Summer Problem Solving Marathon Question #3

[Value = 7 points]

How many ordered four-tuples of integers (a,b,c,d) are there such that:
(i) 0 < a < b < c < d < 500
(ii) a + d = b + c
(iii) bc - ad = 93

Summer Problem Solving Marathon Question #2

[Value = 3 points]

Each side of triangle ABC is length 12. D is the foot of the perpendicular dropped from A on BC, and E is the midpoint of AD. What is the length of BE?

Summer Problem Solving Marathon Solution #1

Let x and y be the two numbers. Then we are told x + y = 10, and xy = 20. We want to find ¹⁄_x + ¹⁄_y.

Converting both fractions to the common denominator xy, we get ^y⁄_xy + ^x⁄_xy, or ^(x+y)⁄_xy.

Substituting our starting values for x + y and xy, we have ¹⁰⁄₂₀, or ¹⁄₂.

[Put any questions, objections, or alternative solutions in the comments.]

Summer Problem Solving Marathon Question #1

[Value = 2 points]

The sum of two numbers is ten, and their product is twenty. What is the sum of their reciprocals?