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2015 MMATHS Mathathon Rounds 1-4 Math Majors of America Tournament for HS

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February 16, 2022
algebrageometrycombinatoricsnumber theoryMMATHS

Problem Statement

Round 1
p1. If this mathathon has 77 rounds of 33 problems each, how many problems does it have in total? (Not a trick!)
p2. Five people, named A,B,C,D,A, B, C, D, and EE, are standing in line. If they randomly rearrange themselves, what’s the probability that nobody is more than one spot away from where they started?
p3. At Barrios’s absurdly priced fish and chip shop, one fish is worth $13\$13, one chip is worth $5\$5. What is the largest integer dollar amount of money a customer can enter with, and not be able to spend it all on fish and chips?
Round 2
p4. If there are 1515 points in 44-dimensional space, what is the maximum number of hyperplanes that these points determine?
p5. Consider all possible values of z1−z2z2−z3⋅z1−z4z2−z4\frac{z_1 - z_2}{z_2 - z_3} \cdot \frac{z_1 - z_4}{z_2 - z_4} for any distinct complex numbers z1z_1, z2z_2, z3z_3, and z4z_4. How many complex numbers cannot be achieved?
p6. For each positive integer nn, let S(n)S(n) denote the number of positive integers k≤nk \le n such that gcd(k,n)=gcd(k+1,n)=1gcd(k, n) = gcd(k + 1, n) = 1. Find S(2015)S(2015).
Round 3
p7. Let P1P_1, P2P_2,......, P2015P_{2015} be 20152015 distinct points in the plane. For any i,j∈{1,2,....,2015}i, j \in \{1, 2, ...., 2015\}, connect PiP_i and PjP_j with a line segment if and only if gcd(i−j,2015)=1gcd(i - j, 2015) = 1. Define a clique to be a set of points such that any two points in the clique are connected with a line segment. Let ω\omega be the unique positive integer such that there exists a clique with ω\omega elements and such that there does not exist a clique with ω+1\omega + 1 elements. Find ω\omega.
p8. A Chinese restaurant has many boxes of food. The manager notices that ∙\bullet He can divide the boxes into groups of MM where MM is 1919, 2020, or 2121. ∙\bullet There are exactly 33 integers xx less than 1616 such that grouping the boxes into groups of xx leaves 33 boxes left over. Find the smallest possible number of boxes of food.
p9. If f(x)=x∣x∣+2f(x) = x|x| + 2, then compute ∑k=−10001000f−1(f(k)+f(−k)+f−1(k))\sum^{1000}_{k=-1000} f^{-1}(f(k) + f(-k) + f^{-1}(k)).
Round 4
p10. Let ABCABC be a triangle with AB=13AB = 13, BC=20BC = 20, CA=21CA = 21. Let ABDEABDE, BCFGBCFG, and CAHICAHI be squares built on sides ABAB, BCBC, and CACA, respectively such that these squares are outside of ABCABC. Find the area of DEHIFGDEHIFG.
p11. What is the sum of all of the distinct prime factors of 7783=65+6+17783 = 6^5 + 6 + 1?
p12. Consider polyhedron ABCDEABCDE, where ABCDABCD is a regular tetrahedron and BCDEBCDE is a regular tetrahedron. An ant starts at point AA. Every time the ant moves, it walks from its current point to an adjacent point. The ant has an equal probability of moving to each adjacent point. After 66 moves, what is the probability the ant is back at point AA?
PS. You should use hide for answers. Rounds 5-7 have been posted [url=https://artofproblemsolving.com/community/c4h2782011p24434676]here. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.
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