Difference between revisions of "2019 AIME I Problems"
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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 sum |
− | < | + | <math> \left| \sum_{1 \le j <k \le 673} z_jz_k \right| </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>. | 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 ABC</math> has side lengths <math>AB=4</math>, <math>AC=6</math>, and <math>BC=5</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 = \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]] |
Revision as of 13:45, 14 October 2020
2019 AIME I (Answer Key) | AoPS Contest Collections | ||
Instructions
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Contents
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 such that and are relatively prime. 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 (including and ). 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 sum 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
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 , 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) | ||
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