4

Part of 2005 Ukraine National Mathematical Olympiad

Problems(4)

Division of Coins

Source:

Mykolka the numismatist possesses

241

coins, each worth an integer number of turgiks. The total value of the coins is

360

turgiks. Is is necessarily true that the coins can be divided into three groups of equal total value ? :maybe:

combinatorics unsolvedcombinatorics

Source: Ukraine 2005 grade 8

For which real numbers

x>1

is there a triangle with side lengths x^4\plus{}x^3\plus{}2x^2\plus{}x\plus{}1, 2x^3\plus{}x^2\plus{}2x\plus{}1, and x^4\minus{}1?

geometry proposedgeometry

Source: Ukraine 2005 grade 10

P

Q

do not lie on the diagonal

AC

of a parallelogram

ABCD

and satisfy \angle ABP\equal{}\angle ADP, \angle CBQ\equal{}\angle CDQ. Prove that \angle PAQ\equal{}\angle PCQ.

geometryparallelogramgeometry proposed

find all functions

Source: Ukraine 2005 grade 11

Find all monotone (not necessarily strictly) functions f: \mathbb{R}^{\plus{}}_0\rightarrow \mathbb{R} such that: f(x\plus{}y)\minus{}f(x)\minus{}f(y)\equal{}f(xy\plus{}1)\minus{}f(xy)\minus{}f(1) \forall x,y \ge 0; f(3)\plus{}3f(1)\equal{}3f(2)\plus{}f(0).

functionvectormodular arithmeticalgebra unsolvedalgebra