2016年AMC12真題及答案

1、2016 AMC12 AProblem 111! 一 10!What is the value of 9!?99(B) 100(C) 110(D) 121(E) 132Soluti onProblem 2For what value of x does 10” 1002；r = 10005?1(B) 2(C) 3(D) 4(E) 5Soluti onProblem 3The remainder can be defined for all real numbers x and V with 9 弄()byrem(x5?/) = x yx一 Mere一ydenotes the greatest

2、integer less than or equal to y.What is the value of rem(8 5)?3i33(A)-8(B) IO (C) ) IoSoluti onProblem 4The mean, median, and mode of the 7 data values 60)100, rr, 40, 50,200. 90 are all equal to x. What is the value of z?(A) 50(B) 60(C) 75(D) 90(E) 100Soluti onProblem 5Goldbachs conjecture states t

3、hat every even integer greater than 2 can be written as the sum of two prime numbers (for example, 2016 = 13 + 2003). 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?

4、an odd integer greater than 2 that can be written as the sum of two prime numbersan odd integer greater than 2 that cannot be written as the sum of two prime numbersan even integer greater than 2 that can be written as the sum of two numbers that are nan even integer greater than 2 that can be writt

5、en as the sum of two prime numbersan even integer greater than 2 that cannot be written as the sum of two prime numbersSoluti onProblem 6A triangular array of 2016 coins has 1 coin in the first row, 2 coins in the second row, 3coins in the third row, and so on up to N coins in the Nth row What is th

6、e sum of the digits of TV ?(A) 6(B) 7(C) 8(D) 9(E) 10Soluti onProblem 7Which of these describes the graph of *(丁 + 9 + 1)=+ g + 1) ?two parallel linestwo intersecting linesthree lines that all pass through a common pointthree lines that do not all pass through a common pointa line and a parabolaSolu

7、ti onProblem 8What is the area of the shaded region of the given 8x5 rectangle?31I(A) 4-(B) 5(C) 5-(D) 6-(E) 8Soluti onProblem 9The five small shaded squares in side this unit square are con grue nt and have disjoint in teriors. The midpoint of each side of the middle square coincides with one of th

8、e vertices of the other a_four small squares as show n. The comm on side length is b, where a and b are positiveintegers. What is a + b ?Soluti on(E)llProblem 10Five trie nds sat in a movie theater in a row containing 5 seats, nu mbered 1 to 5 from left to right. (The directions left” and right” are

9、 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 popeom. 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. I

10、n which seat had Ada been sitting before she got up?(A) 1(B) 2(C) 3(D) 4(E) 5Soluti onProblem 11Each of the 100 students in a certain summer camp can either sing dance, or act. Some stude nts have more tha n one tale nt, but no stude nt has all three talents. There are 42 students who cannot sing, 6

11、5 students who cannot dance, and 29 students who cannot act. How many students have two of these talents?(A) 16(B) 25(C) 36(D) 49(E) 64Soluti onProblem 12In /ABC、AB = 6, BC = 7, and CA = 8. Point D lies on BC、and AD bisects ABAC. Point Elies on AC, and FEbisects /.ABC. The bisectors intersect at F.

12、What is the ratio AF: FD?(A) 3:2(B) 5:3(C)2: 1(D) 7:3(E)5 : 2SolutionProblem 13Let N be a positive multiple of 5. One red ball and Ngreen balls are arranged in a line in 3random order. Let P(N) be the probability that at least 5 of the green balls are on the same side of the red ball. Observe that)(

13、5) 1 and that 廠(approaches 百 as N grows large.What is the sum of the digits of the least value of N such that 400?(A) 12(B) 14(C) 16(D) 18(E) 20Soluti onProblem 14Each vertex of a cube is to be labeled with an integer from 1 through 8, with each integer being used once, in such a way that the sum of

14、 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?(A) 1(B) 3(C) 6(D) 12(E) 24Soluti onProblem 15Circles with centers P)Q and

15、 R、having radii 1, 2 and 3, respectively, lie on the same side of line I and are tangent to I at P、Q and R respectively, with Q/ between Pf and 7?；Thecircle with center Q is externally tangent to each of the other two circles. What is the area of triangle PQR?(B)(C) 1(D) v/6-辺(E)Soluti onProblem 16T

16、he graphs of 3 勺弘 5 and 3 are plotted on thesame set of axes How many points in the plane with positive T-coordinates lie on two or more of the graphs?(A) 2(B) 3(C) 4(D) 5(E) 6Soluti onProblem 17Let ABCD be a square Let E)F、G and H be the centers, respectively, of equilateral triangles with basesEC)

17、CD、aneach exterior to the square What is the ratioof the area of square EFGH to the area of square ABCD?(A) 1(B)脊0(C) V2 (D)血；(E)箱Soluti onProblem 18For some positive integer nUhe number 11 On3 has 110 positive integer divisors, including 1 and the number 11 On3 How many positive integer divisors do

18、es the number 81n4 have?(A) 110(B) 191(C) 261(D) 325(E) 425Soluti onProblem 19Jerry starts at Oon the real number line. He tosses a fair coin 8 times When he gets heads, he moves 1 unit in the positive di recti on; when he gets tails, he moves 1 unit in the n egative directi on. The probability that

19、 he reaches 4 at some time duri ng this processais b where a and b are relatively prime positive integers What is a + 6? (For example, he succeeds if his sequence of tosses is HTHHHHHH)(A) 69(B) 151(C) 257(D) 293(E) 313Soluti onProblem 20A binary operation has the properties that。 (b c) = (d b) c an

20、dthat a a = 1 for all non zero real numbers a, b and c. (Here the dot. represe nts the usual multiplication operation.) The solution to the equation 2016 (6 爼)=100 can bepwritte n as Q where P a nd Q are relatively prime positive in tegers What is P + g?(A) 109(B) 201(C) 301(D) 3049(E) 33,601Soluti

21、onProblem 21A quadrilateral is in scribed in a circle of radius 200 /2. Three of the sides of this quadrilateral have length 200. What is the length of its fourth side?(A) 200(B) 200/2(C) 200箱 (D) 300邁 (E) 500Soluti onProblem 22How many ordered triples 仗S z) of positive integers satisfy lcm y) = 72,

22、 lcm z) = 600 and lcm(g, z) = 900?(A) 15(B) 16(C) 24(D) 27(E) 64Soluti onProblem 23Three numbers in the interval丄are chosen independently and at random. What is theprobability that the chosen numbers are the side lengths of a triangle with positive area?11125(A) 6(B) 3(C) 2(D) 3(E) 6Soluti onProblem

23、 24There is a smallest positive real number a such that there exists a positive realnumber b such that all the roots of the polynomial ax bx a are real. In fact, for this value of a the value of b is unique. What is the value of d?(A) 8(B) 9(C) 10(D) 11(E) 12Soluti onProblem 25Let k be a positive integer. Bemardo and Silvia take turns writing and erasing numbers on a black