# Nick's Mathematical Puzzles: 91 to 100

## 91. Nonagon diagonals

In regular nonagon ABCDEFGHI, show that AF = AB + AC.

Hint - Solution

## 92. Consecutive integer sums

In how many ways can 50! be expressed as a sum of two or more consecutive positive integers?

Hint - Answer - Solution

## 93. Pascal's triangle

Show that any two elements (both greater than one) drawn from the same row of Pascal's triangle have greatest common divisor greater than one. For example, the greatest common divisor of 28 and 70 is 14.

1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1
1 6 15 20 15 6 1
1 7 21 35 35 21 7 1
1 8 28 56 70 56 28 8 1

Hint - Solution

## 94. Square endings

Find all 8-digit natural numbers n such that n^{2} ends in the same 8 digits as n. Numbers are written in standard decimal notation, with no leading zeroes.

Hint - Answer - Solution

## 95. Integer polynomial

Let P be a polynomial with integer coefficients. If a, b, c are distinct integers, show that

- P(a) = b,
- P(b) = c,
- P(c) = a,

cannot be satisfied simultaneously.

Hint - Solution

## 96. Five real numbers

The sum of five real numbers is 7; the sum of their squares is 10. Find the minimum and maximum possible values of any one of the numbers.

Hint - Answer - Solution

## 97. Two squares

Find all pairs of positive integers, x, y, such that x^{2} + 3y and y^{2} + 3x are both perfect squares.

Hint - Answer - Solution

## 98. Three powers

Find all solutions of 3^{x} + 4^{y} = 5^{z}, for integers x, y, and z.

Hint - Answer - Solution

## 99. Two similar triangles

Two similar triangles with integral sides have two of their sides the same. If the third sides differ by 20141, find all of the sides.

Hint - Answer - Solution

## 100. Perpendicular medians

Suppose the medians AA' and BB' of triangle ABC intersect at right angles. If BC = 3 and AC = 4, what is the length of side AB?

Hint - Answer - Solution

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Nick Hobson

nickh@qbyte.org
*Last updated: March 4, 2005*