MathDB

2

Part of 1997 Ukraine National Mathematical Olympiad

Problems(4)

game

Source: Ukraine 1997

7/18/2009
There are n n candidates on a table. Petrik and Mikola alternately take candies from the table according to the following rule. Petrik starts by taking one candy; then Mikola takes i i candies, where i i divides 2 2, then Petrik takes j j candies, where j j divides 3 3, and so on. The player who takes the last candy wins the game. Which player has a winning strategy?
combinatorics proposedcombinatorics
solve the equation

Source: Ukraine 1997 grade 9

7/18/2009
Solve in real numbers the equation 9^x\plus{}4^x\plus{}1\equal{}6^x\plus{}3^x\plus{}2^x.
inequalitiesfunctionalgebra proposedalgebra
marked points

Source: Ukraine 1997 grade 10

7/21/2009
Each side of an equilateral triangle is divided into 1997 1997 equal parts, and each vertex of the triangle is joined by a segment to each of the division points on the opposite side. All the intersection points of these segments inside the triangle are marked. How many marked points are there?
combinatorics proposedcombinatorics
trigonometry

Source: Ukraine 1997 grade 11

7/22/2009
Prove that among any four distinct numbers from the interval (0,π2) (0,\frac{\pi}{2}) there are two, say x,y, x,y, such that: 8 \cos x \cos y \cos (x\minus{}y)\plus{}1>4(\cos ^2 x\plus{}\cos ^2 y).
trigonometryalgebra unsolvedalgebra