2015 AMC 10A 考题及答案
Problem 1
What is the value of
Problem 2
A box contains a collection of triangular and square tiles. There are
tiles in the box, containing
edges total. How many square tiles are there in the box?
Problem 3
Ann made a 3-step staircase using 18 toothpicks as shown in the figure. How many toothpicks does she need to add to complete a 5-step staircase?
Problem 4
Pablo, Sofia, and Mia got some candy eggs at a party. Pablo had three times as many eggs as Sofia, and Sofia had twice as many eggs as Mia. Pablo decides to give some of his eggs to Sofia and Mia so that all three will have the same number of eggs. What fraction of his eggs should Pablo give to Sofia?
Problem 5
Mr. Patrick teaches math to
students. He was grading tests and found that when he graded everyone's test except Payton's, the average grade for the class was
. After he graded Payton's test, the test average became
. What was Payton's score on the test?
Problem 6
The sum of two positive numbers is
times their difference. What is the ratio of the larger number to the smaller number?
Problem 7
How many terms are there in the arithmetic sequence
,
,
, . . .,
,
?
Problem 8
Two years ago Pete was three times as old as his cousin Claire. Two years before that, Pete was four times as old as Claire. In how many years will the ratio of their ages be
:
?
Problem 9
Two right circular cylinders have the same volume. The radius of the second cylinder is
more than the radius of the first. What is the relationship between the heights of the two cylinders?
Problem 10
How many rearrangements of
are there in which no two adjacent letters are also adjacent letters in the alphabet? For example, no such rearrangements could include either
or
.
Problem 11
The ratio of the length to the width of a rectangle is
:
. If the rectangle has diagonal of length
, then the area may be expressed as
for some constant
. What is
?
Problem 12
Points
and
are distinct points on the graph of
. What is
?
Problem 13
Claudia has 12 coins, each of which is a 5-cent coin or a 10-cent coin. There are exactly 17 different values that can be obtained as combinations of one or more of her coins. How many 10-cent coins does Claudia have?
Problem 14
The diagram below shows the circular face of a clock with radius
cm and a circular disk with radius
cm externally tangent to the clock face at
o'clock. The disk has an arrow painted on it, initially pointing in the upward vertical direction. Let the disk roll clockwise around the clock face. At what point on the clock face will the disk be tangent when the arrow is next pointing in the upward vertical direction?
Problem 15
Consider the set of all fractions
where
and
are relatively prime positive integers. How many of these fractions have the property that if both numerator and denominator are increased by
, the value of the fraction is increased by
?
Problem 16
If
, and
, what is the value of
?
Problem 17
A line that passes through the origin intersects both the line
and the line
. The three lines create an equilateral triangle. What is the perimeter of the triangle?
Problem 18
Hexadecimal (base-16) numbers are written using numeric digits
through
as well as the letters
through
to represent
through
. Among the first
positive integers, there are
whose hexadecimal representation contains only numeric digits. What is the sum of the digits of
?
Problem 19
The isosceles right triangle
has right angle at
and area
. The rays trisecting
intersect
at
and
. What is the area of
?
Problem 20
A rectangle with positive integer side lengths in
has area
and perimeter
. Which of the following numbers cannot equal
?
NOTE: As it originally appeared in the AMC 10, this problem was stated incorrectly and had no answer; it has been modified here to be solvable.
Problem 21
Tetrahedron
has
,
,
,
,
, and
. What is the volume of the tetrahedron?
Problem 22
Eight people are sitting around a circular table, each holding a fair coin. All eight people flip their coins and those who flip heads stand while those who flip tails remain seated. What is the probability that no two adjacent people will stand?
Problem 23
The zeroes of the function
are integers. What is the sum of the possible values of
?
Problem 24
For some positive integers
, there is a quadrilateral
with positive integer side lengths, perimeter
, right angles at
and
,
, and
. How many different values of
are possible?
Problem 25
Let
be a square of side length
. Two points are chosen at random on the sides of
. The probability that the straight-line distance between the points is at least
is
, where
,
, and
are positive integers with
. What is
?
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