The terms of a sequence of positive integers satisfy a_{n+3} = a_{n+2}(a_{n+1} + a_{n}), for n = 1, 2, 3, ... .

If a_{6} = 8820, what is a_{7}?

Hint - Answer - Solution

If the numbers 2^{n} and 5^{n} (where n is a positive integer) start with the same digit, what is this digit? The numbers are written in decimal notation, with no leading zeroes.

Hint - Answer - Solution

Let ABC be a triangle, with AB AC. Drop a perpendicular from A to BC, meeting at O. Let AD be the median joining A to BC. If OAB = CAD, show that CAB is a right angle.

Hint - Solution

A ladder, leaning against a building, rests upon the ground and just touches a box, which is flush against the wall and the ground. The box has a height of 64 units and a width of 27 units.

Find the length of the ladder so that there is only one position in which it can touch the ground, the box, and the wall.

Hint - Answer - Solution

Show that, for all integers m and n, mn(m^{420} − n^{420}) is divisible by 446617991732222310.

Hint - Solution

The sides of two squares (not necessarily of the same size) intersect in eight distinct points: A, B, C, D, E, F, G, and H. These eight points form an octagon. Join opposite pairs of vertices to form two non-adjacent diagonals. (For example, diagonals AE and CG.) Show that these two diagonals are perpendicular.

Hint - Solution

Let P = {p_{1}, ... , p_{n}} be the set of the first n prime numbers. Let S be an arbitrary (possibly empty) subset of P. Let A be the product of the elements of S, and B the product of the elements of S', the complement of S. (An empty product is assigned the value of 1.)

Prove that each of A + B and |A − B| is prime, provided that it is less than p_{n+1}^{2} and greater than 1.

For example, if P = {2, 3, 5, 7}, the table below shows all the distinct possibilities for A + B and |A − B|. Values of A + B and |A − B| that are less than p_{5}^{2} = 121 and greater than 1, shown in bold, are all prime.

S | S' | A | B | A + B | |A − B| |
---|---|---|---|---|---|

Empty set | {2, 3, 5, 7} | 1 | 210 | 211 | 209 |

{2} | {3, 5, 7} | 2 | 105 | 107 | 103 |

{3} | {2, 5, 7} | 3 | 70 | 73 | 67 |

{5} | {2, 3, 7} | 5 | 42 | 47 | 37 |

{7} | {2, 3, 5} | 7 | 30 | 37 | 23 |

{2, 3} | {5, 7} | 6 | 35 | 41 | 29 |

{2, 5} | {3, 7} | 10 | 21 | 31 | 11 |

{2, 7} | {3, 5} | 14 | 15 | 29 | 1 |

Hint - Solution

For how many integers n > 1 is x^{49} x (modulo n) true for all integers x?

Hint - Answer - Solution

Let G be a group with the following two properties:

- (i) For all x, y in G, (xy)
^{2}= (yx)^{2}, - (ii) G has no element of order 2.

Prove that G is abelian.

Hint - Solution

Let p be a polynomial of degree n with complex coefficients. Is there a value of n such that the equations

- p(1) = 1/1,
- p(2) = 1/2,

... - p(n) = 1/n,
- p(n + 1) = 1/(n + 1),
- p(n + 2) = 1/(n + 2),

can be satisfied simultaneously?

Hint - Answer - Solution

Nick Hobson

nickh@qbyte.org