1973 Canadian MO Problems
Contents
[hide]Problem 1
Solve the simultaneous inequalities, and ; i.e. find a single inequality equivalent to the two simultaneous inequalities.
What is the greatest integer that satisfies both inequalities and .
Give a rational number between and .
Express as a product of two integers neither of which is an integral multiple of .
Without the use of logarithm tables evaluate .
Problem 2
Find all real numbers that satisfy the equation . (Note: if if .)
Problem 3
Prove that if and are prime integers greater than , then is a factor of .
Problem 4
The figure shows a (convex) polygon with nine vertices. The six diagonals which have been drawn dissect the polygon into the seven triangles: . In how many ways can these triangles be labeled with the names so that is a vertex of triangle for ? Justify your answer.
Problem 5
For every positive integer , let .
For example, .
Prove that for
Problem 6
If and are fixed points on a given circle not collinear with center of the circle, and if is a variable diameter, find the locus of (the intersection of the line through and and the line through and ).
Problem 7
Observe that $\frac{1}{1}= \frac{1}{2}+\frac{1}{2};\quad \frac{1}{2}=\frac{1}{3}+\frac{1}{6};\quad \frac{1}{3}=\frac{1}{4}+\frac{1}{12};\qu...$ (Error compiling LaTeX. Unknown error_msg) State a general law suggested by these examples, and prove it.
Prove that for any integer greater than there exist positive integers and such that