2022 AMC 10A Exam Problems
Scroll down and press Start to try the exam! Or, go to the printable PDF, answer key, or solutions.
Used with permission of the Mathematical Association of America.
Time Left:
1:15:00
1:15:00
1.
What is the value of
Answer: D
Solution(s):
We can simplify this expression as follows:
Thus, D is the correct answer.
2.
Mike cycled laps in minutes. Assume he cycled at a constant speed throughout. Approximately how many laps did he complete in the first minutes?
Answer: B
Solution(s):
We can set up a proportion to solve this problem:
Cross multiplying, we get
Thus, B is the correct answer.
3.
The sum of three numbers is The first number is times the third number, and the third number is less than the second number. What is the absolute value of the difference between the first and second numbers?
Answer: E
Solution(s):
Let and be the three numbers. The conditions from the problem give us the following relations:
\begin{gather*} x + y + z = 96 \tag*{(1)} \\ x = 6z \tag*{(2)} \\ z = y - 40 \tag*{(3)}. \end{gather*}
Rearranging we get Plugging this new equation and into we get
From this, we get that and
Therefore,
Thus, E is the correct answer.
4.
In some countries, automobile fuel efficiency is measured in liters per kilometers while other countries use miles per gallon. Suppose that kilometer equals miles, and gallon equals liters. Which of the following gives the fuel efficiency in liters per kilometers for a car that gets miles per gallon?
Answer: E
Solution(s):
We can do the following conversions to get the desired answer.
Taking the reciprocal, we find that the equivalent fuel efficiency would be
Thus, E is the correct answer.
5.
Square has side length Points and each lie on a side of such that is an equilateral convex hexagon with side length What is
Answer: C
Solution(s):
Consider the diagram:
Since we know that This shows that is a right triangle. Using the Pythagorean theorem, we get that
We also know that
This equation simplifies to Which implies that
We can rationalize this fraction to get
Thus, C is the correct answer.
6.
Which expression is equal to for
Answer: A
Solution(s):
By the definition of square root, we get that
Since we get that which means that
The whole expression therefore simplifies to Since we know that This means that
Thus, A is the correct answer.
7.
The least common multiple of a positive integer and is and the greatest common divisor of and is What is the sum of the digits of
Answer: B
Solution(s):
We can see that and
This means that must include a factor of and We also know that and which means that has a factor of and Therefore, we can set
Thus, B is the correct answer.
8.
A data set consists of (not distinct) positive integers: and The average (arithmetic mean) of the numbers equals a value in the data set. What is the sum of all positive values of
Answer: D
Solution(s):
The average of the numbers is
This value can equal any of the terms in the set, so we can case on what it equals.
Adding up all the positive values for we get
Thus, D is the correct answer.
9.
A rectangle is partitioned into regions as shown. Each region is to be painted a solid color - red, orange, yellow, blue, or green - so that regions that touch are painted different colors, and colors can be used more than once. How many different colorings are possible?
Answer: D
Solution(s):
There are choices for the color of the bottom left rectangle. This forces there to be choices for the top left rectangle. The middle bottom rectangle touches both of the previous ones, so there are color options for this rectangle.
The rectangle in the top right is also limited to colors since it touches the two previous rectangles. Finally, the rectangle in the bottom right also has color options.
Multiplying these together, we get total colorings.
Thus, D is the correct answer.
10.
Daniel finds a rectangular index card and measures its diagonal to be centimeters. Daniel then cuts out equal squares of side cm at two opposite corners of the index card and measures the distance between the two closest vertices of these squares to be centimeters, as shown below. What is the area of the original index card?
Answer: E
Solution(s):
We can label and as the width and height as in the diagram. Then we get that and
The latter expression simplifies to which is the same as From this we get
Squaring this, we get which gets us that which means that the area is
Thus, E is the correct answer.
11.
Ted mistakenly wrote as
What is the sum of all real numbers for which these two expressions have the same value?
Answer: C
Solution(s):
We can rewrite as so Then if we equate the given expressions, we get Equating the exponents, we get
Multiplying by we get and so
Therefore, we can see that the sum of the solutions is
Thus, C is the correct answer.
12.
On Halloween children walked into the principal's office asking for candy. They can be classified into three types: Some always lie; some always tell the truth; and some alternately lie and tell the truth. The alternaters arbitrarily choose their first response, either a lie or the truth, but each subsequent statement has the opposite truth value from its predecessor. The principal asked everyone the same three questions in this order.
"Are you a truth-teller?" The principal gave a piece of candy to each of the children who answered yes.
"Are you an alternater?" The principal gave a piece of candy to each of the children who answered yes.
"Are you a liar?" The principal gave a piece of candy to each of the children who answered yes.
How many pieces of candy in all did the principal give to the children who always tell the truth?
Answer: A
Solution(s):
For the first question, the truth-tellers will respond yes, the liars will respond yes, and the alternaters who decided to lie first will say yes. The alternaters who decide to tell the truth first will say no. Denote this as
For the second questions, the liars will respond yes, and the alternaters who decided to lie first will say yes (they are forced to tell the truth for this question). The truth-tellers will respond no, and the alternaters who told the truth first would lie this round, responding no. Denote this as
From this, we get that The principal only gives candy to children who always tell the truth in the first round, therefore only giving them candies total.
Thus, A is the correct answer.
13.
Let be a scalene triangle. Point lies on so that bisects The line through perpendicular to intersects the line through parallel to at point Suppose and What is
Answer: C
Solution(s):
Consider the following diagram:
By the Angle Bisector Theorem, we can label as and as
We also get that is isosceles since Therefore,
Since and are parallel, we know that
so
Using similar triangles, we get that
Thus, C is the correct answer.
14.
How many ways are there to split the integers through into pairs such that in each pair, the greater number is at least times the lesser number?
Answer: E
Solution(s):
We can see that the numbers must be in different pairs. must also be paired with since no other number is at least twice
Now let's look at what the other numbers can pair with. and can pair with any number and can pair with any number and and can pair with any number
can pair with numbers, but then only has options since took one. then has options, since choices are taken, but it has one more to choose from then has options, has options, and only has
Multiplying these together yields
Thus, E is the correct answer.
15.
Quadrilateral with side lengths is inscribed in a circle. The area interior to the circle but exterior to the quadrilateral can be written in the form where and are positive integers such that and have no common prime factor. What is
Answer: D
Solution(s):
Notice that and are both the same. This forces since otherwise and would both be acute or obtuse, violating the fact that their sum is
Also since is right, we know that is the diameter of the circle. The area of the circle is then
To find the area of the quadrilateral, we can find the area of each of the triangles, which is
To find the area outside the quadrilateral, we subtract to get
Therefore,
Thus, D is the correct answer.
16.
The roots of the polynomial are the height, length, and width of a rectangular box (right rectangular prism). A new rectangular box is formed by lengthening each edge of the original box by units. What is the volume of the new box?
Answer: D
Solution(s):
Let and be the dimensions of the old box. Then the volume of the new box is Expanding, we get We can use Vieta's formulas to find the terms in this expression. We get that and
Plugging these values into the expression, we get
Thus, D is the correct answer.
17.
How many three-digit positive integers are there whose nonzero digits and satisfy (The bar indicates repetition, thus in the infinite repeating decimal )
Answer: D
Solution(s):
Let's find a closed form expression for each of the repeating decimals. We can write as
From this, we can see that this an infinite geometric sequence with first term and ratio
Using the formula for the sum of a infinite geometric sequence, we get that this equals
Similarly, note that we can write as
As above, this equals Therefore,
Substituting all these values into the condition, we get Multiplying through by yields
Note that we can express as Substituting this in, we get which simplifies to
All the solutions where work. The expression remains constant if we increase by and decrease by We could also decrease by and increase by
Applying this principles to the first triples yields as more solutions. Therefore, there are a total of solutions.
Thus, D is the correct solution.
18.
Let be the transformation of the coordinate plane that first rotates the plane degrees counterclockwise around the origin and then reflects the plane across the -axis. What is the least positive integer such that performing the sequence of transformations returns the point back to itself?
Answer: A
Solution(s):
Since we are working with angles and reflections, working with polar coordinates would make this problem easier to deal with.
Let be a polar coordinate. Rotating this by degrees counterclockwise maps the point to and then reflecting it maps it to
Therefore, we have that
From this, we can see that
Now, let's analyze what happens to the point
After we get
After we get
After we get
After we get
After we get
After we get
From this, we can see that the first time the angle is back to is when and
Thus, A is the correct answer.
19.
Define as the least common multiple of all the integers from to inclusive. There is a unique integer such that What is the remainder when is divided by
Answer: C
Solution(s):
We can combine all the addends on the left side into one fraction by making all their denominators
Therefore,
We want to find mod so we can analyze the expression for mod
Each of do not have a factor of in the denominator. This means that when we multiply each of them by the resulting product will have a factor of ( divides ). That means all of these terms evaluate to mod
Now we just need to evaluate mod
To calculate note that it must contain the highest power of every prime less than or equal to Therefore,
Now we simplify:
Thus, C is the correct answer.
20.
A four-term sequence is formed by adding each term of a four-term arithmetic sequence of positive integers to the corresponding term of a four-term geometric sequence of positive integers. The first three terms of the resulting four-term sequence are and What is the fourth term of this sequence?
Answer: E
Solution(s):
Let the arithmetic sequence be and the geometric sequence be
Then and
Subtracting from and from we get and
Subtracting these, we get
Since every variable is an integer, we get that or
If then and This forces the arithmetic sequence to be which is a contradiction.
Therefore, Then and The arithmetic sequence is and the geometric sequence is
The desired answer is
Thus, E is the correct answer.
21.
A bowl is formed by attaching four regular hexagons of side to a square of side The edges of the adjacent hexagons coincide, as shown in the figure. What is the area of the octagon obtained by joining the top eight vertices of the four hexagons, situated on the rim of the bowl?
Answer: B
Solution(s):
We can extend line segments and as follows.
They are concurrent since and intersect and and intersect (they are on the same plane). Since only intersects the plane of and once, it must intersect them both at that one point.
The dashed red lines create equilateral triangles on the lateral faces of the bowl, which all have side length In the top plane, we know that so the dashed red lines create an isosceles right triangle with leg length
The octagon looks like the diagram below.
The area of the octagon is the area of the square minus each of the four corner triangles. This is equal to
Thus, B is the correct answer.
22.
Suppose that cards numbered are arranged in a row. The task is to pick them up in numerically increasing order, working repeatedly from left to right. In the example below, cards are picked up on the first pass, and on the second pass, on the third pass, on the fourth pass, and on the fifth pass. For how many of the possible orderings of the cards will the cards be picked up in exactly two passes?
Answer: D
Solution(s):
Let be the number of cards picked up on the first pass, where
If we choose the spaces that the cards occupy, the positions of the remaining cards are determined since they must be placed in order.
There are ways to choose where the cards go, but if the cards are placed at the very beginning, then all the cards will be picked up on the first pass.
Therefore, for a given there are ways to arrange the cards.
We need to now find the sum over all possible which equals
Thus, D is the correct answer.
23.
Isosceles trapezoid has parallel sides and with and There is a point in the plane such that and What is
Answer: B
Solution(s):
Let be the reflection of across the perpendicular bisector of
This forms two new isosceles trapezoids: and
Therefore, we get \begin{gather*} P'A = PD = 4 \\ P'D = PA = 1 \\ P'C = PB = 2 \\ P'B = PC = 3. \end{gather*}
Using Ptolemy's theorem, we know that the product of the diagonals is equal to the sum of the products of the opposite sides. Therefore: \begin{gather*} PP' \cdot AD + 1 = 16 \\ PP' \cdot BC + 4 = 9. \end{gather*}
This gets us and Dividing these two equations yields
Thus, B is the correct answer.
24.
How many strings of length formed from the digits are there such that for each at least of the digits are less than
(For example, satisfies this condition because it contains at least digit less than at least digits less than at least digits less than and at least digits less than The string does not satisfy the condition because it does not contain at least digits less than )
Answer: E
Solution(s):
Note that there must be at least one to satisfy the condition. We can proceed by casework on the number of distinct digits in the string.
digit
The only possible digit is just This can be done in way.
digits
This gives us a total of strings.
digits
This gives us a total of strings.
digits
This gives us a total of strings.
digits
This gives us strings.
All together, we have strings.
Thus, E is the correct answer.
25.
Let and be squares that have vertices at lattice points (i.e., points whose coordinates are both integers) in the coordinate plane, together with their interiors.
The bottom edge of each square is on the -axis. The left edge of and the right edge of are on the -axis, and contains as many lattice points as does The top two vertices of are in and contains of the lattice points contained in See the figure (not drawn to scale).
The fraction of lattice points in that are in is times the fraction of lattice points in that are in What is the minimum possible value of the edge length of plus the edge length of plus the edge length of
Answer: E
Solution(s):
Let be the number of lattice points on the side length of Similarly define for and for Note that the number of lattice points in a rectangle is the product of the number of lattice points along its width and the number of lattice points along its length.
The first conditions gives us that
The number of lattice points in is the sum of the lattice points in each of the regions, but there is overlap along the -axis where touches it.
The second condition, therefore, yields From we get that is a multiple of We can substitute with to get For the product to be divisible by must be divisible by We can again substitute with to get
Let be the number of lattice points along the bottom of the rectangle formed by and be the number of lattice points along the bottom of the rectangle formed by
Using these variable, we get that the number of lattice points in is and in is
The third condition gives us that
We also know that (accounting for overlap), and this yields
gives us that
However, by we get that
By we also get that is a perfect square since it is relatively prime to and they must multiply to a perfect square.
Using these restrictions on we can try to find the smallest that works. We get that satisfies both conditions.
From this value of we get that and We can also find that Therefore, The question, however, asked for the sum of the side lengths. The side lengths of the squares are less than the number of lattice points on the side, so we have to subtract
Therefore, the desired answer is
Thus, B is the correct answer.