Expert: Sherry Wallin - 3/31/2009

Question
plss...answer this some question... if i dont answer this i just kick on my school for the first time... plssss help me to answer this...

1. A sequence ..., a(-2), a(-1), a(0), a(1), a(2), ..., is such that a(n) + (n+3)[a(2-n)] = (n+3)^2. (a) Determine a(1). (b) Determine a(2).

2. How many positive three-digit integers are there such that if you divide it by the sum of their digits, the result is 19?

3. Find all k such that (x + 2)/(kx - 1) = x has exactly one solution.

4. What are the last two digits (in order) of 11^(11^(11^(11^..., where there are 2009 11's?

5. What is the maximum number of regions by which twelve circles can divide a plane?

6. Determine the roots of (x^2 + 3)^2 = x - 3.

7. How many possible ways can a 10x1 slab be formed from connecting 1x1, 2x1 or 3x1 slabs?

8. Circles C1 and C2, with C1 smaller than C2, intersect at A and B. P is on minor arc AB on C1 and Q and R are on major arc AB on C2 such that P, A, Q and P, B, R are threewise collinear. If PA=3, PB=2, AQ=7 and AB=4, what is the length of QR?

9. Suppose a, b, c are positive real numbers such that a^2 + b^2 + c^2 = 2000. What is the greatest possible value of 3a + 4b + 5c?

10. Determine the largest root of 2x^2 + 3x + 3 = 7xsqrt(x+1).

11. (Yes/No) Let T1 be a transformation such that ax^3 + bx^2 + cx + d becomes dx^3 + cx^2 + bx + a, and let T2 be a transformation that sends ax^3 + bx^2 + cx + d to a(x+t)^3 + b(x+t)^2 + c(x+t) + d where t is any real number. Is it possible to transform x^3 + x^2 - 2x - 2 to x^3 - 5x^2 + 8x - 4 using repeated applications of T1 and T2?

Answer
Here I will choose one to answer and explain for you:
3)  The only singleton answer is when k = 0 and then x = -1 is the singleton solution. Multiply both sides by kx-1 getting x+2 = kx^2 -x now add -x and -2 to both sides getting kx^2 -2x -2 = 0. You know you are looking for a singleton solution so you can't have a quadratic equation so k must be 0 now simple solve -2x-2 = 0 and find that 2x = -2 so that x = -1. Good luck on finding other solutions on your other problems.


Math Prof