Difference between revisions of "2019 AIME II Problems"
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==Problem 6== | ==Problem 6== | ||
− | In a Martian civilization, all logarithms whose bases are not specified | + | In a Martian civilization, all logarithms whose bases are not specified are assumed to be base <math>b</math>, for some fixed <math>b\ge2</math>. A Martian student writes down |
<cmath>3\log(\sqrt{x}\log x)=56</cmath> | <cmath>3\log(\sqrt{x}\log x)=56</cmath> | ||
<cmath>\log_{\log x}(x)=54</cmath> | <cmath>\log_{\log x}(x)=54</cmath> | ||
Line 66: | Line 66: | ||
==Problem 12== | ==Problem 12== | ||
− | For <math>n\ge1</math> call a finite sequence <math>(a_1,a_2,\ldots,a_n)</math> of positive integers <i> | + | For <math>n\ge1</math> call a finite sequence <math>(a_1,a_2,\ldots,a_n)</math> of positive integers <i>progressive</i> if <math>a_i<a_{i+1}</math> and <math>a_i</math> divides <math>a_{i+1}</math> for <math>1\le i\le n-1</math>. Find the number of progressive sequences such that the sum of the terms in the sequence is equal to <math>360.</math> |
[[2019 AIME II Problems/Problem 12 | Solution]] | [[2019 AIME II Problems/Problem 12 | Solution]] | ||
Line 81: | Line 81: | ||
==Problem 15== | ==Problem 15== | ||
− | In acute triangle <math>ABC</math> points <math>P</math> and <math>Q</math> are the feet of the perpendiculars from <math>C</math> to <math>\overline{AB}</math> and from <math>B</math> to <math>\overline{AC}</math>, respectively. Line <math>PQ</math> intersects the circumcircle of <math>\triangle ABC</math> in two distinct points, <math>X</math> and <math>Y</math>. Suppose <math>XP=10</math>, <math>PQ=25</math>, and <math>QY=15</math>. The value of <math>AB\cdot AC</math> can be written in the form <math>m\sqrt n</math> where <math>m</math> and <math>n</math> are positive | + | In acute triangle <math>ABC,</math> points <math>P</math> and <math>Q</math> are the feet of the perpendiculars from <math>C</math> to <math>\overline{AB}</math> and from <math>B</math> to <math>\overline{AC}</math>, respectively. Line <math>PQ</math> intersects the circumcircle of <math>\triangle ABC</math> in two distinct points, <math>X</math> and <math>Y</math>. Suppose <math>XP=10</math>, <math>PQ=25</math>, and <math>QY=15</math>. The value of <math>AB\cdot AC</math> can be written in the form <math>m\sqrt n</math> where <math>m</math> and <math>n</math> are positive integers, and <math>n</math> is not divisible by the square of any prime. Find <math>m+n</math>. |
[[2019 AIME II Problems/Problem 15 | Solution]] | [[2019 AIME II Problems/Problem 15 | Solution]] | ||
− | {{AIME box|year=2019|n=II|before=[[2019 AIME I]]|after=[[2020 AIME I]]}} | + | {{AIME box|year=2019|n=II|before=[[2019 AIME I Problems]]|after=[[2020 AIME I Problems]]}} |
{{MAA Notice}} | {{MAA Notice}} |
Latest revision as of 14:45, 21 August 2023
2019 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
Two different points, and
, lie on the same side of line
so that
and
are congruent with
, and
. The intersection of these two triangular regions has area
, where
and
are relatively prime positive integers. Find
.
Problem 2
Lily pads lie in a row on a pond. A frog makes a sequence of jumps starting on pad
. From any pad
the frog jumps to either pad
or pad
chosen randomly with probability
and independently of other jumps. The probability that the frog visits pad
is
, where
and
are relatively prime positive integers. Find
.
Problem 3
Find the number of -tuples of positive integers
that satisfy the following system of equations:
Problem 4
A standard six-sided fair die is rolled four times. The probability that the product of all four numbers rolled is a perfect square is , where
and
are relatively prime positive integers. Find
.
Problem 5
Four ambassadors and one advisor for each of them are to be seated at a round table with chairs numbered in order
to
. Each ambassador must sit in an even-numbered chair. Each advisor must sit in a chair adjacent to his or her ambassador. There are
ways for the
people to be seated at the table under these conditions. Find the remainder when
is divided by
.
Problem 6
In a Martian civilization, all logarithms whose bases are not specified are assumed to be base , for some fixed
. A Martian student writes down
and finds that this system of equations has a single real number solution
. Find
.
Problem 7
Triangle has side lengths
, and
. Lines
, and
are drawn parallel to
, and
, respectively, such that the intersections of
, and
with the interior of
are segments of lengths
, and
, respectively. Find the perimeter of the triangle whose sides lie on lines
, and
.
Problem 8
The polynomial has real coefficients not exceeding
and
. Find the remainder when
is divided by
.
Problem 9
Call a positive integer
-pretty if
has exactly
positive divisors and
is divisible by
. For example,
is
-pretty. Let
be the sum of the positive integers less than
that are
-pretty. Find
.
Problem 10
There is a unique angle between
and
such that for nonnegative integers
the value of
is positive when
is a multiple of
, and negative otherwise. The degree measure of
is
, where
and
are relatively prime positive integers. Find
.
Problem 11
Triangle has side lengths
and
Circle
passes through
and is tangent to line
at
Circle
passes through
and is tangent to line
at
Let
be the intersection of circles
and
not equal to
Then
where
and
are relatively prime positive integers. Find
Problem 12
For call a finite sequence
of positive integers progressive if
and
divides
for
. Find the number of progressive sequences such that the sum of the terms in the sequence is equal to
Problem 13
Regular octagon is inscribed in a circle of area
Point
lies inside the circle so that the region bounded by
and the minor arc
of the circle has area
while the region bounded by
and the minor arc
of the circle has area
There is a positive integer
such that the area of the region bounded by
and the minor arc
of the circle is equal to
Find
Problem 14
Find the sum of all positive integers such that, given an unlimited supply of stamps of denominations
and
cents,
cents is the greatest postage that cannot be formed.
Problem 15
In acute triangle points
and
are the feet of the perpendiculars from
to
and from
to
, respectively. Line
intersects the circumcircle of
in two distinct points,
and
. Suppose
,
, and
. The value of
can be written in the form
where
and
are positive integers, and
is not divisible by the square of any prime. Find
.
2019 AIME II (Problems • Answer Key • Resources) | ||
Preceded by 2019 AIME I Problems |
Followed by 2020 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.