NL-NWO 2006

A palindrome is a word that doesn't matter if you read it from left to right or from right to left. Examples: OMO, lepel and parterretrap. How many palindromes can you make with the five letters $a, b, c, d$ and $e$ under the conditions:

• each letter may appear no more than twice in each palindrome,
• the length of each palindrome is at least $3$ letters. (Any possible combination of letters is considered a word.)

Given is a acute angled triangle $ABC$ . The lengths of the altitudes from $A, B$ and $C$ are successively $h_A, h_B$ and $h_C$ . Inside the triangle is a point $P$ . The distance from $P$ to $BC$ is $1/3 h_A$ and the distance from $P$ to $AC$ is $1/4 h_B$ . Express the distance from $P$ to $AB$ in terms of $h_C$ .

$1+2+3+4+5+6=6+7+8$ . What is the smallest number $k$ greater than $6$ for which: $1 + 2 +...+ k = k + (k+1) +...+ n$ , with $n$ an integer greater than $k$ ?

Given is triangle $ABC$ with an inscribed circle with center $M$ and radius $r$ . The tangent to this circle parallel to $BC$ intersects $AC$ in $D$ and $AB$ in $E$ . The tangent to this circle parallel to $AC$ intersects $AB$ in $F$ and $BC$ in $G$ . The tangent to this circle parallel to $AB$ intersects $BC$ in $H$ and $AC$ in $K$ . Name the centers of the inscribed circles of triangle $AED$ , triangle $FBG$ and triangle $KHC$ successively $M_A, M_B, M_C$ and the rays successively $r_A, r_B$ and $r_C$ . Prove that $r_A + r_B + r_C = r$ .

Player $A$ and player $B$ play the next game on an $8$ by $8$ square chessboard. They in turn color a field that is not yet colored. One player uses red and the other blue. Player $A$ starts. The winner is the first person to color the four squares of a square of $2$ by $2$ squares with his color somewhere on the board. Prove that player $B$ can always prevent player $A$ from winning.