Nick's Mathematical Puzzles: 141 to 150

141. Alternating series

Consider the alternating series f(x) = x − x² + x⁴ − x⁸ + ... + (−1)ⁿx^(2ⁿ) + ... , which converges for |x| < 1. Does the limit of f(x) as x approaches 1 from below exist, and if so what is it?

Hint  -  Answer  -  Solution

142. Sum of two nth powers

Let a, b, n, and m be positive integers, with n > 1. Show that aⁿ + bⁿ = 2^m a = b.

Hint  -  Solution

143. Semicircle in a triangle

In ABC, side AB = 20, AC = 11, and BC = 13. Find the diameter of the semicircle inscribed in ABC, whose diameter lies on AB, and that is tangent to AC and BC.

Triangle ABC, with inscribed semicircle.

Hint  -  Answer  -  Solution

144. Difference of two nth powers

Let a, b, and n be positive integers, with a b. Show that n divides aⁿ − bⁿ n divides (aⁿ − bⁿ)/(a − b).

Hint  -  Solution

145. Heads and tails

A fair coin is tossed n times and the outcome of each toss is recorded. Find the probability that in the resulting sequence of tosses a head immediately follows a head exactly h times and a tail immediately follows a tail exactly t times. (For example, for the sequence HHHTTHTHH, we have n = 9, h = 3, and t = 1.)

Hint  -  Answer  -  Solution

146. Odds and evens

A and B play a game in which they alternate calling out positive integers less than or equal to n, according to the following rules:

A goes first and always calls out an odd number.
B always calls out an even number.
Each player must call out a number which is greater than the previous number. (Except for A's first turn.)
The game ends when one player cannot call out a number.

Some example games (for n = 8):

1, 8
3, 4, 5, 8
1, 2, 3, 4, 5, 6, 7, 8

The length of a game is defined as the number of numbers called out. For example, the game 1, 8, above, has length 2.

How many different possible games are there?
How many different possible games of length k are there?

Hint  -  Answer  -  Solution

147. Prime or composite 2

Is the number prime or composite?

Hint  -  Answer  -  Solution

148. Power series

Find the power series (expanded about x = 0) for .

Hint  -  Answer  -  Solution

149. Ones and nines

Show that all the divisors of any number of the form 19...9 (with an odd number of nines) end in 1 or 9. For example, the numbers 19, 1999, 199999, and 19999999 are prime (so clearly the property holds), and the (positive) divisors of 1999999999 are 1, 31, 64516129 and 1999999999 itself.

Show further that this property continues to hold if we insert an equal number of zeroes before the nines. For example, the numbers 109, 1000999, 10000099999, 100000009999999, and 1000000000999999999 are prime, and the (positive) divisors of 10000000000099999999999 are 1, 19, 62655709, 1190458471, 8400125030569, 159602375580811, 526315789478947368421, and 10000000000099999999999 itself.

(Dario Alpern's Java applet Factorization using the Elliptic Curve Method may be useful in obtaining divisors of large numbers.)

Hint  -  Solution

150. Isosceles apex angle

Triangle ABC is isosceles with AB = AC. Point D on AB is such that BCD = 15° and BC = AD. Find, with proof, the measure of CAB.

Isosceles triangle ABC, with AB = AC, and as described above.

Hint  -  Answer  -  Solution