MathDB

1

Part of 2016 IMC

Problems(2)

IMC 2016, Problem 1

Source: IMC 2016

7/27/2016
Let f:[a,b]→Rf : \left[ a, b\right]\rightarrow\mathbb{R} be continuous on [a,b]\left[ a, b\right] and differentiable on (a,b)\left( a, b\right). Suppose that ff has infinitely many zeros, but there is no x∈(a,b)x\in \left( a, b\right) with f(x)=f′(x)=0f(x)=f'(x)=0. (a) Prove that f(a)f(b)=0f(a)f(b)=0. (b) Give an example of such a function on [0,1]\left[ 0, 1\right].
(Proposed by Alexandr Bolbot, Novosibirsk State University)
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IMC 2016, Problem 6

Source: IMC 2016

7/28/2016
Let (x1,x2,…)(x_1,x_2,\ldots) be a sequence of positive real numbers satisfying ∑n=1∞xn2n−1=1{\displaystyle \sum_{n=1}^{\infty}\frac{x_n}{2n-1}=1}. Prove that ∑k=1∞∑n=1kxnk2≤2. \displaystyle \sum_{k=1}^{\infty} \sum_{n=1}^{k} \frac{x_n}{k^2} \le2.
(Proposed by Gerhard J. Woeginger, The Netherlands)
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