▪ Austria Regional Mathematical Competition For Advanced Students 2012

1. Prove that the inequality

holds for all real numbers . 

2. Determine all integer solutions  of the equation

3. In an arithmetic sequence, the di fference of consecutive terms in constant. We consider sequences of integers in which the di fference of consecutive terms equals the sum of the differences of all preceding consecutive terms.

Which of these sequences with and contain square numbers? 

4. In a triangle , let , and denote the base points of the altitudes on the sides , and , respectively.

Determine for which triangles two of the lengths , and are equal.

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