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Part of 1996 Estonia National Olympiad

Problems(4)

hat fell from a fisherman , whole sailing in a rowing boat , in a river

Source: 1996 Estonia National Olympiad Final Round grade 9 p1

A fisherman, who was sailing in a rowing boat against the current of the river, had a hat falling from the bow of the boat into the water. After half an hour, the fisherman noticed the loss of his cap and immediately turned back. Find the speed of the river if the fisherman caught up with the cap at a distance of

a

km from the place where it fell into the water (the speed of the river and the movement of the boat relative to the water is considered constant).

19/x+ 96/ y = (19/x) (96 y) diophantine , sum = product

Source: 1996 Estonia National Olympiad Final Round grade 10 p1

Find all pairs of integers

(x, y)

such that ths sum of the fractions

\frac{19}{x}

\frac{96}{y}

would be equal to their product.

SumProductDiophantine equationdiophantinenumber theory

x^xy^y >= x^yy^x for any x,y>0

Source: 1996 Estonia National Olympiad Final Round grade 11 p1

Prove that for any positive numbers

x,y

x^xy^y \ge x^yy^x

inequalitiesalgebra

p(x-y) = xy diophantine with prime

Source: 1996 Estonia National Olympiad Final Round grade 12 p1

p

be a fixed prime. Find all pairs

(x,y)

of positive numbers satisfying

p(x-y) = xy

primeDiophantine equationdiophantinenumber theory