1

Part of 1997 Ukraine National Mathematical Olympiad

Problems(4)

which number is greater

Source: Ukraine 1997

Which number is greater:

4^{4^{4^{4^4}}}

5^{5^{5^5}}

logarithmsinequalities proposedinequalities

Source: Ukraine 1997 grade 9

There are four apples on a plate, of weights

600,400,300,

250

grammes. Two pupils take an apple each (one by one) and start eating simultaneously. A pupil can take another apple only after he has eaten the previous apple. It is assumed that they equaly fast (measured in grammes per second). How should the first pupil do in order to eat as large amount as possible?

combinatorics unsolvedcombinatorics

Source: Ukraine 1997 grade 10

Find the maximum value of the function: f(x)\equal{}\frac{x}{x^2\plus{}9}\plus{}\frac{1}{x^2\minus{}6x\plus{}21}\plus{}\cos {2 \pi x}, x>0.

functiontrigonometryalgebra unsolvedalgebra

determine the length

Source: Ukraine 1997 grade 11

ABCD

be a parallelogram with AB\equal{}1. Suppose that

K

is a point on the side

AD

such that KD\equal{}1, \angle ABK\equal{}90^{\circ} and \angle DBK\equal{}30^{\circ}. Determine

AD

geometryparallelogramgeometry proposed