# Difference between revisions of "2014 AMC 10A Problems"

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Nonzero real numbers <math>x</math>, <math>y</math>, <math>a</math>, and <math>b</math> satisfy <math>x < a</math> and <math>y < b</math>. How many of the following inequalities must be true? | Nonzero real numbers <math>x</math>, <math>y</math>, <math>a</math>, and <math>b</math> satisfy <math>x < a</math> and <math>y < b</math>. How many of the following inequalities must be true? | ||

− | <math>\textbf{(I)} x+y < a+b\qquad</math> | + | <math>\textbf{(I)}\ x+y < a+b\qquad</math> |

<math> | <math> | ||

− | \textbf{(II)} x-y < a-b\qquad</math> | + | \textbf{(II)}\ x-y < a-b\qquad</math> |

<math> | <math> | ||

− | \textbf{(III)} xy < ab\qquad</math> | + | \textbf{(III)}\ xy < ab\qquad</math> |

<math> | <math> | ||

− | \textbf{(IV)} \frac{x}{y} < \frac{a}{b}</math> | + | \textbf{(IV)}\ \frac{x}{y} < \frac{a}{b}</math> |

<math> \textbf{(A)}\ 0\qquad\textbf{(B)}\ 1\qquad\textbf{(C)}\ 2\qquad\textbf{(D)}}\ 3\qquad\textbf{(E)}\ 4</math> | <math> \textbf{(A)}\ 0\qquad\textbf{(B)}\ 1\qquad\textbf{(C)}\ 2\qquad\textbf{(D)}}\ 3\qquad\textbf{(E)}\ 4</math> | ||

[[2014 AMC 10A Problems/Problem 7|Solution]] | [[2014 AMC 10A Problems/Problem 7|Solution]] | ||

+ | |||

==Problem 8== | ==Problem 8== | ||

## Revision as of 11:09, 9 February 2014

## 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
- 26 See also

## Problem 1

What is

$\textbf{(A)}\ 3\qquad\textbf{(B)}\ 8\qquad\textbf{(C)}\ \frac{25}{2}\qquad\textbf{(D)}}\ \frac{170}{3}\qquad\textbf{(E)}\ 170$ (Error compiling LaTeX. ! Extra }, or forgotten $.)

## Problem 2

Roy's cat eats of a can of cat food every morning and of a can of cat food every evening. Before feeding his cat on Monday morning, Roy opened a box containing cans of cat food. On what day of the week did the cat finish eating all the cat food in the box?

$\textbf{(A)}\ \text{Tuesday}\qquad\textbf{(B)}\ \text{Wednesday}\qquad\textbf{(C)}\ \text{Thursday}\qquad\textbf{(D)}}\ \text{Friday}\qquad\textbf{(E)}\ \text{Saturday}$ (Error compiling LaTeX. ! Extra }, or forgotten $.)

## Problem 3

Bridget bakes 48 loaves of bread for her bakery. She sells half of them in the morning for each. In the afternoon she sells two thirds of what she has left, and because they are not fresh, she charges only half price. In the late afternoon she sells the remaining loaves at a dollar each. Each loaf costs for her to make. In dollars, what is her profit for the day?

$\textbf{(A)}\ 24\qquad\textbf{(B)}\ 36\qquad\textbf{(C)}\ 44\qquad\textbf{(D)}}\ 48\qquad\textbf{(E)}\ 52$ (Error compiling LaTeX. ! Extra }, or forgotten $.)

## Problem 4

Walking down Jane Street, Ralph passed four houses in a row, each painted a different color. He passed the orange house before the red house, and he passed the blue house before the yellow house. The blue house was not next to the yellow house. How many orderings of the colored houses are possible?

$\textbf{(A)}\ 2\qquad\textbf{(B)}\ 3\qquad\textbf{(C)}\ 4\qquad\textbf{(D)}}\ 5\qquad\textbf{(E)}\ 6$ (Error compiling LaTeX. ! Extra }, or forgotten $.)

## Problem 5

On an algebra quiz, of the students scored points, scored points, scored points, and the rest scored points. What is the difference between the mean and median score of the students' scores on this quiz?

$\textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 3\qquad\textbf{(D)}}\ 4\qquad\textbf{(E)}\ 5$ (Error compiling LaTeX. ! Extra }, or forgotten $.)

## Problem 6

Suppose that cows give gallons of milk in days. At this rate, how many gallons of milk will cows give in days?

$\textbf{(A)}\ \frac{bde}{ac}\qquad\textbf{(B)}\ \frac{ac}{bde}\qquad\textbf{(C)}\ \frac{abde}{c}\qquad\textbf{(D)}}\ \frac{bcde}{a}\qquad\textbf{(E)}\ \frac{abc}{de}$ (Error compiling LaTeX. ! Extra }, or forgotten $.)

## Problem 7

Nonzero real numbers , , , and satisfy and . How many of the following inequalities must be true?

$\textbf{(A)}\ 0\qquad\textbf{(B)}\ 1\qquad\textbf{(C)}\ 2\qquad\textbf{(D)}}\ 3\qquad\textbf{(E)}\ 4$ (Error compiling LaTeX. ! Extra }, or forgotten $.)

## Problem 8

Which of the following number is a perfect square?

## Problem 9

The two legs of a right triangle, which are altitudes, have lengths and . How long is the third altitude of the triangle?

## Problem 10

Five positive consecutive integers starting with have average . What is the average of consecutive integers that start with ?

$\textbf{(A)}\ a+3\qquad\textbf{(B)}\ a+4\qquad\textbf{(C)}\ a+5\qquad\textbf{(D)}}\ a+6\qquad\textbf{(E)}\ a+7$ (Error compiling LaTeX. ! Extra }, or forgotten $.)

## Problem 11

A customer who intends to purchase an appliance has three coupons, only one of which may be used:

Coupon 1: off the listed price if the listed price is at least

Coupon 2: off the listed price if the listed price is at least

Coupon 3: off the amount by which the listed price exceeds

For which of the following listed prices will coupon offer a greater price reduction than either coupon or coupon ?

## Problem 12

A regular hexagon has side length 6. Congruent arcs with radius 3 are drawn with the center at each of the vertices, creating circular sectors as shown. The region inside the hexagon but outside the sectors is shaded as shown What is the area of the shaded region?

## Problem 13

Equilateral has side length , and squares , , lie outside the triangle. What is the area of hexagon ?

## Problem 14

The -intercepts, and , of two perpendicular lines intersecting at the point have a sum of zero. What is the area of ?

## Problem 15

David drives from his home to the airport to catch a flight. He drives miles in the first hour, but realizes that he will be hour late if he continues at this speed. He increases his speed by miles per hour for the rest of the way to the airport and arrives minutes early. How many miles is the airport from his home?

## Problem 16

In rectangle , , , and points , , and are midpoints of , , and , respectively. Point is the midpoint of . What is the area of the shaded region?

## Problem 17

Three fair six-sided dice are rolled. What is the probability that the values shown on two of the dice sum to the value shown on the remaining die?

## Problem 18

A square in the coordinate plane has vertices whose -coordinates are , , , and . What is the area of the square?

## Problem 19

Four cubes with edge lengths , , , and are stacked as shown. What is the length of the portion of contained in the cube with edge length ?

## Problem 20

The product , where the second factor has digits, is an integer whose digits have a sum of . What is ?

$\textbf{(A)}\ 901\qquad\textbf{(B)}\ 911\qquad\textbf{(C)}\ 919\qquad\textbf{(D)}}\ 991\qquad\textbf{(E)}\ 999$ (Error compiling LaTeX. ! Extra }, or forgotten $.)

## Problem 21

Positive integers and are such that the graphs of and intersect the -axis at the same point. What is the sum of all possible -coordinates of these points of intersection?

## Problem 22

In rectangle , and . Let be a point on such that . What is ?

## Problem 23

A rectangular piece of paper whose length is times the width has area . The paper is divided into three sections along the opposite lengths, and then a dotted line is drawn from the first divider to the second divider on the opposite side as shown. The paper is then folded flat along this dotted line to create a new shape with area . What is the ratio ?

## Problem 24

A sequence of natural numbers is constructed by listing the first , then skipping one, listing the next , skipping , listing , skipping , and, on the th iteration, listing and skipping . The sequence begins . What is the th number in the sequence?

## Problem 25

The number is between and . How many pairs of integers are there such that and

## See also

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.