# Nick's Mathematical Puzzles: 111 to 120

## 113. Ant in a field

An ant, located in a square field, is 13 meters from one of the corner posts of the field, 17 meters from the corner post diagonally opposite that one, and 20 meters from a third corner post. Find the area of the field. Assume the land is flat.

Hint - Answer - Solution

## 114. Sums of squares and cubes

Let a, b, and c be positive real numbers such that abc = 1. Show that a^{2} + b^{2} + c^{2} a^{3} + b^{3} + c^{3}.

Hint - Solution

## 115. Sum of sines

Let *f*(x) = sin(x) + sin(x°), with domain the real numbers. Is *f* a periodic function?

(Note: sin(x) is the sine of a real number, x, (or, equivalently, the sine of x radians), while sin(x°) is the sine of x degrees.)

Hint - Answer - Solution

## 116. Factorial divisors

Show that, for each n 3, n! can be represented as the sum of n distinct divisors of itself. (For example, 3! = 1 + 2 + 3.)

Hint - Solution

## 117. Random point in an equilateral triangle

A point P is chosen at random inside an equilateral triangle of side length 1. Find the expected value of the sum of the (perpendicular) distances from P to the three sides of the triangle.

Hint - Answer - Solution

## 118. Powers of 2: deleted digit

Find all powers of 2 such that, after deleting the first digit, another power of 2 remains. (For example, 2^{5} = 32. On deleting the initial 3, we are left with 2 = 2^{1}.) Numbers are written in standard decimal notation, with no leading zeroes.

Hint - Answer - Solution

## 119. Three sines

A triangle has two acute angles, A and B. Show that the triangle is right-angled if, and only if, sin^{2}A + sin^{2}B = sin(A + B).

Hint - Solution

## 120. Factorial plus one

Let n be a positive integer. Prove that n! + 1 is composite for infinitely many values of n.

Hint 1 - Hint 2 - Solution

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Nick Hobson

nickh@qbyte.org
*Last updated: September 19, 2005*