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\noindent {\bf 34th IMO 1993}
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\noindent {\bf A1}. Let $f(x)=x^n+5x^{n-1}+3$, where $n>1$ is an integer. Prove that $f(x)$ cannot be expressed as the produce of two non-constant polynomials with integer coefficients.
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\noindent {\bf A2}. Let $D$ be a point inside the acute-angled triangle $ABC$ such that $\angle ADB=\angle ACB+90^o$, and $AC\cdot BD=AD\cdot BC$.

(a) Calculate the ratio $AB\cdot CD/(AC\cdot BD)$.

(b) Prove that the tangents at $C$ to the circumcircles of $ACD$ and $BCD$ are perpendicular.
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\noindent {\bf A3}. On an infinite chessboard a game is played as follows. At the start $n^2$ pieces are arranged in an $n\times n$ block of adjoining squares, one piece on each square. A move in the game is a jump in a horizontal or vertical direction over an adjacent occupied square to an unoccupied square immediately beyond. The piece which has been jumped over is removed. Find those values of $n$ for which the game can end with only one piece remaining on the board.
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\noindent {\bf B1}. For the points $P,Q,R$ in the plane define $m(PQR)$ as the minimum length of the three altitudes of the triangle $PQR$ (or zero if the points are collinear). Prove that for any points $A,B,C,X$: $$m(ABC)\le m(ABX)+m(AXC)+m(XBC).$$
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\noindent {\bf B2}. Does there exist a function $f$ from the positive integers to the positive integers such that $f(1)=2,f(f(n))=f(n)+n$ for all $n$, and $f(n)<f(n+1)$ for all $n$?
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\noindent {\bf B3}. There are $n>1$ lamps $L_0,L_1,\ldots ,L_{n-1}$ in a circle. We use $L_{n+k}$ to mean $L_k$. A lamp is at all times either on or off. Initially they are all on. Perform steps $s_0,s_1,\ldots$ as follows: at step $s_i$, if $L_{i-1}$ is lit, then switch $L_i$ from on to off or vice versa, otherwise do nothing. Show that:

(a) There is a positive integer $M(n)$ such that after $M(n)$ steps all the lamps are on again;

(b) If $n=2^k$, then we can take $M(n)=n^2-1$;

(c) If $n=2^k+1$, then we can take $M(n)=n^2-n+1$.
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\noindent \copyright John Scholes

\noindent jscholes@kalva.demon.co.uk

\noindent 19 August 2003

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