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AHSMC 2011

AHSMC 2011. Part II. Problem 1. A cross shaped figure is made up of five unit squares. Determine which has the larger area: the square containing the cross or the circle containing the cross. Problem 2. If there is exactly one triplet satisfying and , determine.

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AHSMC 2011

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  1. AHSMC 2011 Part II

  2. Problem 1.A cross shaped figure is made up of five unit squares. Determine which has the larger area: the square containing the cross or the circle containing the cross.

  3. Problem 2.If there is exactly one triplet satisfying and , determine .

  4. Problem 3.On side of triangle , points and exist such that is closer to than and moreover and are on lines and respectively. Suppose that and . Prove that

  5. Problem 4.Determine all functions where for every , .

  6. Problem 5.Seven teams gather to play one of three sports, and no set of three teams play the same sport among themselves. A triplet is considered diverse if all three sports are played among themselves. What is the maximum possible number of diverse triplets? • In the above configuration, we notice there are 14 diverse triangles. • We prove that this is actually the maximum possible configuration. • Suppose a vertex has black, green, and red edges where . • Then at least non-diverse triangles exist containing this vertex. • We are not overcounting: if the same triangle is counted twice then it is monochromatic. • The answer is .

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