Difference between revisions of "2019 AIME I Problems"
Line 1: | Line 1: | ||
− | The | + | ==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>. | ||
+ | |||
+ | ==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>\frac{m}{n}</math> where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m+n</math>. | ||
+ | |||
+ | ==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>. | ||
+ | |||
+ | ==Problem 4== | ||
+ | |||
+ | A soccer team has 22 available players. A fixed set of 11 players starts the game, while the other 11 are available as substitutes. During the game, the coach may make as many as 3 substitutions, where any one of the 11 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 1000. | ||
+ | |||
+ | ==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>\frac{1}{3}</math>, independently of its previous moves. The probability that it will hit the coordinate axes at <math>(0,0)</math> is <math>\frac{m}{3^n}</math>, where <math>m</math> and <math>n</math> are positive integers. Find <math>m + n</math>. | ||
+ | |||
+ | ==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>. | ||
+ | |||
+ | ==Problem 7== | ||
+ | |||
+ | There are positive integers <math>x</math> and <math>y</math> that satisfy the system of equations | ||
+ | \begin{align*} | ||
+ | \log_{10} x + 2 \log_{10} (\gcd(x,y)) &= 60 \ \log_{10} y + 2 \log_{10} (\text{lcm}(x,y)) &= 570. | ||
+ | \end{align*}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>. | ||
+ | |||
+ | ==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> |
Revision as of 17:48, 14 March 2019
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 22 available players. A fixed set of 11 players starts the game, while the other 11 are available as substitutes. During the game, the coach may make as many as 3 substitutions, where any one of the 11 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 1000.
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. 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
Problem 8
Let be a real number such that . Then where and are relatively prime positive integers. Find