Mock AMC 10B Problems
- 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 Solution 1
- 26 Problem 25
What is the difference between and ?
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?
Which of these numbers is a rational number?
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 .
The three medians of the unit equilateral triangle intersect at point . Find .
Mark rolled two standard dice. Given that he rolled two distinct values, find the probability that he rolled two primes.
What is the sum of the solutions to ?, where is a positive integer?
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 .
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 .
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 ?
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 .
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 .
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?
Let be the number of distinct triangles that can be formed from coplanar points. Find the sum of all possible values of .
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 .
For a particular positive integer , the number of ordered sextuples of positive integers that satisfy is exactly . Find .
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?
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?
What is the largest power of that divides ?
Define a permutation of the set to be if for all . Find the number of permutations.
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 .
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 .
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 .
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 .
Let's label the people from three schools with A,B,C. We can consider the case in which A is in seat number 1. Now, we consider to space between two successive A's. There are totally 8 spaces, which can be broken into sum of four positive integers:
For case of , there are four possible orders. For each order, we can arrange 5 people in the space with 5 seats as or , and then arrange two B's or two C's in two of three remaining spaces, making a total of .
For case of , there are twelve possible orders, , , , , , , , , , , , . For each order, we can arrange 4 people in the space with 4 seats as or , and then arrange the space with two seats as or , and then choose 1 space for and one space for , making a total of .
For case of , there are twelve possible orders, similar to the case above. For each order, we can arrange 3 people in the space with 3 seats as or , and then arrange the space with two seats as or , and then arrange the last people in the space with 1 seat, making a total of .
For case of , there are six possible orders, , , , , , . For each order, we can arrange 3 people in the two spaces with 3 seats as or or or , and then arrange the space with one seats for the people left, so there six arrangements in total: , , , , , , making a total of .
For case of , there is only one order, and each space can be arranged as or , making a total of 16.
Considering the three possibilities for seat 1, the final result is
Let . Find the remainder when is divided by .