# Nick's Mathematical Puzzles: 111 to 120

## 111. Trigonometric progression

`Hint  -  Solution`

## 112. Angle bisector

ABC is right-angled at A.  The angle bisector from A meets BC at D, so that DAB = 45°.  If CD = 1 and BD = AD + 1, find the lengths of AC and AD.

`Hint  -  Answer  -  Solution`

## 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 a2 + b2 + c2 a3 + b3 + c3.

`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, 25 = 32.  On deleting the initial 3, we are left with 2 = 21.)  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, sin2A + sin2B = 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`