MathDB

1

Part of 2011 Mongolia Team Selection Test

Problems(4)

Mongolia TST 2011 Test 2 #1

Source: Mongolia TST 2011 Test 2 #1

11/8/2011
A group of the pupils in a class are called dominant if any other pupil from the class has a friend in the group. If it is known that there exist at least 100 dominant groups, prove that there exists at least one more dominant group.
(proposed by B. Batbayasgalan, inspired by Komal problem)
combinatorics unsolvedcombinatorics
Mongolia TST 2011 Test 1 #1

Source:

11/7/2011
Let v(n)v(n) be the order of 22 in n!n!. Prove that for any positive integers aa and mm there exists nn (n>1n>1) such that v(n)≡a(mod  m)v(n) \equiv a (\mod m).
I have a book with Mongolian problems from this year, and this problem appeared in it. Perhaps I am terribly misinterpreting this problem, but it seems like it is wrong. Any ideas?
modular arithmeticnumber theory unsolvednumber theory
Mongolia TST 2011 Test 3 #1

Source: Mongolia TST 2011 Test 3 #1

11/8/2011
Let A={a2+13b2∣a,b∈Z,b≠0}A=\{a^2+13b^2 \mid a,b \in\mathbb{Z}, b\neq0\}. Prove that there a) exist b) exist infinitely many x,yx,y integer pairs such that x13+y13∈Ax^{13}+y^{13} \in A and x+y∉Ax+y \notin A.
(proposed by B. Bayarjargal)
quadraticsmodular arithmeticRing TheoryDiophantine equationnumber theory unsolvednumber theory
Mongolia TST 2011 Test 4 #1

Source: Mongolia TST 2011 Test 4 #1

11/8/2011
Let t,k,mt,k,m be positive integers and t>kmt>\sqrt{km}. Prove that (2m0)+(2m1)+⋯+(2mm−t−1)<22m2k\dbinom{2m}{0}+\dbinom{2m}{1}+\cdots+\dbinom{2m}{m-t-1}<\dfrac{2^{2m}}{2k}
(proposed by B. Amarsanaa, folklore)
inequalities unsolvedinequalities