MathDB

3

Part of 2008 IMC

Problems(2)

IMC 2008 Day 1 P3

Source:

7/26/2014
Let pp be a polynomial with integer coefficients and let a1<a2<⋯<aka_1<a_2<\cdots <a_k be integers. Given that p(ai)≠0∀  i=1,2,⋯ ,kp(a_i)\ne 0\forall\; i=1,2,\cdots, k.
(a) Prove ∃  a∈Z\exists\; a\in \mathbb{Z} such that p(ai)∣p(a)    ∀i=1,2,…,k p(a_i)\mid p(a)\;\;\forall i=1,2,\dots ,k (b) Does there exist a∈Za\in \mathbb{Z} such that ∏i=1kp(ai)∣p(a) \prod_{i=1}^{k}p(a_i)\mid p(a)
algebrapolynomialquadraticsIMCcollege contests
IMC 2008 Day 2 P3 - Power of 2 divides binomial sum

Source: Problem 3

7/28/2008
Let n n be a positive integer. Prove that 2^{n\minus{}1} divides \sum_{0\leq k < n/2} \binom{n}{2k\plus{}1}5^k.
IMCcollege contests