Difference between revisions of "2017 AMC 10A Problems"
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==Problem 15== | ==Problem 15== | ||
− | Chloé chooses a real number uniformly at random from the interval <math>[0, 2017]</math>. Independently, Laurent chooses a real number uniformly at random from the interval <math>[0, 4034]</math>. What is the probability that Laurent's number is greater than Chloé's number? | + | Chloé chooses a real number uniformly at random from the interval <math>[0, 2017]</math>. Independently, Laurent chooses a real number uniformly at random from the interval <math>[0, 4034]</math>. What is the probability that Laurent's number is greater than Chloé's number? |
<math>\textbf{(A)}\ \frac{1}{2}\qquad\textbf{(B)}\ \frac{2}{3}\qquad\textbf{(C)}\ \frac{3}{4}\qquad\textbf{(D)}\ \frac{5}{6}\qquad\textbf{(E)}\ \frac{7}{8}</math> | <math>\textbf{(A)}\ \frac{1}{2}\qquad\textbf{(B)}\ \frac{2}{3}\qquad\textbf{(C)}\ \frac{3}{4}\qquad\textbf{(D)}\ \frac{5}{6}\qquad\textbf{(E)}\ \frac{7}{8}</math> | ||
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[[2017 AMC 10A Problems/Problem 23|Solution]] | [[2017 AMC 10A Problems/Problem 23|Solution]] | ||
− | |||
==Problem 24== | ==Problem 24== | ||
For certain real numbers <math>a</math>, <math>b</math>, and <math>c</math>, the polynomial <cmath>g(x) = x^3 + ax^2 + x + 10</cmath>has three distinct roots, and each root of <math>g(x)</math> is also a root of the polynomial <cmath>f(x) = x^4 + x^3 + bx^2 + 100x + c.</cmath>What is <math>f(1)</math>? | For certain real numbers <math>a</math>, <math>b</math>, and <math>c</math>, the polynomial <cmath>g(x) = x^3 + ax^2 + x + 10</cmath>has three distinct roots, and each root of <math>g(x)</math> is also a root of the polynomial <cmath>f(x) = x^4 + x^3 + bx^2 + 100x + c.</cmath>What is <math>f(1)</math>? |
Latest revision as of 18:02, 28 October 2023
2017 AMC 10A (Answer Key) Printable versions: • AoPS Resources • PDF | ||
Instructions
| ||
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 |
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
- 26 See also
Problem 1
What is the value of ?
Problem 2
Pablo buys popsicles for his friends. The store sells single popsicles for each,
-popsicle boxes for
each, and
-popsicle boxes for
. What is the greatest number of popsicles that Pablo can buy with
?
Problem 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?
Problem 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?
Problem 5
The sum of two nonzero real numbers is times their product. What is the sum of the reciprocals of the two numbers?
Problem 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 of these statements necessarily follows logically?
Problem 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?
Problem 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?
Problem 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?
Problem 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?
Problem 11
The region consisting of all points in three-dimensional space within units of line segment
has volume
. What is the length
?
Problem 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
Problem 13
Define a sequence recursively by and
the remainder when
is divided by
for all
Thus the sequence starts
What is
Problem 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?
Problem 15
Chloé 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 Chloé's number?
Problem 16
There are 10 horses, named Horse 1, Horse 2, , Horse 10. 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 0 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 10 horses will again simultaneously be at the starting point is
. Let
be the least time, in minutes, such that at least 5 of the horses are again at the starting point. What is the sum of the digits of
?
Problem 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
?
Problem 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
?
Problem 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 5 chairs under these conditions?
Problem 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
?
Problem 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
?
Problem 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?
Problem 23
How many triangles with positive area have all their vertices at points in the coordinate plane, where
and
are integers between
and
, inclusive?
Problem 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
?
Problem 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.
See also
2017 AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by 2016 AMC 10B Problems |
Followed by 2017 AMC 10B Problems | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
All AMC 10 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.