Practice Test

Problem 1
 

Let \( n \) and \( k \) be positive integers such that \( n> k \). Prove that the numbers \( \binom nk \), \( \binom{n+1}k \), \( \binom{n+2}k \), \( \dots \), \( \binom{n+k}k \) do not have a common factor greater than 1.

Problem 2
 

Prove that there exists a function \( f:\mathbb R\to\mathbb R \) and a sequence \( (x_k)_{k=1}^{\infty} \) that satisfy the following relations: \begin{eqnarray*} &&\lim_{k\to\infty}x_k=+\infty,\\ &&\lim_{x\to+\infty}\frac{f(x)}{x}=+\infty, \mbox{ and }\\ &&\left|f(2x_k)-2f(x_k)\right|\leq 1. \end{eqnarray*}

Problem 3
 

Let \( n \) be a positive integer. Determine the largest integer \( k \) for which there exists a \( 4n\times 4n \) matrix with \( 0 \) and \( 1 \) entries such that the following three conditions are satisfied:

  • (i) The sum of the numbers in each row is \( k \);

  • (ii) The sum of the numbers in each column is \( k \);

  • (iii) The product of numbers from any two adjacent cells is \( 0 \). (Two cells are adjacent if they share an edge or a corner.)

Problem 4
 
Assume that \( P \) is a polynomial with real coefficients such that \( P(x)\geq 0 \) for all \( x\geq 0 \). Prove that there are two polynomials \( A \) and \( B \) with real coefficients such that \[ P(x)=A(x)^2+x\cdot B(x)^2.\]


2005-2017 IMOmath.com | imomath"at"gmail.com | Math rendered by MathJax
Home | Olympiads | Book | Training | IMO Results | Forum | Links | About | Contact us