50-th Mathematical Olympiad in Poland Final Round, April 14–15, 1999 First Day

1. Point D lies on the side BC of the triangle ABC and is chosen so that AD > BC . Point E

lies on the side AC and satisfies

\frac{AE}{EC} = \frac{BD}{AD − BC} .

Prove that AD > BE .

2. Given positive integers a_{1} < a_{2} < a_{3} < … < a_{101} less than 5050. Prove that there exist four different numbers a_{k} , a_{l} , a_{m} , a_{n} such that the number a_{k} + a_{l} − a_{m} − a_{n} is divisible by 5050.

3. Prove that there exist positive integers n_{1} < n_{2} < … < n_{50} such that

n_{1} + S(n_{1}) = n_{2} + S(n_{2}) = n_{3} + S(n_{3}) = … = n_{50} + S(n_{50}) ,

where S(n) denotes the sum of the digits of n.

Second Day

4. Find out for which integers n ≥ 2 the system of equations

\{ \begin{matrix}x_{1}^{2} + x_{2}^{2} + 50 = 16x_{1} + 12x_{2} \\ x_{2}^{2} + x_{3}^{2} + 50 = 16x_{2} + 12x_{3} \\ x_{3}^{2} + x_{4}^{2} + 50 = 16x_{3} + 12x_{4} \\ . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . \\ x_{n−1}^{2} + x_{n}^{2} + 50 = 16x_{n−1} + 12x_{n} \\ x_{n}^{2} + x_{1}^{2} + 50 = 16x_{n} + 12x_{1} \end{matrix}

has integer solutions x_{1}, x_{2}, x_{3}, …, x_{n}.

5. Let a_{1}, a_{2}, ..., a_{n}, b_{1}, b_{2}, ..., b_{n} be given integers. Prove that

∑_{1≤i<j≤n}(|a_{i} − a_{j}| + |b_{i} − b_{j}|) ≤ ∑_{1≤i,j≤n}|a_{i} − b_{j}| .

6. In a convex hexagon ABCDEF the following equalities hold:

A̸ +^{}C̸ +^{}E̸ = 360^{∘} , \frac{AB}{BC} · \frac{CD}{DE} · \frac{EF}{F A} = 1 .

Prove that \frac{AB}{BF} · \frac{F D}{DE} · \frac{EC}{CA} = 1 .