2019 AMC 10C Problems

Here are the problems from the 2019 AMC 10C, a mock contest created by the AoPS user fidgetboss_4000.

Problem 1

Given that $a+m=2017$ , $m+c=2018$ , and $a+c=2019$ , find the value of $a+m+c+10$ .

$\mathrm{(A) \ } 3027\qquad \mathrm{(B) \ } 3037\qquad \mathrm{(C) \ } 4037\qquad \mathrm{(D) \ } 6054\qquad \mathrm{(E) \ } 6064$

Solution

Problem 2

Three distinct vertices of a regular pentagon are chosen at random. What is the probability that they form an obtuse triangle?

$\mathrm{(A) \ } \frac{1}{10}\qquad \mathrm{(B) \ } \frac{1}{5}\qquad \mathrm{(C) \ } \frac{1}{2}\qquad \mathrm{(D) \ } \frac{4}{5}\qquad \mathrm{(E) \ } \frac{9}{10}$

Solution

Problem 3

The numbers from 1 to 10 are written on 10 slips of paper, with 1 number on each slip of paper, and put in a bag. Maria will randomly draw 2 slips from the bag without replacement. What is the probability that she will obtain one prime number and one composite number?

$\mathrm{(A) \ } \frac{2}{15}\qquad \mathrm{(B) \ } \frac{2}{5}\qquad \mathrm{(C) \ } \frac{4}{9}\qquad \mathrm{(D) \ } \frac{5}{9}\qquad \mathrm{(E) \ } \frac{8}{9}$

Solution

Problem 4

In the figure below, the part of the 1st square that is not contained in the 2nd square will be shaded, the part of the 3rd square that is not contained in the 4th square will be shaded, the part of the 5th square that is not contained in the 6th square will be shaded, and so on to infinity. What is the ratio of the shaded area to the non-shaded area?

$\mathrm{(A) \ } \frac{1}{3}\qquad \mathrm{(B) \ } \frac{1}{2}\qquad \mathrm{(C) \ } \frac{3}{2}\qquad \mathrm{(D) \ } 2\qquad \mathrm{(E) \ } 3$

Solution

Problem 5

The numeral representation of the product $99...9 \times 999...9$ , where the former number has $n$ 9’s and the latter number has $n+1$ 9’s, where $n$ is some positive integer, has a sum of digits of 630. Find $n$ .

$\mathrm{(A) \ } 68\qquad \mathrm{(B) \ } 69\qquad \mathrm{(C) \ } 70\qquad \mathrm{(D) \ } 71\qquad \mathrm{(E) \ } 72$

Solution

Problem 6

A function $f(x)$ is defined for all positive integers $n>2$ as $f(n)=f(n-1) \times f(n-2)$ . Given that $f(1)=2$ and $f(2)=3$ , the value of $f(12)$ can be expressed in the form of $2^{p}3^{q}$ ,where $p$ and $q$ are positive integers. Find $|p-q|$ .

$\mathrm{(A) \ } 21\qquad \mathrm{(B) \ } 34\qquad \mathrm{(C) \ } 55\qquad \mathrm{(D) \ } 89\qquad \mathrm{(E) \ } 144$

Solution

Problem 7

Given that $n^4 = 22,667,121$ for a positive integer $n$ , find the sum of the digits of $n$ .

$\textbf{(A)}\ 6 \qquad\textbf{(B)}\ 9 \qquad\textbf{(C)}\ 12 \qquad\textbf{(D)}\ 15 \qquad\textbf{(E)}\ 18$

Solution

Problem 8

When rolling 12 standard 6-sided dice, the probability that the sum of the numbers rolled on the 12 dice is 69 can be expressed in $\frac{N}{6^{12}}$ . Find the sum of the digits of $N$ .

$\mathrm{(A) \ } 4\qquad \mathrm{(B) \ } 10\qquad \mathrm{(C) \ } 13\qquad \mathrm{(D) \ } 14\qquad \mathrm{(E) \ } 15$

Solution

Problem 9

A gear with radius 34.5 rolls around another the circumference of a larger gear with radius 103.5. How many revolutions around the larger gear would the smaller gear have completed by the time it makes 3 complete rotations?

$\mathrm{(A) \ } \frac{1}{2}\qquad \mathrm{(B) \ } \frac{2}{3}\qquad \mathrm{(C) \ } \frac{3}{4}\qquad \mathrm{(D) \ } 1\qquad \mathrm{(E) \ } \frac{6}{5}$

Solution

Problem 10

Al and Bob play a game of Bottle Flip. Each of them takes turns flipping a water bottle and the first person to not land the bottle is deemed the loser. Al has a $\frac{4}{5}$ chance of landing the bottle every time he flips it, and Bob has a $\frac{3}{4}$ chance of landing the bottle every time he flips it. Given that Al goes first, what is the probability that he wins the game?

$\mathrm{(A) \ } \frac{1}{3}\qquad \mathrm{(B) \ } \frac{3}{8}\qquad \mathrm{(C) \ } \frac{1}{2}\qquad \mathrm{(D) \ } \frac{5}{8} \qquad \mathrm{(E) \ } \frac{2}{3}$

Solution

Problem 11

How many base 3 positive integers $n$ with nonzero digits have the property that the sum of the digits of $n$ is 10?

$\mathrm{(A) \ } 74\qquad \mathrm{(B) \ } 79\qquad \mathrm{(C) \ } 88\qquad \mathrm{(D) \ } 89\qquad \mathrm{(E) \ } 97$

Solution

Problem 12

In music, a whole octave consists of 12 semitones: C, C#, D, D#, E, F, F#, G, G#, A, A#, and B. Consist a certain four-note chord to be pretty if all four notes are from the same octave and no two notes are either $1$ semitone or $11$ semitones apart. How many four-note chords are not pretty?

$\textbf{(A)}\ 117 \qquad\textbf{(B)}\ 369 \qquad\textbf{(C)}\ 390 \qquad\textbf{(D)}\ 425 \qquad\textbf{(E)}\ 440$

Solution

Problem 13

How many positive integers $N$ where $1 \leq N \leq 20$ cannot be the number of 0s at the end of the decimal representation of $x!$ , where $x$ is a positive integer?

$\mathrm{(A) \ } 2\qquad \mathrm{(B) \ } 3\qquad \mathrm{(C) \ } 4\qquad \mathrm{(D) \ } 5\qquad \mathrm{(E) \ } 6$

Solution

Problem 14

Let $M$ denote the largest integer power of 16 that allows $100^{2019}-16^{M}$ to be a positive integer and let $N$ denote the smallest integer power of 100 that allows $16^{2019}-100^{N}$ to be a negative integer. In which of the following ranges does $M+N$ lie?

$\mathrm{(A) \ } [2000,3000]\qquad \mathrm{(B) \ } [3000,4000]\qquad \mathrm{(C) \ } [4000,5000]\qquad \mathrm{(D) \ } [5000,6000]\qquad \mathrm{(E) \ } [6000,7000]$

Solution

Problem 15

Let $N$ be $2019^{2019^{2019^{2019...}}}$ . (2019 $2019$ 's) Find the remainder when $N$ is divided by $7$ .

$\mathrm{(A) \ } 1\qquad \mathrm{(B) \ } 2\qquad \mathrm{(C) \ } 3\qquad \mathrm{(D) \ } 4\qquad \mathrm{(E) \ } 6$

Solution

Problem 16

Given that : 1.The sum of the roots of $ax^3+bx^2+cx+d=0$ , where $a, b, c,$ and $d$ are integers no more than 1000 and no less than $-1000$ , is $7$ . 2.The product of the roots in $ax^4+bx^3+cx+abcd=0$ , where $a, b, c,$ and $d$ are integers no more than 1000 and no less than $-1000$ , is $420$ . Find the number of ordered pairs $(a, b, c, d)$ that satisfy these conditions.

$\mathrm{(A) \ } 27\qquad \mathrm{(B) \ } 54\qquad \mathrm{(C) \ } 108\qquad \mathrm{(D) \ } 162\qquad \mathrm{(E) \ } 216$

Solution

Problem 17

Define a subset of the first ten positive integers {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} to be spicy if the sum of the squares of the elements in the subset is divisible by 5. How many spicy subsets are there?

$\mathrm{(A) \ } 35\qquad \mathrm{(B) \ } 70\qquad \mathrm{(C) \ } 140\qquad \mathrm{(D) \ } 210\qquad \mathrm{(E) \ } 280$

Solution

Problem 18

A tripod has three legs of length 3 feet, 4 feet, and 4 feet. It is set up so that the angle between any two legs is $90^{\circ}$ . The height from the top of the tripod to the ground, in feet, can be expressed in the form $\dfrac{a\sqrt{b}}{c}$ , where $a$ , $b$ , and $c$ are positive integers, $a$ and $c$ are relatively prime, and $b$ is not divisible by the square of a prime. What is $a+b+c$ ?

$\textbf{(A)}\ 53 \qquad\textbf{(B)}\ 57 \qquad\textbf{(C)}\ 69 \qquad\textbf{(D)}\ 80 \qquad\textbf{(E)}\ 89$

Solution

Problem 19

Triangle $ABC$ has $AB=10$ , $BC=14$ , and $AC=22$ . The line $AN$ , where $N$ is a point on $BC$ , divides the triangle into two halves with equal perimeters. Let $N$ be the length of $AN$ . Find the greatest integer less or equal to $N$ .

$\mathrm{(A) \ } 18\qquad \mathrm{(B) \ } 19\qquad \mathrm{(C) \ } 20\qquad \mathrm{(D) \ } 21\qquad \mathrm{(E) \ } 22$

Solution

Problem 20

How many five digit positive integers have the property that every two adjacent digits are consecutive? For example, the integer 32345 has this property, while the integer 98623 does not.

$\mathrm{(A) \ } 116\qquad \mathrm{(B) \ } 120\qquad \mathrm{(C) \ } 125\qquad \mathrm{(D) \ } 127\qquad \mathrm{(E) \ } 144$

Solution

Problem 21

Jack eats candy while abiding to the following rule: On Day $n$ , he eats exactly $n^3$ pieces of candy if $n$ is odd and exactly $n^2$ pieces of candy if $n$ is even. Let $N$ be the number of candies Jack has eaten after Day 2019. Find the last two digits of $N$ .

$\textbf{(A)}\ 16 \qquad\textbf{(B)}\ 30 \qquad\textbf{(C)}\ 40 \qquad\textbf{(D)}\ 80 \qquad\textbf{(E)}\ 90$

Solution

Problem 22

$100$ chicks are sitting in a circle. The first chick in the circle says the number $1$ , then the chick $2$ seats away from the first chick says the number $2$ , then the chick $3$ seats away from the chicken that said the number $2$ says the number $3$ , and so on. The process will always go clockwise. Some of the chicks in the circle will say more than one number while others might not even say a number at all. The process stops when the $1001$ th number is said. How many numbers would the chick that said $1001$ have said by that point (including $1001$ )?

$\mathrm{(A) \ } 20\qquad \mathrm{(B) \ } 21\qquad \mathrm{(C) \ } 30\qquad \mathrm{(D) \ } 31\qquad \mathrm{(E) \ } 41$

Solution

Problem 23

Bernado has an infinite amount of red, blue, orange, pink, yellow, purple, and black blocks. He puts them in the 2 by 2019 grid such that adjacent blocks are of different colors. What is the hundreds digit of the number of ways he can put the blocks in?

$\textbf{(A)}\ 0\qquad\textbf{(B)}\ 1\qquad\textbf{(C)}\ 2\qquad\textbf{(D)}\ 3\qquad\textbf{(E)}\ 4$

Solution

Problem 24

Let $S$ be the set of points $(x,y,z)$ in the 3-dimensional coordinate plane such that $x$ , $y$ , and $z$ are integers, $0 \le x \le 2$ , $0 \le y \le 2$ , and $0 \le z \le 1$ . How many tetrahedrons of positive volume can be formed by choosing four points in $S$ as vertices of the tetrahedron?

$\mathrm{(A) \ } 2436\qquad \mathrm{(B) \ } 2472\qquad \mathrm{(C) \ } 2580\qquad \mathrm{(D) \ } 2664\qquad \mathrm{(E) \ } 3060$

Solution

Problem 25

Let $N$ be the least positive integer $x$ such that $\left\lfloor \frac{x^{8}}{x-1}\right\rfloor$ is a multiple of 10000. Find the sum of the digits of $N$ . (Note: $\lfloor x \rfloor$ denotes the greatest integer less than or equal to $x$ . )