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A1. How many paths are there from A to the line BC if the path does not go through any vertex twice and always moves to the left?
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| A2. ABC is a triangle with ∠B = 90o and altitude BH. The inradii of ABC, ABH, CBH are r, r1, r2. Find a relation between them. |
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| A3. Show that nn-1 - 1 is divisible by (n-1)2 for n > 2. |
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| B1. Find 0/1 + 1/1 + 0/2 + 1/2 + 2/2 + 0/3 + 1/3 + 2/3 + 3/3 + 0/4 + 1/4 + 2/4 + 3/4 + 4/4 + 0/5 + 1/5 + 2/5 + 3/5 + 4/5 + 5/5 + 0/6 + 1/6 + 2/6 + 3/6 + 4/6 + 5/6 + 6/6. |
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| B2. Given 19 points in the plane with integer coordinates, no three collinear, show that we can always find three points whose centroid has integer coordinates. |
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B3. ABC is a triangle with ∠C = 90o. E is a point on AC, and F is the midpoint of EC. CH is an altitude. I is the circumcenter of AHE, and G is the midpoint of BC. Show that ABC and IGF are similar.
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To avoid possible copyright problems, I have changed the wording, but not the substance, of the problems.
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© John Scholes
jscholes@kalva.demon.co.uk
24 February 2004
Last corrected/updated 24 Feb 04