Home > Uncategorized > IMC 2017 – Day 1

## IMC 2017 – Day 1

Problem 1. Determine all complex numbers ${\lambda}$ for which there exists a positive integer ${n}$ and a real ${n\times n}$ matrix ${A}$ such that ${A^2 = A^T}$ and ${\lambda}$ is an eigenvalue of ${A}$.

Problem 2. Let ${f: \Bbb{R} \rightarrow (0,\infty)}$ be a differentiable function and suppose that there exists a constant ${L>0}$ such that

$\displaystyle |f'(x)-f'(y)|\leq L|x-y|$

for all ${x,y}$. Prove that

$\displaystyle (f'(x))^2 < 2Lf(x)$

holds for all ${x}$.

Problem 3. For any positive integer ${m}$ denote by ${P(m)}$ the product of positive divisors of ${m}$. For every positive integer ${n}$ define the sequence

$\displaystyle a_1(n)=n, \ \ \ a_{k+1}(n) = P(a_k(n)) \ \ (k=1,2,...,2016).$

Determine whether for every set ${S \subseteq \{1,2,...,2017\}}$ there exists a positive integer ${n}$ such that the following condition is satisfied:

For every ${k}$ with ${1\leq k \leq 2017}$ the number ${a_k(n)}$ is a perfect square if and only if ${k \in S}$.

Problem 4. There are ${n}$ people in a city and each of them has exactly ${1000}$ friends (friendship is mutual). Prove that it is possible to select a group ${S}$ of people such that at least ${n/2017}$ persons in ${S}$ have exactly two friends in ${S}$.

Problem 5. Let ${k}$ and ${n}$ be positive integers with ${n \geq k^2-3k+4}$ and let

$\displaystyle f(z) = z^{n-1}+c_{n-2}z^{n-2}+...+c_0$

be a polynomial with complex coefficients such that

$\displaystyle c_0c_{n-2} = c_1c_{n-3} = ... = c_{n-2}c_0 = 0.$

Prove that ${f(z)}$ and ${z^n-1}$ have at most ${n-k}$ common roots.