1 OAA OMC Orange Math Competitions Orange Math Competitions OMC 10 Orange Mathematics Competitions Saturday, October 30, 2021 INSTRUCTIONS 1. DO NOT LOOK AT THE PROBLEMS UNTIL YOU ARE READY TO BEGIN. 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 however you want. If you would like to create a more realistic test experience, then you may obtain an AMC 10 Answer Sheet from https://www. maa.org/math-competitions/amc-1012/ and mark you answer to each problem on the AMC 10 Answer Sheet with a number 2 pencil. To simulate the real test, check the blackened circles for accuracy and erase errors and stray marks completely. Only answers properly marked on the answer sheet will be graded in a real test. For the OMC, you must submit your answers using the Submission Form found at https://tinyurl.com/omc10submission. Only answers submitted to the Submission 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. Only pencils, erasers, rulers, and scratch paper are allowed as aids. No calculators, smartwatches, phones, computing devices, or resources such as Wolfram Alpha are allowed. No problems on the exam require the use of a calculator. 6. Figures are not necessarily drawn to scale. 7. Before beginning the exam, you will ask yourself to record certain information on the answer form if you chose to obtain an AMC 10/12 Answer Sheet from https://www.maa.org/ math-competitions/amc-1012/. You will have 75 MINUTES to complete the test. 8. When you finish the exam, sign your name in the space provided at the top of the Answer Sheet should you choose to obtain one from https://www.maa.org/math-competitions/ amc-1012/. 9. Enjoy the problems! The Committee on the Orange Math Competitions reserves the right to disqualify scores from an individual if it determines that the required security procedures were not followed. 2 2021 OMC 10 Problems 1. Jonus is required to take a pill at least once every two consecutive days. What is the least number of times he could have taken the pill in a week? (A) 2 (B) 3 (C) 4 (D) 5 (E) 6 2. Annie is playing a game with 3 red and 3 green marbles in a container. Every second, she selects two marbles from the container at random without replacement. She continues to draw pairs of marbles until the two selected are of different colors, in which case she wins. What is the least number of marbles that Annie must draw to guarantee a win? (A) 2 (B) 3 (C) 4 (D) 6 3. What value of x satisfies (A) 190 (B) 210 (E) Annie is not guaranteed to win 20 2 19 + = ? x 20 21 (C) 420 (D) 760 (E) 840 4. Elysia and her three cousins are sharing fish tacos. If all four people ate a distinct composite number of fish tacos, what is the least number of fish tacos they could have eaten in total? (Note: 1 is not considered to be composite.) (A) 20 (B) 23 (C) 25 (D) 27 (E) 28 5. Rounded to the nearest percent, Joshua got a 44% on his true/false test. Given that the test contained at most 10 questions, how many questions were on the test? (A) 5 (B) 6 (C) 7 (D) 8 (E) 9 6. For which of the following pairs (a, b) is a2 + b = b2 + a? (A) (−2022, 2021) (E) (2021, −2021) (B) (−2022, 2023) (C) (−2021, 2020) (D) (2021, −2022) 7. There are 10 integers written on a blackboard with mean n for some positive integer n. When a positive integer k is added on the list, the mean of the eleven numbers is 2n − 1. Which of the following numbers could be k? (A) 2027 (B) 2028 (C) 2029 (D) 2030 (E) 2031 8. Three real numbers a, b, c are in an arithmetic progression in that order. If ab = 13 and bc = 17, what is the value of b? (A) √ 13 (B) √ 4 221 (C) √ 15 (D) √ 17 (E) √ 30 9. Five sisters, Ann, Emily, Lauren, Mia, and Sophia, have ages 4, 7, 9, 14, and 16 respectively. On one summer day, a group of sisters whose ages sum up to 30 went to watch a movie while the other sisters stayed at home. Which two sisters must have been together? (A) Ann and Emily (B) Ann and Lauren (C) Ann and Sophia (D) Emily and Lauren (E) Mia and Sophia 10. How many positive integers N less than or equal to 1000 are there such that 75% of N ’s divisors are multiples of 3? (A) 24 (B) 25 (C) 36 (D) 37 (E) 38 11. What is the perimeter of the quadrilateral formed by the tange