# Nick's Mathematical Puzzles: 71 to 80

## 71. Consecutive cubes and squares

Show that if the difference of the cubes of two consecutive integers is the square of an integer, then this integer is the sum of the squares of two consecutive integers.

(The smallest non-trivial example is: 8^{3} − 7^{3} = 169. This is the square of an integer, namely 13, which can be expressed as 2^{2} + 3^{2}.)

Hint - Solution

## 72. Depleted harmonic series

It is well known that the harmonic series, 1/1 + 1/2 + 1/3 + 1/4 + ... , diverges. Consider a *depleted* harmonic series; see below; which contains only terms whose denominator does not contain a 9. (In decimal representation.) Does this series diverge or converge?

S = 1/1 + 1/2 + ... + 1/8 + 1/10 + ... + 1/18 + 1/20 + ... + 1/88 + 1/100 + 1/101 + ...

Hint - Answer - Solution

## 73. Unobtuse triangle

A triangle has internal angles A, B, and C, none of which exceeds 90°. Show that

- sin A + sin B + sin C > 2
- cos A + cos B + cos C > 1
- tan (A/2) + tan (B/2) + tan (C/2) < 2

Hint - Solution

## 74. Sum of 9999 consecutive squares

Show that the sum of 9999 consecutive squares cannot be a perfect power.

That is, show that (n + 1)^{2} + ... + (n + 9999)^{2} = m^{r} has no solution in integers n, m, r > 1.

Hint - Solution

## 75. Car journey

A car travels downhill at 72 mph (miles per hour), on the level at 63 mph, and uphill at only 56 mph The car takes 4 hours to travel from town A to town B. The return trip takes 4 hours and 40 minutes.

Find the distance between the two towns.

Hint - Answer - Solution

## 76. Square inscribed in a triangle

A triangle has sides 10, 17, and 21. A square is inscribed in the triangle. One side of the square lies on the longest side of the triangle. The other two vertices of the square touch the two shorter sides of the triangle. What is the length of the side of the square?

Hint - Answer - Solution

## 78. Perfect square

Find all integer solutions of y^{2} = x^{3} − 432.

Hint - Answer - Solution

## 79. Sum of fourth powers

The sum of three numbers is 6, the sum of their squares is 8, and the sum of their cubes is 5. What is the sum of their fourth powers?

Hint - Answer - Solution

## 80. Sixes and sevens

Does there exist a (base 10) 67-digit multiple of 2^{67}, written exclusively with the digits 6 and 7?

Hint - Answer - Solution

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Nick Hobson

nickh@qbyte.org
*Last updated: March 17, 2004*