AMC 12C 2020

Problem 1

What is the sum of the solutions to the equation $(x + 5)(x + 4) - (x + 5)(x - 6) = 0$ ?

$\mathrm{(A) \ } -10\qquad \mathrm{(B) \ } -3\qquad \mathrm{(C) \ } 5\qquad \mathrm{(D) \ } 10\qquad \mathrm{(E) \ } 15$

Problem 2

How many increasing subsets of $\{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}$ contain no $2$ consecutive prime numbers?

Problem 3

A field is on the real $xy$ plane in the shape of a circle, centered at $(5, 6)$ with a a radius of $8$ . The area that is in the field but above the line $y = x$ is planted. What fraction of the field is planted?

Problem 4

What is the numerical value of $1^{3} + 2^{3} + 3^{3} + … + 11^{3}$ ?

$\mathrm{(A) \ } -1000\qquad \mathrm{(B) \ } 1290\qquad \mathrm{(C) \ } 4356\qquad \mathrm{(D) \ } 7840\qquad \mathrm{(E) \ } 8764$

Problem 5

$10$ cows can consume $20$ kilograms of grass in $5$ days. How many more cows are required such that it takes all of the cows to consume $80$ kilograms of grass in $8$ days?

$\mathrm{(A) \ } 15\qquad \mathrm{(B) \ } 16\qquad \mathrm{(C) \ } 18\qquad \mathrm{(D) \ } 19\qquad \mathrm{(E) \ } 22$

Problem 6

$10$ candy canes and $9$ lollipops are to be distributed among $8$ children such that each child gets atleast $1$ candy. What is the probability that once the candies are distributed, no child has both types of candies?

Problem 7

Persons $A$ and $B$ can plough a field in $10$ days, persons $B$ and $C$ can plough the same field in $7$ days, and persons $A$ and $C$ can plough the same field in $15$ days. In how many days can all of them plough the field together?

Problem 8

The real value of $n$ that satisfies the equation $ln(n) + ln(n^{2} - 34) = ln(72)$ can be written in the form $a + \sqrt{b}$ where $a$ and $b$ are integers. What is $a + b$ ?

$\mathrm{(A) \ } -12\qquad \mathrm{(B) \ } 0\qquad \mathrm{(C) \ } 5\qquad \mathrm{(D) \ } 16\qquad \mathrm{(E) \ } 24$

Problem 9

On a summer evening stargazing, the probability of seeing a shooting star in any given hour on a sunny day is $\frac{3}{5}$ and the probability of seeing a shooting star on a rainy day is $\frac{1}{3}$ . Both rainy and sunny days happen with equal chances. What is the probability of seeing a shooting star in the second $15$ minutes of an hour stargazing on a random night?

Problem 10

Let $R(x)$ denote the number of trailing $0$ s in the numerical value of the expression $x!$ , for example, $R(5) = 1$ since $5! = 120$ which has $1$ trailing zero. What is the sum

$R(100) + R(99) + R(98) + R(97) + … + R(3) + R(2) + R(1) + R(0)$ ?

Problem 11

A line of hunters walk into a jungle where the distance between the first and last hunter is $125$ meters which maintains constant throughout their walk as the hunters walk at a constant speed of $5$ meters per second. A butterfly starts from the front of their line and flies to the back as they come forward and then turns and comes back as soon as it reaches the back of the line. When the butterfly is back at the front of the line, the hunter finds out that the butterfly has travelled a distance of $625$ meters. What was the speed of the butterfly?

Problem 12

How many positive base $- 4$ integers are divisible by $4$ but the sum of their digits is not divisible by $4$ ?

Problem 13

The pentagon $ABCDE$ rolls on a straight line as each side of the pentagon touches the ground at $1$ stage in the entire cycle. What is the length of the path that vertex $C$ travels throughout $1$ whole cycle?

Problem 14

Let $P(x)$ be a polynomial with integral coefficients and $S(x) = \frac{P^{2}(x)}{x(2 - x)}$ for all nonzero values of $x$ . If $P(2) = P(3) = 8$ , what is the sum of the digits in the numerical value of $P(100)$ ?

Problem 15

Let $N$ be $10^{100^{1000^{10000…}}}$ . (All the way till the number consisting of $100$ zeroes starting with a $1$ . What is the remainder when N is divided by $629$ ?

Problem 16

$5$ red balls and $5$ blue balls are to be placed in a grid of $25$ squares in which each ball must be placed in $1$ square in which each square contains at most $1$ ball. In how many ways can the gals be places such that each row contains exactly $1$ of each a red ball and a blue ball and each column also contains exactly $1$ of each a red ball and a blue ball?

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