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Asian Pacific Math Olympiad Problems

The 1994 Asian Pacific Mathematical Olympiad consisted of 5 questions worth 7 points each. Question 1 involved finding all functions f(x) that satisfy three properties relating to addition and values between 0 and 1. Question 2 asked to prove that the distance between a triangle's orthocentre and circumcentre is always less than 3 times the circumradius. Question 3 involved determining all integers of the form a^2 + b^2 where a and b are relatively prime and satisfy a certain divisibility property. Question 4 asked if there exists an infinite set of points in the plane where no three are collinear and all distances are rational. Question 5 involved proving a property about numbers in base 10, 2 and 5 representations.

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68 views1 page

Asian Pacific Math Olympiad Problems

The 1994 Asian Pacific Mathematical Olympiad consisted of 5 questions worth 7 points each. Question 1 involved finding all functions f(x) that satisfy three properties relating to addition and values between 0 and 1. Question 2 asked to prove that the distance between a triangle's orthocentre and circumcentre is always less than 3 times the circumradius. Question 3 involved determining all integers of the form a^2 + b^2 where a and b are relatively prime and satisfy a certain divisibility property. Question 4 asked if there exists an infinite set of points in the plane where no three are collinear and all distances are rational. Question 5 involved proving a property about numbers in base 10, 2 and 5 representations.

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george
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© © All Rights Reserved
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THE 1994 ASIAN PACIFIC MATHEMATICAL OLYMPIAD

Time allowed: 4 hours


NO calculators are to be used.
Each question is worth seven points.

Question 1
Let f : R ! R be a function such that
(i) For all x; y 2 R,
f (x) + f (y) + 1  f (x + y)  f (x) + f (y);
(ii) For all x 2 [0; 1), f (0)  f (x),
(iii) ;f (;1) = f (1) = 1.
Find all such functions f .
Question 2
Given a nondegenerate triangle ABC , with circumcentre O, orthocentre H , and circumradius
R, prove that jOH j < 3R.
Question 3
Let n be an integer of thepform a2 + b2, where a and b are relatively prime integers and such
that if p is a prime, p  n, then p divides ab. Determine all such n.
Question 4
Is there an in nite set of points in the plane such that no three points are collinear, and the
distance between any two points is rational?
Question 5
You are given three lists A, B, and C. List A contains the numbers of the form 10k in base
10, with k any integer greater than or equal to 1. Lists B and C contain the same numbers
translated into base 2 and 5 respecitvely:
A B C
10 1010 20
100 1100100 400
1000 1111101000 13000
... ... ...
Prove that for every integer n > 1, there is exactly one number in exactly one of the sets B
or C that has exactly n digits.

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