## China Girls Math Olympiad 2004

1. 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 $\{11, 13, 15, 17, 19\}$.
2. Let $a, b, c$ be positive reals. Find the smallest value of $\frac {a + 3c}{a + 2b + c} + \frac {4b}{a + b + 2c} - \frac {8c}{a + b + 3c}.$
3. 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 $\sqrt {2} + 1$.
4. 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 $2004$.
5. Let $u, v, w$ be positive real numbers such that $$u\sqrt {vw} + v\sqrt {wu} + w\sqrt {uv} \geq 1.$$ Find the smallest value of $u + v + w$.
6. 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 $F$.
7. 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 $i + j \leq n$.
8. 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.)