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Part of 2010 IberoAmerican

Problems(2)

Problem 1, Iberoamerican Olympiad 2010

Source:

There are ten coins a line, which are indistinguishable. It is known that two of them are false and have consecutive positions on the line. For each set of positions, you may ask how many false coins it contains. Is it possible to identify the false coins by making only two of those questions, without knowing the answer to the first question before making the second?

vectoranalytic geometryfunctioncombinatorics proposedcombinatorics

Problem 4, Iberoamerican Olympiad 2010

Source:

The arithmetic, geometric and harmonic mean of two distinct positive integers are different numbers. Find the smallest possible value for the arithmetic mean.

algebra proposedalgebra