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Part of 2014 Tuymaada Olympiad

Problems(2)

g(a,b,c)\ge \sqrt{2abc}

Source: Tuymaada 2014, Day 2, Problem 4, Senior League

Let positive integers

a,\ b,\ c

be pairwise coprime. Denote by

g(a, b, c)

the maximum integer not representable in the form

xa+yb+zc

with positive integral

x,\ y,\ z

g(a, b, c)\ge \sqrt{2abc}

(M. Ivanov)
[hide="Remarks (containing spoilers!)"] 1. It can be proven that

g(a,b,c)\ge \sqrt{3abc}

. 2. The constant

3

is the best possible, as proved by the equation

g(3,3k+1,3k+2)=9k+5

calculusintegrationanalytic geometrygeometrymodular arithmeticnumber theoryTuymaada

Islands and numbered feriboats

Source: Tuymaada 2014, Day 2, Problem 4, Junior League

m

villages on the left bank of the Lena,

n

villages on the right bank and one village on an island. It is known that

(m+1,n+1)>1

. Every two villages separated by water are connected by ferry with positive integral number. The inhabitants of each village say that all the ferries operating in their village have different numbers and these numbers form a segment of the series of the integers. Prove that at least some of them are wrong.
(K. Kokhas)

combinatorics unsolvedcombinatoricsTuymaada