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Mock 2018 AIME II 2 1. Find the largest positive integer value of n < 1000 such that φ (36 n ) = φ (25 n ) + φ (16 n ), where φ ( n ) denotes the number of positive integers less than n that are relatively prime to n . 2. Garfield and Odie are situated at (0 , 0) and (25 , 0), respectively. Suddenly, Garfield and Odie dash in the direction of the point (9 , 12) at speeds of 7 and 10 units per minute, respectively. During this chase, the minimum distance between Garfield and Odie can be written as m n for relatively prime positive integers m and n . Find m + n . 3. Let ABCD be a convex quadrilateral with positive area such that every side has a positive integer length and AC = BC = AD = 25. If P max and P min are the quadrilaterals with maximum and minimum possible perimeter, the ratio of the area of P max and P min can be expressed in the form a b c for some positive integers a, b, c , where a, c are relatively prime and b is not divisible by the square of any integer. Find a + b + c . 4. Three fair six-sided dice are rolled. The expected value of the median of the numbers rolled can be written as m n , where m and n are relatively prime integers. Find m + n . 5. Laurie plays a game called bash where she picks two distinct numbers be- tween 1 and 10, inclusive, at random. She then finds their sum, product, and non-negative difference. At random, she picks two of these three numbers and tells them to Ali. If the probability that Ali is able to logically deduce the orig- inal numbers can be written as m n , with m and n relatively prime, find m + n .

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• Fall '19
• Prime number, 36N, AIME II

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