CGMO 2004

We say a positive integer $n$ is good if there exists a permutation $a_1, a_2, \ldots, a_n$ of $1, 2, \ldots, n$ such that $k + a_k$ is perfect square for all $1\le k\le n$ . Determine all the good numbers in the set $\{11, 13, 15, 17, 19\}$ .

Let $a, b, c$ be positive reals. Find the smallest value of $$\frac {a + 3c}{a + 2b + c} + \frac {4b}{a + b + 2c} - \frac {8c}{a + b + 3c}.$$

Let $ABC$ be an obtuse inscribed in a circle of radius $1$ . Prove that $\triangle ABC$ can be covered by an isosceles right-angled triangle with hypotenuse of length $\sqrt {2} + 1$ .

A deck of $32$ cards has $2$ different jokers each of which is numbered $0$ . There are $10$ red cards numbered $1$ through $10$ and similarly for blue and green cards. One chooses a number of cards from the deck. If a card in hand is numbered $k$ , then the value of the card is $2^k$ , and the value of the hand is sum of the values of the cards in hand. Determine the number of hands having the value $2004$ .

Given an acute triangle $ABC$ with $O$ as its circumcenter. Line $AO$ intersects $BC$ at $D$ . Points $E$ , $F$ are on $AB$ , $AC$ respectively such that $A$ , $E$ , $D$ , $F$ are concyclic. Prove that the length of the projection of line segment $EF$ on side $BC$ does not depend on the positions of $E$ and $F$ .

Let $p$ and $q$ be two coprime positive integers, and $n$ be a non-negative integer. Determine the number of integers that can be written in the form $ip + jq$ , where $i$ and $j$ are non-negative integers with $i + j \leq n$ .

When the unit squares at the four corners are removed from a three by three squares, the resulting shape is called a cross. What is the maximum number of non-overlapping crosses placed within the boundary of a $10\times 11$ chessboard? (Each cross covers exactly five unit squares on the board.)