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1975 Canadian MO Problems

Problem 1

Simplify \[\left(\frac{1\cdot2\cdot4+2\cdot4\cdot8+\cdots+n\cdot2n\cdot4n}{1\cdot3\cdot9+2\cdot6\cdot18+\cdots+n\cdot3n\cdot9n}\right)^{1/3}\].

Solution

Problem 2

A sequence of numbers $a_1, a_2, a_3, \dots$ satisfies

(i) $a_1 = \frac{1}{2}$
(ii) $a_1+a_2+\cdots+a_n=n^2a_n\quad(n\ge1).$

Determine the value of $a_n\quad(n\ge1).$

Solution

Problem 3

For each real number $r$, $[r]$ denotes the largest integer less than or equal to $r$, $e.g.,$ $[6] = 6, [\pi] = 3, [-1.5] = -2.$ Indicate on the $(x,y)$-plane the set of all points $(x,y)$ for which $[x]^2+[y]^2 = 4$.

Solution

Problem 4

For a positive number such as $3.27$, $3$ is referred to as the integral part of the number and $.27$ as the decimal part. Find a positive number such that its decimal part, its integral part, and the number itself form a geometric progression.

Solution

Problem 5

$A, B, C, D$ are four "consecutive" points on the circumference of a circle and $P, Q, R, S$ are points on the circumference which are respectively the midpoints of the arcs $AB, BC, CD, DA.$ Prove that $PR$ is perpendicular to $QS$.

Solution

Problem 6

(i)$15$ chairs are equally place around a circular table on which are name cards for $15$ guests. The guests fail to notice these cards until after they have sat down, and it turns out that no one is sitting in the correct seat. Prove that the table can be rotated so that at least two of the guests are simultaneously correctly seated.
(ii) Give an example of an arrangement in which just one of the 15 quests is correctly seated and for which no rotation correctly places more than one person.

Solution

Problem 7

A function $f(x)$ is $\textit{periodic}$ if there is a positive integer such that $f(x+p) = f(x)$ for all $x$. For example, $\sin x$ is periodic with period $2\pi$. Is the function $\sin(x^2)$ periodic? Prove your assertion.

Solution

Problem 8

Let $k$ be a positive integer. Find all polynomials \[P(x) = a_0+a_1x+\cdots+a_nx^n\] where the $a_i$ are real, which satisfy the equation \[P(P(x)) = \{P(x)\}^k\].

Solution

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