INMO 2005

Let $M$ be the midpoint of side $B C$ of a triangle $A B C$. Let the median $A M$ intersect the incircle of $A B C$ at $K$ and $L, K$ being nearer to $A$ than $L$. If $A K=K L=L M$, prove that the sides of triangle $A B C$ are in the ratio $5: 10: 13$ in some order.

Let $\alpha$ and $\beta$ be positive integers such that

$$\frac{43}{197}<\frac{\alpha}{\beta}<\frac{17}{77}$$

Find the minimum possible value of $\beta$.

Let $p, q, r$ be positive real numbers, not all equal, such that some two of the equations

$$p x^{2}+2 q x+r=0, q x^{2}+2 r x+p=0, r x^{2}+2 p x+q=0$$

have a common root, say $\alpha$. Prove that

(a) $\alpha$ is real and negative; and

(b) the third equation has non-real roots.

All possible 6-digit numbers, in each of which the digits occur in non-increasing order (from left to right, e.g., 877550) are written as a sequence in increasing order. Find the 2005 -th number in this sequence.

Let $x_{1}$ be a given positive integer. A sequence $\left\langle x_{n}\right\rangle_{n=1}^{\infty}=\left\langle x_{1}, x_{2}, x_{3}, \cdots\right\rangle$ of positive integers is such that $x_{n}$, for $n \geq 2$, is obtained from $x_{n-1}$ by adding some nonzero digit of $x_{n-1}$. Prove that

(a) the sequence has an even number;

(b) the sequence has infinitely many even numbers.

Find all functions $f: \mathbf{R} \rightarrow \mathbf{R}$ such that

$$f\left(x^{2}+y f(z)\right)=x f(x)+z f(y)$$

for all $x, y, z$ in $\mathbf{R}$. (Here $\mathbf{R}$ denotes the set of all real numbers.)