Difference between revisions of "2016 AIME I Problems"
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==Problem 3== | ==Problem 3== | ||
− | A | + | A ''regular icosahedron'' is a <math>20</math>-faced solid where each face is an equilateral triangle and five triangles meet at every vertex. The regular icosahedron shown below has one vertex at the top, one vertex at the bottom, an upper pentagon of five vertices all adjacent to the top vertex and all in the same horizontal plane, and a lower pentagon of five vertices all adjacent to the bottom vertex and all in another horizontal plane. Find the number of paths from the top vertex to the bottom vertex such that each part of a path goes downward or horizontally along an edge of the icosahedron, and no vertex is repeated. |
− | + | <asy> | |
− | ( | + | size(3cm); |
+ | pair A=(0.05,0),B=(-.9,-0.6),C=(0,-0.45),D=(.9,-0.6),E=(.55,-0.85),F=(-0.55,-0.85),G=B-(0,1.1),H=F-(0,0.6),I=E-(0,0.6),J=D-(0,1.1),K=C-(0,1.4),L=C+K-A; | ||
+ | draw(A--B--F--E--D--A--E--A--F--A^^B--G--F--K--G--L--J--K--E--J--D--J--L--K); | ||
+ | draw(B--C--D--C--A--C--H--I--C--H--G^^H--L--I--J^^I--D^^H--B,dashed); | ||
+ | dot(A^^B^^C^^D^^E^^F^^G^^H^^I^^J^^K^^L); | ||
+ | </asy> | ||
[[2016 AIME I Problems/Problem 3 | Solution]] | [[2016 AIME I Problems/Problem 3 | Solution]] | ||
==Problem 4== | ==Problem 4== | ||
− | A right prism with height <math>h</math> has bases that are regular hexagons with sides of length <math>12</math>. A vertex <math>A</math> of the prism and its three adjacent vertices are the vertices of a triangular pyramid. The dihedral angle (the angle between the two planes) formed by the face of the pyramid that lies in a base of the prism and the face of the pyramid that does not contain <math>A</math> measures <math>60^ | + | A right prism with height <math>h</math> has bases that are regular hexagons with sides of length <math>12</math>. A vertex <math>A</math> of the prism and its three adjacent vertices are the vertices of a triangular pyramid. The dihedral angle (the angle between the two planes) formed by the face of the pyramid that lies in a base of the prism and the face of the pyramid that does not contain <math>A</math> measures <math>60^\circ</math>. Find <math>h^2</math>. |
[[2016 AIME I Problems/Problem 4 | Solution]] | [[2016 AIME I Problems/Problem 4 | Solution]] | ||
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==Problem 7== | ==Problem 7== | ||
− | For integers <math>a</math> and <math>b</math> consider the complex number <cmath>\frac{\sqrt{ab+2016}}{ab+100}-\left(\frac{\sqrt{|a+b|}}{ab+100}\right) i.</cmath> Find the number of ordered pairs of integers <math>(a,b)</math> such that this complex number is a real number. | + | For integers <math>a</math> and <math>b</math> consider the complex number <cmath>\frac{\sqrt{ab+2016}}{ab+100}-\left(\frac{\sqrt{|a+b|}}{ab+100}\right)i.</cmath> Find the number of ordered pairs of integers <math>(a,b)</math> such that this complex number is a real number. |
[[2016 AIME I Problems/Problem 7 | Solution]] | [[2016 AIME I Problems/Problem 7 | Solution]] | ||
==Problem 8== | ==Problem 8== | ||
− | For a permutation <math>p = (a_1,a_2,\ldots,a_9)</math> of the digits <math>1,2,\ldots,9</math>, let <math>s(p)</math> denote the sum of the three <math>3</math>-digit numbers <math>a_1a_2a_3</math>, <math>a_4a_5a_6</math>, and <math>a_7a_8a_9</math>. Let <math>m</math> be the minimum value of <math>s(p)</math> subject to the condition that the units digit of <math>s(p)</math> is <math>0</math>. Let <math>n</math> denote the number of permutations <math>p</math> with <math>s(p) = m</math>. Find <math>|m - n|</math>. | + | For a permutation <math>p = (a_1,a_2,\ldots,a_9)</math> of the digits <math>1,2,\ldots,9</math>, let <math>s(p)</math> denote the sum of the three <math>3</math>-digit numbers <math>a_1a_2a_3</math> , <math>a_4a_5a_6</math>, and <math>a_7a_8a_9</math>. Let <math>m</math> be the minimum value of <math>s(p)</math> subject to the condition that the units digit of <math>s(p)</math> is <math>0</math>. Let <math>n</math> denote the number of permutations <math>p</math> with <math>s(p) = m</math>. Find <math>|m - n|</math>. |
[[2016 AIME I Problems/Problem 8 | Solution]] | [[2016 AIME I Problems/Problem 8 | Solution]] | ||
==Problem 9== | ==Problem 9== | ||
+ | Triangle <math>ABC</math> has <math>AB=40,AC=31,</math> and <math>\sin{A}=\frac{1}{5}</math>. This triangle is inscribed in rectangle <math>AQRS</math> with <math>B</math> on <math>\overline{QR}</math> and <math>C</math> on <math>\overline{RS}</math>. Find the maximum possible area of <math>AQRS</math>. | ||
+ | |||
[[2016 AIME I Problems/Problem 9 | Solution]] | [[2016 AIME I Problems/Problem 9 | Solution]] | ||
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==Problem 12== | ==Problem 12== | ||
− | Find the least positive integer <math>m</math> such that <math>m^2 - m + 11</math> is a product of at least | + | Find the least positive integer <math>m</math> such that <math>m^2 - m + 11</math> is a product of at least four not necessarily distinct primes. |
[[2016 AIME I Problems/Problem 12 | Solution]] | [[2016 AIME I Problems/Problem 12 | Solution]] | ||
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[[2016 AIME I Problems/Problem 15 | Solution]] | [[2016 AIME I Problems/Problem 15 | Solution]] | ||
− | {{AIME box|year=2016|n=I|before=[[2015 AIME II]]|after=[[2016 AIME II]]}} | + | {{AIME box|year=2016|n=I|before=[[2015 AIME II Problems]]|after=[[2016 AIME II Problems]]}} |
{{MAA Notice}} | {{MAA Notice}} |
Latest revision as of 12:19, 29 January 2022
2016 AIME I (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
For , let
denote the sum of the geometric series
Let
between
and
satisfy
. Find
.
Problem 2
Two dice appear to be normal dice with their faces numbered from to
, but each die is weighted so that the probability of rolling the number
is directly proportional to
. The probability of rolling a
with this pair of dice is
, where
and
are relatively prime positive integers. Find
.
Problem 3
A regular icosahedron is a -faced solid where each face is an equilateral triangle and five triangles meet at every vertex. The regular icosahedron shown below has one vertex at the top, one vertex at the bottom, an upper pentagon of five vertices all adjacent to the top vertex and all in the same horizontal plane, and a lower pentagon of five vertices all adjacent to the bottom vertex and all in another horizontal plane. Find the number of paths from the top vertex to the bottom vertex such that each part of a path goes downward or horizontally along an edge of the icosahedron, and no vertex is repeated.
Problem 4
A right prism with height has bases that are regular hexagons with sides of length
. A vertex
of the prism and its three adjacent vertices are the vertices of a triangular pyramid. The dihedral angle (the angle between the two planes) formed by the face of the pyramid that lies in a base of the prism and the face of the pyramid that does not contain
measures
. Find
.
Problem 5
Anh read a book. On the first day she read pages in
minutes, where
and
are positive integers. On the second day Anh read
pages in
minutes. Each day thereafter Anh read one more page than she read on the previous day, and it took her one more minute than on the previous day until she completely read the
page book. It took her a total of
minutes to read the book. Find
.
Problem 6
In let
be the center of the inscribed circle, and let the bisector of
intersect
at
. The line through
and
intersects the circumscribed circle of
at the two points
and
. If
and
, then
, where
and
are relatively prime positive integers. Find
.
Problem 7
For integers and
consider the complex number
Find the number of ordered pairs of integers
such that this complex number is a real number.
Problem 8
For a permutation of the digits
, let
denote the sum of the three
-digit numbers
,
, and
. Let
be the minimum value of
subject to the condition that the units digit of
is
. Let
denote the number of permutations
with
. Find
.
Problem 9
Triangle has
and
. This triangle is inscribed in rectangle
with
on
and
on
. Find the maximum possible area of
.
Problem 10
A strictly increasing sequence of positive integers ,
,
,
has the property that for every positive integer
, the subsequence
,
,
is geometric and the subsequence
,
,
is arithmetic. Suppose that
. Find
.
Problem 11
Let be a nonzero polynomial such that
for every real
, and
. Then
, where
and
are relatively prime positive integers. Find
.
Problem 12
Find the least positive integer such that
is a product of at least four not necessarily distinct primes.
Problem 13
Freddy the frog is jumping around the coordinate plane searching for a river, which lies on the horizontal line . A fence is located at the horizontal line
. On each jump Freddy randomly chooses a direction parallel to one of the coordinate axes and moves one unit in that direction. When he is at a point where
, with equal likelihoods he chooses one of three directions where he either jumps parallel to the fence or jumps away from the fence, but he never chooses the direction that would have him cross over the fence to where
. Freddy starts his search at the point
and will stop once he reaches a point on the river. Find the expected number of jumps it will take Freddy to reach the river.
Problem 14
Centered at each lattice point in the coordinate plane are a circle radius and a square with sides of length
whose sides are parallel to the coordinate axes. The line segment from
to
intersects
of the squares and
of the circles. Find
.
Problem 15
Circles and
intersect at points
and
. Line
is tangent to
and
at
and
, respectively, with line
closer to point
than to
. Circle
passes through
and
intersecting
again at
and intersecting
again at
. The three points
,
,
are collinear,
,
, and
. Find
.
2016 AIME I (Problems • Answer Key • Resources) | ||
Preceded by 2015 AIME II Problems |
Followed by 2016 AIME II 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.