# AMC 12C 2020 Problems

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==Problem 1

A plane flies at a speed of miles/hour. How many miles in two hours

## Contents

- 1 Problem 3
- 2 Problem 4
- 3 Problem 5
- 4 Problem 6
- 5 Problem 7
- 6 Problem 8
- 7 Problem 9
- 8 Problem 10
- 9 Problem 11
- 10 Problem 12
- 11 Problem 13
- 12 Problem 14
- 13 Problem 15
- 14 Problem 16
- 15 Problem 17
- 16 Problem 18
- 17 Problem 19
- 18 Problem 20
- 19 Problem 21
- 20 Problem 22
- 21 Problem 23
- 22 Problem 24
- 23 Problem 25

## Problem 3

In a bag are marbles consisting of blue marbles and red marbles. If each marble is pulled out at a time, what is the probability that the marble pulled out red?

## Problem 4

A spaceship flies in space at a speed of miles/hour and the spaceship is paid dollars for each miles traveled. It’s only expense is fuel in which it pays dollars per gallon, while going at a rate of hours per gallon. Traveling miles, how much money would the spaceship have gained?

## Problem 5

Let be a function satisfying for all real numbers and . Let What is ?

## Problem 6

How many increasing(lower to higher numbered) subsets of contain no consecutive prime numbers?

## Problem 7

A Regular Octagon has an area of . What is the sum of the lengths of the diagonals of the octagon?

## Problem 8

What is the value of ?

## Problem 9

Let denote the sum of the even digits of a positive integer and let denote the sum of the odd digits of a positive integer. For some positive integer , = . What is the product of the digits of the least possible such ?

## Problem 10

In how many ways can candy canes and lollipops be split between children if each child must receive atleast candy but no child receives both types?

## Problem 11

Let be an isosceles trapezoid with being parallel to and , , and . If is the intersection of and , and is the circumcenter of , what is the length of ? Source: JHMMC 2019

## Problem 12

For some positive integer , let satisfy the equation

. What is the sum of the digits of ?

## Problem 13

An alien walks horizontally on the real number line starting at the origin. On each move, the alien can walk or numbers the right or left of it. What is the expected distance from the alien to the origin after moves?

## Problem 14

Let be the set of solutions to the equation on the complex plane, where . points from are chosen, such that a circle passes through both points. What is the least possible area of ?

## Problem 15

Let . What is the remainder when is divided by ?

## Problem 16

Let and be the vertex and focus of the Parabola respectively. For a point lying on the directrix of , and a point lying on , and Quadrilateral is cyclic. If has integral side lengths, what is the minimum possible area of ?

## Problem 17

Let denote the nonzero digit from the right in the base - expansion of , for example, . What is the sum of the digits of ?

## Problem 18

lays flat on the ground and has side lengths , and . Vertex is then lifted up creating an elevation angle with the triangle and the ground of . A wooden pole is dropped from perpendicular to the ground, making an altitude of a Dimensional figure. Ropes are connected from the foot of the pole, , to form other segments, and . What is the volume of ?

## Problem 19

Let be a cubic polynomial with integral coefficients and roots , , and . What is the least possible sum of the coefficients of ?

## Problem 20

What is the maximum value of as varies through all real numbers to the nearest integer?

## Problem 21

Let denote the greatest integer less than or equal to . How many positive integers , satisfy the equation

?

## Problem 22

A convex hexagon is inscribed in a circle. . . The measure of can be written as where and are relatively prime positive integers. What is ?

## Problem 23

## Problem 24

A sequence is defined as , , and for all ,

What is ?

## Problem 25

Let and let . Let be the sum of the power of the roots of . It is given that the least positive integer , such that is . What is ?