2nd Austrian-Polish Mathematics Competition 1979 Problems
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].
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