SEEMOUS 2008

Let $f:[1,\infty)\to(0,\infty)$ be a continuous function. Assume that for every $a>0$ , the equation $f(x)=ax$ has at least one solution in the interval $[1,\infty)$ . (a) Prove that for every $a>0$ , the equation $f(x)=ax$ has infinitely many solutions. (b) Give an example of a strictly increasing continuous function $f$ with these properties.

Let $P_0,P_1,P_2,\ldots$ be a sequence of convex polygons such that, for each $k\ge0$ , the vertices of $P_{k+1}$ are the midpoints of all sides of $P_k$ . Prove that there exists a unique point lying inside all these polygons.

Let $\mathcal M_n(\mathbb R)$ denote the set of all real $n\times n$ matrices. Find all surjective functions $f:\mathcal M_n(\mathbb R)\to\{0,1,\ldots,n\}$ which satisfy $$f(XY)\le\min\{f(X),f(Y)\}$$ for all $X,Y\in\mathcal M_n(\mathbb R)$ .

Let $n$ be a positive integer and $f:[0,1]\to\mathbb R$ be a continuous function such that $$\int^1_0x^kf(x)dx=1$$ for every $k\in\{0,1,\ldots,n-1\}$ . Prove that $$\int^1_0f(x)^2dx\ge n^2.$$