In how many ways, counting ties, can eight horses cross the finishing line?

(For example, two horses, A and B, can finish in three ways: A wins, B wins, A and B tie.)

Hint - Answer - Solution

In ABC, draw AD, where D is the midpoint of BC.

If ACB = 30° and ADB = 45°, find ABC.

Hint - Answer - Solution

A sequence of integers is defined by

- a
_{0}= p, where p > 0 is a prime number, - a
_{n+1}= 2a_{n}+ 1, for n = 0, 1, 2, ... .

Is there a value of p such that the sequence consists entirely of prime numbers?

Hint - Answer - Solution

If the equation x^{4} − x^{3} + x + 1 = 0 has roots a, b, c, d, show that 1/a + 1/b is a root of x^{6} + 3x^{5} + 3x^{4} + x^{3} − 5x^{2} − 5x − 2 = 0.

Hint - Solution

The minute hand of a clock is twice as long as the hour hand. At what time, between 00:00 and when the hands are next aligned (just after 01:05), is the distance between the tips of the hands increasing at its greatest rate?

Hint - Answer - Solution

Point P lies inside ABC, and is such that PAC = 18°, PCA = 57°, PAB = 24°, and PBA = 27°.

Show that ABC is isosceles.

Hint - Solution

Observe that

- (2 − 1)! + 1 = 2
^{1}, - (3 − 1)! + 1 = 3
^{1}, - (5 − 1)! + 1 = 5
^{2}.

Are there any other primes p such that (p − 1)! + 1 is a power of p?

Hint 1 - Hint 2 - Answer - Solution

Find all positive real numbers x such that both + 1/ and + 1/ are integers.

Hint - Answer - Solution

The towns of Alpha, Beta, and Gamma are equidistant from each other. If a car is three miles from Alpha and four miles from Beta, what is the maximum possible distance of the car from Gamma? Assume the land is flat.

Hint - Answer - Solution

The smallest distance between any two of six towns is m miles. The largest distance between any two of the towns is M miles. Show that M/m . Assume the land is flat.

Hint - Solution

Nick Hobson

nickh@qbyte.org