angle BHP.
inequalitiesgeometrycircumcircle
Consecutive integers, none divisible by n, whose product is

Source: XVII Tuymaada Mathematical Olympiad (2010), Senior Level

7/31/2011
For a given positive integer nn, it's known that there exist 20102010 consecutive positive integers such that none of them is divisible by nn but their product is divisible by nn. Prove that there exist 20042004 consecutive positive integers such that none of them is divisible by nn but their product is divisible by nn.
number theory unsolvednumber theory