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Difference of descending and ascending triples

Source: Romania 3rd JBMO TST 2023 P2

May 20, 2023

combinatorics

Problem Statement

Given is a positive integer

n \geq 2

and three pairwise disjoint sets

A, B, C

, each of

n

distinct real numbers. Denote by

a

the number of triples

(x, y, z) \in A \times B \times C

satisfying

x<y<z

and let

b

denote the number of triples

(x, y, z) \in A \times B \times C

such that

x>y>z

. Prove that

n

divides

a-b

.

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