# 2013 AMC 10B 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
- 26 See also

## Problem 1

What is ?

## Problem 2

Mr. Green measures his rectangular garden by walking two of the sides and finding that it is steps by steps. Each of Mr. Green's steps is feet long. Mr. Green expects a half a pound of potatoes per square foot from his garden. How many pounds of potatoes does Mr. Green expect from his garden?

## Problem 3

On a particular January day, the high temperature in Lincoln, Nebraska, was degrees higher than the low temperature, and the average of the high and the low temperatures was . In degrees, what was the low temperature in Lincoln that day?

## Problem 4

When counting from to , is the number counted. When counting backwards from to , is the number counted. What is ?

## Problem 5

Positive integers and are each less than . What is the smallest possible value for ?

## Problem 6

The average age of 33 fifth-graders is 11. The average age of 55 of their parents is 33. What is the average age of all of these parents and fifth-graders?

## Problem 7

Six points are equally spaced around a circle of radius 1. Three of these points are the vertices of a triangle that is neither equilateral nor isosceles. What is the area of this triangle?

## Problem 8

Ray's car averages 40 miles per gallon of gasoline, and Tom's car averages 10 miles per gallon of gasoline. Ray and Tom each drive the same number of miles. What is the cars' combined rate of miles per gallon of gasoline?

## Problem 9

Three positive integers are each greater than , have a product of , and are pairwise relatively prime. What is their sum?

## Problem 10

A basketball team's players were successful on 50% of their two-point shots and 40% of their three-point shots, which resulted in 54 points. They attempted 50% more two-point shots than three-point shots. How many three-point shots did they attempt?

## Problem 11

Real numbers and satisfy the equation . What is ?

## Problem 12

Let be the set of sides and diagonals of a regular pentagon. A pair of elements of are selected at random without replacement. What is the probability that the two chosen segments have the same length?

## Problem 13

Jo and Blair take turns counting from to one more than the last number said by the other person. Jo starts by saying "", so Blair follows by saying "" . Jo then says "" , and so on. What is the 53rd number said?

## Problem 14

Define . Which of the following describes the set of points for which ?

## Problem 15

A wire is cut into two pieces, one of length and the other of length . The piece of length is bent to form an equilateral triangle, and the piece of length is bent to form a regular hexagon. The triangle and the hexagon have equal area. What is ?

## Problem 16

In triangle , medians and intersect at , , , and . What is the area of ?

## Problem 17

Alex has red tokens and blue tokens. There is a booth where Alex can give two red tokens and receive in return a silver token and a blue token, and another booth where Alex can give three blue tokens and receive in return a silver token and a red token. Alex continues to exchange tokens until no more exchanges are possible. How many silver tokens will Alex have at the end?

## Problem 18

The number has the property that its units digit is the sum of its other digits, that is . How many integers less than but greater than share this property?

## Problem 19

The real numbers form an arithmetic sequence with . The quadratic has exactly one root. What is this root?

## Problem 20

The number is expressed in the form where and are positive integers and is as small as possible. What is ?

## Problem 21

Two non-decreasing sequences of nonnegative integers have different first terms. Each sequence has the property that each term beginning with the third is the sum of the previous two terms, and the seventh term of each sequence is . What is the smallest possible value of N?

## Problem 22

The regular octagon has its center at . Each of the vertices and the center are to be associated with one of the digits through , with each digit used once, in such a way that the sums of the numbers on the lines , , , and are all equal. In how many ways can this be done?

## Problem 23

In triangle , , , and . Distinct points , , and lie on segments , , and , respectively, such that , , and . The length of segment can be written as , where and are relatively prime positive integers. What is ?

## Problem 24

A positive integer is *nice* if there is a positive integer with exactly four positive divisors (including and ) such that the sum of the four divisors is equal to . How many numbers in the set are nice?

## Problem 25

Bernardo chooses a three-digit positive integer and writes both its base-5 and base-6 representations on a blackboard. Later LeRoy sees the two numbers Bernardo has written. Treating the two numbers as base-10 integers, he adds them to obtain an integer . For example, if , Bernardo writes the numbers and , and LeRoy obtains the sum . For how many choices of are the two rightmost digits of , in order, the same as those of ?

## See also

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