# Difference between revisions of "User:Rowechen"

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==Problem 12== | ==Problem 12== | ||

<math>ABC</math> is a scalene triangle. The circle with diameter <math>AB</math> intersects <math>BC</math> at <math>D</math>, and <math>E</math> is the foot of the altitude from <math>C</math>. <math>P</math> is the intersection of <math>AD</math> and <math>CE</math>. Given that <math>AP = 136</math>, <math>BP = 80</math>, and <math>CP = 26</math>, determine the circumradius of <math>ABC</math>. | <math>ABC</math> is a scalene triangle. The circle with diameter <math>AB</math> intersects <math>BC</math> at <math>D</math>, and <math>E</math> is the foot of the altitude from <math>C</math>. <math>P</math> is the intersection of <math>AD</math> and <math>CE</math>. Given that <math>AP = 136</math>, <math>BP = 80</math>, and <math>CP = 26</math>, determine the circumradius of <math>ABC</math>. | ||

+ | |||

+ | ==Problem 13== | ||

+ | Point <math>D</math> lies on side <math>\overline{BC}</math> of <math>\triangle ABC</math> so that <math>\overline{AD}</math> bisects <math>\angle BAC.</math> The perpendicular bisector of <math>\overline{AD}</math> intersects the bisectors of <math>\angle ABC</math> and <math>\angle ACB</math> in points <math>E</math> and <math>F,</math> respectively. Given that <math>AB=4,BC=5,</math> and <math>CA=6,</math> the area of <math>\triangle AEF</math> can be written as <math>\tfrac{m\sqrt{n}}p,</math> where <math>m</math> and <math>p</math> are relatively prime positive integers, and <math>n</math> is a positive integer not divisible by the square of any prime. Find <math>m+n+p.</math> | ||

+ | |||

+ | [[2020 AIME I Problems/Problem 13 | Solution]] | ||

+ | ==Problem 15== | ||

+ | Let <math>\triangle ABC</math> be an acute triangle with circumcircle <math>\omega,</math> and let <math>H</math> be the intersection of the altitudes of <math>\triangle ABC.</math> Suppose the tangent to the circumcircle of <math>\triangle HBC</math> at <math>H</math> intersects <math>\omega</math> at points <math>X</math> and <math>Y</math> with <math>HA=3,HX=2,</math> and <math>HY=6.</math> The area of <math>\triangle ABC</math> can be written as <math>m\sqrt{n},</math> where <math>m</math> and <math>n</math> are positive integers, and <math>n</math> is not divisible by the square of any prime. Find <math>m+n.</math> | ||

+ | |||

+ | [[2020 AIME I Problems/Problem 15 | Solution]] | ||

+ | |||

+ | ==Problem 14== | ||

+ | Let <math>P(x)</math> be a quadratic polynomial with complex coefficients whose <math>x^2</math> coefficient is <math>1.</math> Suppose the equation <math>P(P(x))=0</math> has four distinct solutions, <math>x=3,4,a,b.</math> Find the sum of all possible values of <math>(a+b)^2.</math> | ||

+ | |||

+ | [[2020 AIME I Problems/Problem 14 | Solution]] | ||

+ | ==Problem 13== | ||

+ | For each integer <math>n\geq3</math>, let <math>f(n)</math> be the number of <math>3</math>-element subsets of the vertices of a regular <math>n</math>-gon that are the vertices of an isosceles triangle (including equilateral triangles). Find the sum of all values of <math>n</math> such that <math>f(n+1)=f(n)+78</math>. | ||

+ | |||

+ | [[2017 AIME II Problems/Problem 13 | Solution]] |

## Revision as of 18:31, 1 June 2020

Here's the AIME compilation I will be doing:

## Contents

## Problem 7

An angle is drawn on a set of equally spaced parallel lines as shown. The ratio of the area of shaded region to the area of shaded region is 11/5. Find the ratio of shaded region to the area of shaded region

## Problem 2

A 100 foot long moving walkway moves at a constant rate of 6 feet per second. Al steps onto the start of the walkway and stands. Bob steps onto the start of the walkway two seconds later and strolls forward along the walkway at a constant rate of 2 feet per second. Two seconds after that, Cy reaches the start of the walkway and walks briskly forward beside the walkway at a constant rate of 4 feet per second. At a certain time, one of these three persons is exactly halfway between the other two. At that time, find the distance in feet between the start of the walkway and the middle person.

## Problem 3

Let be the product of the first 100 positive odd integers. Find the largest integer such that is divisible by .

## Problem 9

The value of the sum can be expressed in the form , for some relatively prime positive integers and . Compute the value of .

## Problem 8

Determine the remainder obtained when the expression is divided by .

## Problem 9

Let where and . Determine the remainder obtained when is divided by .

## Problem 11

A sequence is defined as follows and, for all positive integers Given that and find the remainder when is divided by 1000.

## Problem 10

, and are positive real numbers such that Compute the value of .

## Problem 11

, , and are complex numbers such that

Let , where . Determine the value of .

## Problem 12

is a scalene triangle. The circle with diameter intersects at , and is the foot of the altitude from . is the intersection of and . Given that , , and , determine the circumradius of .

## Problem 13

Point lies on side of so that bisects The perpendicular bisector of intersects the bisectors of and in points and respectively. Given that and the area of can be written as where and are relatively prime positive integers, and is a positive integer not divisible by the square of any prime. Find

## Problem 15

Let be an acute triangle with circumcircle and let be the intersection of the altitudes of Suppose the tangent to the circumcircle of at intersects at points and with and The area of can be written as where and are positive integers, and is not divisible by the square of any prime. Find

## Problem 14

Let be a quadratic polynomial with complex coefficients whose coefficient is Suppose the equation has four distinct solutions, Find the sum of all possible values of

## Problem 13

For each integer , let be the number of -element subsets of the vertices of a regular -gon that are the vertices of an isosceles triangle (including equilateral triangles). Find the sum of all values of such that .