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?

Hint 1 - Hint 2 - Answer - Solution

Show that 1 + x + x^{2}/2! + x^{3}/3! + ... + x^{2n}/(2n)! is positive for all real values of x.

Hint - Solution

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

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: a_{1} < a_{2} < ... < a_{n}; the numbers in the second group are sorted in descending order: b_{1} > b_{2} > ... > b_{n}.

Find, with proof, the value of the sum |a_{1} − b_{1}| + |a_{2} − b_{2}| + ... + |a_{n} − b_{n}|.

Hint - Answer - Solution

Let x, y, n be positive integers, with n > 1. How many solutions are there to the equation x^{n} − y^{n} = 2^{100}?

Hint - Answer - Solution

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

A sequence of positive real numbers is defined by

- a
_{0}= 1, - a
_{n+2}= 2a_{n}− a_{n+1}, for n = 0, 1, 2, ... .

Find a_{2005}.

Hint - Answer - Solution

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.

Hint - Solution

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

Hint - Answer - Solution

Let n be a positive integer, and let S_{n} = {n^{2} + 1, n^{2} + 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, S_{2} = {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

Nick Hobson

nickh@qbyte.org