# Let p q and r be distinct prime numbers such that p 2

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11. Letp, q,andrbe distinct prime numbers such thatp2+1,q2+2,andr2+ 3, form an arithmetic sequence in that order. What is theminimum possible value ofp+q+r? (A)15(B)25(C)30(D)37(E)40 12. 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? 13. What is the sum of all integers 1m100 such that thereexists an integernwhich has the property that exactlym% of thepositive integer divisors ofnare perfect squares?
Mock 2018 AMC 10A Problems 4 14. Suppose that the graphs ofy=x2-4x-7 andx= (y+a)2 intersect at an odd number of points. What is the nearest integer to the sum of all possible values of a ? 15. Equilateral triangle4ABChasAB=BC=AC= 1. SquaresABDE,BCFG, andACHIare constructed outside4ABCwithsides coinciding withAB,BC, andAC, respectively. ConnectEF,GI, andDH, so that they intersect4ABCat six points, forming ahexagon. Then, the area of the hexagon can be expressed in the form a - b c d for some positive integers a, b, c, d , where c is not divisible by the square of any prime and gcd( a, b, d ) = 1. Compute a + b + c + d . (A)10(B)12(C)16(D)22(E)26 A B C D E F G H I 16.A positive integer is calledtriangular numberif it can be ex-pressed in the formn(n+1)2for a positive integern. How many or-dered pairs (x, y) of triangular numbers satisfyx-y= 31,500? 17. Consider all positive integersn, with the property that 35×nhas exactly 3500 digits. LetNbe the largest integer with such prop-erty. What is the sum of the digits ofN? Mock 2018 AMC 10A Problems 5

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• Fall '19
• Prime number

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