AMC 12C 2020

Problem 1

What is the sum of the solutions to the equation $(x + 5)(x + 4) - (x + 5)(x - 6) = 100$ ?

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

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

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