MathDB

P2

Part of 2017 Romania Team Selection Test

Problems(3)

Combinatorics from "The editors"

Source: Romania 2017 IMO TST 1, problem 2

3/18/2018
Consider a finite collection of 3-element sets AiA_i, no two of which share more than one element, whose union has cardinality 2017. Show that the elements of this union can be coloured with two colors, blue and red, so that at least 64 elements are blue and each AiA_i has at least one red element.
combinatorics
Classic Problem

Source: Romania 2017 IMO TST 3, problem 2

3/18/2018
Determine all intergers n≥2n\geq 2 such that a+2a+\sqrt{2} and an+2a^n+\sqrt{2} are both rational for some real number aa depending on nn
irrational numberpolynomialnumber theory
Romania TST 5 2017 P2

Source:

2/25/2021
Let nn be a positive integer, and let SnS_n be the set of all permutations of 1,2,...,n1,2,...,n. let kk be a non-negative integer, let an,ka_{n,k} be the number of even permutations σ\sigma in SnS_n such that ∑i=1n∣σ(i)−i∣=2k\sum_{i=1}^{n}|\sigma(i)-i|=2k and bn,kb_{n,k} be the number of odd permutations σ\sigma in SnS_n such that ∑i=1n∣σ(i)−i∣=2k\sum_{i=1}^{n}|\sigma(i)-i|=2k. Evaluate an,k−bn,ka_{n,k}-b_{n,k}.
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combinatoricspermutationsRomanian TST2017