CR-MO 2010

Consider points $D,E$ and $F$ on sides $BC,AC$ and $AB$ , respectively, of a triangle $ABC$ , such that $AD, BE$ and $CF$ concurr at a point $G$ . The parallel through $G$ to $BC$ cuts $DF$ and $DE$ at $H$ and $I$ , respectively. Show that triangles $AHG$ and $AIG$ have the same areas.

Consider the sequence $x_n>0$ defined with the following recurrence relation:

$$x_1 = 0$$

and for $n>1$ $$(n+1)^2x_{n+1}^2 + (2^n+4)(n+1)x_{n+1}+ 2^{n+1}+2^{2n-2} = 9n^2x_n^2+36nx_n+32.$$

Show that if $n$ is a prime number larger or equal to $5$ , then $x_n$ is an integer.

Christian Reiher and Reid Barton want to open a security box, they already managed to discover the algorithm to generate the key codes and they obtained the following information: $i)$ In the screen of the box will appear a sequence of $n+1$ numbers, $C_0 = (a_{0,1},a_{0,2},...,a_{0,n+1})$ $ii)$ If the code $K = (k_1,k_2,...,k_n)$ opens the security box then the following must happen:

a) A sequence $C_i = (a_{i,1},a_{i,2},...,a_{i,n+1})$ will be asigned to each $k_i$ defined as follows: $a_{i,1} = 1$ and $a_{i,j} = a_{i-1,j}-k_ia_{i,j-1}$ , for $i,j \ge 1$ b) The sequence $(C_n)$ asigned to $k_n$ satisfies that $S_n = \sum_{i=1}^{n+1}|a_i|$ has its least possible value, considering all possible sequences $K$ .

The sequence $C_0$ that appears in the screen is the following: $a_{0,1} = 1$ and $a_0,i$ is the sum of the products of the elements of each of the subsets with $i-1$ elements of the set $A =$ { $1,2,3,...,n$ }, $i\ge 2$ , such that $a_{0, n+1} = n!$ Find a sequence $K = (k_1,k_2,...,k_n)$ that satisfies the conditions of the problem and show that there exists at least $n!$ of them.

Find all integer solutions $(a,b)$ of the equation $$(a+b+3)^2 + 2ab = 3ab(a+2)(b+2)$$

Let $C_1$ be a circle with center $O$ and let $B$ and $C$ be points in $C_1$ such that $BOC$ is an equilateral triangle. Let $D$ be the midpoint of the minor arc $BC$ of $C_1$ . Let $C_2$ be the circle with center $C$ that passes through $B$ and $O$ . Let $E$ be the second intersection of $C_1$ and $C_2$ . The parallel to $DE$ through $B$ intersects $C_1$ for second time in $A$ . Let $C_3$ be the circumcircle of triangle $AOC$ . The second intersection of $C_2$ and $C_3$ is $F$ . Show that $BE$ and $BF$ trisect the angle $\angle ABC$ .

Let $F$ be the family of all sets of positive integers with $2010$ elements that satisfy the following condition: The difference between any two of its elements is never the same as the difference of any other two of its elements. Let $f$ be a function defined from $F$ to the positive integers such that $f(K)$ is the biggest element of $K \in F$ . Determine the least value of $f(K)$ .