2023年12月10日发(作者：新津小升初数学试卷)
2018 AIME I Problems
Problem 1
Let be the number of ordered pairs of
integers with and such that the
polynomial can be factored into the product of two (not necessarily
distinct) linear factors with integer coefficients. Find the remainder when is
divided by .
Problem 2
The number can be written in base as , can be written in
base as , and can be written in base as , where
base- representation of .
. Find the
Problem 3
Kathy has red cards and green cards. She shuffles the cards and lays
out of the cards in a row in a random order. She will be happy if and only if all
the red cards laid out are adjacent and all the green cards laid out are adjacent.
For example, card orders RRGGG, GGGGR, or RRRRR will make Kathy happy,
but RRRGR will not. The probability that Kathy will be happy is
where and are relatively prime positive integers. Find
,
.
Problem 4
In
between and on
and . Point lies strictly
so and point lies strictly between and on
that . Then can be expressed in the form ,
where and are relatively prime positive integers. Find .
Problem 5
For each ordered pair of real numbers satisfyingthere is a real number such thatFind the product of all possible
values of .
Problem 6
Let
that
divided by
be the number of complex numbers with the properties
and
.
is a real number. Find the remainder when is
Problem 7
A right hexagonal prism has height . The bases are regular hexagons with side
length . Any of the vertices determine a triangle. Find the number of these
triangles that are isosceles (including equilateral triangles).
Problem 8
Let be an equiangular hexagon such
that , and . Denote the diameter of
the largest circle that fits inside the hexagon. Find .
Problem 9
Find the number of four-element subsets of
that two distinct elements of a subset have a sum of
elements of a subset have a sum of . For
example, and
with the property
, and two distinct
are two such subsets.
Problem 10
The wheel shown below consists of two circles and five spokes, with a label at
each point where a spoke meets a circle. A bug walks along the wheel, starting at
point . At every step of the process, the bug walks from one labeled point to an
adjacent labeled point. Along the inner circle the bug only walks in a
counterclockwise direction, and along the outer circle the bug only walks in a
clockwise direction. For example, the bug could travel along the
path , which has steps. Let be the number of paths
with steps that begin and end at point . Find the remainder when is
divided by . Problem 11
Find the least positive integer such that when
right-most digits in base are .
is written in base , its two
Problem 12
For every subset of
elements of , with
, let be the sum of the
defined to be . If is chosen at random among all
, where and are subsets of , the probability that is divisible by is
relatively prime positive integers. Find .
Problem 13
Let
Point
of
of
have side lengths , , and .
lies in the interior of , and points and are the incenters
and , respectively. Find the minimum possible area
as varies along .
Problem 14
Let be a heptagon. A frog starts jumping at vertex . From any
vertex of the heptagon except , the frog may jump to either of the two adjacent vertices. When it reaches vertex , the frog stops and stays there. Find the
number of distinct sequences of jumps of no more than jumps that end at .
Problem 15
David found four sticks of different lengths that can be used to form three non-congruent convex cyclic quadrilaterals, , which can each be inscribed
in a circle with radius . Let denote the measure of the acute angle made by
the diagonals of quadrilateral , and define and similarly. Suppose
that , , and . All three quadrilaterals have the
, where and are relatively same area , which can be written in the form
prime positive integers. Find . 2018 AMC 8 Problems
Problem 1
An amusement park has a collection of scale models, with ratio , of
buildings and other sights from around the country. The height of the United
States Capitol is 289 feet. What is the height in feet of its replica to the nearest
whole number?
Problem 2
What is the value of the productProblem 3
Students Arn, Bob, Cyd, Dan, Eve, and Fon are arranged in that order in a circle.
They start counting: Arn first, then Bob, and so forth. When the number contains
a 7 as a digit (such as 47) or is a multiple of 7 that person leaves the circle and
the counting continues. Who is the last one present in the circle?
Problem 4
The twelve-sided figure shown has been drawn on
What is the area of the figure in ?
graph paper. Problem 5
What is the value
of ?
Problem 6
On a trip to the beach, Anh traveled 50 miles on the highway and 10 miles on a
coastal access road. He drove three times as fast on the highway as on the
coastal road. If Anh spent 30 minutes driving on the coastal road, how many
minutes did his entire trip take?
Problem 7
The -digit number
number is divided by ?
is divisible by . What is the remainder when this
Problem 8
Mr. Garcia asked the members of his health class how many days last week they
exercised for at least 30 minutes. The results are summarized in the following bar
graph, where the heights of the bars represent the number of students. What was the mean
number of days of exercise last week, rounded to the nearest hundredth,
reported by the students in Mr. Garcia\'s class?
Problem 9
Tyler is tiling the floor of his 12 foot by 16 foot living room. He plans to place one-foot by one-foot square tiles to form a border along the edges of the room and to
fill in the rest of the floor with two-foot by two-foot square tiles. How many tiles will
he use?
Problem 10
The of a set of non-zero numbers is the reciprocal of the
average of the reciprocals of the numbers. What is the harmonic mean of 1, 2,
and 4?
Problem 11
Abby, Bridget, and four of their classmates will be seated in two rows of three for
a group picture, as shown.
If the seating positions are assigned randomly, what is the probability that Abby
and Bridget are adjacent to each other in the same row or the same column? Problem 12
The clock in Sri\'s car, which is not accurate, gains time at a constant rate. One
day as he begins shopping he notes that his car clock and his watch (which is
accurate) both say 12:00 noon. When he is done shopping, his watch says 12:30
and his car clock says 12:35. Later that day, Sri loses his watch. He looks at his
car clock and it says 7:00. What is the actual time?
Problem 13
Laila took five math tests, each worth a maximum of 100 points. Laila\'s score on
each test was an integer between 0 and 100, inclusive. Laila received the same
score on the first four tests, and she received a higher score on the last test. Her
average score on the five tests was 82. How many values are possible for Laila\'s
score on the last test?
Problem 14
Let be the greatest five-digit number whose digits have a product of
is the sum of the digits of ?
. What
Problem 15
In the diagram below, a diameter of each of the two smaller circles is a radius of
the larger circle. If the two smaller circles have a combined area of square unit,
then what is the area of the shaded region, in square units? Problem 16
Professor Chang has nine different language books lined up on a bookshelf: two
Arabic, three German, and four Spanish. How many ways are there to arrange
the nine books on the shelf keeping the Arabic books together and keeping the
Spanish books together?
Problem 17
Bella begins to walk from her house toward her friend Ella\'s house. At the same
time, Ella begins to ride her bicycle toward Bella\'s house. They each maintain a
constant speed, and Ella rides 5 times as fast as Bella walks. The distance
between their houses is miles, which is feet, and Bella covers feet
with each step. How many steps will Bella take by the time she meets Ella?
Problem 18
How many positive factors does have?
Problem 19
In a sign pyramid a cell gets a \"+\" if the two cells below it have the same sign,
and it gets a \"-\" if the two cells below it have different signs. The diagram below
illustrates a sign pyramid with four levels. How many possible ways are there to
fill the four cells in the bottom row to produce a \"+\" at the top of the pyramid? Problem 20
In
on
a point is on
so that
with and
so that
Point is
What is the and point is on
to the area of ratio of the area of
Problem 21
How many positive three-digit integers have a remainder of 2 when divided by 6,
a remainder of 5 when divided by 9, and a remainder of 7 when divided by 11?
Problem 22
Point is the midpoint of side in square
diagonal at The area of quadrilateral
of
and meets
is What is the area Problem 23
From a regular octagon, a triangle is formed by connecting three randomly
chosen vertices of the octagon. What is the probability that at least one of the
sides of the triangle is also a side of the octagon?
Problem 24
In the cube
midpoints of edges
the cross-section
and
with opposite vertices and and are the
respectively. Let be the ratio of the area of
to the area of one of the faces of the cube. What is Problem 25
How many perfect cubes lie between and , inclusive? 2018 AMC 10A Problems
Problem 1
What is the value ofProblem 2
Liliane has more soda than Jacqueline, and Alice has more soda than Jacqueline. What
is the relationship between the amounts of soda that Liliane and Alice have?
Liliane has
Liliane has
Liliane has
Liliane has
Liliane has
more soda than Alice.
more soda than Alice.
more soda than Alice.
more soda than Alice.
more soda than Alice.
Problem 3
A unit of blood expires after seconds. Yasin donates a unit of blood at
noon of January 1. On what day does his unit of blood expire?
Problem 4
How many ways can a student schedule 3 mathematics courses -- algebra, geometry, and number
theory -- in a 6-period day if no two mathematics courses can be taken in consecutive periods? (What
courses the student takes during the other 3 periods is of no concern here.)
Problem 5 Alice, Bob, and Charlie were on a hike and were wondering how far away the nearest town was. When
Alice said, \"We are at least 6 miles away,\" Bob replied, \"We are at most 5 miles away.\" Charlie then
remarked, \"Actually the nearest town is at most 4 miles away.\" It turned out that none of the three
statements were true. Let be the distance in miles to the nearest town. Which of the following
intervals is the set of all possible values of ?
Problem 6
Sangho uploaded a video to a website where viewers can vote that they like or dislike a video. Each
video begins with a score of 0, and the score increases by 1 for each like vote and decreases by 1 for
each dislike vote. At one point Sangho saw that his video had a score of 90, and that
point?
of the
votes cast on his video were like votes. How many votes had been cast on Sangho\'s video at that
Problem 7
For how many (not necessarily positive) integer values of is the value of
integer?
an
Problem 8
Joe has a collection of 23 coins, consisting of 5-cent coins, 10-cent coins, and 25-cent coins. He has 3
more 10-cent coins than 5-cent coins, and the total value of his collection is 320 cents. How many
more 25-cent coins does Joe have than 5-cent coins?
Problem 9
All of the triangles in the diagram below are similar to iscoceles triangle
which . Each of the 7 smallest triangles has area 1, and
?
, in
has area 40. What
is the area of trapezoid Problem 10
Suppose that real number satisfiesof ?
. What is the value
Problem 11
When fair standard -sided die are thrown, the probability that the sum of the numbers on the top
faces is can be written as, where is a positive integer. What is ?
Problem 12
How many ordered pairs of real numbers
equations? satisfy the following system of Problem 13
A paper triangle with sides of lengths 3, 4, and 5 inches, as shown, is folded so that point falls on
point . What is the length in inches of the crease?Problem 14
What is the greatest integer less than or equal toProblem 15
Two circles of radius 5 are externally tangent to each other and are internally tangent to a circle of
radius 13 at points
form , where
and , as shown in the diagram. The distance can be written in the
? and are relatively prime positive integers. What is Problem 16
Right triangle has leg lengths and . Including
to a point on
and ,
how many line segments with integer length can be drawn from vertex
hypotenuse ?
Problem 17
Let be a set of 6 integers taken from
with
with the property that if and are
elements of
element in
, then is not a multiple of . What is the least possible values of an
Problem 18
How many nonnegative integers can be written in the
formwhere for ? Problem 19
A number
that
is randomly selected from the set , and a number is
. What is the probability randomly selected from
has a units digit of ?
Problem 20
A scanning code consists of a grid of squares, with some of its squares colored black and the
squares. A rest colored white. There must be at least one square of each color in this grid of
scanning code is called
by a multiple of
if its look does not change when the entire square is rotated
counterclockwise around its center, nor when it is reflected across a line joining
opposite corners or a line joining midpoints of opposite sides. What is the total number of possible
symmetric scanning codes?
Problem 21
Which of the following describes the set of values of for which the
curves and in the real -plane intersect at exactly points?
Problem 22
Let
that
and
and be positive integers such
, , ,
. Which of the following must be a divisor of ?
Problem 23
Farmer Pythagoras has a field in the shape of a right triangle. The right triangle\'s legs have lengths 3
and 4 units. In the corner where those sides meet at a right angle, he leaves a small unplanted square so that from the air it looks like the right angle symbol. The rest of the field is planted. The
to the hypotenuse is 2 units. What fraction of the field is planted? shortest distance from
Problem 24
Triangle
of
of
quadrilateral
, and let
with and has area
. The angle bisector
at and , respectively. What is the area of
. Let be the midpoint
be the midpoint of
and
?
intersects
Problem 25
For a positive integer and nonzero digits , , and , let
digits is equal to ; let
the
be the -digit integer each of whose
be be the -digit integer each of whose digits is equal to , and let
-digit (not -digit) integer each of whose digits is equal to . What is the greatest possible
for which there are at least two values of such that ? value of 2018 AMC 10B Problems
Problem 1
Kate bakes a 20-inch by 18-inch pan of cornbread. The cornbread is cut into pieces that measure 2
inches by 2 inches. How many pieces of cornbread does the pan contain?
Problem 2
Sam drove 96 miles in 90 minutes. His average speed during the first 30 minutes was 60 mph (miles
per hour), and his average speed during the second 30 minutes was 65 mph. What was his average
speed, in mph, during the last 30 minutes?
Problem 3
In the expression
the digits
obtained?
or
each blank is to be filled in with one of
with each digit being used once. How many different values can be
Problem 4
A three-dimensional rectangular box with dimensions
are 24, 24, 48, 48, 72, and 72 square units. What is
, , and has faces whose surface areas
?
Problem 5
How many subsets of contain at least one prime number?
Problem 6 A box contains 5 chips, numbered 1, 2, 3, 4, and 5. Chips are drawn randomly one at a time without
replacement until the sum of the values drawn exceeds 4. What is the probability that 3 draws are
required?
Problem 7
In the figure below, congruent semicircles are drawn along a diameter of a large semicircle, with
be the combined their diameters covering the diameter of the large semicircle with no overlap. Let
area of the small semicircles and
the small semicircles. The ratio
be the area of the region inside the large semicircle but outside
is 1:18. What is ?
Problem 8
Sara makes a staircase out of toothpicks as shown: This is a 3-step staircase and uses 18 toothpicks. How many steps would be in a staircase that used
180 toothpicks?
Problem 9
The faces of each of 7 standard dice are labeled with the integers from 1 to 6. Let be the probability
that when all 7 dice are rolled, the sum of the numbers on the top faces is 10. What other sum occurs
with the same probability ?
Problem 10
In the rectangular parallelepiped shown,
midpoint of
, , and . Point is the
? . What is the volume of the rectangular pyramid with base and apex Problem 11
Which of the following expressions is never a prime number when is a prime number?
Problem 12
Line segment
the circle. As point
is a diameter of a circle with . Point , not equal to or , lies on
moves around the circle, the centroid (center of mass) of traces out
a closed curve missing two points. To the nearest positive integer, what is the area of the region
bounded by this curve?
Problem 13
How many of the first
divisible by ?
numbers in the sequence are Problem 14
A list of positive integers has a unique mode, which occurs exactly times. What is the least
number of distinct values that can occur in the list?
Problem 15
A closed box with a square base is to be wrapped with a square sheet of wrapping paper. The box is
centered on the wrapping paper with the vertices of the base lying on the midlines of the square sheet
of paper, as shown in the figure on the left. The four corners of the wrapping paper are to be folded up
over the sides and brought together to meet at the center of the top of the box, point in the figure on
the right. The box has base length and height . What is the area of the sheet of wrapping paper?
Problem 16
Let be a strictly increasing sequence of positive integers such
thatWhat is the remainder
when is divided by ? Problem 17
In rectangle , and . Points and lie on ,
points
that
and lie on , points and lie on , and points and lie on so
and the convex octagon is equilateral. The length
, where , , and are
?
of a side of this octagon can be expressed in the form
integers and is not divisible by the square of any prime. What is
Problem 18
Three young brother-sister pairs from different families need to take a trip in a van. These six children
will occupy the second and third rows in the van, each of which has three seats. To avoid disruptions,
siblings may not sit right next to each other in the same row, and no child may sit directly in front of his
or her sibling. How many seating arrangements are possible for this trip?
Problem 19
Joey and Chloe and their daughter Zoe all have the same birthday. Joey is 1 year older than Chloe,
and Zoe is exactly 1 year old today. Today is the first of the 9 birthdays on which Chloe\'s age will be
an integral multiple of Zoe\'s age. What will be the sum of the two digits of Joey\'s age the next time his
age is a multiple of Zoe\'s age?
Problem 20
A function is defined recursively
by
integers . What is
and?
for all Problem 21
Mary chose an even -digit number . She wrote down all the divisors of in increasing order from
left to right: . At some moment Mary wrote as a divisor of . What is the
? smallest possible value of the next divisor written to the right of
Problem 22
Real numbers and are chosen independently and uniformly at random from the interval
Which of the following numbers is closest to the probability that
obtuse triangle?
.
and are the side lengths of an
Problem 23
How many ordered pairs
equationgreatest common divisor of and , and
of positive integers satisfy the
where denotes the
denotes their least common multiple?
Problem 24
Let
of sides ,
be a regular hexagon with side length . Denote by
, and
, , and the midpoints
, respectively. What is the area of the convex hexagon whose interior
and ? is the intersection of the interiors of
Problem 25 Let denote the greatest integer less than or equal to . How many real numbers satisfy the
equation ? 2018 AMC 12A Problems
Problem 1
A large urn contains
be removed.)
balls, of which are red and the rest are blue. How many of the blue
? (No red balls are to balls must be removed so that the percentage of red balls in the urn will be
Problem 2
While exploring a cave, Carl comes across a collection of -pound rocks worth
rocks worth
the cave?
each, and -pound rocks worth each. There are at least
each, -pound
of each size. He
can carry at most pounds. What is the maximum value, in dollars, of the rocks he can carry out of
Problem 3
How many ways can a student schedule 3 mathematics courses -- algebra, geometry, and number
theory -- in a 6-period day if no two mathematics courses can be taken in consecutive periods? (What
courses the student takes during the other 3 periods is of no concern here.)
Problem 4
Alice, Bob, and Charlie were on a hike and were wondering how far away the nearest town was. When
Alice said, \"We are at least 6 miles away,\" Bob replied, \"We are at most 5 miles away.\" Charlie then
remarked, \"Actually the nearest town is at most 4 miles away.\" It turned out that none of the three
statements were true. Let be the distance in miles to the nearest town. Which of the following
intervals is the set of all possible values of ?
Problem 5 What is the sum of all possible values of for which the polynomials
have a root in common?
and
Problem 6
For positive integers
the set
and such that , both the mean and the median of
are equal to . What is ?
Problem 7
For how many (not necessarily positive) integer values of is the value of
integer?
an
Problem 8
All of the triangles in the diagram below are similar to iscoceles triangle
. Each of the 7 smallest triangles has area 1, and
area of trapezoid ?
, in which
has area 40. What is the Problem 9
Which of the following describes the largest subset of values of within the closed interval
whichfor every between and , inclusive? for
Problem 10
How many ordered pairs of real numbers satisfy the following system of equations?Problem 11 A paper triangle with sides of lengths 3,4, and 5 inches, as shown, is folded so that point falls on
point . What is the length in inches of the crease?Problem 12
Let be a set of 6 integers taken from
with
with the property that if and are
elements of
element in
, then is not a multiple of . What is the least possible value of an
Problem 13
How many nonnegative integers can be written in the formwhere
for ?
Problem 14
The solutions to the equation , where is a positive real number other than
? or , can be written as where and are relatively prime positive integers. What is Problem 15
A scanning code consists of a grid of squares, with some of its squares colored black and the
squares. A rest colored white. There must be at least one square of each color in this grid of
scanning code is called
a multiple of
if its look does not change when the entire square is rotated by
counterclockwise around its center, nor when it is reflected across a line joining
opposite corners or a line joining midpoints of opposite sides. What is the total number of possible
symmetric scanning codes?
Problem 16
Which of the following describes the set of values of for which the curves
in the real -plane intersect at exactly points?
and
Problem 17
Farmer Pythagoras has a field in the shape of a right triangle. The right triangle\'s legs have lengths 3
and 4 units. In the corner where those sides meet at a right angle, he leaves a small unplanted square
so that from the air it looks like the right angle symbol. The rest of the field is planted. The shortest
distance from to the hypotenuse is 2 units. What fraction of the field is planted? Problem 18
Triangle
, and let
and
with
be the midpoint of
and has area . Let be the midpoint of
and at . The angle bisector of intersects
? , respectively. What is the area of quadrilateral
Problem 19
Let be the set of positive integers that have no prime factors other than , , or . The infinite sumof the
reciprocals of the elements of
positive integers. What is
can be expressed as
?
, where and are relatively prime
Problem 20
Triangle
hypotenuse
and
is an isosceles right triangle with
. Points and lie on sides and
. Let be the midpoint of
, respectively, so that
has area , the length can is a cyclic quadrilateral. Given that triangle
be written as , where , , and are positive integers and is not divisible by the square
? of any prime. What is the value of
Problem 21
Which of the following polynomials has the greatest real root?
Problem 22 The solutions to the equations and where
form the vertices of a parallelogram in the complex plane. The area of this
parallelogram can be written in the form where and are positive integers
and neither nor is divisible by the square of any prime number. What is
Problem 23
In
and
respectively, so that
and
Let
Points
and
and lie on sides
be the midpoints of segments
and
and
respectively. What is the degree measure of the acute angle formed by lines
Problem 24
Alice, Bob, and Carol play a game in which each of them chooses a real number between 0 and 1. The
winner of the game is the one whose number is between the numbers chosen by the other two
players. Alice announces that she will choose her number uniformly at random from all the numbers
between 0 and 1, and Bob announces that he will choose his number uniformly at random from all the
numbers between and Armed with this information, what number should Carol choose to
maximize her chance of winning?
Problem 25 For a positive integer and nonzero digits , , and , let
digits is equal to ; let
the
be the -digit integer each of whose
be be the -digit integer each of whose digits is equal to , and let
-digit (not -digit) integer each of whose digits is equal to . What is the greatest possible
for which there are at least two values of such that ? value of 2018 AMC 12B Problems
Problem 1
Kate bakes 20-inch by 18-inch pan of cornbread. The cornbread is cut into pieces that measure 2
inches by 2 inches. How many pieces of cornbread does the pan contain?
Problem 2
Sam drove 96 miles in 90 minutes. His average speed during the first 30 minutes was 60 mph (miles
per hour), and his average speed during the second 30 minutes was 65 mph. What was his average
speed, in mph, during the last 30 minutes?
Problem 3
A line with slope 2 intersects a line with slope 6 at the point
the -intercepts of these two lines?
. What is the distance between
Problem 4
A circle has a chord of length
is the area of the circle?
, and the distance from the center of the circle to the chord is . What
Problem 5
How many subsets of contain at least one prime number?
Problem 6 Suppose cans of soda can be purchased from a vending machine for quarters. Which of the
following expressions describes the number of cans of soda that can be purchased for dollars,
where 1 dollar is worth 4 quarters?
Problem 7
What is the value ofProblem 8
Line segment is a diameter of a circle with . Point , not equal to or , lies on
the circle. As point moves around the circle, the centroid (center of mass) of traces out
a closed curve missing two points. To the nearest positive integer, what is the area of the region
bounded by this curve?
Problem 9
What isProblem 10
A list of positive integers has a unique mode, which occurs exactly times. What is the least
number of distinct values that can occur in the list?
Problem 11 A closed box with a square base is to be wrapped with a square sheet of wrapping paper. The box is
centered on the wrapping paper with the vertices of the base lying on the midlines of the square sheet
of paper, as shown in the figure on the left. The four corners of the wrapping paper are to be folded up
over the sides and brought together to meet at the center of the top of the box, point in the figure on
the right. The box has base length and height . What is the area of the sheet of wrapping paper?
Problem 12
Side of has length . The bisector of angle meets at , and
. The set of all possible values of is an open interval . What is ?
Problem 13
Square has side length . Point lies inside the square so that and
. The centroids of , , , and are the vertices
of a convex quadrilateral. What is the area of that quadrilateral? Problem 14
Joey and Chloe and their daughter Zoe all have the same birthday. Joey is 1 year older than Chloe,
and Zoe is exactly 1 year old today. Today is the first of the 9 birthdays on which Chloe\'s age will be
an integral multiple of Zoe\'s age. What will be the sum of the two digits of Joey\'s age the next time his
age is a multiple of Zoe\'s age?
Problem 15
How many odd positive 3-digit integers are divisible by 3 but do not contain the digit 3?
Problem 16
The solutions to the equation are connected in the complex plane to form a
and . What is the least convex regular polygon, three of whose vertices are labeled
possible area of Problem 17
Let and be positive integers such that?
and is as small as possible. What is
Problem 18
A function is defined recursively by andfor all integers . What is ?
Problem 19
Mary chose an even -digit number . She wrote down all the divisors of in increasing order from
left to right: . At some moment Mary wrote as a divisor of . What is the
? smallest possible value of the next divisor written to the right of
Problem 20
Let
of sides ,
be a regular hexagon with side length . Denote by
, and
, , and the midpoints
, respectively. What is the area of the convex hexagon whose interior
and ? is the intersection of the interiors of
Problem 21
In with side lengths , , and , let and denote
and the circumcenter and incenter, respectively. A circle with center
and to the circumcircle of . What is the area of
is tangent to the legs
? Problem 22
Consider polynomials of degree at most , each of whose coefficients is an element of
. How many such polynomials satisfy ?
Problem 23
Ajay is standing at point
standing at point
near Pontianak, Indonesia, latitude and
latitude and
longitude. Billy is
longitude.
?
near Big Baldy Mountain, Idaho, USA,
Assume that Earth is a perfect sphere with center . What is the degree measure of
Problem 24
Let denote the greatest integer less than or equal to . How many real numbers satisfy the
equation ?
Problem 25
Circles , , and each have radius and are placed in the plane so that each circle is
, , and lie on , , and
for each
can be written in the form
?
respectively such
,
externally tangent to the other two. Points
that
where
and line
. See the figure below. The area of
for positive integers and . What is
is tangent to
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