# We define a real number x to be a semi integer if 2 x

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7. We define a real numberxto be asemi-integerif2·xis an integer.How manyreal numbers 0x100 aresemi-integers? (A)71(B)140(C)141(D)142(E)211 8. For real numbersmandn,we define the operatorasmn=m-1-(mn)-1+n-1.What is the value of 1(2(· · ·(20182019)))? Christmas Mathematics Competitions Questions and comments about problems and solutions for this exam should be sent by PM to: AOPS12142015, Blast S1, djmathman, eisirrational, FedeX333X, illogical 21, novus677, Th3Numb3rThr33, TheUltimate123, and WannabeCharmander Send questions and comments about administrative arrangements by PM to AOPS12142015, Blast S1, djmathman, eisirrational, FedeX333X, illogical 21, novus677, Th3Numb3rThr33, TheUltimate123, and WannabeCharmander The problems and solutions for this CMC 12 were prepared by MAC’s Subcommittee on the CMC10/CMC12 Exams. 2019 CIME The 2nd Annual CIME will be held on Friday, December 28, 2018, with the alternate on Friday, February 8, 2019. It is a 15-question, 3-hour, integer-answer exam. You will be invited to participate regardless of your score on this competition. All students will be selected to take the 2nd Annual Christmas Mathematical Olympiad (CMO) on January 4-25, 2019.
9. Regular hexagonABCDEFhas side length 1.PointsPandQare placed in theinterior of the hexagon such thatABPQis a square.IfMandNare the midpointsof sidesABandDE,respectively,what is the area of the shaded region? A B C D E F P Q M N 10. Consider a polynomialp(x) of degree 1 such that for a real numbera, p(a) = 2,p(p(a)) = 17 andp(p(p(a))) = 167.What is the value ofa? 11. Let 0< x <100 be a randomly chosen real number.What is the expectedvalue ofx{x},wherexdenotes the greatest integer less than or equal toxand{x}=x-xdenotes the fractional part ofx? (A)24.25(B)24.75(C)25(D)49.5(E)50 12. Geoff the frog is standing at the origin in the coordinate plane.For each move,Geoff can only move one unit to the right or one unit upwards; also,every up movemust be immediately followed by a right move (except for the last move).What is thenumber of distinct sequences of moves that end at the point (9,5)?

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