OSN 2007

Let $ABC$ be a triangle with $\angle ABC=\angle ACB=70^{\circ}$ . Let point $D$ on side $BC$ such that $AD$ is the altitude, point $E$ on side $AB$ such that $\angle ACE=10^{\circ}$ , and point $F$ is the intersection of $AD$ and $CE$ . Prove that $CF=BC$ .

For every positive integer $n$ , $b(n)$ denote the number of positive divisors of $n$ and $p(n)$ denote the sum of all positive divisors of $n$ . For example, $b(14)=4$ and $p(14)=24$ . Let $k$ be a positive integer greater than $1$ .

(a) Prove that there are infinitely many positive integers $n$ which satisfy $b(n)=k^2-k+1$ .

(b) Prove that there are finitely many positive integers $n$ which satisfy $p(n)=k^2-k+1$ .

Let $a,b,c$ be positive real numbers which satisfy $5(a^2+b^2+c^2)<6(ab+bc+ca)$ . Prove that these three inequalities hold: $a+b>c$ , $b+c>a$ , $c+a>b$ .

A 10-digit arrangement $0,1,2,3,4,5,6,7,8,9$ is called beautiful if (i) when read left to right, $0,1,2,3,4$ form an increasing sequence, and $5,6,7,8,9$ form a decreasing sequence, and (ii) $0$ is not the leftmost digit. For example, $9807123654$ is a beautiful arrangement. Determine the number of beautiful arrangements.

Let $r$ , $s$ be two positive integers and $P$ a 'chessboard' with $r$ rows and $s$ columns. Let $M$ denote the maximum value of rooks placed on $P$ such that no two of them attack each other.

(a) Determine $M$ .

(b) How many ways to place $M$ rooks on $P$ such that no two of them attack each other?

[Note: In chess, a rook moves and attacks in a straight line, horizontally or vertically.]

Find all triples $(x,y,z)$ of real numbers which satisfy the simultaneous equations

$$x = y^3 + y - 8$$

$$y = z^3 + z - 8$$

$$z = x^3 + x - 8.$$

Points $A,B,C,D$ are on circle $S$ , such that $AB$ is the diameter of $S$ , but $CD$ is not the diameter. Given also that $C$ and $D$ are on different sides of $AB$ . The tangents of $S$ at $C$ and $D$ intersect at $P$ . Points $Q$ and $R$ are the intersections of line $AC$ with line $BD$ and line $AD$ with line $BC$ , respectively.

(a) Prove that $P$ , $Q$ , and $R$ are collinear.

(b) Prove that $QR$ is perpendicular to line $AB$ .

Let $m$ and $n$ be two positive integers. If there are infinitely many integers $k$ such that $k^2+2kn+m^2$ is a perfect square, prove that $m=n$ .