1^st Annual Eastern Oregon University Mathematics Competition Exam – 2004

1. Two circles of equal radius r intersect. The area of intersection has width x. Suppose the dark shaded area (the area of intersection of the two circles) is twice as large as the light shaded area (the area bounded by the two circles, together with the horizontal tangent line intersecting both). Determine the ratio (x/r).

2a. Determine the exact values of all real solutions of x³ – 4x² + 3x + 2 = 0.

2b. Determine the exact values of all real solutions of x³ – 3x² + 3x – 5 = 0.

3. Suppose a, b, and c are inputs for a function f(x) and that the following three equations hold:

f(a) = b; f(b) = c; f(c) = a.

In this case we call the triple {a,b,c} a 3-cycle for the function f(x). Let

	{
g(x) =		2x if x £ 1
		1 – x^2 if x > 1

Find (with proof) all 3-cycles of the function g(x).

4. To what value does the following sequence converge?

2, (2+ 1/2), (2 + 1/(2 + 1/2)), (2 + 1/(2 + 1/(2 + 1/2))), …

[You may assume, without proof, that the sequence does, in fact, converge to something].

5. Suppose f(x) is a twice-differentiable function such that f(x) f ‘(x) f ‘’(x) > 0 for all real values of x. Suppose f(2) = –2. Is it possible that f(0) = –1? If so, find a specific example. If not, prove your claim.

6. Prove that the equation 8x⁴ – 6x + 1 = 0 has two solutions in the interval [0,1].

7. Suppose for n ³ 4, the numbers (n² – 14) and (n² + 4) are both prime. Prove that n must be a multiple of five.