# 2016 AMC 10A Problems

## Contents

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

## Problem 1

What is the value of ?

## Problem 2

For what value of does ?

## Problem 3

For every dollar Ben spent on bagels, David spent cents less. Ben paid more than David. How much did they spend in the bagel store together?

## Problem 4

The remainder can be defined for all real numbers and with by where denotes the greatest integer less than or equal to . What is the value of ?

## Problem 5

A rectangular box has integer side lengths in the ratio . Which of the following could be the volume of the box?

## Problem 6

Ximena lists the whole numbers through once. Emilio copies Ximena's numbers, replacing each occurrence of the digit by the digit . Ximena adds her numbers and Emilio adds his numbers. How much larger is Ximena's sum than Emilio's?

## Problem 7

The mean, median, and mode of the data values are all equal to . What is the value of ?

## Problem 8

Trickster Rabbit agrees with Foolish Fox to double Fox's money every time Fox crosses the bridge by Rabbit's house, as long as Fox pays coins in toll to Rabbit after each crossing. The payment is made after the doubling, Fox is excited about his good fortune until he discovers that all his money is gone after crossing the bridge three times. How many coins did Fox have at the beginning?

## Problem 9

A triangular array of coins has coin in the first row, coins in the second row, coins in the third row, and so on up to coins in the th row. What is the sum of the digits of ?

## Problem 10

A rug is made with three different colors as shown. The areas of the three differently colored regions form an arithmetic progression. The inner rectangle is one foot wide, and each of the two shaded regions is foot wide on all four sides. What is the length in feet of the inner rectangle?

## Problem 11

What is the area of the shaded region of the given rectangle?

## Problem 12

Three distinct integers are selected at random between and , inclusive. Which of the following is a correct statement about the probability that the product of the three integers is odd?

## Problem 13

Five friends sat in a movie theater in a row containing seats, numbered to from left to right. (The directions "left" and "right" are from the point of view of the people as they sit in the seats.) During the movie Ada went to the lobby to get some popcorn. When she returned, she found that Bea had moved two seats to the right, Ceci had moved one seat to the left, and Dee and Edie had switched seats, leaving an end seat for Ada. In which seat had Ada been sitting before she got up?

## Problem 14

How many ways are there to write as the sum of twos and threes, ignoring order? (For example, and are two such ways.)

## Problem 15

Seven cookies of radius inch are cut from a circle of cookie dough, as shown. Neighboring cookies are tangent, and all except the center cookie are tangent to the edge of the dough. The leftover scrap is reshaped to form another cookie of the same thickness. What is the radius in inches of the scrap cookie?

## Problem 16

A triangle with vertices , , and is reflected about the -axis, then the image is rotated counterclockwise about the origin by to produce . Which of the following transformations will return to ?

counterclockwise rotation about the origin by .

clockwise rotation about the origin by .

reflection about the -axis

reflection about the line

reflection about the -axis.

## Problem 17

Let be a positive multiple of . One red ball and green balls are arranged in a line in random order. Let be the probability that at least of the green balls are on the same side of the red ball. Observe that and that approaches as grows large. What is the sum of the digits of the least value of such that ?

## Problem 18

Each vertex of a cube is to be labeled with an integer through , with each integer being used once, in such a way that the sum of the four numbers on the vertices of a face is the same for each face. Arrangements that can be obtained from each other through rotations of the cube are considered to be the same. How many different arrangements are possible?

## Problem 19

In rectangle ABCD, and . Point between and , and point between and are such that . Segments and intersect at and , respectively. The ration can be written as where the greatest common factor of and is 1. What is ?

## Problem 20

For some particular value of , when is expanded and like terms are combined, the resulting expression contains exactly terms that include all four variables and , each to some positive power. What is ?

## Problem 21

Circles with centers and , having radii and , respectively, lie on the same side of line and are tangent to at and , respectively, with between and . The circle with center is externally tangent to each of the other two circles. What is the area of triangle ?

## Problem 22

For some positive integer , the number has positive integer divisors, including and the number . How many positive integer divisors does the number have?

## Problem 23

A binary operation has the properties that and that for all nonzero real numbers and . (Here represents multiplication). The solution to the equation can be written as , where and are relatively prime positive integers. What is

## Problem 24

A quadrilateral is inscribed in a circle of radius . Three of the sides of this quadrilateral have length . What is the length of the fourth side?

## Problem 25

How many ordered triples of positive integers satisfy and ?