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?

Splitting Points

From Five Hundred Mathematical Challenges:

Prove that if an even number of points are placed in a plane, it is always possible to draw a line such that half of the points are on one side of the line, and half the points are on the other side.

Dealing Cards

Mildly adapted from the 1895 Eötvös competition:

You have a deck of n cards, to be dealt to two players. The players do not have to receive the same number of cards. How many ways are there to do the dealing?

Some More Number Theory

From the 1901 Eötvös Contest:

Let n be a positive integer. Prove that 1ⁿ + 2ⁿ + 3ⁿ + 4ⁿ is divisible by 5 if and only if n is not divisible by 4.