1

Part of 2009 ITAMO

Problems(2)

Determine all values of 'e'

Source: ItaMO 2009, p1

a < b < c < d < e

be real numbers. We calculate all possible sums in pairs of these 5 numbers. Of these 10 sums, the three smaller ones are 32, 36, 37, while the two larger ones are 48 and 51. Determine all possible values that

e

number theory proposednumber theory

A flea jumps n times from origin

Source:

A flea is initially at the point

(0, 0)

in the Cartesian plane. Then it makes

n

jumps. The direction of the jump is taken in a choice of the four cardinal directions. The first step is of length

1

, the second of length

2

, the third of length

4

, and so on. The

n^{th}

-jump is of length

2^{n-1}

. Prove that, if you know the final position flea, then it is possible to uniquely determine its position after each of the

n

combinatorics unsolvedcombinatorics