26th Austrian-Polish 2003

1. Find all real polynomials p(x) such that p(x-1)p(x+1) ≡ p(x²-1).

2. The sequence a₀, a₁, a₂, ... is defined by a₀ = a, a_n+1 = a_n + L(a_n), where L(m) is the last digit of m (eg L(14) = 4). Suppose that the sequence is strictly increasing. Show that infinitely many terms must be divisible by d = 3. For what other d is this true?

3. ABC is a triangle. Take a = BC etc as usual. Take points T₁, T₂ on the side AB so that AT₁ = T₁T₂ = T₂B. Similarly, take points T₃, T₄ on the side BC so that BT₃ = T₃T₄ = T₄C, and points T₅, T₆ on the side CA so that CT₅ = T₅T₆ = T₆A. Show that if a' = BT₅, b' = CT₁, c' = AT₃, then there is a triangle A'B'C' with sides a', b', c' (a' = B'C' etc). In the same way we take points T_i' on the sides of A'B'C' and put a" = B'T₆', b" = C'T₂', c" = A'T₄'. Show that there is a triangle A"B"C" with sides a", b", c" and that it is similar to ABC. Find a"/a.

4. A positive integer m is alpine if m divides 2²ⁿ⁺¹ + 1 for some positive integer n. Show that the product of two alpine numbers is alpine.

5. A triangle with sides a, b, c has area F. The distances of its centroid from the vertices are x, y, z. Show that if (x + y + z)² ≤ (a² + b² + c²)/2 + 2F√3, then the triangle is equilateral.

6. ABCD is a tetrahedron such that we can find a sphere k(A,B,C) through A, B, C which meets the plane BCD in the circle diameter BC, meets the plane ACD in the circle diameter AC, and meets the plane ABD in the circle diameter AB. Show that there exist spheres k(A,B,D), k(B,C,D) and k(C,A,D) with analogous properties.

7. Put f(n) = (nⁿ - 1)/(n - 1). Show that n!^f(n) divides (nⁿ)! . Find as many positive integers as possible for which n!^f(n)+1 does not divide (nⁿ)! .

8. Given reals x₁ ≥ x₂ ≥ ... ≥ x₂₀₀₃ ≥ 0, show that x₁ⁿ - x₂ⁿ + x₃ⁿ - ... - x₂₀₀₂ⁿ + x₂₀₀₃ⁿ ≥ (x₁ - x₂ + x₃ - x₄ + ... - x₂₀₀₂ + x₂₀₀₃)ⁿ for any positive integer n.

9. Take any 26 distinct numbers from {1, 2, ... , 100}. Show that there must be a non-empty subset of the 26 whose product is a square.

10. What is the smallest number of 5x1 tiles which must be placed on a 31x5 rectangle (each covering exactly 5 unit squares) so that no further tiles can be placed? How many different ways are there of placing the minimal number (so that further tiles are blocked)? What are the answers for a 52x5 rectangle?

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


1. Find all real polynomials p(x) such that p(x-1)p(x+1) ≡ p(x²-1).
2. The sequence a₀, a₁, a₂, ... is defined by a₀ = a, a_n+1 = a_n + L(a_n), where L(m) is the last digit of m (eg L(14) = 4). Suppose that the sequence is strictly increasing. Show that infinitely many terms must be divisible by d = 3. For what other d is this true?
3. ABC is a triangle. Take a = BC etc as usual. Take points T₁, T₂ on the side AB so that AT₁ = T₁T₂ = T₂B. Similarly, take points T₃, T₄ on the side BC so that BT₃ = T₃T₄ = T₄C, and points T₅, T₆ on the side CA so that CT₅ = T₅T₆ = T₆A. Show that if a' = BT₅, b' = CT₁, c' = AT₃, then there is a triangle A'B'C' with sides a', b', c' (a' = B'C' etc). In the same way we take points T_i' on the sides of A'B'C' and put a" = B'T₆', b" = C'T₂', c" = A'T₄'. Show that there is a triangle A"B"C" with sides a", b", c" and that it is similar to ABC. Find a"/a.
4. A positive integer m is alpine if m divides 2²ⁿ⁺¹ + 1 for some positive integer n. Show that the product of two alpine numbers is alpine.
5. A triangle with sides a, b, c has area F. The distances of its centroid from the vertices are x, y, z. Show that if (x + y + z)² ≤ (a² + b² + c²)/2 + 2F√3, then the triangle is equilateral.
6. ABCD is a tetrahedron such that we can find a sphere k(A,B,C) through A, B, C which meets the plane BCD in the circle diameter BC, meets the plane ACD in the circle diameter AC, and meets the plane ABD in the circle diameter AB. Show that there exist spheres k(A,B,D), k(B,C,D) and k(C,A,D) with analogous properties.
7. Put f(n) = (nⁿ - 1)/(n - 1). Show that n!^f(n) divides (nⁿ)! . Find as many positive integers as possible for which n!^f(n)+1 does not divide (nⁿ)! .
8. Given reals x₁ ≥ x₂ ≥ ... ≥ x₂₀₀₃ ≥ 0, show that x₁ⁿ - x₂ⁿ + x₃ⁿ - ... - x₂₀₀₂ⁿ + x₂₀₀₃ⁿ ≥ (x₁ - x₂ + x₃ - x₄ + ... - x₂₀₀₂ + x₂₀₀₃)ⁿ for any positive integer n.
9. Take any 26 distinct numbers from {1, 2, ... , 100}. Show that there must be a non-empty subset of the 26 whose product is a square.
10. What is the smallest number of 5x1 tiles which must be placed on a 31x5 rectangle (each covering exactly 5 unit squares) so that no further tiles can be placed? How many different ways are there of placing the minimal number (so that further tiles are blocked)? What are the answers for a 52x5 rectangle?