# Nick's Mathematical Puzzles: 121 to 130

## 121. Integer sequence

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

If a6 = 8820, what is a7?

`Hint  -  Answer  -  Solution`

## 122. Powers of 2 and 5

If the numbers 2n and 5n (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`

## 123. Right angle and median

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`

## 125. Divisibility by 446617991732222310

Show that, for all integers m and n, mn(m420 − n420) is divisible by 446617991732222310.

`Hint  -  Solution`

## 126. Intersecting squares

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`

## 127. Prime number generator

Let P = {p1, ... , pn} 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 pn+12 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 p52 = 121 and greater than 1, shown in bold, are all prime.

Prime number generator example
SS'ABA + B|A − B|
Empty set{2, 3, 5, 7}1210211209
{2}{3, 5, 7}2105107103
{3}{2, 5, 7}3707367
{5}{2, 3, 7}5424737
{7}{2, 3, 5}7303723
{2, 3}{5, 7}6354129
{2, 5}{3, 7}10213111
{2, 7}{3, 5}1415291
`Hint  -  Solution`

## 128. Modular equation

For how many integers n > 1 is x49 x (modulo n) true for all integers x?

`Hint  -  Answer  -  Solution`

## 129. Abelian group

Let G be a group with the following two properties:

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

Prove that G is abelian.

`Hint  -  Solution`

## 130. Reciprocal polynomial?

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`