1991 Swedish Mathematical Competition

Part of Swedish Mathematical Competition

Subcontests

(6)

1

x_0 = 0, x_{k+1} = [(n - \sum_0^k x_i)/2

x_0, x_1, x_2, ...

x_0 = 0

x_{k+1} = [(n - \sum_0^k x_i)/2]

x_k = 0

for all sufficiently large

k

and that the sum of the non-zero terms

x_k

n-1

1

y - \sqrt{y} < x - 1/4 < y + \sqrt{y} if x - \sqrt{x} < y - 1/4 <= x + \sqrt{x}

x, y

are positive reals such that

x - \sqrt{x} \le y - 1/4 \le x + \sqrt{x}

y - \sqrt{y} \le x - 1/4 \le y + \sqrt{y}

1

moves on a permutation of 1, 2, ..., 8

x_1, x_2, ... , x_8

is a permutation of

1, 2, ..., 8

. A move is to take

x_3

x_8

and place it at the start to from a new sequence. Show that by a sequence of moves we can always arrive at

1, 2, ..., 8

1

exists equilateral with area <=1/4, inscribed in each triangle

Given any triangle, show that we can always pick a point on each side so that the three points form an equilateral triangle with area at most one quarter of the original triangle.

1

odd positive integers n such that in binary n has more 1s than n^2

Show that there are infinitely many odd positive integers

n

such that in binary

n

1

n^2

1

1/m + 1/n - 1/(mn) = 2/5

Find all positive integers

m, n

\frac{1}{m} + \frac{1}{n} - \frac{1}{mn} =\frac{2}{5}