## IMC 2016 – Day 2 – Problem 8

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

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

## IMC 2016 – Day 2 – Problem 7

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.

## IMC 2016 – Day 2 – Problem 6

July 28, 2016 1 comment

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

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

## 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 ${x \in \Bbb{C}^n}$.)

Categories: Olympiad, Uncategorized Tags: , ,

## IMC 2016 – Day 1 – Problem 2

Problem 2. Let ${k}$ and ${n}$ be positive integers. A sequence ${(A_1,...,A_k)}$ of ${n\times n}$ matrices is preferred by Ivan the Confessor if ${A_i^2 \neq 0}$ for ${1\leq i \leq k}$, but ${A_iA_j = 0}$ for ${1\leq i,j \leq k}$ with ${i \neq j}$. Show that if ${k \leq n}$ in al preferred sequences and give an example of a preferred sequence with ${k=n}$ for each ${n}$.

## IMC 2016 – Day 1 – Problem 3

Problem 3. Let ${n}$ be a positive integer. Also let ${a_1,a_2,...,a_n}$ and ${b_1,b_2,...,b_n}$ be reap numbers such that ${a_i+b_i >0}$ for ${i = 1,2,...,n}$. Prove that

$\displaystyle \sum_{i=1}^n \frac{a_ib_i -b_i^2}{a_i+b_i} \leq \frac{\sum_{i=1}^n a_i \cdot \sum_{i=1}^n b_i - \left(\sum_{i=1}^n b_i \right)^2 }{\sum_{i=1}^n (a_i+b_i)}.$

## IMC 2016 – Day 1 – Problem 1

Problem 1. Let ${f:[a,b] \rightarrow \Bbb{R}}$ be continuous on ${[a,b]}$ and differentiable on ${(a,b)}$. Suppose that ${f}$ has infinitely many zeros, but there is no ${x \in (a,b)}$ with ${f(x)=f'(x) = 0}$.
• (a) Prove that ${f(a)f(b)=0}$.