MathDB

4

Part of 2011 IMC

Problems(2)

IMC Day 1 Problem 4

Source: combinaotrics/algebra [?]

8/1/2011
Let A1,A2,…,AnA_1,A_2,\dots, A_n be finite, nonempty sets. Define the function f(t)=∑k=1n∑1≤i1<i2<⋯<ik≤n(−1)k−1t∣Ai1∪Ai2∪⋯∪Aik∣.f(t)=\sum_{k=1}^n \sum_{1\leq i_1<i_2<\dots<i_k\leq n} (-1)^{k-1}t^{|A_{i_1}\cup A_{i_2}\cup \dots\cup A_{i_k}|}. Prove that ff is nondecreasing on [0,1].[0,1]. (∣A∣|A| denotes the number of elements in A.A.)
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IMC 2011 Day 2, Problem 4

Source:

8/1/2011
Let ff be a polynomial with real coefficients of degree nn. Suppose that f(x)−f(y)x−y\displaystyle \frac{f(x)-f(y)}{x-y} is an integer for all 0≤x<y≤n0 \leq x<y \leq n. Prove that a−b∣f(a)−f(b)a-b | f(a)-f(b) for all distinct integers a,ba,b.
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