MathDB

3

Part of 2015 Vietnam National Olympiad

Problems(2)

VMO 2015 Problem 3

Source: VMO 2015

7/30/2016
Given m∈Z+m\in\mathbb{Z}^+. Find all natural numbers nn that does not exceed 10m10^m satisfying the following conditions: i) 3∣n.3|n. ii) The digits of nn in decimal representation are in the set {2,0,1,5}\{2,0,1,5\}.
number theory
VMO 2015 Problem 7

Source: VMO 2015

7/31/2016
There are mm boys and nn girls participate in a duet singing contest (m,n≥2m,n\geq 2). At the contest, each section there will be one show. And a show including some boy-girl duets where each boy-girl couple will sing with each other no more than one song and each participant will sing at least one song. Two show AA and BB are considered different if there is a boy-girl couple sing at show AA but not show BB. The contest will end if and only if every possible shows are performed, and each show is performed exactly one time.
a) A show is called depend on a participant XX if we cancel all duets that XX perform, then there will be at least one another participant will not sing any song in that show. Prove that among every shows that depend on XX, the number of shows with odd number of songs equal to the number of shows with even number of songs.
b) Prove that the organizers can arrange the shows in order to make sure that the numbers of songs in two consecutive shows have different parity.
combinatorics