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Nick's Mathematical Puzzles: 131 to 140

131. Horse race (3 star)

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

132. Triangular angle (1 star)

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

Triangle ABC, with median AD, and angles as described below.

If angleACB = 30° and angleADB = 45°, find angleABC.

Hint  -  Answer  -  Solution

133. Prime sequence? (2 star)

A sequence of integers is defined by

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

Hint  -  Answer  -  Solution

134. Sum of reciprocal roots (3 star)

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.

Hint  -  Solution

135. Clock hands (2 star)

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

136. Point in a triangle (3 star)

Point P lies inside triangleABC, and is such that anglePAC = 18°, anglePCA = 57°, anglePAB = 24°, and anglePBA = 27°.

Triangle ABC, with internal point P, and angles as described above.

Show that triangleABC is isosceles.

Hint  -  Solution

137. Factorial plus one equals prime power? (4 star)

Observe that

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

Hint 1  -  Hint 2  -  Answer  -  Solution

138. Integer sum of roots (2 star)

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

Hint  -  Answer  -  Solution

139. Three towns (2 star)

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

140. Six towns (4 star)

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 greater than or equal to root 3.  Assume the land is flat.

Hint  -  Solution

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Nick Hobson
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Last updated: April 26, 2006