4

Part of 2011 Moldova Team Selection Test

Problems(2)

Integer part of an expresion

Source: Moldova TST 2011, day 1, problem 4

n

be an integer satisfying

n\geq2

. Find the greatest integer not exceeding the expression: E=1+\sqrt{1+\frac{2^2}{3!}}+\sqrt[3]{1+\frac{3^2}{4!}}+\dots+\+\sqrt[n]{1+\frac{n^2}{(n+1)!}}

algebra unsolvedalgebra

increase a number by 1 and decrease another number by 1.

Source: Moldova TST 2011, day 2, problem 4

Initially, on the blackboard are written all natural numbers from

1

20

. A move consists of selecting

2

a<b

written on the blackboard such that their difference is at least

2

, erasing these numbers and writting

a+1

b-1

instead. What is the maximum numbers of moves one can perform?

combinatorics unsolvedcombinatorics