ZA-SAMO 2006

Reduce the fraction $$\frac{2121212121210}{1121212121211}$$ to its simplest form.

Triangle $ABC$ has $BC=1$ and $AC=2$ . What is the maximum possible value of $\hat{A}$ .

Determine all positive integers whose squares end in $196$ .

In triangle $ABC$ , $AB=AC$ and $B\hat{A}C=100^\circ$ . Let $D$ be on $AC$ such that $A\hat{B}D=C\hat{B}D$ . Prove that $AD+DB=BC$ .

Find the number of subsets $X$ of $\{1,2,\dots,10\}$ such that $X$ contains at least two elements and such that no two elements of $X$ differ by $1$ .

Consider the function $f$ defined by $$f(n)=\frac{1}{n}\left (\left \lfloor\frac{n}{1}\right \rfloor+\left \lfloor\frac{n}{2}\right \rfloor+\cdots+\left \lfloor\frac{n}{n}\right \rfloor \right )$$ for all positive integers $n$ . (Here $\lfloor x\rfloor$ denotes the greatest integer less than or equal to $x$ .) Prove that

(a) $f(n+1)>f(n)$ for infinitely many $n$ .

(b) $f(n+1)<f(n)$ for infinitely many $n$ .