Difference between revisions of "1975 Canadian MO Problems"
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− | <cmath>\left(\frac{1\cdot2\ | + | <cmath>\left(\frac{1\cdot2\cdot4+2\cdot4\cdot8+\cdots+n\cdot2n\cdot4n}{1\cdot3\cdot9+2\cdot6\cdot18+\cdots+n\cdot3n\cdot9n}\right)^{1/3}</cmath>. |
[[1975 Canadian MO Problems/Problem 1 | Solution]] | [[1975 Canadian MO Problems/Problem 1 | Solution]] |
Revision as of 14:47, 5 August 2016
Contents
[hide]Problem 1
Simplify .
Problem 2
A sequence of numbers satisfies
Determine the value of
Problem 3
For each real number , denotes the largest integer less than or equal to , Indicate on the -plane the set of all points for which .
Problem 4
For a positive number such as , is referred to as the integral part of the number and as the decimal part. Find a positive number such that its decimal part, its integral part, and the number itself form a geometric progression.
Problem 5
are four "consecutive" points on the circumference of a circle and are points on the circumference which are respectively the midpoints of the arcs Prove that is perpendicular to .
Problem 6
Problem 7
A function is if there is a positive integer such that for all . For example, is periodic with period . Is the function periodic? Prove your assertion.
Problem 8
Let be a positive integer. Find all polynomials where the are real, which satisfy the equation .