Combinatorics
Four distinct circles are drawn in a plane. What is the maximum number of points where at least two of the circles intersect?
A) 8 B) 9 C) 10 D) 12 E) 16
D) 12
Samia set off on her bicycle to visit her friend, traveling at an average speed of 17 kilometers per hour. When she had gone half the distance to her friend's house, a tire went flat, and she walked the rest of the way at 5 kilometers per hour. In all, it took her 44 minutes to reach her friend's house. In kilometers rounded to the nearest tenth, how far did Samia walk?
A) 2.0 B) 2.2 C) 2.8 D) 3.4 E) 4.4
C) 2.8
In multiplying two positive integers a and b, Ron reversed the digits of the two-digit number a. His erroneous product was 161. What is the correct value of the product a and b?
A) 116 B) 161 C) 204 D) 214 E) 224
E) 224
Chloe chooses a real number uniformly at random from the interval [0, 2017]. Independently, Laurent chooses a real number uniformly at random from the interval [0, 4034]. What is the probability that Laurent’s number is greater than Chloe’s number?
(A) 1/2 (B) 2/3 (C) 3/4 (D) 5/6 (E) 7/8
C) 3/4
There are 24 different complex numbers z such that z^24 = 1. For how many of these is z^6 a real number? (A) 1 (B) 3 (C) 6 (D) 12 (E) 24
D) 12
How many different integers can be expressed as the sum of three distinct members of the set {1,4,7,10,13,16,19}?
A) 13 B) 16 C) 24 D) 30 E) 35
A) 13
Five positive consecutive integers starting with a have average b. What is the average of 5 consecutive integers that start with b?
A) a+3 B) a+4 C) a+5 D) a+6 E) a+7
B) a + 4
What is the sum of the exponents of the prime factors of the square root of the largest perfect square that divides 12!?
A) 5 B) 7 C) 8 D) 10 E) 12
C) 8
A number m is randomly selected from the set {11, 13, 15, 17, 19}, and a number n is randomly selected from {1999, 2000, 2001, . . . , 2018}. What is the probability that m^n has a units digit of 1?
(A) 1/5 (B) 1/4 (C) 3/10 (D) 7/20 (E) 2/5
The polynomial f(x) = x 4 + ax3 + bx2 + cx + d has real coefficients, and f(2i) = f(2 + i) = 0.
What is a + b + c + d?
(A) 0 (B) 1 (C) 4 (D) 9 (E) 16
D) 9
Alice has 24 apples. In how many ways can she share them with Becky and Chris so that each of the three people has at least two apples?
A) 105 B) 114 C) 190 D) 210 E) 380
C) 190
A function f is defined recursively by f(1) = f(2) = 1 and f(n) = f(n-1) - f(n-2) + n for all integers n > 2. What is f(2018)?
A) 2016 B) 2017 C) 2018 D) 2019 E) 2020
B) 2017
A positive integer n is nice if there is a positive integer m with exactly four positive divisors (including 1 and m) such that the sum of the four divisors is equal to n. How many numbers in the set {2010, 2011, 2012, ..., 2019} are nice?
A) 1 B) 2 C) 3 D) 4 E) 5
A) 1
An ”unfair” coin has a 2/3 probability of turning up heads. If this coin is tossed 50 times, what is the probability that the total number of heads is even?
(A) 25(2/3)^50 (B) 1/2*(1 - 3^-50) (C) 1/2
(D) 1/2*(1 + 3^-50) (E) 2/3
(D) 1/2*(1 + 3^-50)
For what value of n is i + 2*i^2 + 3*i^3 + · · · + n*i^n = 48 + 49i? Note: here i = √ −1.
(A) 24 (B) 48 (C) 49 (D) 97 (E) 98
(D) 97
How many four-digit positive integers have at least one digit that is a 2 or a 3?
A) 2439 B) 4096 C) 4903 D) 4904
E) 5416
E) 5416
How many non-similar triangles have angles whose degree measures are distinct positive integers in arithmetic progression?
A) 0 B) 1 C) 59 D) 89 E) 178
C) 59
Let S(n) equal the sum of the digits of positive integer n. For example, S(1507) = 13. For a particular positive integer n, S(n) = 1274. Which of the following could be the value of S(n+1)?
A) 1 B) 3 C) 12 D) 1239 E) 1265
A) D
Amelia has a coin that lands heads with probability 1 3 , and Blaine has a coin that lands on heads with probability 2 5 . Amelia and Blaine alternately toss their coins until someone gets a head; the first one to get a head wins. All coin tosses are independent. Amelia goes first. The probability that Amelia wins is p q , where p and q are relatively prime positive integers. What is q − p?
(A) 1 (B) 2 (C) 3 (D) 4 (E) 5
D) 4
How many complex numbers 𝑧 satisfy 𝑧^5 = |𝑧|^2/z, where |𝑧| is the magnitude of the complex number 𝑧?
A) 2 B) 3 C) 5 D) 6 E) 7
E) 7
For some particular value of N, when (a+b+c+d+1)^N is expanded and like terms are combined, the resulting expression contains exactly 1001 terms that include all four variables a, b, c, and d, each to some positive power. What is N?
A) 9 B) 14 C) 16 D) 17 E) 19
B) 14
Suppose that sin a + sin b = sqrt(5/3) and cos a + cos b = 1. What is cos(a-b)?
A) sqrt(5/3) - 1 B) 1/3 C) 1/2 D) 2/3 E) 1
B) 1/3
The number obtained from the last two nonzero digits of 90! is equal to n. What is n?
A) 12 B) 32 C) 48 D) 52 E) 68
A) 12
The number 21! = 51, 090, 942, 171, 709, 440, 000 has over 60, 000 positive integer divisors. One of them is chosen at random. What is the probability that it is odd?
(A) 1/21 (B) 1/19 (C) 1/18 (D) 1/2 (E) 11/21
B) 1/19
Find the number of ordered pairs of real numbers (a, b) such that
(a + bi)^2002 = a − bi.
(A) 1001 (B) 1002 (C) 2001 (D) 2002 (E) 2004
E) 2004