IMC 2001

Let $n$ be a positive integer. Consider an $n\times n$ matrix with entries $1,2,...,n^2$ written in order, starting at the top left and moving along each row in turn left-to-right. (e.g. for $n = 3$ we get $\left[\begin{array}{ccc}1 & 2 & 3 4 & 5 & 6 7 & 8 & 9\end{array}\right]$ ) We choose $n$ entries of the matrix such that exactly one entry is chosen in each row and each column. What are the possible values of the sum of the selected entries?

Let $r,s,t$ positive integers which are relatively prime and $a,b \in G$ , $G$ a commutative multiplicative group with unit element $e$ , and $a^r=b^s=(ab)^t=e$ . (a) Prove that $a=b=e$ . (b) Does the same hold for a non-commutative group $G$ ?

Suppose that the differentiable functions $a, b, f, g:\mathbb{R} \rightarrow \mathbb{R}$ satisfy $$f(x)\geq 0, f'(x) \geq 0,g(x)\geq 0, g'(x) \geq 0 \text{ for all } x \in \mathbb{R},$$ $$\lim_{x\rightarrow \infty} a(x)=A\geq 0,\lim_{x\rightarrow \infty} b(x)=B\geq 0, \lim_{x\rightarrow \infty} f(x)=\lim_{x\rightarrow \infty} g(x)=\infty,$$ and $$\frac{f'(x)}{g'(x)}+a(x)\frac{f(x)}{g(x)}=b(x).$$ Prove that $\lim_{x\rightarrow\infty}\frac{f(x)}{g(x)}=\frac{B}{A+1}$ .

Let $A=(a_{k,l})_{k,l=1,...,n}$ be a complex $n \times n$ matrix such that for each $m \in \{1,2,...,n\}$ and $1 \leq j_{1} <...<j_{m}$ the determinant of the matrix $(a_{j_{k},j_{l}})_{k,l=1,...,n}$ is zero. Prove that $A^{n}=0$ and that there exists a permutation $\sigma \in S_{n}$ such that the matrix $(a_{\sigma(k),\sigma(l)})_{k,l=1,...,n}$ has all of its nonzero elements above the diagonal.

For each positive integer $n$ , let $f_{n}(\vartheta)=\sin(\vartheta)\cdot \sin(2\vartheta) \cdot \sin(4\vartheta)\cdots \sin(2^{n}\vartheta)$ . For each real $\vartheta$ and all $n$ , prove that $$|f_{n}(\vartheta)| \leq \frac{2}{\sqrt{3}}|f_{n}(\frac{\pi}{3})|$$