International Mathematics Competition
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IMC 2019
  Problems & Solutions

IMC2016: Problems on Day 2

6. Let $(x_1,x_2,\ldots)$ be a sequence of positive real numbers satisfying ${\displaystyle \sum_{n=1}^{\infty}\frac{x_n}{2n-1}=1}$. Prove that $$ \displaystyle \sum_{k=1}^{\infty} \sum_{n=1}^{k} \frac{x_n}{k^2} \le2. $$

Proposed by Gerhard J. Woeginger, The Netherlands


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

Proposed by Fedor Petrov, St. Petersburg State University


8. Let $n$ be a positive integer, and denote by $\mathbb{Z}_n$ the ring of integers modulo $n$. Suppose that there exists a function $f:\mathbb{Z}_n\to\mathbb{Z}_n$ 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 $x\in\mathbb{Z}_n$.

Prove that $n\equiv 2 \pmod4$.

Proposed by Ander Lamaison Vidarte, Berlin Mathematical School, Germany


9. Let $k$ be a positive integer. For each nonnegative integer $n$, let $f(n)$ be the number of solutions $(x_1,\ldots,x_k)\in\mathbb{Z}^k$ of the inequality $|x_1|+...+|x_k|\leq n$. Prove that for every $n\ge1$, we have $f(n-1)f(n+1)\leq f(n)^2$.

Proposed by Esteban Arreaga, Renan Finder and José Madrid, IMPA, Rio de Janeiro


10. Let $A$ be a $n\times n$ complex matrix whose eigenvalues have absolute value at most $1$. Prove that $$ \|A^n\|\le \dfrac{n}{\ln 2} \|A\|^{n-1}. $$ (Here $\|B\|=\sup\limits_{\|x\|\leq 1} \|Bx\|$ for every $n\times n$ matrix $B$ and $\|x\|=\sqrt{\sum\limits_{i=1}^n |x_i|^2}$ for every complex vector $x\in\mathbb{C}^n$.)

Proposed by Ian Morris and Fedor Petrov, St. Petersburg State University