MathDB

1

Part of 2008 International Zhautykov Olympiad

Problems(2)

Nice geometry problem

Source: Problem 1 from ZIMO 2008

1/21/2008
Points K,L,M,N K,L,M,N are repectively the midpoints of sides AB,BC,CD,DA AB,BC,CD,DA in a convex quadrliateral ABCD ABCD.Line KM KM meets dioganals AC AC and BD BD at points P P and Q Q,respectively.Line LN LN meets dioganals AC AC and BD BD at points R R and S S,respectively. Prove that if AP\cdot PC\equal{}BQ\cdot QD,then AR\cdot RC\equal{}BS\cdot SD.
geometryrhombusgeometry proposed
The sum of digits

Source: Problem 4 from ZIMO 2008

1/21/2008
For each positive integer n n,denote by S(n) S(n) the sum of all digits in decimal representation of n n. Find all positive integers n n,such that n\equal{}2S(n)^3\plus{}8.
inequalitiesmodular arithmeticnumber theory proposednumber theory