Difference between revisions of "2019 AIME I Problems"
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==Problem 5== | ==Problem 5== | ||
− | A moving particle starts at the point <math>(4,4)</math> and moves until it hits one of the coordinate axes for the first time. When the particle is at the point <math>(a,b)</math>, it moves at random to one of the points <math>(a-1,b)</math>, <math>(a,b-1)</math>, or <math>(a-1,b-1)</math>, each with probability <math>\tfrac{1}{3}</math>, independently of its previous moves. The probability that it will hit the coordinate axes at <math>(0,0)</math> is <math>\tfrac{m}{3^n}</math>, where <math>m</math> and <math>n</math> are positive integers | + | A moving particle starts at the point <math>(4,4)</math> and moves until it hits one of the coordinate axes for the first time. When the particle is at the point <math>(a,b)</math>, it moves at random to one of the points <math>(a-1,b)</math>, <math>(a,b-1)</math>, or <math>(a-1,b-1)</math>, each with probability <math>\tfrac{1}{3}</math>, independently of its previous moves. The probability that it will hit the coordinate axes at <math>(0,0)</math> is <math>\tfrac{m}{3^n}</math>, where <math>m</math> and <math>n</math> are positive integers, and <math>m</math> is not divisible by <math>3</math>. Find <math>m + n</math>. |
[[2019 AIME I Problems/Problem 5 | Solution]] | [[2019 AIME I Problems/Problem 5 | Solution]] | ||
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==Problem 6== | ==Problem 6== | ||
− | In convex quadrilateral <math>KLMN</math> side <math>\overline{MN}</math> is perpendicular to diagonal <math>\overline{KM}</math>, side <math>\overline{KL}</math> is perpendicular to diagonal <math>\overline{LN}</math>, <math>MN = 65</math>, and <math>KL = 28</math>. The line through <math>L</math> perpendicular to side <math>\overline{KN}</math> intersects diagonal <math>\overline{KM}</math> at <math>O</math> with <math>KO = 8</math>. Find <math>MO</math>. | + | In convex quadrilateral <math>KLMN</math>, side <math>\overline{MN}</math> is perpendicular to diagonal <math>\overline{KM}</math>, side <math>\overline{KL}</math> is perpendicular to diagonal <math>\overline{LN}</math>, <math>MN = 65</math>, and <math>KL = 28</math>. The line through <math>L</math> perpendicular to side <math>\overline{KN}</math> intersects diagonal <math>\overline{KM}</math> at <math>O</math> with <math>KO = 8</math>. Find <math>MO</math>. |
[[2019 AIME I Problems/Problem 6 | Solution]] | [[2019 AIME I Problems/Problem 6 | Solution]] | ||
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==Problem 9== | ==Problem 9== | ||
− | Let <math>\tau(n)</math> denote the number of positive integer divisors of <math>n</math> | + | Let <math>\tau(n)</math> denote the number of positive integer divisors of <math>n</math>. Find the sum of the six least positive integers <math>n</math> that are solutions to <math>\tau (n) + \tau (n+1) = 7</math>. |
[[2019 AIME I Problems/Problem 9 | Solution]] | [[2019 AIME I Problems/Problem 9 | Solution]] | ||
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For distinct complex numbers <math>z_1,z_2,\dots,z_{673}</math>, the polynomial | For distinct complex numbers <math>z_1,z_2,\dots,z_{673}</math>, the polynomial | ||
<cmath> (x-z_1)^3(x-z_2)^3 \cdots (x-z_{673})^3 </cmath> | <cmath> (x-z_1)^3(x-z_2)^3 \cdots (x-z_{673})^3 </cmath> | ||
− | can be expressed as <math>x^{2019} + 20x^{2018} + 19x^{2017}+g(x)</math>, where <math>g(x)</math> is a polynomial with complex coefficients and with degree at most <math>2016</math>. The | + | can be expressed as <math>x^{2019} + 20x^{2018} + 19x^{2017}+g(x)</math>, where <math>g(x)</math> is a polynomial with complex coefficients and with degree at most <math>2016</math>. The value of |
− | < | + | <cmath> \left| \sum_{1 \le j <k \le 673} z_jz_k \right| </cmath> |
can be expressed in the form <math>\tfrac{m}{n}</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m+n</math>. | can be expressed in the form <math>\tfrac{m}{n}</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m+n</math>. | ||
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==Problem 13== | ==Problem 13== | ||
− | <math> | + | Triangle <math>ABC</math> has side lengths <math>AB=4</math>, <math>BC=5</math>, and <math>CA=6</math>. Points <math>D</math> and <math>E</math> are on ray <math>AB</math> with <math>AB<AD<AE</math>. The point <math>F \neq C</math> is a point of intersection of the circumcircles of <math>\triangle ACD</math> and <math>\triangle EBC</math> satisfying <math>DF=2</math> and <math>EF=7</math>. Then <math>BE</math> can be expressed as <math>\tfrac{a+b\sqrt{c}}{d}</math>, where <math>a</math>, <math>b</math>, <math>c</math>, and <math>d</math> are positive integers such that <math>a</math> and <math>d</math> are relatively prime, and <math>c</math> is not divisible by the square of any prime. Find <math>a+b+c+d</math>. |
[[2019 AIME I Problems/Problem 13 | Solution]] | [[2019 AIME I Problems/Problem 13 | Solution]] |
Latest revision as of 05:57, 4 January 2021
2019 AIME I (Answer Key) | AoPS Contest Collections • PDF | ||
Instructions
| ||
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Contents
[hide]Problem 1
Consider the integer Find the sum of the digits of
.
Problem 2
Jenn randomly chooses a number from
. Bela then randomly chooses a number
from
distinct from
. The value of
is at least
with a probability that can be expressed in the form
, where
and
are relatively prime positive integers. Find
.
Problem 3
In ,
,
, and
. Points
and
lie on
, points
and
lie on
, and points
and
lie on
, with
. Find the area of hexagon
.
Problem 4
A soccer team has available players. A fixed set of
players starts the game, while the other
are available as substitutes. During the game, the coach may make as many as
substitutions, where any one of the
players in the game is replaced by one of the substitutes. No player removed from the game may reenter the game, although a substitute entering the game may be replaced later. No two substitutions can happen at the same time. The players involved and the order of the substitutions matter. Let
be the number of ways the coach can make substitutions during the game (including the possibility of making no substitutions). Find the remainder when
is divided by
.
Problem 5
A moving particle starts at the point and moves until it hits one of the coordinate axes for the first time. When the particle is at the point
, it moves at random to one of the points
,
, or
, each with probability
, independently of its previous moves. The probability that it will hit the coordinate axes at
is
, where
and
are positive integers, and
is not divisible by
. Find
.
Problem 6
In convex quadrilateral , side
is perpendicular to diagonal
, side
is perpendicular to diagonal
,
, and
. The line through
perpendicular to side
intersects diagonal
at
with
. Find
.
Problem 7
There are positive integers and
that satisfy the system of equations
Let
be the number of (not necessarily distinct) prime factors in the prime factorization of
, and let
be the number of (not necessarily distinct) prime factors in the prime factorization of
. Find
.
Problem 8
Let be a real number such that
. Then
where
and
are relatively prime positive integers. Find
.
Problem 9
Let denote the number of positive integer divisors of
. Find the sum of the six least positive integers
that are solutions to
.
Problem 10
For distinct complex numbers , the polynomial
can be expressed as
, where
is a polynomial with complex coefficients and with degree at most
. The value of
can be expressed in the form
, where
and
are relatively prime positive integers. Find
.
Problem 11
In , the sides have integer lengths and
. Circle
has its center at the incenter of
. An excircle of
is a circle in the exterior of
that is tangent to one side of the triangle and tangent to the extensions of the other two sides. Suppose that the excircle tangent to
is internally tangent to
, and the other two excircles are both externally tangent to
. Find the minimum possible value of the perimeter of
.
Problem 12
Given , there are complex numbers
with the property that
,
, and
are the vertices of a right triangle in the complex plane with a right angle at
. There are positive integers
and
such that one such value of
is
. Find
.
Problem 13
Triangle has side lengths
,
, and
. Points
and
are on ray
with
. The point
is a point of intersection of the circumcircles of
and
satisfying
and
. Then
can be expressed as
, where
,
,
, and
are positive integers such that
and
are relatively prime, and
is not divisible by the square of any prime. Find
.
Problem 14
Find the least odd prime factor of .
Problem 15
Let be a chord of a circle
, and let
be a point on the chord
. Circle
passes through
and
and is internally tangent to
. Circle
passes through
and
and is internally tangent to
. Circles
and
intersect at points
and
. Line
intersects
at
and
. Assume that
,
,
, and
, where
and
are relatively prime positive integers. Find
.
2019 AIME I (Problems • Answer Key • Resources) | ||
Preceded by 2018 AIME II Problems |
Followed by 2019 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.