Mock AMC 10/12 djmathman November 23, 2013 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 the Google Form to djmathman with a working computer. Check your work for accuracy and mis-selections completely. Only answered properly marked in the form 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 form. 8. When your proctor or timer gives the signal, begin working on the problems. You will have 75 minutes to complete the test. 9. When you finish the exam, make sure djmathman knows you are not a troll. 1
djmathman s Mock AMC10 Page 2 Mock AMC 10 1. What is the value of 1 + 2 + 3 2 + 3 + 4 + 3 + 4 + 5 4 + 5 + 6? (A) 5 4 (B) 22 15 (C) 25 12 (D) 147 60 (E) 5 2 2. Through an online internet service named Likes, the amount of money in Sam s back account increases by $0.03 for every person who accesses his page through a link. Suppose that at midnight on Monday Sam s account was worth $10.56, and by midnight on Tuesday Sam s account was worth $11.25. How many people clicked on Sam s link on Monday? (A) 15 (B) 18 (C) 20 (D) 22 (E) 23 3. Suppose that x is a real number such that 7x + 19 = 2013. What is 7x 19? (A) 1971 (B) 1973 (C) 1975 (D) 1977 (E) 1994 4. The ratio of the area of a circle to the circumference of the same circle is 6. What is the radius of the circle? (A) 6π (B) 12 (C) 6 (D) 3 (E) 6 π 5. How many positive integers from 1 to 1000 have an odd number of digits? (A) 500 (B) 700 (C) 900 (D) 909 (E) 1000 6. Define an ordered triplet of positive integers (a, b, c) to be triangular if it is possible to contruct a triangle with side lengths a, b, and c. What is the maximum value of x + y if the ordered triplet (5, x, y) is triangular, given that x 20 and y 13? (A) 28 (B) 29 (C) 30 (D) 31 (E) 32 7. The probability that it will rain on any given day is 30%. What is the probability that it will rain on exactly one of two consecutive days? (A) 15% (B) 21% (C) 30% (D) 42% (E) 50% 8. A certain number of people attended a mathematics convention. Of the total attendees, 3 5 of them 15 participated in a math competition; 22 of the remaining instead decided to listen to a seminar on Topology. What is the smallest possible number of attendees that neither participated in the competition nor listened to the lecture? (A) 5 (B) 7 (C) 14 (D) 15 (E) 22 9. Let p and q be real numbers with p < 1 and q < 1 such that p + pq + pq 2 + pq 3 + = 2 and q + qp + qp 2 + qp 3 + = 3. What is 100pq? (A) 24 (B) 32 (C) 36 (D) 40 (E) 48
djmathman s Mock AMC10 Page 3 10. A regular 2013-gon is placed in the plane in general position. A family of lines is drawn on this plane such that no two lines are parallel, and for every vertex A of the 2013-gon, there exists a line that passes through A. What is the smallest number of possible lines in the family? (A) 504 (B) 1006 (C) 1007 (D) 2012 (E) 2013 11. In a small private science-oriented school, there are eight classes to choose from: Earth Systems, Biology, Chemistry, Physics, Art, Music, History, and Spanish. David is trying to fill his four-period schedule with classes. He can take at most three science courses, and he has to take exactly one of art or music. If the order of the courses does not matter, how many possible schedules can David make? (A) 40 (B) 44 (C) 48 (D) 52 (E) 56 12. Suppose a and b are real numbers that satisfy the system of equations a + b = 9, a(a 2) + b(b 2) = 21. What is ab? (A) 15 (B) 20 (C) 21 (D) 25 (E) 27 13. Rectangle ABCD has AB = 5 and BC = 12. CD is extended to a point E such that ACE is isosceles with base AC. What is the area of triangle ADE? (A) 15 (B) 131 5 (C) 30 (D) 357 5 (E) 125 14. Rachel went to the local convienece store to buy some packs of seeds. In the convienence store, large packs cost $1.35 apiece and small packs cost $0.85 apiece. If Rachel paid $20.15 for her order, which type of pack did she buy more of, and by how many? (A) 1 more small pack (B) 2 more small packs (C) 3 more small packs (D) 1 more large pack (E) 2 more large packs 15. Let ABCD be a rectangle such that AB = 3 and BC = 4. Suppose that M and N are the centers of the circles inscribed inside triangles ABC and ADC respectively. What is MN 2? B C M N A D (A) 2 (B) 3 (C) 4 (D) 5 (E) 6
djmathman s Mock AMC10 Page 4 16. A rhombus has height 15 and diagonals with lengths in a ratio of 3 : 4. What is the length of the longer diagonal of the rhombus? (A) 10 (B) 25 (C) 15 (D) 20 (E) 25 2 17. Mr. Mebane s Preschool WOOT class is taking a class photograph. There are five boys in the class, and they are 48, 49, 50, 51, and 52 inches tall. In the class photograph, they are to sit themselves in five chairs positioned in a row. In how many ways can Mr. Mebane s students seat themselves such that no boy sits next to another boy who is exactly one inch taller than himself? (A) 6 (B) 8 (C) 10 (D) 12 (E) 14 18. Define the sequence {a k } such that a k = 1! + 2! + 3! + + k! for each positive integer k. (Note: the factorial function x! is defined as the product of the first x positive integers.) For how many positive integers 1 n 2013 does the quantity a 1 + a 2 + a 3 + + a n end in the digit 3? (A) 100 (B) 201 (C) 202 (D) 203 (E) 403 19. For each positive integer n, let a n = n2. Furthermore, let P and Q be real numbers such that 2n + 1 P = a 1 a 2... a 2013, Q = (a 1 + 1)(a 2 + 1)... (a 2013 + 1). What is the sum of the digits of Q P? (Note: x denotes the largest integer x.) (A) 27 (B) 29 (C) 31 (D) 33 (E) 35 20. One deleted scene from the movie Inception involved a square hanging in mid-air at a certain angle as shown. Scientists tried to determine the dimensions of the square directly, but to no avail. However, working her magic, a passerby named Ellen was able to extend the sides of the square until they met the ground at points P, A, G, and E in that order. Simple measurements showed that P A = 2 and GE = 3. What was the area of the square? P A G E (A) 36 13 (B) 3 (C) 48 13 (D) 23 4 (E) 4 21. In ABC, AB = 13, AC = 14, and BC = 15. Let M denote the midpoint of AC. Point P is placed on line segment BM such that AP P C. Suppose that p, q, and r are positive integers with p and r relatively prime and q squarefree such that the area of AP C can be written in the form p q r. What is p + q + r? (A) 221 (B) 257 (C) 302 (D) 368 (E) 391
djmathman s Mock AMC10 Page 5 22. Five weight scales, labeled A, B, C, D, and E, are placed side-by-side on a table. Richard wants to distribute eight identical 5 ounce weights among these five scales. However, if the combined weight on any one of the scales exceeds 15 ounces, the scale will break. In how many ways can Richard distribute the weights such that none of the five scales collapse? (A) 155 (B) 160 (C) 165 (D) 175 (E) 190 23. An integer is said to be two-separable if it can be represented as the product of two positive integers that differ by two. (For example, since 8 = 2 4, 8 is two-separable.) Similarly, an integer is said to be nine-separable if it can be represented as the product of two positive integers that differ by 9. What is the sum of all positive integers that are both two-separable and nine-separable? (A) 460 (B) 424 (C) 396 (D) 375 (E) 360 24. For any set X, denote by X the number of elements in the set, and let the statement a X mean that the element a can be found in set X. How many disjoint subsets A and B of {1, 2, 3, 4, 5, 6, 7, 8} are there such that A + B A and A B B? (A) 294 (B) 316 (C) 342 (D) 363 (E) 375 25. Let P (x) = (x 2 x + 1)(x 4 x 2 + 1)(x 8 x 4 + 1)... (x 1024 x 512 + 1). When P (x) is expanded, how many terms does it contain? (A) 1025 (B) 1365 (C) 1679 (D) 1885 (E) 2043