Home > Olympiad, Uncategorized > IMC 2016 Problems – Day 2

IMC 2016 Problems – Day 2

Problem 6. Let {(x_1,x_2,...)} be a sequence of positive real numbers satisfying {\displaystyle \sum_{n=1}^\infty \frac{x_n}{2n-1}=1}. Prove that

\displaystyle \sum_{k=1}^\infty \sum_{n=1}^k \frac{x_n}{k^2} \leq 2.

Problem 7. Today, Ivan the Confessor prefers continuous functions {f:[0,1]\rightarrow \Bbb{R}} satisfying {f(x)+f(y) \geq |x-y|} for all {x,y \in [0,1]}. Fin the minimum of {\int_0^1 f} over all preferred functions.

Problem 8. Let {n} be a positive integer and denote by {\Bbb{Z}_n} the ring of integers modulo {n}. Suppose that there exists a function {f:\Bbb{Z}_n \rightarrow \Bbb{Z}_n} satisfying the following three properties:

  • (i) {f(x) \neq x},
  • (ii) {f(f(x))=x},
  • (iii) {f(f(f(x+1)+1)+1) = x} for all {x \in \Bbb{Z}_n}.

Prove that {n \equiv 2} modulo {4}.

Problem 9. Let {k} be a positive integer. For each nonnegative integer {n} let {f(n)} be the number of solutions {(x_1,...,x_k) \in \Bbb{Z}^k} of the inequality {|x_1|+...+|x_k| \leq n}. Prove that for every {n \geq 1} we have {f(n-1)f(n+1) \leq f(n)^2}.

Problem 10. Let {A} be a {n \times n} complex matrix whose eigenvalues have absolute value at most {1}. Prove that

\displaystyle \|A^n\| \leq \frac{n}{\ln 2} \|A\|^{n-1}.

(Here {\|B\| = \sup_{\|x\|\leq 1} \|Bx\|} for every {n \times n} matrix {B} and {\|x\| = \sqrt{\sum_{i=1}^n |x_i|^2 }} for every complex vector {x \in \Bbb{C}^n}.)

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