Home > Olympiad, Problem Solving, Uncategorized > IMC 2017 – Day 2 – Problems

## IMC 2017 – Day 2 – Problems

Problem 6. Let ${f: [0,\infty) \rightarrow \Bbb{R}}$ be a continuous function such that ${\lim_{x \rightarrow \infty}f(x) = L}$ exists (finite or infinite).

Prove that

$\displaystyle \lim_{n \rightarrow \infty} \int_0^1 f(nx) dx = L.$

Problem 7. Let ${p(x)}$ be a nonconstant polynomial with real coefficients. For every positive integer ${n}$ let

$\displaystyle q_n(x) = (x+1)^n p(x)+x^n p(x+1).$

Prove that there are only finitely many numbers ${n}$ such that all roots of ${q_n(x)}$ are real.

Problem 8. Define the sequence ${A_1,A_2,...}$ of matrices by the following recurrence

$\displaystyle A_1 = \begin{pmatrix} 0& 1 \\ 1& 0 \end{pmatrix}, \ A_{n+1} = \begin{pmatrix} A_n & I_{2^n} \\ I_{2^n} & A_n \end{pmatrix} \ \ (n=1,2,...)$

where ${I_m}$ is the ${m\times m}$ identity matrix.

Prove that ${A_n}$ has ${n+1}$ distinct integer eigenvalues ${\lambda_0<\lambda_1<...<\lambda_n}$ with multiplicities ${{n \choose 0},\ {n\choose 1},...,{n \choose n}}$, respectively.

Problem 9. Define the sequence ${f_1,f_2,... : [0,1) \rightarrow \Bbb{R}}$ of continuously differentiable functions by the following recurrence

$\displaystyle f_1 = 1; f'_{n+1} = f_nf_{n+1} \text{ on } (0,1) \text{ and } f_{n+1}(0)=1.$

Show that ${\lim_{n\rightarrow \infty}f_n(x)}$ exists for every ${x \in [0,1)}$ and determine the limit function.

Problem 10. 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 ${T_1,...,T_n}$ are non-overlapping triangles inside ${K}$ such that each of them is homothetic to ${K}$ with a negative ratio and

$\displaystyle \sum_{\ell =1}^n \text{area}(T_\ell) > \text{area} (K)-\varepsilon,$

then

$\displaystyle \sum_{\ell =1}^n \text{perimeter} (T_\ell) > p.$