4th Mexican Mathematical Olympiad Problems 1990

4th Mexican Mathematical Olympiad Problems 1990

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?

A2.  ABC is a triangle with ∠B = 90^o and altitude BH. The inradii of ABC, ABH, CBH are r, r₁, r₂. Find a relation between them.
A3.  Show that n^n-1 - 1 is divisible by (n-1)² for n > 2.
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.
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.
B3.  ABC is a triangle with ∠C = 90^o. 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.


Solutions

Problem 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?

Answer
252
Solution
The simplest approach is to start with the vertices next to BC and to mark the number of paths for each, then move the vertices one further away, and so on.

2 
  4
2   8
  4   16
2   8    32
  4   16    62
2   8    30    112
  4   14    50     182
2   6    20     70     252
  2    6    20      70    252 

Problem A2
ABC is a triangle with ∠B = 90^o and altitude BH. The inradii of ABC, ABH, CBH are r, r₁, r₂. Find a relation between them.
Answer
r² = r₁² + r₂²
Solution
The triangles are similar so r: r₁ : r₂ = AC : AB : BC. Hence r² = r₁² + r₂².

Problem A3
Show that n^n-1 - 1 is divisible by (n-1)² for n > 2.
Solution
n^n-1 - 1 = (n-1)(n^n-2 + n^n-3 + ... + 1). Subtract 1 + 1 + ... + 1 = n-1 from (n^n-2 + n^n-3 + ... + 1) and we get ∑ n^k - 1 = (n-1) ∑ (n^k-1 + ... + 1).

Problem 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.
Answer
13½

Problem 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.
Solution
Put the points into nine sets according to the residues of the coordinates mod 3, so the sets are {(a,b): a = b = 0 mod 3}, {(a,b): a = 0, b = 1 mod 3} etc. Then one set must have three points. Take these three points (a,b), (a',b'), (a",b"), then the centroid coordinates ((a+a'+a")/3, (b+b'+b")/3) are integers.