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Nick's Mathematical Puzzles: 111 to 120

111. Trigonometric progression (3 star)

Show that tan [(n+1)a/2] = (sin a + sin 2a + ... + sin na)/(cos a + cos 2a + ... + cos na)
Hint  -  Solution

112. Angle bisector (3 star)

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

Right triangle ABC, with angle bisector and lengths as described above.
Hint  -  Answer  -  Solution

113. Ant in a field (2 star)

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.

Square field, containing an ant, as described above.
Hint  -  Answer  -  Solution

114. Sums of squares and cubes (4 star)

Let a, b, and c be positive real numbers such that abc = 1.  Show that a2 + b2 + c2 less than or equal to a3 + b3 + c3.

Hint  -  Solution

115. Sum of sines (1 star)

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 (2 star)

Show that, for each n greater than or equal to 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 (2 star)

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.

Equilateral triangle containing point P.
Hint  -  Answer  -  Solution

118. Powers of 2: deleted digit (3 star)

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 (3 star)

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 (2 star)

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
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Last updated: September 19, 2005