# Difference between revisions of "User:Rowechen"

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Here's the AIME compilation I will be doing: | Here's the AIME compilation I will be doing: | ||

− | == Problem | + | == Problem 3 == |

− | + | Sarah intended to multiply a two-digit number and a three-digit number, but she left out the multiplication sign and simply placed the two-digit number to the left of the three-digit number, thereby forming a five-digit number. This number is exactly nine times the product Sarah should have obtained. What is the sum of the two-digit number and the three-digit number? | |

− | [[ | + | [[1997 AIME Problems/Problem 3|Solution]] |

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− | [[ | + | == Problem 5 == |

− | == Problem | + | For certain real values of <math>a, b, c,</math> and <math>d_{},</math> the equation <math>x^4+ax^3+bx^2+cx+d=0</math> has four non-real roots. The product of two of these roots is <math>13+i</math> and the sum of the other two roots is <math>3+4i,</math> where <math>i=\sqrt{-1}.</math> Find <math>b.</math> |

− | The | + | |

− | < | + | [[1995 AIME Problems/Problem 5|Solution]] |

− | are | + | |

+ | == Problem 4 == | ||

+ | In triangle <math>ABC</math>, angles <math>A</math> and <math>B</math> measure <math>60</math> degrees and <math>45</math> degrees, respectively. The bisector of angle <math>A</math> intersects <math>\overline{BC}</math> at <math>T</math>, and <math>AT=24</math>. The area of triangle <math>ABC</math> can be written in the form <math>a+b\sqrt{c}</math>, where <math>a</math>, <math>b</math>, and <math>c</math> are positive integers, and <math>c</math> is not divisible by the square of any prime. Find <math>a+b+c</math>. | ||

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+ | [[2001 AIME I Problems/Problem 4|Solution]] | ||

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+ | == Problem 9 == | ||

+ | A solitaire game is played as follows. Six distinct pairs of matched tiles are placed in a bag. The player randomly draws tiles one at a time from the bag and retains them, except that matching tiles are put aside as soon as they appear in the player's hand. The game ends if the player ever holds three tiles, no two of which match; otherwise the drawing continues until the bag is empty. The probability that the bag will be emptied is <math>p/q,\,</math> where <math>p\,</math> and <math>q\,</math> are relatively prime positive integers. Find <math>p+q.\,</math> | ||

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+ | [[1994 AIME Problems/Problem 9|Solution]] | ||

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== Problem 8 == | == Problem 8 == | ||

− | + | For how many ordered pairs of positive integers <math>(x,y),</math> with <math>y<x\le 100,</math> are both <math>\frac xy</math> and <math>\frac{x+1}{y+1}</math> integers? | |

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+ | [[1995 AIME Problems/Problem 8|Solution]] | ||

− | + | == Problem 9 == | |

− | == Problem | + | Triangle <math>ABC</math> is isosceles, with <math>AB=AC</math> and altitude <math>AM=11.</math> Suppose that there is a point <math>D</math> on <math>\overline{AM}</math> with <math>AD=10</math> and <math>\angle BDC=3\angle BAC.</math> Then the perimeter of <math>\triangle ABC</math> may be written in the form <math>a+\sqrt{b},</math> where <math>a</math> and <math>b</math> are integers. Find <math>a+b.</math> |

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− | [[ | + | [[Image:AIME_1995_Problem_9.png]] |

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− | [[ | + | [[1995 AIME Problems/Problem 9|Solution]] |

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== Problem 11 == | == Problem 11 == | ||

− | + | Let <math>\mathrm {P}</math> be the product of the roots of <math>z^6+z^4+z^3+z^2+1=0</math> that have a positive imaginary part, and suppose that <math>\mathrm {P}=r(\cos{\theta^{\circ}}+i\sin{\theta^{\circ}})</math>, where <math>0<r</math> and <math>0\leq \theta <360</math>. Find <math>\theta</math>. | |

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+ | [[1996 AIME Problems/Problem 11|Solution]] | ||

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+ | == Problem 13 == | ||

+ | Let <math>S</math> be the set of points in the Cartesian plane that satisfy <center><math>\Big|\big| |x|-2\big|-1\Big|+\Big|\big| |y|-2\big|-1\Big|=1.</math></center> If a model of <math>S</math> were built from wire of negligible thickness, then the total length of wire required would be <math>a\sqrt{b}</math>, where <math>a</math> and <math>b</math> are positive integers and <math>b</math> is not divisible by the square of any prime number. Find <math>a+b</math>. | ||

+ | |||

+ | [[1997 AIME Problems/Problem 13|Solution]] | ||

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

− | + | Let <math>ABC</math> be [[equilateral triangle|equilateral]], and <math>D, E,</math> and <math>F</math> be the [[midpoint]]s of <math>\overline{BC}, \overline{CA},</math> and <math>\overline{AB},</math> respectively. There exist [[point]]s <math>P, Q,</math> and <math>R</math> on <math>\overline{DE}, \overline{EF},</math> and <math>\overline{FD},</math> respectively, with the property that <math>P</math> is on <math>\overline{CQ}, Q</math> is on <math>\overline{AR},</math> and <math>R</math> is on <math>\overline{BP}.</math> The [[ratio]] of the area of triangle <math>ABC</math> to the area of triangle <math>PQR</math> is <math>a + b\sqrt {c},</math> where <math>a, b</math> and <math>c</math> are integers, and <math>c</math> is not divisible by the square of any [[prime]]. What is <math>a^{2} + b^{2} + c^{2}</math>? | |

− | [[ | + | [[1998 AIME Problems/Problem 12|Solution]] |

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− | + | == Problem 11 == | |

− | == Problem | + | Given that <math>\sum_{k=1}^{35}\sin 5k=\tan \frac mn,</math> where angles are measured in degrees, and <math>m_{}</math> and <math>n_{}</math> are relatively prime positive integers that satisfy <math>\frac mn<90,</math> find <math>m+n.</math> |

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− | [[ | + | [[1999 AIME Problems/Problem 11|Solution]] |

== Problem 14 == | == Problem 14 == | ||

− | + | Point <math>P_{}</math> is located inside triangle <math>ABC</math> so that angles <math>PAB, PBC,</math> and <math>PCA</math> are all congruent. The sides of the triangle have lengths <math>AB=13, BC=14,</math> and <math>CA=15,</math> and the tangent of angle <math>PAB</math> is <math>m/n,</math> where <math>m_{}</math> and <math>n_{}</math> are relatively prime positive integers. Find <math>m+n.</math> | |

− | [[ | + | [[1999 AIME Problems/Problem 14|Solution]] |

− | == Problem | + | == Problem 14 == |

− | In triangle <math>ABC</math> | + | In triangle <math>ABC,</math> it is given that angles <math>B</math> and <math>C</math> are congruent. Points <math>P</math> and <math>Q</math> lie on <math>\overline{AC}</math> and <math>\overline{AB},</math> respectively, so that <math>AP = PQ = QB = BC.</math> Angle <math>ACB</math> is <math>r</math> times as large as angle <math>APQ,</math> where <math>r</math> is a positive real number. Find the greatest integer that does not exceed <math>1000r</math>. |

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− | [[ | + | [[2000 AIME I Problems/Problem 14|Solution]] |

== Problem 15 == | == Problem 15 == | ||

− | + | A stack of <math>2000</math> cards is labelled with the integers from <math>1</math> to <math>2000,</math> with different integers on different cards. The cards in the stack are not in numerical order. The top card is removed from the stack and placed on the table, and the next card is moved to the bottom of the stack. The new top card is removed from the stack and placed on the table, to the right of the card already there, and the next card in the stack is moved to the bottom of the stack. The process - placing the top card to the right of the cards already on the table and moving the next card in the stack to the bottom of the stack - is repeated until all cards are on the table. It is found that, reading from left to right, the labels on the cards are now in ascending order: <math>1,2,3,\ldots,1999,2000.</math> In the original stack of cards, how many cards were above the card labeled <math>1999</math>? | |

+ | [[2000 AIME I Problems/Problem 15|Solution]] | ||

+ | == Problem 14 == | ||

+ | Every positive integer <math>k</math> has a unique factorial base expansion <math>(f_1,f_2,f_3,\ldots,f_m)</math>, meaning that <math>k=1!\cdot f_1+2!\cdot f_2+3!\cdot f_3+\cdots+m!\cdot f_m</math>, where each <math>f_i</math> is an integer, <math>0\le f_i\le i</math>, and <math>0<f_m</math>. Given that <math>(f_1,f_2,f_3,\ldots,f_j)</math> is the factorial base expansion of <math>16!-32!+48!-64!+\cdots+1968!-1984!+2000!</math>, find the value of <math>f_1-f_2+f_3-f_4+\cdots+(-1)^{j+1}f_j</math>. | ||

− | [[ | + | [[2000 AIME II Problems/Problem 14|Solution]] |

== Problem 13 == | == Problem 13 == | ||

− | + | In a certain circle, the chord of a <math>d</math>-degree arc is 22 centimeters long, and the chord of a <math>2d</math>-degree arc is 20 centimeters longer than the chord of a <math>3d</math>-degree arc, where <math>d < 120.</math> The length of the chord of a <math>3d</math>-degree arc is <math>- m + \sqrt {n}</math> centimeters, where <math>m</math> and <math>n</math> are positive integers. Find <math>m + n.</math> | |

− | [[ | + | [[2001 AIME I Problems/Problem 13|Solution]] |

## Revision as of 13:35, 26 May 2020

Hey how did you get to this page? If you aren't me then I have to say hello. If you are me then I must be pretty conceited to waste my time looking at my own page. If you aren't me, seriously, how did you get to this page? This is pretty cool. Well, nice meeting you! I'm going to stop wasting my time typing this up and do some math. Gtg. Bye.

Here's the AIME compilation I will be doing:

## Contents

## Problem 3

Sarah intended to multiply a two-digit number and a three-digit number, but she left out the multiplication sign and simply placed the two-digit number to the left of the three-digit number, thereby forming a five-digit number. This number is exactly nine times the product Sarah should have obtained. What is the sum of the two-digit number and the three-digit number?

## Problem 5

For certain real values of and the equation has four non-real roots. The product of two of these roots is and the sum of the other two roots is where Find

## Problem 4

In triangle , angles and measure degrees and degrees, respectively. The bisector of angle intersects at , and . The area of triangle can be written in the form , where , , and are positive integers, and is not divisible by the square of any prime. Find .

## Problem 9

A solitaire game is played as follows. Six distinct pairs of matched tiles are placed in a bag. The player randomly draws tiles one at a time from the bag and retains them, except that matching tiles are put aside as soon as they appear in the player's hand. The game ends if the player ever holds three tiles, no two of which match; otherwise the drawing continues until the bag is empty. The probability that the bag will be emptied is where and are relatively prime positive integers. Find

## Problem 8

For how many ordered pairs of positive integers with are both and integers?

## Problem 9

Triangle is isosceles, with and altitude Suppose that there is a point on with and Then the perimeter of may be written in the form where and are integers. Find

## Problem 11

Let be the product of the roots of that have a positive imaginary part, and suppose that , where and . Find .

## Problem 13

Let be the set of points in the Cartesian plane that satisfy

If a model of were built from wire of negligible thickness, then the total length of wire required would be , where and are positive integers and is not divisible by the square of any prime number. Find .

## Problem 12

Let be equilateral, and and be the midpoints of and respectively. There exist points and on and respectively, with the property that is on is on and is on The ratio of the area of triangle to the area of triangle is where and are integers, and is not divisible by the square of any prime. What is ?

## Problem 11

Given that where angles are measured in degrees, and and are relatively prime positive integers that satisfy find

## Problem 14

Point is located inside triangle so that angles and are all congruent. The sides of the triangle have lengths and and the tangent of angle is where and are relatively prime positive integers. Find

## Problem 14

In triangle it is given that angles and are congruent. Points and lie on and respectively, so that Angle is times as large as angle where is a positive real number. Find the greatest integer that does not exceed .

## Problem 15

A stack of cards is labelled with the integers from to with different integers on different cards. The cards in the stack are not in numerical order. The top card is removed from the stack and placed on the table, and the next card is moved to the bottom of the stack. The new top card is removed from the stack and placed on the table, to the right of the card already there, and the next card in the stack is moved to the bottom of the stack. The process - placing the top card to the right of the cards already on the table and moving the next card in the stack to the bottom of the stack - is repeated until all cards are on the table. It is found that, reading from left to right, the labels on the cards are now in ascending order: In the original stack of cards, how many cards were above the card labeled ?

## Problem 14

Every positive integer has a unique factorial base expansion , meaning that , where each is an integer, , and . Given that is the factorial base expansion of , find the value of .

## Problem 13

In a certain circle, the chord of a -degree arc is 22 centimeters long, and the chord of a -degree arc is 20 centimeters longer than the chord of a -degree arc, where The length of the chord of a -degree arc is centimeters, where and are positive integers. Find