MathDB

2

Part of 2008 Moldova Team Selection Test

Problems(3)

Minimum Value of a Non-symmetric Expression

Source: Moldova 2008 IMO-BMO Second TST Problem 2

3/29/2008
Let a1,…,an a_1,\ldots,a_n be positive reals so that a_1\plus{}a_2\plus{}\ldots\plus{}a_n\le\frac n2. Find the minimal value of \sqrt{a_1^2\plus{}\frac1{a_2^2}}\plus{}\sqrt{a_2^2\plus{}\frac1{a_3^2}}\plus{}\ldots\plus{}\sqrt{a_n^2\plus{}\frac1{a_1^2}}.
inequalitiesfunctioninequalities proposed
Again Warm-Up Prob for a TST-Partition of a set into triples

Source: Moldova 2008 IMO-BMO First TST Problem 2

3/3/2008
We say the set {1,2,…,3k} \{1,2,\ldots,3k\} has property D D if it can be partitioned into disjoint triples so that in each of them a number equals the sum of the other two. (a) Prove that {1,2,…,3324} \{1,2,\ldots,3324\} has property D D. (b) Prove that {1,2,…,3309} \{1,2,\ldots,3309\} hasn't property D D.
combinatorics proposedcombinatorics
Prove that a binomial coefficient is not divisible by p.

Source: Moldova 2008 IMO-BMO Third TST Problem 2

3/30/2008
Let p p be a prime number and k,n k,n positive integers so that \gcd(p,n)\equal{}1. Prove that (n⋅pkpk) \binom{n\cdot p^k}{p^k} and p p are coprime.
number theory proposednumber theory