2nd Annual Eastern Oregon University Mathematics Competition Exam – 2005

 

                                                           

1. Given that the two indicated angles are congruent, determine the length Z as a function of length X – (it should be a function of X and no other variable, further it should not involve any trigonometric functions).

 

2. Is it possible to find a cubic polynomial that passes through the points (0,1) and (1,0) and that has horizontal tangent lines at both these points?  If so, find an example.  If not, explain why not.

 

3. Determine all real solutions to the following system of equations

               x2 + y2 + x – y   = 10

               xy        – x + y   =   5

 

4. In trigonometry you learn the identity                         sin2(x) + cos2(x) = 1.

It is also easy to see that                                               sin0(x) + cos0(x) = 2

holds for all x for which we don’t need to worry about raising zero to the zeroth power.

This can be avoided by noting that both identities hold for x Ξ (0, p/2).

Besides n=0 and n=2 are there any other non-negative integers  n  for which  

sinn(x) + cosn(x)   is constant for xΞ(0, p/2)?

If so, find all such integers.  If not, explain why not.

 

5. Zig-Zag and Xeno are standing in a large open field.  Zig-Zag walks 1 mile north, then 1/2 mile west, then 1/4 mile north, then 1/8 mile west, then 1/16 mile north, continuing in this manner until, in the limit, he reaches his destination.  Xeno begins and ends at the same locations as does Zig-Zag, but he walks a straight line to get to the destination.  If each of them walks at a speed of 1 mile per hour, how much longer (to the nearest minute) does it take Zig-Zag to walk his route than it takes Xeno?

 

6.  For k ³ 1 let Ek be the number of positive k-digit integers containing the digit “8” and let Nk be the number of positive k-digit integers not containing an “8”.  For example, there are nine positive 1-digit integers {1, 2, …, 9} and only one of them contains an “8”, so E1 = 1 and N1 = 8.  There are ninety positive 2-digit integers {10, 11, …, 99} eighteen of which contain an “8” digit and seventy-two of which do not, so E2 = 18 and N2 = 72.

Determine (with proof) the limit of (Ek / Nk) as k ΰ ₯.

 

7. What are the last two digits of the quantity (41)2005?  Prove your claim.