Here problems from first series of Polish Olympiad 2004:
1. Solve in real numbers x,y,z:
2. Find all natural n>1 for which value of the sum 2^2+3^2+...+n^2 equals p^k where p is prime and k is natural
3. In acute-angled triangle ABC point D is perpendicular projection of C on a side AB. Point E is a perpendicular projection od D on a side BC. Point F lies on a side DE and:
EF\FD = AD\DB
Prove that
4. Given is n in Z and positive reals a,b. Find possible maximal value of the sum: x_1*y_1 + x_2*y_2 + ... + x_n*y_n when x_1,x_2,...,x_n and y_1,y_2,...,y_n are in <0;1> and satisfies: x_1 + x_2 + ... + x_n < a
y_1 + y_2 + ... + y_n < b
Inequality invented by me:
a,b,c are non-negative reals:
Article "Inequalities" written by me in Polish
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