Mock Aime II

Mock AIME II vincenthuang75025, FlakeLCR, mathtastic, bobthesmartypants March 2015

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Overview

This test is designed to simulate an American Invitational Mathematics Examination (AIME), and so should be taken under the same guidelines: • This test consists of 15 questions. • You have 3 hours to complete the test. • No calculators are allowed on the AIME. No aids other than scratch paper, graph paper, ruler, compass, and protractor are permitted. In particular, calculators and computers are not permitted. • All answers will be integers between 0 and 999 inclusive. This test consists of 15 original problems designed to be roughly the same difficulty as questions found on the AIME, although more trash. Any problems similar to those found on past contests are purely coincidental.

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Problems 1. (FlakeLCR) Bob the ant lives on the Cartesian plane, and he only ever steps on lattice points in this plane. However, he does not like lattice points that are far away from the origin, so he calls a lattice point ”accept√ 4 able” if and only if it’s distance to the origin is less than 2π 2 . Bob will only ever move to other lattice points in a very particular way; he will move to another lattice point if and only if it is 1 unit away from his current lattice point, and it is an ”acceptable” lattice point. If there are multiple directions he can travel in, he will pick any one of them with equal likelihood. If the probability that, starting at the origin, Bob ends up back m at the origin after moving 6 times, can be expressed as , where m and n are relatively prime, find m + n. n

2. (FlakeLCR) How many ways are there to color the squares of a 2 by 6 rectangular grid red, blue, and yellow, so that there are four squares of each color, and that no two adjacent squares have the same color? Rotations and reflections count as distinct grids.

3. (vincenthuang75025 ) The positive x, y, z satisfy x + y + z = 1. If the minimum possible value of m 1−y 1−z + equals , find 10m + n. 1+y 1+z n

1−x + 1+x

4. (vincenthuang75025 ) f, g, h, i are four functions who’s domain and range are the set of lattice points in the plane. f ((x, y)) = (x, 2y), g((x, y)) = (2x, y), h((x, y)) = (x + 1, y), i((x, y)) = (x, y + 1). Determine the number of ways to apply the functions f, g, h, i in some order (you may use a function more than once) on the point (1, 1) to end on the point (5, 5).

5. (mathtastic) Suppose for integers a and b that the equation x4 + abx3 − (ab3 + a3 b)x − (a4 + 2a2 b2 + b4 ) = 0 has exactly two real roots. Find the number of ordered pairs (a, b) such that −10 ≤ a, b ≤ 10.

6. (vincenthuang75025 ) The graph of y = x3 − 3x2 − 4 is drawn. Lines l1 , l2 , l3 are drawn such that they each intersect this graph at three distinct points with x-coordinates a1 , b1 , c1 , a2 , b2 , c2 , and a3 , b3 , c3 respectively (for each line). Determine the maximum possible value of (a1 + b1 + c1 )(a2 + b2 + c2 )(a3 + b3 + c3 ).

7. (mathtastic) Let 4ABC be a triangle with incenter I and circumcenter O such that AB = 104, BC = 112, CA = 120. A sphere S with center O passes through A, B, and C. A line through I perpendicular to the plane of 4ABC √ intersects S at distinct points D and E. The volume of tetrahedron ABCD can be expressed in the form a b for positive integers a and b where b is square free. Compute the remainder when a + b is divided by 1000.

8. (FlakeLCR) Let f (a, b) be the number of natural numbers less than b which are relatively prime to a. For n · f (n, n + 6) n < 1000, determine the largest possible integer value of . f (n, n + 1)

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9. (vincenthuang75025 ) Determine the largest prime p under 1000 such that 2n ≡ 1 mod p for some odd integer n.

10. (FlakeLCR) Let n = 1000.

11. (mathtastic) If value of |ab|.

j √

20 + 4


4

+

√

30 + 5


4

+

√

40 + 6


4 k . Find the remainder when n is divided by

1 1 1 1 + 3 + 3 + 3 + · · · = aπ 2 + b, for rational a and b, find the 3 3 3 3 3 1 1 +2 1 +2 +3 1 + 2 + 33 + 43

12. (vincenthuang75025 ) Consider triangle ABC, with angles a, b, c on vertices A, B, C respecctively. Suppose some acute angle λ satisfies cot λ = cot a + cot b + cot c. Suppose also that angle λ is half of angle a. Find 10 sin2 λ the maximum value of sin(b − λ) sin(c − λ)

13. (FlakeLCR) Let p = F1337−k · F1337+k , where Fn is the nth term of the Fibonacci sequence (F1 = 1, F2 = 1). √ √ Find the largest integral value k < 1000 such that at least one of b pc and d pe is equal to F1337 .

14. (vincenthuang75025 ) DetermineX the number of positiveX integers 3 < n < 1000 such that for the set S = {1, 2, 3, ..n}, the inequality n · |A|n |B|n |A ∩ B| ≥ |D||D|+1 holds. A,B∈S

D∈S

15. (bobthesmartypants) If the probability that the GCD of two randomly chosen positive integers is a perfect aπ 2 square is , where a and b are relatively prime, then find the value of a + b. b

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