AMC 12 Mock Exam B Unofficial practice paper AMC 12 Mock Exam B 25 problems · 75 minutes Instructions. Each problem has exactly one correct answer among the five choices. Scoring: 6 points for each correct answer, 1 5 points for each problem left blank, and 0 points for each incorrect answer. No calculators, phones, or other aids are allowed; scratch paper, graph paper, ruler, compass, and protractor are permitted. Figures are not necessarily drawn to scale. This is an unofficial practice paper and is not affiliated with or endorsed by the MAA. 1. Ines can jog the length of a moving walkway in 12 seconds, or stroll along it in 24 seconds; in both cases she travels in the direction the walkway moves. One evening the walkway is switched off, and Ines jogs its length in 18 seconds. How many seconds would it take her to stroll the length of the switched-off walkway? (A) 36 (B) 48 (C) 54 (D) 60 (E) 72 2. The side lengths of the triangle below are n 2 − 5, 3 n + 13, and 4 n , where n is an integer. What is the sum of all values of n for which the triangle is isosceles? 3 n + 13 4 n n 2 − 5 (A) 11 (B) 13 (C) 18 (D) 19 (E) 24 3. How many ordered pairs ( a, b ) of integers with 2 ≤ a ≤ 2026 and 2 ≤ b ≤ 2026 satisfy log a b + 6 log b a = 5 ? (A) 43 (B) 44 (C) 54 (D) 55 (E) 56 4. Rectangle ABCD with AB = 6 and BC = 8 is inscribed in a circle. Point M is the midpoint of minor arc AB , and point N is the midpoint of minor arc CD , as shown. What is AM · AN ? A B C D M N (A) 24 (B) 25 (C) 30 (D) 10 √ 10 (E) 32 1 AMC 12 Mock Exam B Unofficial practice paper 5. Let P ( x ) be a polynomial of degree 2 with real coefficients such that P ( x ) = P (0) + P (1) x + P (2) x 2 for all real x , and P (4) = 22. What is P ( − 2)? (A) − 10 (B) − 2 (C) 2 (D) 6 (E) 10 6. Let a , b , c be integers with 0 ≤ a, b, c ≤ 63, each written in base 2 with exactly six digits (leading zeros allowed). Consider the ordered triples ( a, b, c ) for which there is no digit position where all three of a , b , c have the digit 1. Among these triples, how many make a + b + c as large as possible? (A) 64 (B) 216 (C) 343 (D) 512 (E) 729 7. Eight friends fill the eight seats of a single row in a cinema. There is an aisle at each end of the row and nowhere else. After the film, the friends leave one at a time in a uniformly random order. When it is their turn, each person may walk toward either aisle. What is the probability that every person can reach an aisle without squeezing past a friend who is still seated? (A) 1 630 (B) 1 315 (C) 1 280 (D) 1 105 (E) 1 40 8. A list of five integers has exactly one mode. The mode is 3 greater than the median, and the mean is 1 less than the median. What is the greatest possible value of the difference between the largest and the smallest numbers in the list? (A) 9 (B) 10 (C) 12 (D) 13 (E) 15 9. Positive integers a and b satisfy a + b = 2025. What is the least possible value of the least common multiple of a and b ? (A) 1350 (B) 1620 (C) 1800 (D) 1944 (E) 2024 10. A spinner is divided into ten equal sectors labelled 1 , 2 , 3 , . . . , 10. It is spun repeatedly. What is the probability that each of the labels 2, 3, 5, and 7 comes up before the first time a label that is not prime comes up? (A) 1 5040 (B) 1 210 (C) 1 120 (D) 2 105 (E) 1 24 11. What is the least possible value of ( 1 x + y + 1 y + z + 1 z + x ) √ x 2 + y 2 + z 2 over all positive real numbers x , y , and z ? (A) 3 √ 3 2 (B) 8 3 (C) 3 (D) 2 √ 3 (E) 9 2 2 AMC 12 Mock Exam B Unofficial practice paper 12. Three tokens sit on a number line at the points 4, 11, and 30. A move consists of choosing the leftmost or the rightmost token and moving it to an unoccupied integer point strictly between the other two tokens (so that it becomes the middle token). The process stops when the three tokens occupy three consecutive integers. Let m be the least number of moves in which the process can stop, and let M be the greatest number of moves the process can last. What is m + M ? 0 5 10 15 20 25 30 (A) 20 (B) 21 (C) 24 (D) 25 (E) 26 13. In triangle ABC , AB = AC Point P lies inside the triangle, and D , E , F are the feet of the perpendiculars from P to BC , CA , and AB , respectively. Given that BD = 14, DC = 4, P E = 2, and P F = 10, what is AB ? A B C D E F P (A) 12 (B) 13 (C) 14 (D) 15 (E) 4 √ 15 14. In triangle ABC , ∠ BAC = 60 ◦ . Point P lies on side BC . The circumradius of triangle ABP is 40, and the circumradius of triangle ACP is 30. What is cos ∠ ACB ? 60 ◦ A B C P (A) √ 13 26 (B) √ 13 13 (C) 1 3 (D) 3 √ 13 26 (E) 2 √ 13 13 3 AMC 12 Mock Exam B Unofficial practice paper 15. Each of the eight vertices of a cube is painted red, white, or blue so that every vertex has the same color as at least two of the three vertices joined to it by an edge. How many such paintings are there? (Two paintings are different if some vertex receives different colors in them; paintings that differ by a rotation of the cube are counted as different.) (A) 21 (B) 24 (C) 27 (D) 30 (E) 36 16. Let a and b be real numbers. A nonreal complex number p is a root of x 2 − ax + b = 0 and satisfies p 5 = 243. What is (2 a + 3) 2 + b 2 ? (A) 45 (B) 81 (C) 90 (D) 117 (E) 126 17. Triangle ABC has a right angle at A and AB = 12. The segment joining the centroid of the triangle to the incenter of the triangle is parallel to AB . What is the perimeter of triangle ABC ? (A) 30 (B) 32 (C) 36 (D) 40 (E) 42 18. Real numbers x and y are chosen independently and uniformly at random from the interval [0 , 1]. What is the probability that the circle with center ( x, y ) and radius 1 2 | x − y | lies entirely inside the square with vertices (0 , 0), (1 , 0), (1 , 1), and (0 , 1)? ( x, y ) (0 , 0) (1 , 1) (A) 1 4 (B) 1 3 (C) 3 8 (D) 1 2 (E) 2 3 19. A robot starts at the origin of the coordinate plane. In each move it travels a distance of exactly 5 in a straight line and stops at a point both of whose coordinates are integers. What is the least number of moves the robot needs to reach the point (2026 , 2026)? (A) 577 (B) 579 (C) 580 (D) 581 (E) 584 4 AMC 12 Mock Exam B Unofficial practice paper 20. Call a line in the coordinate plane tilted if it is not parallel to the x -axis, the y -axis, or the line x + y = 0. Three distinct lines are drawn so that each of the six points ( a, b ) shown below, where a and b are positive integers with a + b ≤ 4, lies on at least one of the three lines. Let k be the number of tilted lines among the three. What is the sum of all possible values of k ? x y 1 1 2 2 3 3 (A) 3 (B) 4 (C) 5 (D) 6 (E) 7 21. Ana and Bo play a game on the tree with 11 vertices shown below. Before the game, one vertex is chosen as the target , and both players know which vertex it is. Ana and Bo then take turns, with Ana going first. On each turn, the player deletes one vertex that is adjacent to at most one remaining vertex, together with its edge (if any). The player who deletes the target wins. For how many of the 11 possible choices of target does Bo have a winning strategy? (A) 5 (B) 6 (C) 7 (D) 9 (E) 11 22. How many non-congruent triangles with integer side lengths and perimeter at most 20 have at least one angle whose measure, in degrees, is an integer? (A) 6 (B) 7 (C) 8 (D) 9 (E) 10 23. How many integers n with 1 ≤ n ≤ 100 can be written in the form A 3 + B 3 + C 3 − 3 ABC for some nonnegative integers A , B , and C ? (A) 67 (B) 72 (C) 75 (D) 78 (E) 89 24. The Fibonacci numbers are defined by F 1 = F 2 = 1 and F n +1 = F n + F n − 1 for n ≥ 2. For how many positive integers n ≤ 1000 is F n divisible by 100? (A) 4 (B) 5 (C) 6 (D) 13 (E) 33 5 AMC 12 Mock Exam B Unofficial practice paper 25. Triangle ABC has AB = 13, BC = 14, and CA = 15. Let I be its incenter, O its circumcenter, and R its circumradius. The incircle touches BC , CA , AB at D , E , F , respectively. Points D ′ , E ′ , F ′ lie on rays ID , IE , IF , respectively, with ID ′ = IE ′ = IF ′ = R Let K be the orthocenter of triangle D ′ E ′ F ′ . What is OK ? (A) √ 65 8 (B) 2 (C) √ 65 4 (D) √ 65 2 (E) 65 16 6