MathDB

3

Part of 2011 Mongolia Team Selection Test

Problems(4)

Mongolia TST 2011 Test 1 #3

Source: Mongolia TST 2011 Test 1 #3

11/7/2011
We are given an acute triangle ABCABC. Let (w,I)(w,I) be the inscribed circle of ABCABC, (Ω,O)(\Omega,O) be the circumscribed circle of ABCABC, and A0A_0 be the midpoint of altitude AHAH. ww touches BCBC at point DD. A0DA_0 D and ww intersect at point PP, and the perpendicular from II to A0DA_0 D intersects BCBC at the point MM. MRMR and MSMS lines touch Ω\Omega at RR and SS respectively [note: I am not entirely sure of what is meant by this, but I am pretty sure it means draw the tangents to Ω\Omega from MM]. Prove that the points R,P,D,SR,P,D,S are concyclic. (proposed by E. Enkzaya, inspired by Vietnamese olympiad problem)
geometrycircumcircleIMO Shortlistpower of a pointradical axisgeometry unsolved
Mongolia TST 2011 Test 2 #3

Source: Mongolia TST 2011 Test 2 #3

11/8/2011
Let GG be a graph, not containing K4K_4 as a subgraph and ∣V(G)∣=3k|V(G)|=3k (I interpret this to be the number of vertices is divisible by 3). What is the maximum number of triangles in GG?
inequalitiespigeonhole principlecombinatorics unsolvedcombinatorics
Mongolia TST 2011 Test 3 #3

Source: Mongolia TST 2011 Test 3 #3

11/8/2011
Let nn and dd be positive integers satisfying d<n2d<\dfrac{n}{2}. There are nn boys and nn girls in a school. Each boy has at most dd girlfriends and each girl has at most dd boyfriends. Prove that one can introduce some of them to make each boy have exactly 2d2d girlfriends and each girl have exactly 2d2d boyfriends. (I think we assume if a girl has a boyfriend, she is his girlfriend as well and vice versa)
(proposed by B. Batbaysgalan, folklore).
functiongraph theorycombinatorics unsolvedcombinatorics
Mongolia TST 2011 Test 4 #3

Source: Mongolia TST 2011 Test 4 #3

11/8/2011
Let mm and nn be positive integers such that m>nm>n and m≡n(mod2)m \equiv n \pmod{2}. If (m2−n2+1)∣n2−1(m^2-n^2+1) \mid n^2-1, then prove that m2−n2+1m^2-n^2+1 is a perfect square.
(proposed by G. Batzaya, folklore)
modular arithmeticalgebrapolynomialVietanumber theory unsolvednumber theory