# 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 $P(x)=A(x)^2+x\cdot B(x)^2.$

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