Читать книгу The Ultimate Mathematical Challenge: Over 365 puzzles to test your wits and excite your mind - Литагент HarperCollins USD, F. M. L. Thompson - Страница 36

Week 20

134. nth term n

The first term of a sequence of positive integers is 6. The other terms in the sequence follow these rules:

 if a term is even then divide it by 2 to obtain the next term;

 if a term is odd then multiply it by 5 and subtract 1 to obtain the next term.

For which values of n is the nth term equal to n?

[SOLUTION]

135. How many numbers?

Rafael writes down a five-digit number whose digits are all distinct, and whose first digit is equal to the sum of the other four digits.

How many five-digit numbers with this property are there?

[SOLUTION]

136. A square area

A regular octagon is placed inside a square, as shown.

The shaded square connects the midpoints of four sides of the octagon.

What fraction of the outer square is shaded?

[SOLUTION]

137. ODD plus ODD is EVEN

Find all possible solutions to the ‘word sum’ shown.

Each letter stands for one of the digits 0−9 and has the same meaning each time it occurs. Different letters stand for different digits. No number starts with a zero.

[SOLUTION]

138. Only odd digits

How many three-digit multiples of 9 consist only of odd digits?

[SOLUTION]

139. How many tests?

Before the last of a series of tests, Sam calculated that a mark of 17 would enable her to average 80 over the series, but that a mark of 92 would raise her average mark over the series to 85.

How many tests were in the series?

[SOLUTION]

140. Three primes

Find all positive integers p such that p, p + 8 and p + 16 are all prime.

[SOLUTION]