2017 AMC 8 Exam Problems
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All of the real AMC 8 and AMC 10 problems in our complete solution collection are used with official permission of the Mathematical Association of America (MAA).
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1.
Which of the following values is largest?
Answer: A
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Written solution:
Option (A) evaluates to
Option (B) evaluates to
Option (C) evalutes to
Option (D) evaluates to
Option (E) evalutes to
Thus, A is the correct answer.
2.
Alicia, Brenda, and Colby were the candidates in a recent election for student president. The pie chart below shows how the votes were distributed among the three candidates. If Brenda received votes, then how many votes were cast all together?
Answer: E
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Written solution:
If votes is of the total votes, then of the total votes is votes. The number of total votes would then be
Thus, E is the correct answer.
3.
What is the value of the expression
Answer: C
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Written solution:
This expression can be reduced as follows:
Thus, C is the correct answer.
4.
When is multiplied by the product is closest to which of the following?
Answer: D
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Written solution:
We can approximate the product as
Thus, D is the correct answer.
5.
What is the value of the expression
Answer: B
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Written solution:
Looking at the denominator independently, so the desired answer is
Thus, B is the correct answer.
6.
If the degree measures of the angles of a triangle are in the ratio what is the degree measure of the largest angle of the triangle?
Answer: D
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Written solution:
We can let the three angles be equal to and Then we know that their sum equals From this we can set and solving this, we get and
The largest angle is
Thus, D is the correct answer.
7.
Let be a 6-digit positive integer, such as 247247, whose first three digits are the same as its last three digits taken in the same order. Which of the following numbers must also be a factor of
Answer: A
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Written solution:
We can let have the form Then, we get that This means that is a factor of
Thus, A is the correct answer.
8.
Malcolm wants to visit Isabella after school today and knows the street where she lives but doesn't know her house number. She tells him, "My house number has two digits, and exactly three of the following four statements about it are true."
(1) It is prime.
(2) It is even.
(3) It is divisible by 7.
(4) One of its digits is 9.
This information allows Malcolm to determine Isabella's house number. What is its units digit?
Answer: D
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Written solution:
If a number is even, it is divisible by This means that the house number is either divisible by or by This rules out the possibility that the house number is prime. This means that the house number is divisible by and by and one of its digits is
The divisibility conditions show that the house number is divisble by so if we look at all the two-digit multiples of the only one with a digit of is Therefore, units digit is
Thus, D is the correct answer.
9.
All of Marcy's marbles are blue, red, green, or yellow. One third of her marbles are blue, one fourth of them are red, and six of them are green. What is the smallest number of yellow marbles that Macy could have?
Answer: D
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Written solution:
If the number of marbles is divisble by both and then the number must be divisible by If we test we get that there are blue marbles and red marbles. This leaves a maximum of green marbles, which is not possible.
If there are marbles, then there are blue marbles and red marbles. To find the number of yellow marbles, we get
Thus, D is the correct answer.
10.
A box contains five cards, numbered 1, 2, 3, 4, and 5. Three cards are selected randomly without replacement from the box. What is the probability that 4 is the largest value selected?
Answer: C
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Written solution:
The number of ways to choose cards from is If is the largest value selected, then the other two cards have to be chosen from There are ways to do this. The probability is then
Thus, C is the correct answer.
11.
A square-shaped floor is covered with congruent square tiles. If the total number of tiles that lie on the two diagonals is 37, how many tiles cover the floor?
Answer: C
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Written solution:
tiles on both diagonals imply that there are tiles on each diagonal, since one tile overlaps in the middle. The total number of tiles would then be since the number of tiles in each row is equal to the number of tiles in one diagonal.
Thus, C is the correct answer.
12.
The smallest positive integer greater than 1 that leaves a remainder of 1 when divided by 4, 5, and 6 lies between which of the following pairs of numbers?
Answer: D
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Written solution:
If a number leaves a remainder of when divided by then it is one more than the least common multiple of these numbers. The least common multiple is so the smallest such positive integer is
Thus, D is the correct answer.
13.
Peter, Emma, and Kyler played chess with each other. Peter won 4 games and lost 2 games. Emma won 3 games and lost 3 games. If Kyler lost 3 games, how many games did he win?
Answer: B
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Written solution:
Since there are no ties, the number of wins has to equal the number of losses. The number of losses is so the number of games Kyler won is
Thus, B is the correct answer.
14.
Chloe and Zoe are both students in Ms. Demeanor's math class. Last night they each solved half of the problems in their homework assignment alone and then solved the other half together. Chloe had correct answers to only of the problems she solved alone, but overall of her answers were correct. Zoe had correct answers to of the problems she solved alone. What was Zoe's overall percentage of correct answers?
Answer: C
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Written solution:
Since the answer should be same regardless of the number of problems, we can assume that there were problems on the assignment. of is so Chloe answered questions correctly alone. of is so Chloe answered questions correctly in total. This means that Chloe answered together with Zoe.
of is so Zoe answered questions correct by herself. We know that she answered questions correct with Chloe, so she answered correctly in total. This means that her overall percentage is
Thus, C is the correct answer.
15.
In the arrangement of letters and numerals below, by how many different paths can one spell AMC8? Beginning at the A in the middle, a path only allows moves from one letter to an adjacent (above, below, left, or right, but not diagonal) letter. One example of such a path is traced in the picture.
Answer: D
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Written solution:
Starting from there are ways to reach an From each there are ways to reach a From each there are ways to reach an Multiplying all these possibilities, we get
Thus, D is the correct answer.
16.
In the figure below, choose point on so that and have equal perimeters. What is the area of
Answer: D
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Written solution:
The only way to split into two parts such that the two triangles have the same perimeter is if and
and have the same altitudes, so their areas are proportional to their bases. This means that the area of is the area of which is
Thus, D is the correct answer.
17.
Starting with some gold coins and some empty treasure chests, I tried to put 9 gold coins in each treasure chest, but that left 2 treasure chests empty. So instead I put 6 gold coins in each treasure chest, but then I had 3 gold coins left over. How many gold coins did I have?
Answer: C
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Written solution:
Let be the number of treasure chests and be the number of gold coins. Then and Solving this system yields so the number of gold coins is
Thus, C is the correct answer.
18.
In the non-convex quadrilateral shown below, is a right angle, and What is the area of quadrilateral
Answer: B
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Written solution:
Since is a right angle, we can apply the Pythagorean theorem to to get that We also get that is right since the sides of form a Pythagorean triple.
Then the area of is equal to
Thus, B is the correct answer.
19.
For any positive integer the notation denotes the product of the integers through What is the largest integer for which is a factor of the sum:
Answer: D
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Written solution:
The expression can be factored as follows:
Each possesses two factors of The number of factors of in is This counts the number of numbers divisible by and the number of numbers divisible by to get both factors of The total number of factors of is
Thus, D is the correct answer.
20.
An integer between and inclusive, is chosen at random. What is the probability that it is an odd integer whose digits are all distinct?
Answer: B
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Written solution:
Since the number is odd, the last digit is odd, giving possibilities. The thousands digit cannot be zero or the the number we already got, so that gives possibilities. Similarly, the hundreds digit has possibilities, and the tens digit has possibilities. This gives a total of making the probability
Thus, B is the correct answer.
21.
Suppose and are nonzero real numbers, and What are the possible value(s) for
Answer: A
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Written solution:
Since the sum is all the numbers cannot be positive or negative. This gives two cases: two are positive and one is negative or vice versa. Also note that if and if
two are positive and one is negative
WLOG, let and Then This means that and Then
two are negative and one is positive
WLOG, let and Then This means that and Then
Either way, equals
Thus, A is the correct answer.
22.
In the right triangle and angle is a right angle. A semicircle is inscribed in the triangle as shown. What is the radius of the semicircle?
Answer: D
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Written solution:
Let be the center of the inscribed semicircle and be the tangent point of the semicircle on Then since and are tangents to the semicircle. Then and is perpendicular to so so Solving this, we get
Thus, D is the correct answer.
23.
Each day for four days, Linda traveled for one hour at a speed that resulted in her traveling one mile in an integer number of minutes. Each day after the first, her speed decreased so that the number of minutes to travel one mile increased by minutes over the preceding day. Each of the four days, her distance traveled was also an integer number of miles. What was the total number of miles for the four trips?
Answer: C
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Written solution:
Linda traveled for minutes every day. Since one mile was traveled in an integer amount of minutes each day, her mph every day must be a factor of The factors of are and The only sequence of four of these numbers that differ by are and For the four days, she traveled miles.
Thus, C is the correct answer.
24.
Mrs. Sanders has three grandchildren, who call her regularly. One calls her every three days, one calls her every four days, and one calls her every five days. All three called her on December 31, 2016. On how many days during the next year did she not receive a phone call from any of her grandchildren?
Answer: D
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Written solution:
In a day period, the first child calls times, the second child calls times, and the third child calls days. overcounts, however. The first and second children call on the same day times. The first and third children call on the same day times. The second and third children call on the same day times. Subtracting these from yields
The is added in thrice and subtracted out thrice, so we need to add it back in. This means that for every days, Mrs. Sanders receives a call days, which means that she does not receive a call on days. There are day periods, and there are no calls the or day, which results in total days with no phone calls.
Thus, D is the correct answer.
25.
In the figure shown, and are line segments each of length 2, and
Arcs and are each one-sixth of a circle with radius 2. What is the area of the region shown?
Answer: B
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Written solution:
We can extend and to form the following picture.
The area of this region is the area of an equilaterial triangle with side length of minus the area of two-sixths of a circle with radius The area for an equilateral triangle with side length is This means that the total area is
Thus, B is the correct answer.