# Nick's Mathematical Puzzles: 131 to 140

## 131. Horse race

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

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

If ACB = 30° and ADB = 45°, find ABC.

`Hint  -  Answer  -  Solution`

## 133. Prime sequence?

A sequence of integers is defined by

• a0 = p, where p > 0 is a prime number,
• an+1 = 2an + 1, for n = 0, 1, 2, ... .

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

`Hint  -  Answer  -  Solution`

## 134. Sum of reciprocal roots

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

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

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`

## 137. Factorial plus one equals prime power?

Observe that

• (2 − 1)! + 1 = 21,
• (3 − 1)! + 1 = 31,
• (5 − 1)! + 1 = 52.

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

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

`Hint  -  Answer  -  Solution`

## 139. Three towns

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

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`