CH-SMO 2011

At a party, there are $2011$ people with a glass of fruit juice each sitting around a circular table. Once a second, they clink glasses obeying the following two rules: (a) They do not clink glasses crosswise. (b) At each point of time, everyone can clink glasses with at most one other person. How many seconds pass at least until everyone clinked glasses with everybody else?

(Swiss Mathematical Olympiad 2011, Final round, problem 1)

Let $\triangle{ABC}$ be an acute-angled triangle and let $D$ , $E$ , $F$ be points on $BC$ , $CA$ , $AB$ , respectively, such that $$\angle{AFE}=\angle{BFD}\text{,} \angle{BDF}=\angle{CDE} \text{and} \angle{CED}=\angle{AEF}\text{.}$$ Prove that $D$ , $E$ and $F$ are the feet of the perpendiculars through $A$ , $B$ and $C$ on $BC$ , $CA$ and $AB$ , respectively.

(Swiss Mathematical Olympiad 2011, Final round, problem 2)

For positive integers $m$ and $n$, find the smalles possible value of $|2011^m-45^n|$.

(Swiss Mathematical Olympiad, Final round, problem 3)

Find all functions $f:\mathbb{R}^+\to\mathbb{R}^+$ such that for any real numbers $a, b, c, d >0$ satisfying $abcd=1$ ,$$(f(a)+f(b))(f(c)+f(d))=(a+b)(c+d)$$ holds true.

(Swiss Mathematical Olympiad 2011, Final round, problem 4)

Let $\triangle{ABC}$ be a triangle with circumcircle $\tau$ . The tangentlines to $\tau$ through $A$ and $B$ intersect at $T$ . The circle through $A$ , $B$ and $T$ intersects $BC$ and $AC$ again at $D$ and $E$ , respectively; $CT$ and $BE$ intersect at $F$ . Suppose $D$ is the midpoint of $BC$ . Calculate the ratio $BF:BE$ .

(Swiss Mathematical Olympiad 2011, Final round, problem 5)

Let $a, b, c, d$ be positive real numbers satisfying $a+b+c+d =1$ . Show that $$\frac{2}{(a+b)(c+d)} \leq \frac{1}{\sqrt{ab}}+ \frac{1}{\sqrt{cd}}\text{.}$$

(Swiss Mathematical Olympiad 2011, Final round, problem 6)

For a given rational number $r$ , find all integers $z$ such that $$2^z + 2 = r^2\text{.}$$

(Swiss Mathematical Olympiad 2011, Final round, problem 7)

Let $ABCD$ be a parallelogram and $H$ the Orthocentre of $\triangle{ABC}$ . The line parallel to $AB$ through $H$ intersects $BC$ at $P$ and $AD$ at $Q$ while the line parallel to $BC$ through $H$ intersects $AB$ at $R$ and $CD$ at $S$ . Show that $P$ , $Q$ , $R$ and $S$ are concyclic.

(Swiss Mathematical Olympiad 2011, Final round, problem 8)

For any positive integer $n$ let $f(n)$ be the number of divisors of $n$ ending with $1$ or $9$ in base $10$ and let $g(n)$ be the number of divisors of $n$ ending with digit $3$ or $7$ in base $10$ . Prove that $f(n)\geqslant g(n)$ for all nonnegative integers $n$ .

(Swiss Mathematical Olympiad 2011, Final round, problem 9)

On each square of an $n\times n$ -chessboard, there are two bugs. In a move, each bug moves to a (vertically of horizontally) adjacent square. Bugs from the same square always move to different squares. Determine the maximal number of free squares that can occur after one move.

(Swiss Mathematical Olympiad 2011, Final round, problem 10)