1985 Canada National Olympiad

Part of Canada National Olympiad

Subcontests

(5)

1

Show that |x_n - √2| < 1/2^n for n>2

1 < x_1 < 2

n = 1

\dots

x_{n + 1} = 1 + x_n - \frac{1}{2} x_n^2

. Prove that, for

n \ge 3

|x_n - \sqrt{2}| < 2^{-n}

1

2^(n-1) | n! if and only if n is a power of 2 [Canada 1985]

2^{n - 1}

n!

n = 2^{k - 1}

for some positive integer

k

1

Sum of the perimeters is at least twice the circumference

P_1

P_2

be regular polygons of 1985 sides and perimeters

x

y

respectively. Each side of

P_1

is tangent to a given circle of circumference

c

and this circle passes through each vertex of

P_2

x + y \ge 2c

. (You may assume that

\tan \theta \ge \theta

0 \le \theta < \frac{\pi}{2}

1

An integer which is doubled by moving its first digit to end

Prove or disprove that there exists an integer which is doubled when the initial digit is transferred to the end.

1

Unique line which bisects perimeter and area [Canada 1985]

The lengths of the sides of a triangle are 6, 8 and 10 units. Prove that there is exactly one straight line which simultaneously bisects the area and perimeter of the triangle.