x1 + x2 + ... + xn = k
x12 + x22 + ... + xn2 = k2
...
x1n + x2n + ... + xnn = kn
|
1. ABCD is a square. E is any point on AB. F is the point on BC such that BF = BE. The perpendicular from B meets EF at G. Show that ∠DGF = 90o.
|
|
| 2. Find all polynomials of degree n with real roots x1 ≤ x2 ≤ ... ≤ xn such that xk belongs to the closed interval [k, k+1] and the product of the roots is (n+1)/(n-1)! . | |
| 3. Find all positive integers n such that for all real numbers x1, x2, ... , xn we have S2S1 - S3 ≥ 6P, where Sk = ∑ xik, and P = x1x2 ... xn. | |
| 4. Let N0 = {0, 1, 2, 3, ... } and R be the reals. Find all functions f: N0 → R such that f(m + n) + f(m - n) = f(3m) for all m, n. | |
| 5. A tetrahedron has circumcenter O and incenter I. If O = I, show that the faces are all congruent. | |
|
6. k is real, n is a positive integer. Find all solutions (x1, x2, ... , xn) to the n equations:
x1 + x2 + ... + xn = k x12 + x22 + ... + xn2 = k2 ... x1n + x2n + ... + xnn = kn |
|
|
7. Find the number of paths from (0, 0) to (n, m) which pass through each node at most once.
|
|
| 8. ABCD is a tetrahedron. M is the midpoint of AC and N is the midpoint of BD. Show that AB2 + BC2 + CD2 + DA2 = AC2 + BD2 + 4 MN2. | |
| 9. Find the largest power of 2 that divides [(3 + √11)2n+1]. |
To avoid possible copyright problems, I have changed the wording, but not the substance, of the problems.
APMC home
© John Scholes
jscholes@kalva.demon.co.uk
17 Dec 2002
Last corrected/updated 17 Dec 2002