Nick's Mathematical Puzzles: 101 to 110

101. Right triangles

ABC is right-angled at A. D is a point on AB such that CD = 1. AE is the altitude from A to BC. If BD = BE = 1, what is the length of AD?

Triangle ABC, with cevians as described above.

Hint 1  -  Hint 2  -  Answer  -  Solution

102. Almost exponential

Show that 1 + x + x²/2! + x³/3! + ... + x²ⁿ/(2n)! is positive for all real values of x.

Hint  -  Solution

103. Root sums

Let a, b, c be rational numbers. Show that each of the following equations can be satisfied only if a = b = c = 0.

a + b + c = 0.
a + b + c = 0.
a + b + c = 0.

Hint  -  Solution

104. An arbitrary sum

The first 2n positive integers are arbitrarily divided into two groups of n numbers each. The numbers in the first group are sorted in ascending order: a₁ < a₂ < ... < a_n; the numbers in the second group are sorted in descending order: b₁ > b₂ > ... > b_n.

Find, with proof, the value of the sum |a₁ − b₁| + |a₂ − b₂| + ... + |a_n − b_n|.

Hint  -  Answer  -  Solution

105. Difference of nth powers

Let x, y, n be positive integers, with n > 1. How many solutions are there to the equation xⁿ − yⁿ = 2¹⁰⁰?

Hint  -  Answer  -  Solution

106. Flying cards

A standard pack of cards is thrown into the air in such a way that each card, independently, is equally likely to land face up or face down. The total value of the cards which landed face up is then calculated. (Card values are assigned as follows: Ace=1, 2=2, ... , 10=10, Jack=11, Queen=12, King=13. There are no jokers.)

What is the probability that the total value is divisible by 13?

Hint 1  -  Hint 2  -  Answer  -  Solution

107. A mysterious sequence

A sequence of positive real numbers is defined by

a₀ = 1,
a_n+2 = 2a_n − a_n+1, for n = 0, 1, 2, ... .

Find a₂₀₀₅.

Hint  -  Answer  -  Solution

108. Eyeball to eyeball

Take two circles, with centers O and P. From the center of each circle, draw two tangents to the circumference of the other circle. Let the tangents from O intersect that circle at A and B, and the tangents from P intersect that circle at C and D. Show that chords AB and CD are of equal length.

Two circles, with tangents and chords as described above.

Hint  -  Solution

109. Nested circular functions

Let x be a real number. Which is greater, sin(cos x) or cos(sin x)?

Hint  -  Answer  -  Solution

110. Pairwise products

Let n be a positive integer, and let S_n = {n² + 1, n² + 2, ... , (n + 1)²}. Find, in terms of n, the cardinality of the set of pairwise products of distinct elements of S_n.

For example, S₂ = {5, 6, 7, 8, 9},

5 × 6 = 6 × 5 = 30,
5 × 7 = 7 × 5 = 35,
5 × 8 = 8 × 5 = 40,
5 × 9 = 9 × 5 = 45,
6 × 7 = 7 × 6 = 42,
6 × 8 = 8 × 6 = 48,
6 × 9 = 9 × 6 = 54,
7 × 8 = 8 × 7 = 56,
7 × 9 = 9 × 7 = 63,
8 × 9 = 9 × 8 = 72,

and the required cardinality is 10.

Hint  -  Answer  -  Solution