Monsters University
Mock AMC12
Author: Various People From My School
January 12, 2014
INSTRUCTIONS: 1. DO NOT OPEN THIS BOOKLET UNTIL YOUR PROCTOR TELLS YOU. 2. This is a twenty-five question multiple choice test. Each question is followed by answers marked A, B, C, D, and E. Only one of these is correct. 3. Mark your answer to each problem via PM to StarlightStarbright with a working computer. Check your work for accuracy and backspace errors completely. Only answered properly marked in the PM will be scored. 4. SCORING: You will receive 6 points for each correct answer, 1.5 points for each problem left unanswered, and 0 points for each incorrect answer. 5. No aids are permitted other than scratch paper, graph paper, rulers, compass, protractors, and erasers. No calculators are permitted. No problems on the test will require the use of a calculator. 6. Figures are not necessarily drawn to scale. 7. Before beginning the test, your proctor will ask you to record certain information on your PM. 8. When your proctor or timer gives the signal, begin working on the problems. You will have 75 minutes to complete the test. Special thanks to thecmd999 for problem 23 Special thanks to everybody who helped check this exam
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Problems 1) Find the difference between the sum of the first 5 odd numbers and the sum of the first 5 prime numbers. (A) 1 (B) 3 (C) 5 (D) 6 (E) 7 2) Find the sum of the last 3 digits of 10! (A) 7 (B) 8 (C) 9 (D) 10 (E) 11 3) If Bob works for 10 dollars an hour and Charles works for 5 dollars an hour, and Charles’ wages increase every 2 hours by 1 dollar, what amount will each of them have when they have an equal amount of money? (A) 200 (B) 210 (C) 220 (D) 230 (E) 240 4) Define a masochistic number to be a sequence of digits such that the last n digits form a number divisible by 5n . How long (in digits) is the longest masochistic number? (A) 2 (B) 3 (C) 4 (D) 5 (E) More Than 5 5) Jar A has 3 red and 5 black balls. Jar B has 6 red and 2 black balls. One ball is randomly chosen from each Jar. Given that at least one ball is black, what is the probability that both are black? 5 8 5 (B) 23 (C) 23 (D) 23 (E) 18 (A) 32 32 23 6) Find the number of subsets of 1, 2, 3, 4, 5, 6, 7, 8 that include a 2 or 5, but not both. (A) 64 (B) 128 (C) 192 (D) 240 (E) 256 7) In regular octagon ABCDEF GH, quadrilateral ACEG is drawn. Each side of octagon ABCDEF GH has side length 12. What is the area of quadrilateral ACEG? √ √ √ (A) 144 (B) 144( 2 + 1) (C) 144(2 + 2) (D) 288 (E) 288 2 8) Let S be the set of all numbers one less than an odd perfect square and less than 1000. Find the probability that an element of S, randomly chosen, is divisible by the product of any two consecutive prime numbers. 1 (A) 0 (B) 16 (C) 18 (D) 14 (E) 21 9) Bob has an unlimited amount of 3 dollar bills, 4 dollar bills, and 12 dollar bills. The number of ways that he can pay for a 1200 dollar dinner can be expressed as a a a! (where b = b!(a−b)! ). Find a + b. b (A) 101 (B) 102 (C) 103 (D) 104 (E) 105 2
1 10) Define a number to have swag if it is in the form (2a )(3 b ) , where a, and b are non-negative integers. Find the sum of all numbers with swag. (A) 56 (B) 73 (C) 12 (D) 3 (E) 4 5 11) Given P (x) = x4 + x − 1. Let α, β, γ, ω be the roots of P (x). Find β4 γ4 α4 ω4 + 1−β + 1−γ + 1−ω . 1−α (A) − 16 (B) − 4 (C) 0 (D) 4 (E) 16 12) Let ABCDE be a pyramid with side length 4 and apex E. Let F be the point equidistant from all the vertices of the pyramid. Find the shortest distance from point F to face ABE. √ √ √ √ √ (B) 2 3 6 (C) 4 3 6 (D) 3 (E) 2 3 (A) 36 13) Brice draws a square ABCD with side length 10. He picks two points on AB and CD, and marks them M and N , respectively. He marks point P as the intersection of CM and BN , and Q as the intersection of AN and M D. He is puzzled and continues redrawing this infinitely many times. Of his infinite drawings, what is the smallest length of P Q that he can make? √ √ (C) 10 (D) 10 2 (E) 20 (A) 5 (B) 5 2 2013 P
(ri − log(ri ).
14) Let r be a root of x2 + x + 1 = 0. Find
i=0
(A) − 2013 (B) − 1 (C) 0 (D) 1 (E) 2013 15)The Smith family, consisting of 8 girls and 3 boys, sit in a line of 11 chairs for a family picture. Find the probability that none of the boys are sitting next to each other. 17 38 (A) 55 (B) 27 (C) 28 (D) 55 (E) 49 55 55 55 16) Let ABC be a triangle such that there is a circle with center ω which touches the triangle in only two places- B and D, which is on side AC, such that AB and AD are tangents to circle ω. Let the radius of circle ω be 3, side AB√ be of length √ 2 10, and 6 ωBC = 30 degrees. If cos(6 BAω) can be expressed as a c b , where b is squarefree and a and c are relatively prime, find a + b + c. (A) 15 (B) 17 (C) 19 (D) 21 (E) 23 17) A rigged 6-sided die comes up with the numbers 1, 2, 3, 4, 5, 6 with probabilities 1 1 1 1 1 , , , , 1 , and 32 , respectively. Mui and Hui take turns rolling the dice: Mui 2 4 8 32 16 rolls first, then Hui rolls, then Mui rolls again, then Hui rolls again, etc. Find the probability that Hui is the first person to roll a prime number. 9 (A) 25 (B) 25 (C) 12 (D) 35 (E) 16 25 18) The equation a + b + c = 50 has 1378 solutions over the positive integers. How 3
many solutions are there for the equation a + b + c = 50 over the positive integers if a is not divisible by 2, b is not even, and c ≥ 5? (A) 253 (B) 323 (C) 403 (D) 473 (E) 553 19) Let P (x) be a monic quartic polynomial such that P (1) = 2, P (4) = 17, P (5) = 26, and P (6) = 37. Find the sum of the squares of the roots of P (x). (A) 0 (B) 27 (C) 48 (D) 76 (E) 94 20) Princeton is a puzzled boy, and constructs a triangle whose coordinates are the origin,(θ, 8), (γ, 11). He calls these points α, β, ω, respectively. He makes an isosceles triangle such that 6 βαω is 30 degrees with αβ = αω . He is computing the value of Harvard, a variable whose value is θ2 + γ 2 . In addition, Harvard can √ also be expressed as a + b c where c is squarefree and a and b are relatively prime. Find |b|. (A) 116 (B) 324 (C) 571 (D) 704 (E) 1204 21) Katniss is in the arena and magically calculates that the distance from her location to the cornucopia is 11103 . She needs some food, and Peeta comes and tells her that if she takes the remainder when she divides her distance by 1000, there is a secret stash of food that is the remainder units away. Find the distance to the secret stash of food. (A) 71 (B) 161 (C) 221 (D) 331 (E) 401 4 9
n . 2 (B) 22 9
22) Let an =
Compute
∞ P ai
i=3 100 81
aai
(C) (D) 2 (E) 3 (A) 23) In bizzaro-world, there is a coordinate grid. There is a darkened box on the grid, with vertices at (0, 0), (12, 0), (0, 12), (12, 12). From every lattice point (x, y), where x ≤ 11, and y ≤ 11, a diagonal is drawn through the square grid to the point (x + 1, y + 1). A dragon is placed on (0, 0). The number of ways it can get to (12, 12) if it can travel to the right, upward, or diagonally only is n. Find the sum of the digits of n. (A) 51 (B) 52 (C) 53 (D) 54 (E) 55 24) Donald Duck makes a triangular grid, which is obtained by tiling an equilateral triangle of side 2013 and each of the smaller triangles are of length 1. In such a grid, there are 20132 such triangles. The total number of parallelograms bounded by the line segments of the grid is a. If the remainder of a and 100 is ω, find the value of ω. (A) 15 (B) 35 (C) 55 (D) 75 (E) 95 4
25) 6 students are having a contest. Each one of them picks a real number from 0 to 1 inclusive. Let a1 , a2 , a3 , a4 , a5 , a6 be the numbers they pick. Also, let ω = (a1 − a2 )(a2 − a3 )(a3 − a4 )(a4 − a5 )(a5 − a6 )(a6 − a1 ). What is the highest ω that they can make? 1 (C) 18 (D) 14 (E) 21 (A) 0 (B) 16
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