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Some kind of pigeonhole principle

Source: Science ON 2021 grade VI/3

March 8, 2021
Setspiepigeonhole principlecombinatorics

Problem Statement

Consider positive integers a<ba<b and the set C⊂{a,a+1,a+2,…,b−2,b−1,b}C\subset\{a,a+1,a+2,\dots ,b-2,b-1,b\}. Suppose CC has more than b−a+12\frac{b-a+1}{2} elements. Prove that there are two elements x,y∈Cx,y\in C that satisfy x+y=a+bx+y=a+b.
(From "Radu Păun" contest, Radu Miculescu)
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