OIMU 2010

Let $f:S\to\mathbb{R}$ be the function from the set of all right triangles into the set of real numbers, defined by $f(\Delta ABC)=\frac{h}{r}$ , where $h$ is the height with respect to the hypotenuse and $r$ is the inscribed circle's radius. Find the image, $Im(f)$ , of the function.

Calculate the sum of the series $\sum_{-\infty}^{\infty}\frac{\sin^33^k}{3^k}$ .

A student adds up rational fractions incorrectly:

$$\frac{a}{b}+\frac{x}{y}=\frac{a+x}{b+y} (\star)$$

Despite that, he sometimes obtains correct results. For a given fraction $\frac{a}{b},a,b\in\mathbb{Z},b>0$ , find all fractions $\frac{x}{y},x,y\in\mathbb{Z},y>0$ such that the result obtained by $(\star)$ is correct.

Let $p(x)=x^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0$ be a monic polynomial of degree $n>2$ , with real coefficients and all its roots real and different from zero. Prove that for all $k=0,1,2,\cdots,n-2$ , at least one of the coefficients $a_k,a_{k+1}$ is different from zero.

Let $A,B$ be matrices of dimension $2010\times2010$ which commute and have real entries, such that $A^{2010}=B^{2010}=I$ , where $I$ is the identity matrix. Prove that if $\operatorname{tr}(AB)=2010$ , then $\operatorname{tr}(A)=\operatorname{tr}(B)$ .

Prove that, for all integer $a>1$ , the prime divisors of $5a^4-5a^2+1$ have the form $20k\pm1,k\in\mathbb{Z}$ .

Proposed by Géza Kós.

(a) Prove that, for any positive integers $m\le \ell$ given, there is a positive integer $n$ and positive integers $x_1,\cdots,x_n,y_1,\cdots,y_n$ such that the equality $$\sum_{i=1}^nx_i^k=\sum_{i=1}^ny_i^k$$ holds for every $k=1,2,\cdots,m-1,m+1,\cdots,\ell$ , but does not hold for $k=m$ .

(b) Prove that there is a solution of the problem, where all numbers $x_1,\cdots,x_n,y_1,\cdots,y_n$ are distinct.

Proposed by Ilya Bogdanov and Géza Kós.