Difference between revisions of "2019 AIME II Problems"
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==Problem 8== | ==Problem 8== | ||
− | The polynomial <math>f(z)=az^{2018}+bz^{2017}+cz^{2016}</math> has real coefficients not exceeding <math>2019</math> and <math>f\left(\tfrac{1+\sqrt3i}{2}\right)=2015+2019\sqrt3i</math>. Find the remainder when <math>f(1)</math> is divided by <math>1000</math>. | + | The polynomial <math>f(z)=az^{2018}+bz^{2017}+cz^{2016}</math> has real coefficients not exceeding <math>2019,</math> and <math>f\left(\tfrac{1+\sqrt3i}{2}\right)=2015+2019\sqrt3i</math>. Find the remainder when <math>f(1)</math> is divided by <math>1000</math>. |
[[2019 AIME II Problems/Problem 8 | Solution]] | [[2019 AIME II Problems/Problem 8 | Solution]] |
Revision as of 16:55, 22 March 2019
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 then 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 as 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
Problem 10
Problem 11
Problem 12
Problem 13
Problem 14
Problem 15
2019 AIME II (Problems • Answer Key • Resources) | ||
Preceded by 2019 AIME I |
Followed by 2020 AIME I | |
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.