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.)

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