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After the contest period, permission to make copies of problems in paper for classroom use only is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear the copyright notice. Electronic copies of any type are strictly prohibited.
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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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