MathDB

1997 Irish Math Olympiad

Part of Ireland National Math Olympiad

Subcontests

(5)
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hard exercise

Let S S be the set of odd integers greater than 1 1. For each x∈S x \in S, denote by δ(x) \delta (x) the unique integer satisfying the inequality 2^{\delta (x)}a,b∈S a,b \in S, define: a \ast b\equal{}2^{\delta (a)\minus{}1} (b\minus{}3)\plus{}a. Prove that if a,b,c∈S a,b,c \in S, then: (a) (a) a∗b∈S a \ast b \in S and (b) (b) (a \ast b)\ast c\equal{}a \ast (b \ast c).

p-partitionable

Let p p be an odd prime number and n n a natural number. Then n n is called p\minus{}partitionable if T\equal{}\{1,2,...,n \} can be partitioned into (disjoint) subsets T1,T2,...,Tp T_1,T_2,...,T_p with equal sums of elements. For example, 6 6 is 3 3-partitionable since we can take T_1\equal{}\{ 1,6 \}, T_2\equal{}\{ 2,5 \} and T_3\equal{}\{ 3,4 \}. (a) (a) Suppose that n n is p p-partitionable. Prove that p p divides n n or n\plus{}1. (b) (b) Suppose that n n is divisible by 2p 2p. Prove that n n is p p-partitionable.
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