# 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 1== | ||

+ | The 8 eyelets for the lace of a sneaker all lie on a rectangle, four equally spaced on each of the longer sides. The rectangle has a width of 50 mm and a length of 80 mm. There is one eyelet at each vertex of the rectangle. The lace itself must pass between the vertex eyelets along a width side of the rectangle and then crisscross between successive eyelets until it reaches the two eyelets at the other width side of the rectangle as shown. After passing through these final eyelets, each of the ends of the lace must extend at least 200 mm farther to allow a knot to be tied. Find the minimum length of the lace in millimeters. | ||

− | == | + | <asy> |

− | + | size(200); | |

+ | defaultpen(linewidth(0.7)); | ||

+ | path laceL=(-20,-30)..tension 0.75 ..(-90,-135)..(-102,-147)..(-152,-150)..tension 2 ..(-155,-140)..(-135,-40)..(-50,-4)..tension 0.8 ..origin; | ||

+ | path laceR=reflect((75,0),(75,-240))*laceL; | ||

+ | draw(origin--(0,-240)--(150,-240)--(150,0)--cycle,gray); | ||

+ | for(int i=0;i<=3;i=i+1) | ||

+ | { | ||

+ | path circ1=circle((0,-80*i),5),circ2=circle((150,-80*i),5); | ||

+ | unfill(circ1); draw(circ1); | ||

+ | unfill(circ2); draw(circ2); | ||

+ | } | ||

+ | draw(laceL--(150,-80)--(0,-160)--(150,-240)--(0,-240)--(150,-160)--(0,-80)--(150,0)^^laceR,linewidth(1));</asy> | ||

+ | [[2014 AIME I Problems/Problem 1|Solution]] | ||

− | + | Compute, to the nearest integer, the area of the region enclosed by the graph of | |

− | |||

− | + | <cmath>13x^2-20xy+52y^2-10x+52y=563</cmath> | |

− | + | ==Problem 6== | |

− | ==Problem | + | A flat board has a circular hole with radius <math>1</math> and a circular hole with radius <math>2</math> such that the distance between the centers of the two holes is <math>7</math>. Two spheres with equal radii sit in the two holes such that the spheres are tangent to each other. The square of the radius of the spheres is <math>\tfrac{m}{n}</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m+n</math>. |

− | + | [[2020 AIME I Problems/Problem 6 | Solution]] | |

+ | == Problem 9 == | ||

+ | Let <math>x</math> and <math>y</math> be real numbers such that <math>\frac{\sin x}{\sin y} = 3</math> and <math>\frac{\cos x}{\cos y} = \frac12</math>. The value of <math>\frac{\sin 2x}{\sin 2y} + \frac{\cos 2x}{\cos 2y}</math> can be expressed in the form <math>\frac pq</math>, where <math>p</math> and <math>q</math> are relatively prime positive integers. Find <math>p+q</math>. | ||

− | [[ | + | [[2012 AIME II Problems/Problem 9|Solution]] |

− | == Problem | + | ==Problem 9== |

− | Let <math> | + | Let <math>x_1< x_2 < x_3</math> be the three real roots of the equation <math>\sqrt{2014} x^3 - 4029x^2 + 2 = 0</math>. Find <math>x_2(x_1+x_3)</math>. |

− | [[ | + | [[2014 AIME I Problems/Problem 9|Solution]] |

− | == Problem | + | ==Problem 9== |

− | + | Let <math>S</math> be the set of all ordered triple of integers <math>(a_1,a_2,a_3)</math> with <math>1 \le a_1,a_2,a_3 \le 10</math>. Each ordered triple in <math>S</math> generates a sequence according to the rule <math>a_n=a_{n-1}\cdot | a_{n-2}-a_{n-3} |</math> for all <math>n\ge 4</math>. Find the number of such sequences for which <math>a_n=0</math> for some <math>n</math>. | |

− | + | [[2015 AIME I Problems/Problem 9|Solution]] | |

− | + | ------------- | |

− | == Problem | + | ==Problem 12== |

− | + | Suppose that the angles of <math>\triangle ABC</math> satisfy <math>\cos(3A)+\cos(3B)+\cos(3C)=1</math>. Two sides of the triangle have lengths 10 and 13. There is a positive integer <math>m</math> so that the maximum possible length for the remaining side of <math>\triangle ABC</math> is <math>\sqrt{m}</math>. Find <math>m</math>. | |

− | < | + | [[2014 AIME II Problems/Problem 12|Solution]] |

− | + | ==Problem 11== | |

− | + | Consider arrangements of the <math>9</math> numbers <math>1, 2, 3, \dots, 9</math> in a <math>3 \times 3</math> array. For each such arrangement, let <math>a_1</math>, <math>a_2</math>, and <math>a_3</math> be the medians of the numbers in rows <math>1</math>, <math>2</math>, and <math>3</math> respectively, and let <math>m</math> be the median of <math>\{a_1, a_2, a_3\}</math>. Let <math>Q</math> be the number of arrangements for which <math>m = 5</math>. Find the remainder when <math>Q</math> is divided by <math>1000</math>. | |

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− | + | [[2017 AIME I Problems/Problem 11 | Solution]] | |

− | + | ==Problem 10== | |

− | + | The wheel shown below consists of two circles and five spokes, with a label at each point where a spoke meets a circle. A bug walks along the wheel, starting at point <math>A</math>. At every step of the process, the bug walks from one labeled point to an adjacent labeled point. Along the inner circle the bug only walks in a counterclockwise direction, and along the outer circle the bug only walks in a clockwise direction. For example, the bug could travel along the path <math>AJABCHCHIJA</math>, which has <math>10</math> steps. Let <math>n</math> be the number of paths with <math>15</math> steps that begin and end at point <math>A.</math> Find the remainder when <math>n</math> is divided by <math>1000</math>. | |

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− | + | <asy> | |

− | + | size(6cm); | |

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− | + | draw(unitcircle); | |

+ | draw(scale(2) * unitcircle); | ||

+ | for(int d = 90; d < 360 + 90; d += 72){ | ||

+ | draw(2 * dir(d) -- dir(d)); | ||

+ | } | ||

− | + | dot(1 * dir( 90), linewidth(5)); | |

+ | dot(1 * dir(162), linewidth(5)); | ||

+ | dot(1 * dir(234), linewidth(5)); | ||

+ | dot(1 * dir(306), linewidth(5)); | ||

+ | dot(1 * dir(378), linewidth(5)); | ||

+ | dot(2 * dir(378), linewidth(5)); | ||

+ | dot(2 * dir(306), linewidth(5)); | ||

+ | dot(2 * dir(234), linewidth(5)); | ||

+ | dot(2 * dir(162), linewidth(5)); | ||

+ | dot(2 * dir( 90), linewidth(5)); | ||

− | + | label("$A$", 1 * dir( 90), -dir( 90)); | |

− | + | label("$B$", 1 * dir(162), -dir(162)); | |

− | + | label("$C$", 1 * dir(234), -dir(234)); | |

+ | label("$D$", 1 * dir(306), -dir(306)); | ||

+ | label("$E$", 1 * dir(378), -dir(378)); | ||

+ | label("$F$", 2 * dir(378), dir(378)); | ||

+ | label("$G$", 2 * dir(306), dir(306)); | ||

+ | label("$H$", 2 * dir(234), dir(234)); | ||

+ | label("$I$", 2 * dir(162), dir(162)); | ||

+ | label("$J$", 2 * dir( 90), dir( 90)); | ||

+ | </asy> | ||

− | + | [[2018 AIME I Problems/Problem 10 | Solution]] | |

+ | ==Problem 11== | ||

+ | Find the least positive integer <math>n</math> such that when <math>3^n</math> is written in base <math>143</math>, its two right-most digits in base <math>143</math> are <math>01</math>. | ||

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− | + | [[2018 AIME I Problems/Problem 11 | Solution]] | |

+ | ==Problem 14== | ||

− | + | In <math>\triangle ABC</math>, <math>AB=10</math>, <math>\measuredangle A=30^{\circ}</math>, and <math>\measuredangle C=45^{\circ}</math>. Let <math>H</math>, <math>D</math>, and <math>M</math> be points on line <math>\overline{BC}</math> such that <math>AH\perp BC</math>, <math>\measuredangle BAD=\measuredangle CAD</math>, and <math>BM=CM</math>. Point <math>N</math> is the midpoint of segment <math>HM</math>, and point <math>P</math> is on ray <math>AD</math> such that <math>PN\perp BC</math>. Then <math>AP^2=\frac{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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− | [[ | + | [[2014 AIME II Problems/Problem 14|Solution]] |

− | ==Problem | + | ==Problem 14== |

− | |||

− | + | For each integer <math>n \ge 2</math>, let <math>A(n)</math> be the area of the region in the coordinate plane defined by the inequalities <math>1\le x \le n</math> and <math>0\le y \le x \left\lfloor \sqrt x \right\rfloor</math>, where <math>\left\lfloor \sqrt x \right\rfloor</math> is the greatest integer not exceeding <math>\sqrt x</math>. Find the number of values of <math>n</math> with <math>2\le n \le 1000</math> for which <math>A(n)</math> is an integer. | |

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− | [[ | + | [[2015 AIME I Problems/Problem 14|Solution]] |

− | |||

− | |||

− | + | ------------- | |

− | + | ==Problem 13== | |

− | ==Problem | + | Let <math>\triangle ABC</math> have side lengths <math>AB=30</math>, <math>BC=32</math>, and <math>AC=34</math>. Point <math>X</math> lies in the interior of <math>\overline{BC}</math>, and points <math>I_1</math> and <math>I_2</math> are the incenters of <math>\triangle ABX</math> and <math>\triangle ACX</math>, respectively. Find the minimum possible area of <math>\triangle AI_1I_2</math> as <math>X</math> varies along <math>\overline{BC}</math>. |

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− | [[ | + | [[2018 AIME I Problems/Problem 13 | Solution]] |

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

− | + | David found four sticks of different lengths that can be used to form three non-congruent convex cyclic quadrilaterals, <math>A,\text{ }B,\text{ }C</math>, which can each be inscribed in a circle with radius <math>1</math>. Let <math>\varphi_A</math> denote the measure of the acute angle made by the diagonals of quadrilateral <math>A</math>, and define <math>\varphi_B</math> and <math>\varphi_C</math> similarly. Suppose that <math>\sin\varphi_A=\frac{2}{3}</math>, <math>\sin\varphi_B=\frac{3}{5}</math>, and <math>\sin\varphi_C=\frac{6}{7}</math>. All three quadrilaterals have the same area <math>K</math>, which can be written in the form <math>\dfrac{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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− | [[ | + | [[2018 AIME I Problems/Problem 15 | Solution]] |

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

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− | + | Misha rolls a standard, fair six-sided die until she rolls 1-2-3 in that order on three consecutive rolls. The probability that she will roll the die an odd number of times is <math>\dfrac{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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− | [[ | + | [[2018 AIME II Problems/Problem 13 | Solution]] |

## Revision as of 08:00, 30 May 2020

Here's the AIME compilation I will be doing:

## Contents

## Problem 1

The 8 eyelets for the lace of a sneaker all lie on a rectangle, four equally spaced on each of the longer sides. The rectangle has a width of 50 mm and a length of 80 mm. There is one eyelet at each vertex of the rectangle. The lace itself must pass between the vertex eyelets along a width side of the rectangle and then crisscross between successive eyelets until it reaches the two eyelets at the other width side of the rectangle as shown. After passing through these final eyelets, each of the ends of the lace must extend at least 200 mm farther to allow a knot to be tied. Find the minimum length of the lace in millimeters.

Compute, to the nearest integer, the area of the region enclosed by the graph of

## Problem 6

A flat board has a circular hole with radius and a circular hole with radius such that the distance between the centers of the two holes is . Two spheres with equal radii sit in the two holes such that the spheres are tangent to each other. The square of the radius of the spheres is , where and are relatively prime positive integers. Find .

## Problem 9

Let and be real numbers such that and . The value of can be expressed in the form , where and are relatively prime positive integers. Find .

## Problem 9

Let be the three real roots of the equation . Find .

## Problem 9

Let be the set of all ordered triple of integers with . Each ordered triple in generates a sequence according to the rule for all . Find the number of such sequences for which for some .

## Problem 12

Suppose that the angles of satisfy . Two sides of the triangle have lengths 10 and 13. There is a positive integer so that the maximum possible length for the remaining side of is . Find .

## Problem 11

Consider arrangements of the numbers in a array. For each such arrangement, let , , and be the medians of the numbers in rows , , and respectively, and let be the median of . Let be the number of arrangements for which . Find the remainder when is divided by .

## Problem 10

The wheel shown below consists of two circles and five spokes, with a label at each point where a spoke meets a circle. A bug walks along the wheel, starting at point . At every step of the process, the bug walks from one labeled point to an adjacent labeled point. Along the inner circle the bug only walks in a counterclockwise direction, and along the outer circle the bug only walks in a clockwise direction. For example, the bug could travel along the path , which has steps. Let be the number of paths with steps that begin and end at point Find the remainder when is divided by .

## Problem 11

Find the least positive integer such that when is written in base , its two right-most digits in base are .

## Problem 14

In , , , and . Let , , and be points on line such that , , and . Point is the midpoint of segment , and point is on ray such that . Then , where and are relatively prime positive integers. Find .

## Problem 14

For each integer , let be the area of the region in the coordinate plane defined by the inequalities and , where is the greatest integer not exceeding . Find the number of values of with for which is an integer.

## Problem 13

Let have side lengths , , and . Point lies in the interior of , and points and are the incenters of and , respectively. Find the minimum possible area of as varies along .

## Problem 15

David found four sticks of different lengths that can be used to form three non-congruent convex cyclic quadrilaterals, , which can each be inscribed in a circle with radius . Let denote the measure of the acute angle made by the diagonals of quadrilateral , and define and similarly. Suppose that , , and . All three quadrilaterals have the same area , which can be written in the form , where and are relatively prime positive integers. Find .

## Problem 13

Misha rolls a standard, fair six-sided die until she rolls 1-2-3 in that order on three consecutive rolls. The probability that she will roll the die an odd number of times is where and are relatively prime positive integers. Find .