MathDB

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Part of 2021 European Mathematical Cup

Problems(2)

10th EMC - Balanced Quadruples

Source: 10th European Mathematical Cup - Problem J1

12/22/2021
We say that a quadruple of nonnegative real numbers (a,b,c,d)(a,b,c,d) is balanced if a+b+c+d=a2+b2+c2+d2.a+b+c+d=a^2+b^2+c^2+d^2. Find all positive real numbers xx such that (x−a)(x−b)(x−c)(x−d)≥0(x-a)(x-b)(x-c)(x-d)\geq 0 for every balanced quadruple (a,b,c,d)(a,b,c,d). \\ \\ (Ivan Novak)
emcEuropean Mathematical CupInequalityalgebra
min # of pairs whose labels differ by at most 1

Source: 10th European Mathematical Cup - Problem S1

12/22/2021
Alice drew a regular 20212021-gon in the plane. Bob then labeled each vertex of the 20212021-gon with a real number, in such a way that the labels of consecutive vertices differ by at most 11. Then, for every pair of non-consecutive vertices whose labels differ by at most 11, Alice drew a diagonal connecting them. Let dd be the number of diagonals Alice drew. Find the least possible value that dd can obtain.
combinatoricsminimum valueEuropean Mathematical Cup