2017 AMC 10A 考试题目
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1.
What is the value of
Answer: C
Solution:
Simplifying yields
Thus, C is the correct answer.
2.
Pablo buys popsicles for his friends. The store sells single popsicles for each, 3-popsicle boxes for each, and 5-popsicle boxes for What is the greatest number of popsicles that Pablo can buy with
Answer: D
Solution:
The $3 boxes give us the most popsicles per dollar, so we want to buy as many of those as possible.
We can buy two of those, getting popsicles with $8 - $6 = $2 remaining.
The $1 single popsicles are the worst deal, so Pablo should spend the rest of his money on the -popsicle box.
He then ends up with popsicles.
Thus, D is the correct answer.
3.
Tamara has three rows of two -feet by -feet flower beds in her garden. The beds are separated and also surrounded by -foot-wide walkways, as shown on the diagram. What is the total area of the walkways, in square feet?
Answer: B
Solution:
We can see that the width of the garden is We can also see that the height is The total area of the garden is therefore The area of all the flower beds is Subtracting this from the area of the garden yields which is the area of the walkways.
Thus, B is the correct answer.
4.
Mia is "helping" her mom pick up toys that are strewn on the floor. Mia’s mom manages to put toys into the toy box every seconds, but each time immediately after those seconds have elapsed, Mia takes toys out of the box. How much time, in minutes, will it take Mia and her mom to put all toys into the box for the first time?
Answer: B
Solution:
Note that after seconds, there are toys added and removed, leaving a net total of toys in the box.
We have to be careful towards the end, however, since it is possible for the box to have toys right after Mia's mom adds the toys and before Mia removes them.
After there are toys in the box, Mia's mom can add leaving toys in the box.
It will take seconds, plus another seconds, which gives us minutes to get toys in the box.
Thus, B is the correct answer.
5.
The sum of two nonzero real numbers is times their product. What is the sum of the reciprocals of the two numbers?
Answer: C
Solution:
Let and be the two numbers. We are given that
Note that
Thus, C is the correct answer.
6.
Ms. Carroll promised that anyone who got all the multiple choice questions right on the upcoming exam would receive an A on the exam. Which one of these statements necessarily follows logically?
If Lewis did not receive an A, then he got all of the multiple choice questions wrong.
If Lewis did not receive an A, then he got at least one of the multiple choice questions wrong.
If Lewis got at least one of the multiple choice questions wrong, then he did not receive an A.
If Lewis received an A, then he got all of the multiple choice questions right.
If Lewis received an A, then he got at least one of the multiple choice questions right.
Answer: B
Solution:
There is no stipulation on how to get an A other than that getting all the multiple choice right guarantees an A.
This means that it is possible to get an A without getting all the multiple choice questions right.
It is also possible to not get an A even if all but one of the multiple choice questions are answered correctly.
This rules out A, C, D, and E.
Thus, B is the correct answer.
7.
Jerry and Silvia wanted to go from the southwest corner of a square field to the northeast corner. Jerry walked due east and then due north to reach the goal, but Silvia headed northeast and reached the goal walking in a straight line. Which of the following is closest to how much shorter Silvia's trip was, compared to Jerry's trip?
Answer: A
Solution:
Let be the side length of the field. Then Jerry traveled and Silvia traveled from the Pythagorean theorem.
The desired value is
Thus, A is the correct answer.
8.
At a gathering of people, there are people who all know each other and people who know no one. People who know each other hug, and people who do not know each other shake hands. How many handshakes occur within the group?
Answer: B
Solution:
Each of the people shake hands with each of the people. This results in handshakes.
There are also handshakes within the people (every pair of people shake hands).
Therefore, the total number of handshakes is
Thus, B is the correct answer.
9.
Minnie rides on a flat road at kilometers per hour (kph), downhill at kph, and uphill at kph. Penny rides on a flat road at kph, downhill at kph, and uphill at kph. Minnie goes from town to town a distance of km all uphill, then from town to town a distance of km all downhill, and then back to town a distance of km on the flat. Penny goes the other way around using the same route. How many more minutes does it take Minnie to complete the -km ride than it takes Penny?
Answer: C
Solution:
It will take Minnie hours to travel the uphill distance. It will take her hours to travel the downhill distance.
Finally, it will take her hour to travel the flat. This will take her a total of minutes.
It will take Penny hours to travel the flat. It will take her another hours to travel the uphill.
Finally, it will take her hours to travel the downhill. This is a total of minutes. The trip takes Minnie more minutes to travel than Penny.
Thus, C is the correct answer.
10.
Joy has thin rods, one each of every integer length from cm through cm. She places the rods with lengths cm, cm, and cm on a table. She then wants to choose a fourth rod that she can put with these three to form a quadrilateral with positive area. How many of the remaining rods can she choose as the fourth rod?
Answer: B
Solution:
Note that no one side can be greater than or equal to the sum of the other side lengths.
Let be the length fourth rod. Then we have that and Simplifying, we know that Counting the number of integers in this range, we are left with values for
The rods with length and are already being used, however, so cannot equal these.
This leaves viable solutions for
Thus, B is the correct solution.
11.
The region consisting of all points in three-dimensional space within units of line segment has volume What is the length
Answer: D
Solution:
Recall that all the points at most a fixed distance away from a point form a sphere.
At the end points of this line segment, we can visualize two hemispheres being formed at each end.
All the points in the middle also have spheres forming around them, but they get merged into the ones right next to them.
This means that the middle section forms a cylinder with radius The two hemispheres form a sphere with radius and therefore a volume of This means that the cylinder has a volume of We know the base is so if is then the volume is
Thus, D is the correct answer.
12.
Let be a set of points in the coordinate plane such that two of the three quantities and are equal and the third of the three quantities is no greater than this common value. Which of the following is a correct description for
a single point
two intersecting lines
three lines whose pairwise intersections are three distinct points
a triangle
three rays with a common endpoint
Answer: E
Solution:
Let us case on which of the values are equal. If then This also tells us that This describes a ray starting at and extending in the negative direction.
Similarly, if then and This also describes a ray starting at but instead extending in the negative direction.
Finally, if then we have the line Furthermore, we have that and Note that if one if these conditions is met, the other is also necessarily true due to the equation of the line.
If then The other points are along the line, where and
This describes another ray that starts at and goes off in some third direction.
All three cases result in rays originating from that all go in different directions.
Thus, E is the correct answer.
13.
Define a sequence recursively by and the remainder when is divided by for all Thus the sequence starts What is
Answer: D
Solution:
Let us list out the first few values to see if we can find a pattern in this sequence.
From this we can see that the pattern repeats every terms.
The desired answer is the sum of consecutive numbers, which is fixed. This sum is Thus, D is the correct answer.
14.
Every week Roger pays for a movie ticket and a soda out of his allowance. Last week, Roger's allowance was dollars. The cost of his movie ticket was of the difference between and the cost of his soda, while the cost of his soda was of the difference between and the cost of his movie ticket. To the nearest whole percent, what fraction of did Roger pay for his movie ticket and soda?
Answer: D
Solution:
Let be the cost of the ticket and be the cost of the soda. Then we get the following equations.
Cross-multiplying the first equation gives us Substituting in the expression for yields Solving yields
This also gives us
Adding together the costs gives us
Thus, D is the correct answer.
15.
Chloe chooses a real number uniformly at random from the interval
Independently, Laurent chooses a real number uniformly at random from the interval
What is the probability that Laurent's number is greater than Chloe's number?
Answer: C
Solution:
If Laurent chooses a number in the interval then there is no way that Chloe can have the greater number.
This means that Laurent has a chance of automatically winning.
Otherwise, Laurent chooses a number in the interval The probability that she gets a greater number than Chloe is the same as Chloe getting a greater number then Laurent.
This means that Laurent has a chance of getting a greater number (when working with real intervals, the probability of a tie is essentially due to the infinite size of the intervals).
Laurent's total chance of getting a greater number is
Thus, C is the correct answer.
16.
There are horses, named Horse Horse . . . , Horse They get their names from how many minutes it takes them to run one lap around a circular race track: Horse runs one lap in exactly minutes. At time all the horses are together at the starting point on the track. The horses start running in the same direction, and they keep running around the circular track at their constant speeds.
The least time in minutes, at which all horses will again simultaneously be at the starting point is Let be the least time, in minutes, such that at least of the horses are again at the starting point. What is the sum of the digits of
Answer: B
Solution:
The time it will take for horses to meet again at the start is the least common multiple of their times.
We want to find the numbers that share lots of prime factors and have small prime factors.
This is because to find the least common multiple, we choose the highest power of a prime that is present among all the numbers.
As such, we can choose Horses and These have prime factors of and which the best we can do.
The least common multiple is The sum of its digits is
Thus, B is the correct answer.
17.
Distinct points lie on the circle and have integer coordinates. The distances and are irrational numbers.
What is the greatest possible value of the ratio
Answer: D
Solution:
Note that the only integer coordinate pairs on this circle are and
To get the greatest possible ratio, we want to maximize and minimize
We can see that the distance between any of these points is irrational as long as it is not a diameter.
There are only logical candidates for the longest distance: and or and
Using the distance formula gives us the two distances as and The first is greater.
There is only one viable choice for the shortest distance: and which give us a distance of
The desired ratio is then
Thus, D is the correct answer.
18.
Amelia has a coin that lands heads with probability and Blaine has a coin that lands on heads with probability Amelia and Blaine alternately toss their coins until someone gets a head; the first one to get a head wins. All coin tosses are independent. Amelia goes first. The probability that Amelia wins is where and are relatively prime positive integers. What is
Answer: D
Solution:
Let be the probability that Amelia wins.
There is a chance Amelia wins off her first flip.
If she gets a tails, we want Blaine to lose, which happens with a chance.
The total probability of this case is
The game then goes back to Amelia, who then again has a chance of winning.
Therefore, we get the following equation.
The difference between the denominator and numerator is
Thus, D is the correct answer.
19.
Alice refuses to sit next to either Bob or Carla. Derek refuses to sit next to Eric. How many ways are there for the five of them to sit in a row of chairs under these conditions?
Answer: C
Solution:
If Alice sets on an edge, then the person next to her cannot be Bob or Carla. This means that it must be Derek or Eric.
WLOG, let the person be Eric. Then the person next to Eric has to be Bob or Carla. After that there are no more restrictions.
This gives us a total of
The first is for both edges. The second is for Derek or Eric. The third is for Bob or Carla. The final is just for the people that are remaining.
Otherwise, let Alice be in the middle. Then the two people next to her have to be Derek and Eric. Bob and Carla are forced to be in the last seats.
There are choices for Alice sets. The side on which Derek is sat has options, and then there are options for where Bob and Carla go.
This gives us configurations.
Therefore, there are a total of total seating arrangements.
Thus, C is the correct answer.
20.
Let equal the sum of the digits of positive integer For example, For a particular positive integer
Which of the following could be the value of
Answer: D
Solution:
Recall that a number is divisible by if and only if the sum of its digits is also divisible by
This means that looking at mod would also give us mod
Let us prove this. If we add to without carrying, it is clear that the sum of the digits increases by and that itself increases by
This would increase both their values mod by
Now, if it does carry, we would be subtracting from some digit and adding on to the next digit.
This would keep the value mod constant. We did, however, add in there, so the value mod still increased by
These are the only two cases, and in both we have shown that the value mod for both and increased by
Therefore, we have that
From this, we can see that
The only answer choice that leaves a remainder of when divided by is
Thus, D is the correct answer.
21.
A square with side length is inscribed in a right triangle with sides of length and so that one vertex of the square coincides with the right-angle vertex of the triangle. A square with side length is inscribed in another right triangle with sides of length and so that one side of the square lies on the hypotenuse of the triangle. What is
Answer: D
Solution:
We can see that and are similar (angle-angle). This gives us
Cross-multiplying yields
Here, we have that and are similar (angle-angle).
This means that and This gives us the equation
Finally, we get that The desired ratio is
Thus, D is the correct answer.
22.
Sides and of equilateral triangle are tangent to a circle at points and respectively. What fraction of the area of lies outside the circle?
Answer: E
Solution:
Let the radius of the circle be
To find the area of the triangle outside of the circle, we can find the area of the triangle inside the circle and subtract it.
We get that since and are right angles.
This means that the area of sector is
Now, we need to find the area of Using the formula for the area of a triangle with sine, we get the area to be
Then the area of the triangle inside the circle is
The area of is
The desired fraction is then
Thus, E is the correct answer.
23.
How many triangles with positive area have all their vertices at points in the coordinate plane, where and are integers between and inclusive?
Answer: B
Solution:
We can use complementary counting to find the total number of triangles and subtract out the ones that don't work.
There are a total of points, so there are possible triangles.
Note that the only way a triangle doesn't work is if all the points are in a straight line.
There are rows, columns, and long diagonals. Each of these lines have points, which means they contribute degenerate triangles.
There are also the diagonal lines with points, such as to There are of these lines, so they have degenerate triangles.
Similarly, they are diagonal lines with points. These give us extra triangles that don't work.
Now, we have to look at the lines with slopes of and
There are such lines for each slope, and they all have points on them. Therefore, they contribute more triangles to discount.
The total number of working triangles is then Thus, B is the correct answer.
24.
For certain real numbers and the polynomial has three distinct roots, and each root of is also a root of the polynomial What is
Answer: C
Solution:
We know that has roots, of which are the roots of This means that we can express as as for some complex number that is the other root of
Plugging in we get equals:
Comparing coefficients, we get We also know that
Finally, we have that equals:
Thus, C is the correct answer.
25.
How many integers between and inclusive, have the property that some permutation of its digits is a multiple of between and For example, both and have this property.
Answer: A
Solution:
We can analyze all the multiple so and see how many permutations each of them contribute. We can do this by casing on the number of unique digits in the number.
Case all the digits are the same
This cannot happen. We can see this by the divisibility rule for which says that the sum of the first and last digit minus the middle digit must be divisible by
If all the digits are the sum, then the above expression evaluates to that digit, which cannot be divisible by
Case two of the digits are the same
We can split this up into the numbers that have the digit and those that don't.
There are multiples of that do not have the digit and
Each of these numbers contributes permutations, so this scenario has numbers.
There are multiples of that have the digit and
For these numbers, cannot be the hundreds digit, so each of them only contributes permutations, for a total of
Case all the digits are different
There are a total of multiples of between and The number of these with all different digits is As in case we have to specially account for the numbers with as a digit. There are and
Each of these gives us permutations, but we overcount by a factor of since flipping the first and last digits creates another number already in the set.
Therefore, these numbers provide a total of unique permutations.
There are now multiples of that we need to account for.
We know that each of these provides permutations. As above, however, note that flipping the first and last digit of any number in this set produces another number in this set.
We can see this by using the divisibility rule for If is divisible by then we have that is divisible by
This means that is divisible by which means that is also divisible by
Therefore, these numbers contribute more permutations.
Over all the cases, we have a total of numbers.
Thus, A is the correct answer.