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.)
In ABC, draw AD, where D is the midpoint of BC.
If ACB = 30° and ADB = 45°, find ABC.
A sequence of integers is defined by
Is there a value of p such that the sequence consists entirely of prime numbers?
If the equation x4 − x3 + x + 1 = 0 has roots a, b, c, d, show that 1/a + 1/b is a root of x6 + 3x5 + 3x4 + x3 − 5x2 − 5x − 2 = 0.
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?
Point P lies inside ABC, and is such that PAC = 18°, PCA = 57°, PAB = 24°, and PBA = 27°.
Show that ABC is isosceles.
Are there any other primes p such that (p − 1)! + 1 is a power of p?
Find all positive real numbers x such that both + 1/ and + 1/ are integers.
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.
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.
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