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\noindent {\bf 2nd IMO 1960}
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\noindent {\bf A1}. Determine all three digit numbers $N$ which are divisible by $11$ and where $N\over11$ is equal to the sum of the squares of the digits of $N$.
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\noindent {\bf A2}. For what real values of $x$ does the following inequality hold: $${4x^2\over(1-\sqrt{1+2x})^2}<2x+9?$$
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\noindent {\bf A3}. In a given right triangle $ABC$, the hypoteneuse $BC$, length $a$, is divided into $n$ equal parts with $n$ an odd integer. The central part subtends an angle $\alpha$ at $A$. $h$ is the perpendicular distance from $A$ to $BC$. Prove that $$\tan\alpha={4nh\over an^2-a}.$$
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\noindent {\bf B1}. Construct a triangle $ABC$ given the lengths of the altitudes from $A$ and $B$ and the length of the median from $A$.
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\noindent {\bf B2}. The cube $ABCDA'B'C'D'$ has $A$ above $A',B$ above $B'$ and so on. $X$ is any point of the face diagonal $AC$ and $Y$ is any point of $B'D'$.

(a) find the locus of the midpoint of $XY$;

(b) find the locus of the point $Z$ which lies one-third of the way along $XY$, so that $ZY=2XZ$.
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\noindent {\bf B3}. A cone of revolution has an inscribed sphere tangent to the base of the cone (and to the sloping surface of the cone). A cylinder is circumscribed about the sphere  so that its base lies in the base of the cone. The volume of the cone is $V_1$ and the volume of the cylinder is $V_2$.

(a) Prove that $V_1\ne V_2$;

(b) Find the smallest possible value of $V_1\over V_2$. For this case construct the half angle of the cone.
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\noindent {\bf B4}. In the isosceles trapezoid $ABCD$ ($AB$ parallel to $DC$, and $BC=AD$), let $AB=a,CD=c$ and let the perpendicular distance from $A$ to $CD$ be $h$. Show how to construct all points $X$ on the axis of symmetry such that $\angle BXC=\angle AXD=90^o$. Find the distance of each such $X$ from $AB$ and from $CD$. What is the condition for such points to exist?
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\noindent \copyright John Scholes

\noindent jscholes@kalva.demon.co.uk

\noindent 19 August 2003

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