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1977 Canadian MO Problems

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The seven problems were all on the same day.

Contents

Problem 1

If \displaystyle f(x)=x^2+x, prove that the equation \displaystyle 4f(a)=f(b) has no solutions in positive integers \displaystyle a and \displaystyle b.

Solution

Problem 2

Let \displaystyle O be the center of a circle and \displaystyle A be a fixed interior point of the circle different from \displaystyle O. Determine all points \displaystyle P on the circumference of the circle such that the angle \displaystyle OPA is a maximum.

Image:CanadianMO-1977-2.jpg


Solution

Problem 3

\displaystyle N is an integer whose representation in base \displaystyle b is \displaystyle 777. Find the smallest positive integer \displaystyle b for which \displaystyle N is the fourth power of an integer.

Solution

Problem 4

Solution

Problem 5

Solution

Problem 6

Solution

Problem 7

Solution

Resources

Looking for a challenging geometry text? Preparing for MATHCOUNTS or the AMC exams? Check out Art of Problem Solving's Introduction to Geometry by Richard Rusczyk.
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