Difference between revisions of "2018 AIME II Problems"
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==Problem 4== | ==Problem 4== | ||
− | In equiangular octagon <math>CAROLINE</math>, <math>CA = RO = LI = NE =</math> <math>\sqrt{2}</math> and <math>AR = OL = IN = EC = 1</math>. The self-intersecting octagon <math>CORNELIA</math> | + | In equiangular octagon <math>CAROLINE</math>, <math>CA = RO = LI = NE =</math> <math>\sqrt{2}</math> and <math>AR = OL = IN = EC = 1</math>. The self-intersecting octagon <math>CORNELIA</math> encloses six non-overlapping triangular regions. Let <math>K</math> be the area enclosed by <math>CORNELIA</math>, that is, the total area of the six triangular regions. Then <math>K = \frac{a}{b}</math>, where <math>a</math> and <math>b</math> are relatively prime positive integers. Find <math>a + b</math>. |
[[2018 AIME II Problems/Problem 4 | Solution]] | [[2018 AIME II Problems/Problem 4 | Solution]] | ||
Line 53: | Line 53: | ||
[[2018 AIME II Problems/Problem 8 | Solution]] | [[2018 AIME II Problems/Problem 8 | Solution]] | ||
− | ==Problem 9== | + | ==Problem 9== |
+ | Octagon <math>ABCDEFGH</math> with side lengths <math>AB = CD = EF = GH = 10</math> and <math>BC = DE = FG = HA = 11</math> is formed by removing 6-8-10 triangles from the corners of a <math>23</math> <math>\times</math> <math>27</math> rectangle with side <math>\overline{AH}</math> on a short side of the rectangle, as shown. Let <math>J</math> be the midpoint of <math>\overline{AH}</math>, and partition the octagon into 7 triangles by drawing segments <math>\overline{JB}</math>, <math>\overline{JC}</math>, <math>\overline{JD}</math>, <math>\overline{JE}</math>, <math>\overline{JF}</math>, and <math>\overline{JG}</math>. Find the area of the convex polygon whose vertices are the centroids of these 7 triangles. | ||
− | + | <asy> | |
+ | unitsize(6); | ||
+ | pair P = (0, 0), Q = (0, 23), R = (27, 23), SS = (27, 0); | ||
+ | pair A = (0, 6), B = (8, 0), C = (19, 0), D = (27, 6), EE = (27, 17), F = (19, 23), G = (8, 23), J = (0, 23/2), H = (0, 17); | ||
+ | draw(P--Q--R--SS--cycle); | ||
+ | draw(J--B); | ||
+ | draw(J--C); | ||
+ | draw(J--D); | ||
+ | draw(J--EE); | ||
+ | draw(J--F); | ||
+ | draw(J--G); | ||
+ | draw(A--B); | ||
+ | draw(H--G); | ||
+ | real dark = 0.6; | ||
+ | filldraw(A--B--P--cycle, gray(dark)); | ||
+ | filldraw(H--G--Q--cycle, gray(dark)); | ||
+ | filldraw(F--EE--R--cycle, gray(dark)); | ||
+ | filldraw(D--C--SS--cycle, gray(dark)); | ||
+ | dot(A); | ||
+ | dot(B); | ||
+ | dot(C); | ||
+ | dot(D); | ||
+ | dot(EE); | ||
+ | dot(F); | ||
+ | dot(G); | ||
+ | dot(H); | ||
+ | dot(J); | ||
+ | dot(H); | ||
+ | defaultpen(fontsize(10pt)); | ||
+ | real r = 1.3; | ||
+ | label("$A$", A, W*r); | ||
+ | label("$B$", B, S*r); | ||
+ | label("$C$", C, S*r); | ||
+ | label("$D$", D, E*r); | ||
+ | label("$E$", EE, E*r); | ||
+ | label("$F$", F, N*r); | ||
+ | label("$G$", G, N*r); | ||
+ | label("$H$", H, W*r); | ||
+ | label("$J$", J, W*r); | ||
+ | </asy> | ||
[[2018 AIME II Problems/Problem 9 | Solution]] | [[2018 AIME II Problems/Problem 9 | Solution]] |
Latest revision as of 00:26, 21 January 2024
2018 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
Points ,
, and
lie in that order along a straight path where the distance from
to
is
meters. Ina runs twice as fast as Eve, and Paul runs twice as fast as Ina. The three runners start running at the same time with Ina starting at
and running toward
, Paul starting at
and running toward
, and Eve starting at
and running toward
. When Paul meets Eve, he turns around and runs toward
. Paul and Ina both arrive at
at the same time. Find the number of meters from
to
.
Problem 2
Let ,
, and
, and for
define
recursively to be the remainder when
is divided by
. Find
.
Problem 3
Find the sum of all positive integers such that the base-
integer
is a perfect square and the base-
integer
is a perfect cube.
Problem 4
In equiangular octagon ,
and
. The self-intersecting octagon
encloses six non-overlapping triangular regions. Let
be the area enclosed by
, that is, the total area of the six triangular regions. Then
, where
and
are relatively prime positive integers. Find
.
Problem 5
Suppose that ,
, and
are complex numbers such that
,
, and
, where
. Then there are real numbers
and
such that
. Find
.
Problem 6
A real number is chosen randomly and uniformly from the interval
. The probability that the roots of the polynomial
are all real can be written in the form , where
and
are relatively prime positive integers. Find
.
Problem 7
Triangle has side lengths
,
, and
. Points
are on segment
with
between
and
for
, and points
are on segment
with
between
and
for
. Furthermore, each segment
,
, is parallel to
. The segments cut the triangle into
regions, consisting of
trapezoids and
triangle. Each of the
regions has the same area. Find the number of segments
,
, that have rational length.
Problem 8
A frog is positioned at the origin of the coordinate plane. From the point , the frog can jump to any of the points
,
,
, or
. Find the number of distinct sequences of jumps in which the frog begins at
and ends at
.
Problem 9
Octagon with side lengths
and
is formed by removing 6-8-10 triangles from the corners of a
rectangle with side
on a short side of the rectangle, as shown. Let
be the midpoint of
, and partition the octagon into 7 triangles by drawing segments
,
,
,
,
, and
. Find the area of the convex polygon whose vertices are the centroids of these 7 triangles.
Problem 10
Find the number of functions from
to
that satisfy
for all
in
.
Problem 11
Find the number of permutations of such that for each
with
, at least one of the first
terms of the permutation is greater than
.
Problem 12
Let be a convex quadrilateral with
,
, and
. Assume that the diagonals of
intersect at point
, and that the sum of the areas of triangles
and
equals the sum of the areas of triangles
and
. Find the area of quadrilateral
.
Problem 13
Misha rolls a standard, fair six-sided die until she rolls 1-2-3 in that order on three consecutive rolls. The probability that she will roll the die an odd number of times is where
and
are relatively prime positive integers. Find
.
Problem 14
The incircle of triangle
is tangent to
at
. Let
be the other intersection of
with
. Points
and
lie on
and
, respectively, so that
is tangent to
at
. Assume that
,
,
, and
, where
and
are relatively prime positive integers. Find
.
Problem 15
Find the number of functions from
to the integers such that
,
, and
for all and
in
.
2018 AIME II (Problems • Answer Key • Resources) | ||
Preceded by 2018 AIME I Problems |
Followed by 2019 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.