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4. The publication, reproduction or communication of the prob- lems or solutions of this test during the period when students are eligible to participate seriously jeopardizes the integrity of the results. Dissemination at any time via copier, telephone, email, internet or media of any type is a violation of the com- petition rules. The CMC American Mathematics Competitions are supported by atmchallenge AOPS12142015 CantonMathGuy eisirrational FedeX333X illogical 21 Th3Numb3rThr33 tree3
Mock 2018 AMC 12A Problems 6 22. Let S be the set of all points ( a, b, c ) in 3-dimensional space with all coordinates nonnegative, such that ax 2 + bx + c = 2016 has real solutions in the interval m x n for positive real numbers m and n . If n - m = 2 mn = 16, then the volume of S can be written as 2016 3 · R for some real number R . What is R ? (A) 35 24 (B) 8 3 (C) 2 2 (D) 35 4 (E) 12 2 23. For how many positive integersn <2018 is the number of termsin the expansion of(x45+x42+x40)n+(x20+x18+x15)2n+(x10+x7+x4)3n+(x3+x2+x)4n n less than a perfect square? (A)0(B)1(C)3(D)5(E)8 24.LetP0, P1, ...be points in the complex plane such thatP0=-1, P1=iandPn=i·Pn-1+Pn-2for everyn2.LetPbethe polygonP0P1...P2017and letPandAbe its perimeter and area,respectively. If the value ofPAcan be written in the formab+cde for positive integers a, b, c, d, and e such that b and d are not divis- ible by the square of any prime and gcd( a, c, e ) = 1, compute the remainder when a + b + c + d + e is divided by 100. 25. A group of 5 students sit around a circle. Each minute, eachperson points at another student and that person is thus removedfrom the circle.What is the expected number of minutes it willtake until either a single person or nobody remains? (Note that twopeople could be pointing at the same person) Mock 2018 AMC 12A Problems 3 8. Two random, not necessarily distinct, permutations of the digits2017 are selected and added together. What is the expected valueof this sum? 9.Three triplets, Tam, Tom, and Tim, work together to build atreehouse. Tam works twice as fast as Tom, but three times slowerthan Tim. After working together for 10 days, Tim decides to quitsince he is tired. After another 10 days, Tom also decides to quit. Ittakes Tam another 6 days to complete the treehouse. The numberof days it would had taken them to complete the treehouse if noneof them had quit can be written asmnfor relatively prime positiveintegersmandn. What ism+n? (A)22(B)47(C)53(D)66(E)91 10. Equilateral triangle 4 ABC has AB = BC = AC = 1. Squares ABDE , BCFG , and ACHI are constructed outside 4 ABC with sides coinciding with AB , BC , and AC , respectively. Connect EF , GI , and DH

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• Fall '19
• Prime number, American Mathematics Competitions, American Invitational Mathematics Examination

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