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Nick's Mathematical Puzzles: 51 to 60

51. Greatest common divisor (3 star)

Let a, m, and n be positive integers, with a > 1, and m odd.
What is the greatest common divisor of am − 1 and an + 1?

Hint  -  Answer  -  Solution

52. Floor function sum (2 star)

Let x be a real number and n be a positive integer.
Show that [x] + [x + 1/n] + ... + [x + (n−1)/n] = [nx], where [x] is the greatest integer less than or equal to x.

Hint  -  Solution

53. The absentminded professor (4 star)

An absentminded professor buys two boxes of matches and puts them in his pocket.  Every time he needs a match, he selects at random (with equal probability) from one or other of the boxes.  One day the professor opens a matchbox and finds that it is empty.  (He must have absentmindedly put the empty box back in his pocket when he took the last match from it.)  If each box originally contained n matches, what is the probability that the other box currently contains k matches?  (Where 0 less than or equal to k less than or equal to n.)

Hint  -  Answer  -  Solution

54. Diophantine squares (3 star)

Find all solutions to c2 + 1 = (a2 − 1)(b2 − 1), in integers a, b, and c.

Hint  -  Answer  -  Solution

55. Area of a trapezoid (1 star)

A trapezoid¹ is divided into four triangles by its diagonals.  Let the triangles adjacent to the parallel sides have areas A and B.  Find the area of the trapezoid in terms of A and B.

Trapezoid, divided into four triangles by its diagonals

(1) A trapezoid is a quadrilateral with at least one pair of parallel sides.  In some countries, such a quadrilateral is known as a trapezium.

Hint  -  Answer  -  Solution

56. Partition identity (4 star)

A partition of a positive integer n is a way if writing n as a sum of positive integers, ignoring the order of the summands.  For example, a partition of 7 is 3 + 2 + 1 + 1.

The table below shows all partitions of 5.  The number of 1s column shows how many times the number 1 occurs in each partition.  The number of distinct parts column shows how many distinct numbers occur in each partition.  The sum for each column, over all the partitions of 5, is shown at the foot of the table.

PartitionNumber of 1sNumber of distinct parts
501
4 + 112
3 + 202
3 + 1 + 122
2 + 2 + 112
2 + 1 + 1 + 132
1 + 1 + 1 + 1 + 151
Total:1212

Let a(n) be the number of 1s in all the partitions of n.  Let b(n) be the sum, over all partitions of n, of the number of distinct parts.  The above table demonstrates that a(5) = b(5).
Show that, for all n, a(n) = b(n).

Hint  -  Solution

57. Binomial coefficient divisibility (3 star)

Show that, for n > 0, the binomial coefficient  C(2n,n) = (2n)! / (n! n!)  is divisible by n + 1 and by 4n − 2.

Hint  -  Solution

58. Fifth power plus five (4 star)

Consecutive fifth powers (or, indeed, any powers) of positive integers are always relatively prime.  That is, for all n > 0, n5 and (n + 1)5 are relatively prime.  Are n5 + 5 and (n + 1)5 + 5 always relatively prime?  If not, for what values of n do they have a common factor, and what is that factor?

Hint  -  Answer  -  Solution

59. Triangle inequality (3 star)

A triangle has sides of length a, b, and c.  Show that   3/2 <= a/(b+c) + b/(c+a) + c/(a+b) < 2.

Hint  -  Solution

60. Sum of reciprocals (3 star)

Find the limit as n tends to infinity of 1/(n+1) + 1/(n+2) + ... + 1/(2n).

Hint  -  Answer  -  Solution

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Nick Hobson
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Last updated: May 6, 2003