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Weird algebra with combinatorial flavour

Source: Kazakhstan National MO 2023 (10-11).2

March 21, 2023

algebra

Problem Statement

Let

n>100

be an integer. The numbers

1,2 \ldots, 4n

are split into

n

groups of

4

. Prove that there are at least

\frac{(n-6)^2}{2}

quadruples

(a, b, c, d)

such that they are all in different groups,

a<b<c<d

and

c-b \leq |ad-bc|\leq d-a

.

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