25th Austrian-Polish 2002

1. Find all triples (a, b, c) of non-negative integers such that 2^a + 2^b + 1 is a multiple of 2^c - 1.

2. Show that any convex polygon with an even number of vertices has a diagonal which is not parallel to any of its edges.

3. A line through the centroid of a tetrahedron meets its surface at X and Y, so the centroid divides the segment XY into two parts. Show that the shorter part is at least one-third of the length of the longer part.

4. Given a positive integer n, find the largest set of real numbers such that n + (x₁ⁿ⁺¹ + x₂ⁿ⁺¹ + ... + x_nⁿ⁺¹) ≥ n x₁ ... x_n + (x₁ + ... + x_n) for all x_i in the set. When do we have equality?

5. p(x) is a polynomial with integer coefficients. Every value p(n) for n an integer is divisible by at least one of 2, 7, 11, 13. Show that every coefficient of p(x) is divisible by 2, or every coefficient is 7, or every coefficient is divisible by 11, or every coefficient is divisible by 13.

6. ABCD is a convex quadrilateral whose diagonals meet at E. The circumcenters of ABE, CDE are U, V respectively, and the orthocenters of ABE, CDE are H, K respectively. Show that E lies on the line UK iff it lies on the line VH.

7. Let N be the set of positive integers and R the set of reals. Find all functions f : N → R such that f(x + 22) = f(x) and f(x²y) = f(x)²f(y) for all x, y.

8. How many real n-tuples (x₁, x₂, ... , x_n) satisfy the equations cos x₁= x₂, cos x₂ = x₃, ... , cos x_n-1 = x_n, cos x_n = x₁?

9. A graph G has 2002 points and at least one edge. Every subgraph of 1001 points has the same number of edges. Find the smallest possible number of edges in the graph, or failing that the best lower bound you can.

10. Is it true that given any positive integer N, we can find X such that (1) any real sequence x₀, x₁, x₂, ... satisfying x_n+1 = X - 1/x_n satisfies x_k = x_k+N for all k, and (2) given a positive integer M < N, we can always find some real sequence x₀, x₁, x₂, ... satisfying x_n+1 = X - 1/x_n such that x_k = x_k+M does not hold for all k?

To avoid possible copyright problems, I have changed the wording, but not the substance, of the problems.


1. Find all triples (a, b, c) of non-negative integers such that 2^a + 2^b + 1 is a multiple of 2^c - 1.
2. Show that any convex polygon with an even number of vertices has a diagonal which is not parallel to any of its edges.
3. A line through the centroid of a tetrahedron meets its surface at X and Y, so the centroid divides the segment XY into two parts. Show that the shorter part is at least one-third of the length of the longer part.
4. Given a positive integer n, find the largest set of real numbers such that n + (x₁ⁿ⁺¹ + x₂ⁿ⁺¹ + ... + x_nⁿ⁺¹) ≥ n x₁ ... x_n + (x₁ + ... + x_n) for all x_i in the set. When do we have equality?
5. p(x) is a polynomial with integer coefficients. Every value p(n) for n an integer is divisible by at least one of 2, 7, 11, 13. Show that every coefficient of p(x) is divisible by 2, or every coefficient is 7, or every coefficient is divisible by 11, or every coefficient is divisible by 13.
6. ABCD is a convex quadrilateral whose diagonals meet at E. The circumcenters of ABE, CDE are U, V respectively, and the orthocenters of ABE, CDE are H, K respectively. Show that E lies on the line UK iff it lies on the line VH.
7. Let N be the set of positive integers and R the set of reals. Find all functions f : N → R such that f(x + 22) = f(x) and f(x²y) = f(x)²f(y) for all x, y.
8. How many real n-tuples (x₁, x₂, ... , x_n) satisfy the equations cos x₁= x₂, cos x₂ = x₃, ... , cos x_n-1 = x_n, cos x_n = x₁?
9. A graph G has 2002 points and at least one edge. Every subgraph of 1001 points has the same number of edges. Find the smallest possible number of edges in the graph, or failing that the best lower bound you can.
10. Is it true that given any positive integer N, we can find X such that (1) any real sequence x₀, x₁, x₂, ... satisfying x_n+1 = X - 1/x_n satisfies x_k = x_k+N for all k, and (2) given a positive integer M < N, we can always find some real sequence x₀, x₁, x₂, ... satisfying x_n+1 = X - 1/x_n such that x_k = x_k+M does not hold for all k?