Olympiad problems

2017 IMC, 9

Define the sequence $f_1,f_2,\ldots :[0,1)\to \mathbb{R}$ of continuously differentiable functions by the following recurrence: $$ f_1=1; \qquad \quad f_{n+1}'=f_nf_{n+1} \quad\text{on $(0,1)$}, \quad \text{and}\quad f_{n+1}(0)=1. $$ Show that $\lim\limits_{n\to \infty}f_n(x)$ exists for every $x\in [0,1)$ and determine the limit function.

2017 IMC, 3

Tags: imc 2017 , IMC , number theory

For any positive integer $m$, denote by $P(m)$ the product of positive divisors of $m$ (e.g $P(6)=36$). For every positive integer $n$ define the sequence $$a_1(n)=n,\qquad a_{k+1}(n)=P(a_k(n))\quad (k=1,2,\dots,2016)$$ Determine whether for every set $S\subset\{1,2,\dots,2017\}$, 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 $k\in S$.

2017 IMC, 4

Tags: imc 2017 , IMC , combinatorics , approximation , Probabilistic Method

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

2017 IMC, 2

Tags: function , IMC , imc 2017 , calculus

Let $f:\mathbb R\to(0,\infty)$ be a differentiabe function, and suppose that there exists a constant $L>0$ such that $$|f'(x)-f'(y)|\leq L|x-y|$$ for all $x,y$. Prove that $$(f'(x))^2<2Lf(x)$$ holds for all $x$.

2017 IMC, 7

Tags: college contests , IMC , imc 2017

Let $p(x)$ be a nonconstant polynomial with real coefficients. For every positive integer~$n$, let $$q_n(x) = (x+1)^np(x)+x^n p(x+1) .$$ Prove that there are only finitely many numbers $n$ such that all roots of $q_n(x)$ are real.

2017 IMC, 6

Tags: college contests , IMC , imc 2017

Let $f:[0;+\infty)\to \mathbb R$ be a continuous function such that $\lim\limits_{x\to +\infty} f(x)=L$ exists (it may be finite or infinite). Prove that $$ \lim\limits_{n\to\infty}\int\limits_0^{1}f(nx)\,\mathrm{d}x=L. $$

2017 IMC, 5

Tags: imc 2017 , IMC , polynomial , complex analysis

Let $k$ and $n$ be positive integers with $n\geq k^2-3k+4$, and let $$f(z)=z^{n-1}+c_{n-2}z^{n-2}+\dots+c_0$$ be a polynomial with complex coefficients such that $$c_0c_{n-2}=c_1c_{n-3}=\dots=c_{n-2}c_0=0$$ Prove that $f(z)$ and $z^n-1$ have at most $n-k$ common roots.

2017 IMC, 10

Tags: college contests , IMC , imc 2017

Let $K$ be an equilateral triangle in the plane. Prove that for every $p>0$ there exists an $\varepsilon>0$ with the following property: If $n$ is a positive integer, and $T_1,\ldots,T_n$ are non-overlapping triangles inside $K$ such that each of them is homothetic to $K$ with a negative ratio, and $$ \sum_{\ell=1}^n \textrm{area}(T_\ell) > \textrm{area}(K)-\varepsilon, $$ then $$ \sum_{\ell=1}^n \textrm{perimeter}(T_\ell) > p. $$

2017 IMC, 1

Tags: linear algebra , matrix , IMC , imc 2017

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 $A$.

2017 IMC, 8

Tags: college contests , imc 2017 , IMC , linear algebra

Define the sequence $A_1,A_2,\ldots$ of matrices by the following recurrence: $$ A_1 = \begin{pmatrix} 0 & 1 \\ 1 & 0 \\ \end{pmatrix}, \quad A_{n+1} = \begin{pmatrix} A_n & I_{2^n} \\ I_{2^n} & A_n \\ \end{pmatrix} \quad (n=1,2,\ldots) $$ where $I_m$ is the $m\times m$ identity matrix. Prove that $A_n$ has $n+1$ distinct integer eigenvalues $\lambda_0< \lambda_1<\ldots <\lambda_n$ with multiplicities $\binom{n}{0},\binom{n}{1},\ldots,\binom{n}{n}$, respectively.