2nd Austrian-Polish Mathematics Competition 1979 Problems

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 = 90^o.

2.  Find all polynomials of degree n with real roots x₁ ≤ x₂ ≤ ... ≤ x_n such that x_k 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, ... , x_n we have S₂S₁ - S₃ ≥ 6P, where S_k = ∑ x_i^k, and P = x₁x₂ ... x_n.

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, ... , x_n) to the n equations:

x₁ + x₂ + ... + x_n = k

x₁² + x₂² + ... + x_n² = k²

...

x₁ⁿ + x₂ⁿ + ... + x_nⁿ = 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].