Difference between revisions of "2014 AIME II Problems"
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==Problem 6== | ==Problem 6== | ||
− | Charles has two six-sided | + | Charles has two six-sided dice. One of the die is fair, and the other die is biased so that it comes up six with probability <math>\frac{2}{3}</math> and each of the other five sides has probability <math>\frac{1}{15}</math>. Charles chooses one of the two dice at random and rolls it three times. Given that the first two rolls are both sixes, the probability that the third roll will also be a six is <math>\frac{p}{q}</math>, where <math>p</math> and <math>q</math> are relatively prime positive integers. Find <math>p+q</math>. |
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==Problem 8== | ==Problem 8== | ||
− | Circle <math>C</math> with radius 2 has diameter <math>\overline{AB}</math>. Circle D is internally tangent to circle <math>C</math> at <math>A</math>. Circle <math>E</math> is internally tangent to circle <math>C</math>, externally tangent to circle <math>D</math>, and tangent to <math>\overline{AB}</math>. The radius of circle <math>D</math> is three times the radius of circle <math>E</math>, and can be written in the form <math>\sqrt{m}-n</math>, where <math>m</math> and <math>n</math> are positive integers. Find <math>m+n</math>. | + | Circle <math>C</math> with radius 2 has diameter <math>\overline{AB}</math>. Circle <math>D</math> is internally tangent to circle <math>C</math> at <math>A</math>. Circle <math>E</math> is internally tangent to circle <math>C</math>, externally tangent to circle <math>D</math>, and tangent to <math>\overline{AB}</math>. The radius of circle <math>D</math> is three times the radius of circle <math>E</math>, and can be written in the form <math>\sqrt{m}-n</math>, where <math>m</math> and <math>n</math> are positive integers. Find <math>m+n</math>. |
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==Problem 11== | ==Problem 11== | ||
− | In <math>\triangle RED</math>, <math>\measuredangle DRE=75^{\circ}</math> and <math>\measuredangle RED=45^{\circ}</math>. <math> | + | In <math>\triangle RED</math>, <math>\measuredangle DRE=75^{\circ}</math> and <math>\measuredangle RED=45^{\circ}</math>. <math> RD=1</math>. Let <math>M</math> be the midpoint of segment <math>\overline{RD}</math>. Point <math>C</math> lies on side <math>\overline{ED}</math> such that <math>\overline{RC}\perp\overline{EM}</math>. Extend segment <math>\overline{DE}</math> through <math>E</math> to point <math>A</math> such that <math>CA=AR</math>. Then <math>AE=\frac{a-\sqrt{b}}{c}</math>, where <math>a</math> and <math>c</math> are relatively prime positive integers, and <math>b</math> is a positive integer. Find <math>a+b+c</math>. |
[[2014 AIME II Problems/Problem 11|Solution]] | [[2014 AIME II Problems/Problem 11|Solution]] | ||
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[[2014 AIME II Problems/Problem 15|Solution]] | [[2014 AIME II Problems/Problem 15|Solution]] | ||
− | {{AIME box|year=2014|n=II|before=[[2014 AIME I]]|after=[[2015 AIME I]]}} | + | {{AIME box|year=2014|n=II|before=[[2014 AIME I Problems]]|after=[[2015 AIME I Problems]]}} |
{{MAA Notice}} | {{MAA Notice}} |
Latest revision as of 00:04, 29 December 2021
2014 AIME II (Answer Key) | AoPS Contest Collections • PDF | ||
Instructions
| ||
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 |
Contents
Problem 1
Abe can paint the room in 15 hours, Bea can paint 50 percent faster than Abe, and Coe can paint twice as fast as Abe. Abe begins to paint the room and works alone for the first hour and a half. Then Bea joins Abe, and they work together until half the room is painted. Then Coe joins Abe and Bea, and they work together until the entire room is painted. Find the number of minutes after Abe begins for the three of them to finish painting the room.
Problem 2
Arnold is studying the prevalence of three health risk factors, denoted by A, B, and C, within a population of men. For each of the three factors, the probability that a randomly selected man in the population has only this risk factor (and none of the others) is 0.1. For any two of the three factors, the probability that a randomly selected man has exactly these two risk factors (but not the third) is 0.14. The probability that a randomly selected man has all three risk factors, given that he has A and B is . The probability that a man has none of the three risk factors given that he does not have risk factor A is
, where
and
are relatively prime positive integers. Find
.
Problem 3
A rectangle has sides of length and 36. A hinge is installed at each vertex of the rectangle, and at the midpoint of each side of length 36. The sides of length
can be pressed toward each other keeping those two sides parallel so the rectangle becomes a convex hexagon as shown. When the figure is a hexagon with the sides of length
parallel and separated by a distance of 24, the hexagon has the same area as the original rectangle. Find
.
Problem 4
The repeating decimals and
satisfy
where ,
, and
are (not necessarily distinct) digits. Find the three digit number
.
Problem 5
Real numbers and
are roots of
, and
and
are roots of
. Find the sum of all possible values of
.
Problem 6
Charles has two six-sided dice. One of the die is fair, and the other die is biased so that it comes up six with probability and each of the other five sides has probability
. Charles chooses one of the two dice at random and rolls it three times. Given that the first two rolls are both sixes, the probability that the third roll will also be a six is
, where
and
are relatively prime positive integers. Find
.
Problem 7
Let . Find the sum of all positive integers
for which
Problem 8
Circle with radius 2 has diameter
. Circle
is internally tangent to circle
at
. Circle
is internally tangent to circle
, externally tangent to circle
, and tangent to
. The radius of circle
is three times the radius of circle
, and can be written in the form
, where
and
are positive integers. Find
.
Problem 9
Ten chairs are arranged in a circle. Find the number of subsets of this set of chairs that contain at least three adjacent chairs.
Problem 10
Let be a complex number with
. Let
be the polygon in the complex plane whose vertices are
and every
such that
. Then the area enclosed by
can be written in the form
, where
is an integer. Find the remainder when
is divided by
.
Problem 11
In ,
and
.
. Let
be the midpoint of segment
. Point
lies on side
such that
. Extend segment
through
to point
such that
. Then
, where
and
are relatively prime positive integers, and
is a positive integer. Find
.
Problem 12
Suppose that the angles of satisfy
. Two sides of the triangle have lengths 10 and 13. There is a positive integer
so that the maximum possible length for the remaining side of
is
. Find
.
Problem 13
Ten adults enter a room, remove their shoes, and toss their shoes into a pile. Later, a child randomly pairs each left shoe with a right shoe without regard to which shoes belong together. The probability that for every positive integer , no collection of
pairs made by the child contains the shoes from exactly
of the adults is
, where
and
are relatively prime positive integers. Find
.
Problem 14
In ,
,
, and
. Let
,
, and
be points on line
such that
,
, and
. Point
is the midpoint of segment
, and point
is on ray
such that
. Then
, where
and
are relatively prime positive integers. Find
.
Problem 15
For any integer , let
be the smallest prime which does not divide
. Define the integer function
to be the product of all primes less than
if
, and
if
. Let
be the sequence defined by
, and
for
. Find the smallest positive integer
such that
.
2014 AIME II (Problems • Answer Key • Resources) | ||
Preceded by 2014 AIME I Problems |
Followed by 2015 AIME I Problems | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.