2020年AMC12真题及答案.docx

2016 AMC12 AProblem 111! -10!What is the value of 1 >>?SolutionProblem 2For what value of :: does ： । t 1SolutionProblem 3denotes the greatestThe remainder can be defined for all realrcni^, y) = x-ynumbers T and . with 飞 L u by integer less than or equal to ■. What is the value of -SolutionProblem 4The mean, median, and mode of the 7 data values ।■11 J - '' ' are all equalto - . What is the value of 二？SolutionProblem 5Goldbach's conjecture states that every even integer greater than 2 can be written as the sum of two prime numbers (for example, 2工 C — 13 —2 上 3).So far, no one has been able to prove that the conjecture is true, and no one has found a counterexample to show that the conjecture is false. What would a counterexample consist of?SolutionProblem 6A triangular array of ； coins has I coin in the first row, _ coins in the second row, :' coins in the third row, and so on up to ' coins in the - th row. What is the sum of the digits of ： 'SolutionProblem 7Which of these describes the graph of1 1 < 1 1 ?SolutionProblem 8K x 5 rectangle?What is the area of the shaded region of the givenSolutionProblem 9The five small shaded squares inside this unit square are congruent and have disjoint interiors.The midpoint of each side of the middle square coincides with one of the vertices of the other7 2four small squares as shown. The common side length is-b —, where a and b are positiveintegers. What is (? + b ?SolutionProblem 10Five friends sat in a movie theater in a row containing 5 seats, numbered 1 to 5 from left to right. (The directions "left" and "right" are from the point of view of the people as they sit in the seats.) During the movie Ada went to the lobby to get some popcorn. When she returned, she found that Bea had moved two seats to the right, Ceci had moved one seat to the left, and Dee and Edie had switched seats, leaving an end seat for Ada. In which seat had Ada been sitting before she got up?SolutionProblem 11Each of the 11 i students in a certain summer camp can either sing, dance, or act. Some students have more than one talent, but no student has all three talents. Thereare 42 students who cannot sing, 65 students who cannot dance, and 29 students who cannot act. How many students have two of these talents?SolutionProblem 12In 一..1/；( , . 1；卜，i, and . '，. Point I ■ lies on 3 t , and AlD bisects /BA。

Point E lies on _4C, and bisects /」4EC. The bisectors intersect at / . What is the ratio -d :3■ .D?SolutionProblem 13Let j be a positive multiple of . One red ball and :green balls are arranged in a line inrandom order. Letbe the probability that at leastof the green balls are on the sameside of the red ball. Observe thatsuch that『V： < 六？a a ' a and that ' ' ' approaches . as :\ grows large.What is the sum of the digits of the least value ofSolutionProblem 14Each vertex of a cube is to be labeled with an integer from I through 、, with each integer being used once, in such a way that the sum of the four numbers on the vertices of a face is the same for each face. Arrangements that can be obtained from each other through rotations of the cube are considered to be the same. How many different arrangements are possible?SolutionProblem 15Circles with centers - 1 and /；', having radii I - and , respectively, lie on the same side of line / and are tangent to at ' • . and ,■ < , respectively, with between I and . The circle with center 「is externally tangent to each of the other two circles. What is the area of triangle ，.…I?SolutionProblem 16力 八 r V = log.. v = logr 3t y = logi r. ..,,,,,The graphs ofand। are plotted on the same set of axes. How many points in the plane with positive > -coordinates lie on two or more of the graphs?SolutionProblem 17Let 一 ［上"be a square. Let -.,…and J be the centers, respectively, of equilateraltriangles with bases： 川 and ，'J 】each exterior to the square. What is the ratio of the area of square / ..，.叮 to the area of square ."？SolutionProblem 18For some positive integer ■ the number I。

一 has 11.0； positive integer divisors,including I and the number I • How many positive integer divisors does the number '，, have?SolutionProblem 19Jerry starts at on the real number line. He tosses a fair coin times. When he gets heads, he moves I unit in the positive direction; when he gets tails, he moves I unit in the negative direction. The probability that he reaches . at some time during this process ais where and . are relatively prime positive integers. What is :: 1 (For example, he succeeds if his sequence of tosses is // ) /；")SolutionProblem 20A binary operation . has the properties that 。

andc । ' - ' can bethat QE - . for all nonzero real numbers ' and - (Here the dot . represents the usual multiplication operation.) The solution to the equation written as where .； and / are relatively prime positiv。