Difference between revisions of "2019 AIME I Problems"
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− | + | {{AIME Problems|year=2019|n=I}} | |
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+ | ==Problem 1== | ||
+ | |||
+ | Consider the integer <cmath>N = 9 + 99 + 999 + 9999 + \cdots + \underbrace{99\ldots 99}_\text{321 digits}.</cmath>Find the sum of the digits of <math>N</math>. | ||
+ | |||
+ | [[2019 AIME I Problems/Problem 1 | Solution]] | ||
+ | |||
+ | ==Problem 2== | ||
+ | |||
+ | Jenn randomly chooses a number <math>J</math> from <math>1, 2, 3,\ldots, 19, 20</math>. Bela then randomly chooses a number <math>B</math> from <math>1, 2, 3,\ldots, 19, 20</math> distinct from <math>J</math>. The value of <math>B - J</math> is at least <math>2</math> with a probability that 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>. | ||
+ | |||
+ | [[2019 AIME I Problems/Problem 2 | Solution]] | ||
+ | |||
+ | ==Problem 3== | ||
+ | |||
+ | In <math>\triangle PQR</math>, <math>PR=15</math>, <math>QR=20</math>, and <math>PQ=25</math>. Points <math>A</math> and <math>B</math> lie on <math>\overline{PQ}</math>, points <math>C</math> and <math>D</math> lie on <math>\overline{QR}</math>, and points <math>E</math> and <math>F</math> lie on <math>\overline{PR}</math>, with <math>PA=QB=QC=RD=RE=PF=5</math>. Find the area of hexagon <math>ABCDEF</math>. | ||
+ | |||
+ | [[2019 AIME I Problems/Problem 3 | Solution]] | ||
+ | |||
+ | ==Problem 4== | ||
+ | |||
+ | A soccer team has <math>22</math> available players. A fixed set of <math>11</math> players starts the game, while the other <math>11</math> are available as substitutes. During the game, the coach may make as many as <math>3</math> substitutions, where any one of the <math>11</math> 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 <math>n</math> be the number of ways the coach can make substitutions during the game (including the possibility of making no substitutions). Find the remainder when <math>n</math> is divided by <math>1000</math>. | ||
+ | |||
+ | [[2019 AIME I Problems/Problem 4 | Solution]] | ||
+ | |||
+ | ==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, and <math>m</math> is not divisible by <math>3</math>. Find <math>m + n</math>. | ||
+ | |||
+ | [[2019 AIME I Problems/Problem 5 | Solution]] | ||
+ | |||
+ | ==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>. | ||
+ | |||
+ | [[2019 AIME I Problems/Problem 6 | Solution]] | ||
+ | |||
+ | ==Problem 7== | ||
+ | |||
+ | There are positive integers <math>x</math> and <math>y</math> that satisfy the system of equations | ||
+ | <cmath>\log_{10} x + 2 \log_{10} (\gcd(x,y)) = 60</cmath> <cmath>\log_{10} y + 2 \log_{10} (\text{lcm}(x,y)) = 570.</cmath> | ||
+ | Let <math>m</math> be the number of (not necessarily distinct) prime factors in the prime factorization of <math>x</math>, and let <math>n</math> be the number of (not necessarily distinct) prime factors in the prime factorization of <math>y</math>. Find <math>3m+2n</math>. | ||
+ | |||
+ | [[2019 AIME I Problems/Problem 7 | Solution]] | ||
+ | |||
+ | ==Problem 8== | ||
+ | |||
+ | Let <math>x</math> be a real number such that <math>\sin^{10}x+\cos^{10} x = \tfrac{11}{36}</math>. Then <math>\sin^{12}x+\cos^{12} x = \tfrac{m}{n}</math> where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m+n</math>. | ||
+ | |||
+ | [[2019 AIME I Problems/Problem 8 | Solution]] | ||
+ | |||
+ | ==Problem 9== | ||
+ | |||
+ | 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]] | ||
+ | |||
+ | ==Problem 10== | ||
+ | |||
+ | 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> | ||
+ | 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>. | ||
+ | |||
+ | [[2019 AIME I Problems/Problem 10 | Solution]] | ||
+ | |||
+ | ==Problem 11== | ||
+ | |||
+ | In <math>\triangle ABC</math>, the sides have integer lengths and <math>AB=AC</math>. Circle <math>\omega</math> has its center at the incenter of <math>\triangle ABC</math>. An ''excircle'' of <math>\triangle ABC</math> is a circle in the exterior of <math>\triangle ABC</math> 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 <math>\overline{BC}</math> is internally tangent to <math>\omega</math>, and the other two excircles are both externally tangent to <math>\omega</math>. Find the minimum possible value of the perimeter of <math>\triangle ABC</math>. | ||
+ | |||
+ | [[2019 AIME I Problems/Problem 11 | Solution]] | ||
+ | |||
+ | ==Problem 12== | ||
+ | |||
+ | Given <math>f(z) = z^2-19z</math>, there are complex numbers <math>z</math> with the property that <math>z</math>, <math>f(z)</math>, and <math>f(f(z))</math> are the vertices of a right triangle in the complex plane with a right angle at <math>f(z)</math>. There are positive integers <math>m</math> and <math>n</math> such that one such value of <math>z</math> is <math>m+\sqrt{n}+11i</math>. Find <math>m+n</math>. | ||
+ | |||
+ | [[2019 AIME I Problems/Problem 12 | Solution]] | ||
+ | |||
+ | ==Problem 13== | ||
+ | |||
+ | 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]] | ||
+ | |||
+ | ==Problem 14== | ||
+ | |||
+ | Find the least odd prime factor of <math>2019^8 + 1</math>. | ||
+ | |||
+ | [[2019 AIME I Problems/Problem 14 | Solution]] | ||
+ | |||
+ | ==Problem 15== | ||
+ | |||
+ | Let <math>\overline{AB}</math> be a chord of a circle <math>\omega</math>, and let <math>P</math> be a point on the chord <math>\overline{AB}</math>. Circle <math>\omega_1</math> passes through <math>A</math> and <math>P</math> and is internally tangent to <math>\omega</math>. Circle <math>\omega_2</math> passes through <math>B</math> and <math>P</math> and is internally tangent to <math>\omega</math>. Circles <math>\omega_1</math> and <math>\omega_2</math> intersect at points <math>P</math> and <math>Q</math>. Line <math>PQ</math> intersects <math>\omega</math> at <math>X</math> and <math>Y</math>. Assume that <math>AP=5</math>, <math>PB=3</math>, <math>XY=11</math>, and <math>PQ^2 = \tfrac{m}{n}</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m+n</math>. | ||
+ | |||
+ | [[2019 AIME I Problems/Problem 15 | Solution]] | ||
+ | |||
+ | {{AIME box|year=2019|n=I|before=[[2018 AIME II Problems]]|after=[[2019 AIME II Problems]]}} | ||
+ | {{MAA Notice}} |
Latest revision as of 05:57, 4 January 2021
2019 AIME I (Answer Key) | AoPS Contest Collections • PDF | ||
Instructions
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1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 |
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 | |
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All AIME Problems and Solutions |
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