# Mock 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

### Problem 1

What is the difference between and ?

### Problem 2

Al, Bob, Clayton, Derek, Ethan, and Frank are six Boy Scouts that will be split up into two groups of three Boy Scouts for a boating trip. How many ways are there to split up the six boys if the two groups are indistinguishable?

### Problem 3

Which of these numbers is a rational number?

### Problem 4

In the diagram below, is an isosceles right triangle with a right angle at and with a hypotenuse of units. Find the greatest integer less than or equal to the value of the radius of the quarter circle inscribed inside .

### Problem 5

The three medians of the unit equilateral triangle intersect at point . Find .

### Problem 6

Mark rolled two standard dice. Given that he rolled two distinct values, find the probability that he rolled two primes.

### Problem 7

What is the sum of the solutions to ?, where is a positive integer?

### Problem 8

In the following diagram, Bob starts at the origin and makes a certain number of moves. A move is defined as him starting at and moves to , , , and with equal probability. The probability that Bob will eventually reach the point is . Find the number of distinct points, including , that satisfy that the probability that he will eventually reach that point is .

### Problem 9

Consider the line segment , which has two endpoints and . is constructed by rotating about the point clockwise degrees, where is a positive integer greater than 2 and . After this operation, the line segments , , , , , are drawn. Let be the sum of the areas of the Triangles . As approaches infinity, approaches a constant . Find .

### Problem 10

A certain period of time starts at exactly 6:09PM on a Tuesday and ends at exactly 6:09AM on a Thursday. Which of these numbers listed in the choices here is a possible length in days for ?

### Problem 11

Consider Square , a square with side length . Let Points , , , be the midpoints of sides , , , and , respectively. Find the area of the square formed by the four line segments , , , and .

### Problem 12

In the figure shown here, the triangle has two legs of length and , and the semicircle has diameter . The area of Region can be expressed as , where are positive integers, is square-free, , and . Find .

### Problem 13

Kevin has a friend named Anna. The two of them are both in the same class, BC Calculus, which is a class that has students. To split the class up into partners that work on a group project involving integrals, the teacher, Mrs. Jannesen, randomly partitions the class into groups of two. If he is assigned to be partners with his friend, he will be happy. What is the probability that Kevin is assigned to be with Anna?

### Problem 14

Let be the number of distinct triangles that can be formed from coplanar points. Find the sum of all possible values of .

### Problem 15

In the figure below, a square of area is inscribed inside a square of area . There are two segments, labeled and . The value of can be expressed as , where are positive integers and is square-free. Find .

### Problem 16

For a particular positive integer , the number of ordered sextuples of positive integers that satisfy is exactly . Find .

### Problem 17

Let be a regular octagon. How many distinct quadrilaterals can be formed from the vertices of given that two quadrilaterals are not distinct if the latter can be obtained by a rotation of the former?

### Problem 18

Two logs of length 10 are laying on the ground touching each other. Their radii are 3 and 1, and the smaller log is fastened to the ground. The bigger log rolls over the smaller log without slipping, and stops as soon as it touches the ground again. What is the volume of the set of points swept out by the larger log as it rolls over the smaller one?

### Problem 19

What is the largest power of that divides ?

### Problem 20

Define a permutation of the set to be if for all . Find the number of permutations.

### Problem 21

There are distinct arrays of integers that satisfy: 1. Each integer in the array is a or . 2. Every row and column contains all the integers and . 3. No row or column contains two of the same number. Find .

### Problem 22

Let be the set of all possible remainders when is divided by , where is a positive integer and is the number of elements in . The sum can be expressed as where are positive integers and and are as small as possible. Find .

### Problem 23

Four real numbers are randomly and independently selected from the range . Let the sets , , , contain all of the real numbers in the range and , respectively. The probability that the four aforementioned sets are disjoint can be expressed as , where and are relatively prime positive integers. Find .

### Problem 24

Four elementary schoolers, four middle schoolers, and four high schoolers sit around a round table with seats. There is a rule that no two people of the same school may sit adjacent to each other. Let be the number of distinct seating arrangements following the rule. Find .

### Problem 25

Let . Find the remainder when is divided by .