2021

MAC CMC

CMC 12B
Christmas Math Competitions

Christmas Math Competitions

4th Annual

CMC 12B
DO NOT OPEN UNTIL SATURDAY, January 30, 2021

Christmas Math Competitions

Christmas Math Contest 12B
Questions and comments about problems and Saturday, January 30, 2021
solutions for this exam should be emailed to:
christmas.math.team@gmail.com INSTRUCTIONS
The 4th Annual CIME will be held on Saturday, January 2, 2021, with the alternate on Saturday, February 13, 1. DO NOT OPEN THIS BOOKLET UNTIL YOU HAVE STARTED YOUR TIMER.
2021. It is a 15-question, 3-hour, integer-answer exam. You will be invited to participate regardless of your score
on this competition. All students will be invited to take the 4th Annual Christmas American Math Olympiad 2. This is a 25-question multiple-choice exam. Each question is followed by answers marked
(CAMO) or the Christmas Junior Math Olympiad (CJMO) on Saturday, January 9, 2021. A, B, C, D, and E. Only one of these is correct.
A complete listing of our previous publications may be found at our web site:
3. Mark your answer to each problem however you want. If you would like to create a more
http://cmc.ericshen.net/ realistic test experience, then you may obtain an AMC 10 Answer Sheet from https:
//www.maa.org/math-competitions/amc-10-12/ and mark you answer to each problem
on the AMC 10 Answer Sheet with a #2 pencil. To simulate the real test, check the
**Administration On An Earlier Date Will Literally Be Impossible** 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 CMC, you must
1. All the information needed to administer this exam is contained in the non-existent CMC 10/12 Teacher’s submit your answers using the Submission Form found at http://cmc.ericshen.
Manual. PLEASE READ THE MANUAL EVERY DAY BEFORE JANUARY 30, 2021.
net/CMC-2021/. Only answers submitted to the Submission Form will be scored.
2. YOU must not verify on the CMC 10/12 COMPETITION CERTIFICATION FORM (found on maa.org/
4. Scoring: You will receive 6 points for each correct answer, 1.5 points for each problem left
amc under “AMC 12B”) that you followed all rules associated with the administration of the exam.
unanswered, and 0 points for each incorrect answer.
3. If you chose to obtain an AMC 10 Answer Sheet from the MAA’s website, it must be returned to yourself the
day after the competition. Ship with inappropriate postage without using a tracking method. FedeX333X
5. Only scratch paper, rulers, compasses, and erasers are allowed as aids. No calculators,
or UPS is strongly recommended. smartwatches, phones, computing devices, graph paper, protractors, or resources such as
Wolfram Alpha are allowed. No problems on the exam require the use of a calculator.
4. The publication, asexual reproduction, sexual reproduction, or communication of the problems or solutions
of this exam during the period when students are eligible to participate seriously jeopardizes the definite 6. Figures are not necessarily not drawn to scale.
(but not indefinite) integrity of the results. Dissemination via phone, email, raven, or digital media of any
type during this period is a violation of the competition rules. 7. Before beginning the exam, you will ask yourself to record certain information on the an-
swer form if you chose to obtain an AMC 10 Answer Sheet from https://www.maa.org/
math-competitions/amc-10-12/.
The Christmas Math Competitions 8. When you give yourself the signal, begin working on the problems. You will have 75
is made possible by the contributions of the minutes to complete the exam.
following problem-writers and test-solvers:
9. 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/
David Altizio, Allen Baranov, Ankan Bhattacharya, Luke Choi,
amc-10-12/.
Federico Clerici, Rishabh Das, Mason Fang, Raymond Feng, Preston Fu
Valentio Iverson, Minjae Kwon, Benjamin Lee, Justin Lee, Kyle Lee,
The Committee on the Christmas Math Competitions reserves the right to disqualify scores from an individual if it determines
Kaiwen Li, Sean Li, Elliott Liu, Eric Shen, Albert Wang, Anthony Wang, that the required security procedures were not followed.
Andrew Wen, Tovi Wen, Nathan Xiong, and Joseph Zhang All students will be invited to take the 4th annual Christmas Invitational Math Examination (CIME) on Saturday,
January 2, 2021 and Saturday, February 13, 2021. More details about the CIME are in the back of this test booklet.
2021 CMC 12B Problems 2

1. What is the value of

(2 + 0 + 2 + 1)2 − (2 − 0 + 2 − 1)(2 + 0 − 2 + 1)?

(A) 0 (B) 13 (C) 22 (D) 23 (E) 25

2. A positive integer is a semiprime if it is the product of two distinct primes. Suppose

m, n are positive integers such that m + 1, m + 2, . . ., m + n are all semiprimes. What
is the largest possible value of n?

(A) 2 (B) 3 (C) 4 (D) 5 (E) 6

3. What is the units digit of |20212 − 22021 |?

(A) 1 (B) 3 (C) 5 (D) 7 (E) 9

4. What is the value of

√ √ √ √ √
21 25 29 213 217
1
+ 5
+ 9
+ 13
+ + · · ·?
2 2 2 2 217
√ √ √
2 2 2 2 √
(A) (B) (C) (D) 1 (E) 2
3 2 3

5. If x, y, z are positive integers such that

x · y · z = 7!,

what is the smallest possible value of max(x, y, z)?

(A) 14 (B) 16 (C) 18 (D) 20 (E) 21

6. Luke the Lizard is crawling along the edges of a cube. He starts at vertex A, and each
minute, he moves to one of the three vertices adjacent to the vertex he is currently on
with equal probability. What is the probability that after three minutes, he is at vertex
B, which is opposite A?
1 2 1
(A) 0 (B) (C) (D) (E) 1
9 9 3

7. How many ways can the letters of the word COM BO be arranged such that the M is
adjacent to both a vowel and a consonant?

(A) 6 (B) 12 (C) 18 (D) 24 (E) 32

 2021 CMC 12B Problems 6 3 2021 CMC 12B Problems

23. Let φ(n) denote the number of positive integers less than or equal to n that are relatively 8. An equilateral triangle is inscribed in another equilateral triangle as in the diagram
prime to n. For example, φ(6) = 2 and φ(15) = 8. For how many positive integers below. If the side length of the outer
√ √
equilateral triangle is 1, the side length of the
1 < n < 100 does φ(n) | (5n + 1)? inner triangle can be represented as a−c b for positive integers a, b, and c such that a
and b are square free. What is a + b + c?
(A) 4 (B) 5 (C) 6 (D) 7 (E) 8

24. Let p1 < p2 < · · · < p24 be the distinct permutations of 1234 in increasing order. For
positive integers 1 ≤ i < j ≤ 24, we define d(i,j) = pj − pi . For example, d(1,24) =
4321 − 1234 = 3087. Over all possible choices of (i, j), how many distinct values of d(i,j) 45◦
are there?

(A) 92 (B) 94 (C) 96 (D) 98 (E) 100

25. Let ABCD be a quadrilateral with AB = 8 and CD = 11. If ABC has area 80 and
DBC has area 72, then what is the least possible area of ABCD?
(A) 10 (B) 12 (C) 13 (D) 15 (E) 16
(A) 96 (B) 108 (C) 112 (D) 128 (E) 152
9. Nine friends play a game. The first person says 1. For 1 ≤ n ≤ 8, if the nth person
says k then the (n + 1)th person will say either k − 1 or k + 1 with equal probability.
What is the probability that the number the 9th person says is positive?
163 41 165 83 167
(A) (B) (C) (D) (E)
256 64 256 128 256

10. On the 31st December 2020, Andre realizes that the number of days until his 21st
birthday is the square of the number of entire months leading to it. When does his
birthday fall?

(A) May 16th (B) Jun 17th (C) Jul 18th (D) Aug 19th (E) Sep 20th

11. Alice and Billy are told by their teacher to compute “log x to the power of log x to
the power of log x” for some constant x, and where log is the base-10 logarithm. Alice
computes ( )
log xlog(x )
log x

while Billy computes

( )log x
(log x)log x ,
but surprisingly, they get the same answer. Let S be the sum of (log x)2 over all possible
values of x. Find S.

(A) 1 (B) 3 (C) 4 (D) 6 (E) 8

12. Suppose n is a composite positive integer, and let f (n) be the third-largest divisor of
n. For how many positive integers n ≤ 720 is f (n) divisible by 4?

(A) 45 (B) 60 (C) 72 (D) 90 (E) 120

 2021 CMC 12B Problems 4 5 2021 CMC 12B Problems

13. Let 2x3 − 11x2 + 18x − a be a polynomial with real roots such that the sum of the 18. In the figure below, ABCDEF GH is a cube with side length 10. If I is the center of
reciprocals of two of the roots equals the third root. What is a? ADHE, J is the center of ABCD, K is the center of CDHG, and L is the midpoint
of CG, then the distance between L and the plane formed by △IJK can be written as
9 9 81 81 m
(A) (B) (C) 9 (D) (E) √
n
for positive integers m, n such that n is square-free. Find m + n.
4 2 4 2
H G
14. Jensen has polynomials (x − 1), (x − 1)(x − 2), . . ., (x − 1)(x − 2) · · · (x − 2020)(x − 2021)
E F
and he decides to split them in two groups. Let P (x) and Q(x) denote the product of
K
all the polynomials in the first and in the second group, respectively. Given that Q(x) I
L

is divisible by P (x), what is the smallest possible degree of Q(x)

P (x)
?

(A) 1 (B) 3 (C) 1009 (D) 1011 (E) 2021 D J C

A B

15. There are n values of x which satisfy

(A) 7 (B) 8 (C) 11 (D) 12 (E) 13
2 2020 2
2
⌊x⌋ = {x} + x.
2021 19. Define a positive integer k ≥ 2 to be artistic if there exists a natural number m such
What is the remainder when n is divided by 5? (Here, ⌊•⌋ is the greatest integer that for any positive integer n that is relatively prime to k, we have
function and {•} is the fractional part function.)
mn + nm ≡ 0 (mod k).
(A) 0 (B) 1 (C) 2 (D) 3 (E) 4
Determine the sum of all artistic numbers.

16. Let ABCD be an isosceles trapezoid with bases AB and CD, such that AB < CD. Let (A) 2 (B) 5 (C) 8 (D) 13 (E) 18
ℓ be the altitude from B to AC. Suppose that AD = BC = 5, AB = 4, and ℓ bisects
the area of ABCD. Compute CD. 20. Let f (n) be a function that takes in a positive integer n and outputs the digit that occurs
√ √ √ √ most frequently in n, with ties going to the smaller digit. For example, f (123) = 1
(A) 3 5 (B) 4 3 (C) 7 (D) 5 2 (E) 2 13
and f (2020) = 0. A three-digit positive integer n is selected at random. What is the
probability that f (n) is even?
17. For a permutation of 1, 2, 3, 4, 5, 6, let ak denote the kth element of the permutation.
Suppose that for all integers 1 ≤ i < j ≤ 5, |ai+1 − ai | ̸= |aj+1 − aj |. How many 31 469 24 5 559
(A) (B) (C) (D) (E)
permutations exist in this manner? 60 900 45 9 900

(A) 18 (B) 24 (C) 28 (D) 30 (E) 36 21. An infinite arithmetic sequence of nonnegative real numbers a1 , a2 , a3 , . . . satisfies
a25 − a22 = 432. Find the minimum value of a26 .

(A) 540 (B) 648 (C) 720 (D) 768 (E) 840

22. Triangle △P QR with P Q = 3, QR = 4, RP = 5 is drawn inside a regular hexagon

ABCDEF with P on segment F A, Q the midpoint of segment AB, and R on segment
CD. Given that AB 2 = m
n
for relatively prime positive integers m and n, find m + n.

(A) 865 (B) 866 (C) 867 (D) 868 (E) 869