1989 Canada National Olympiad

Part of Canada National Olympiad

Subcontests

(5)

1

Monkeys and Ladders

There are 5 monkeys and 5 ladders and at the top of each ladder there is a banana. A number of ropes connect the ladders, each rope connects two ladders. No two ropes are attached to the same rung of the same ladder. Each monkey starts at the bottom of a different ladder. The monkeys climb up the ladders but each time they encounter a rope they climb along it to the other ladder at the end of the rope and then continue climbing upwards. Show that, no matter how many ropes there are, each monkey gets a banana.

1

Sum of the Digit Sequence

Define \{ a_n \}_{n\equal{}1} as follows: a_1 \equal{} 1989^{1989}; \ a_n, n > 1, is the sum of the digits of a_{n\minus{}1}. What is the value of

a_5

1

Reflecting Points

ABC

be a right angled triangle of area 1. Let

A'B'C'

be the points obtained by reflecting

A,B,C

respectively, in their opposite sides. Find the area of

\triangle A'B'C'.

1

Ordering Integers

1,2,...,n

are placed in order so that each value is either strictly bigger than all the preceding values or is strictly smaller than all preceding values. In how many ways can this be done?

1

Permutations of the powers of two

Given the numbers 1,2,2^2, \ldots ,2^{n\minus{}1}, for a specific permutation \sigma \equal{} x_1,x_2, \ldots, x_n of these numbers we define S_1(\sigma) \equal{} x_1, S_2(\sigma)\equal{}x_1\plus{}x_2,

\ldots

and Q(\sigma)\equal{}S_1(\sigma)S_2(\sigma)\cdot \cdot \cdot S_n(\sigma). Evaluate

\sum 1/Q(\sigma)

, where the sum is taken over all possible permutations.