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Print55rd Ukrainian National Mathematical Olympiad - Third Round
Ukraine algebra
Problem
1. Compare to , where: In each question, the signs go as follows: "+" before the first term, then two "-" and two "+" signs in turn, and, finally, a "+" sign before the last term.
Solution
a. If we split all numbers into groups of , from left to right, each group will have numbers of the type: As we can see, the sum of numbers in each group is , therefore, .
b. Like in the previous question, split all numbers into groups of : . Then the sum of numbers in each group is positive:
b. Like in the previous question, split all numbers into groups of : . Then the sum of numbers in each group is positive:
Final answer
a) A = 0; b) A > 0
Techniques
Sums and productsFractions