# Difference between revisions of "User:Rowechen"

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== Problem 3 == | == Problem 3 == | ||

− | + | Let <math>P_1^{}</math> be a regular <math>r~\mbox{gon}</math> and <math>P_2^{}</math> be a regular <math>s~\mbox{gon}</math> <math>(r\geq s\geq 3)</math> such that each interior angle of <math>P_1^{}</math> is <math>\frac{59}{58}</math> as large as each interior angle of <math>P_2^{}</math>. What's the largest possible value of <math>s_{}^{}</math>? | |

− | [[ | + | [[1990 AIME Problems/Problem 3|Solution]] |

+ | |||

+ | == Problem 5 == | ||

+ | Given a rational number, write it as a fraction in lowest terms and calculate the product of the resulting numerator and denominator. For how many rational numbers between 0 and 1 will <math>20_{}^{}!</math> be the resulting product? | ||

+ | |||

+ | [[1991 AIME Problems/Problem 5|Solution]] | ||

== Problem 4 == | == Problem 4 == | ||

− | + | In Pascal's Triangle, each entry is the sum of the two entries above it. In which row of Pascal's Triangle do three consecutive entries occur that are in the ratio <math>3: 4: 5</math>? | |

− | [[ | + | [[1992 AIME Problems/Problem 4|Solution]] |

− | == Problem | + | == Problem 9 == |

− | + | Suppose that <math>\sec x+\tan x=\frac{22}7</math> and that <math>\csc x+\cot x=\frac mn,</math> where <math>\frac mn</math> is in lowest terms. Find <math>m+n^{}_{}.</math> | |

− | [[ | + | [[1991 AIME Problems/Problem 9|Solution]] |

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

− | + | For any sequence of real numbers <math>A=(a_1,a_2,a_3,\ldots)</math>, define <math>\Delta A^{}_{}</math> to be the sequence <math>(a_2-a_1,a_3-a_2,a_4-a_3,\ldots)</math>, whose <math>n^{th}</math> term is <math>a_{n+1}-a_n^{}</math>. Suppose that all of the terms of the sequence <math>\Delta(\Delta A^{}_{})</math> are <math>1^{}_{}</math>, and that <math>a_{19}=a_{92}^{}=0</math>. Find <math>a_1^{}</math>. | |

+ | |||

+ | [[1992 AIME Problems/Problem 8|Solution]] | ||

== Problem 7 == | == Problem 7 == | ||

− | + | Three numbers, <math>a_1\,</math>, <math>a_2\,</math>, <math>a_3\,</math>, are drawn randomly and without replacement from the set <math>\{1, 2, 3, \dots, 1000\}\,</math>. Three other numbers, <math>b_1\,</math>, <math>b_2\,</math>, <math>b_3\,</math>, are then drawn randomly and without replacement from the remaining set of 997 numbers. Let <math>p\,</math> be the probability that, after a suitable rotation, a brick of dimensions <math>a_1 \times a_2 \times a_3\,</math> can be enclosed in a box of dimensions <math>b_1 \times b_2 \times b_3\,</math>, with the sides of the brick parallel to the sides of the box. If <math>p\,</math> is written as a fraction in lowest terms, what is the sum of the numerator and denominator? | |

− | [[ | + | [[1993 AIME Problems/Problem 7|Solution]] |

− | == Problem | + | == Problem 12 == |

− | + | Let <math>ABCD^{}_{}</math> be a tetrahedron with <math>AB=41^{}_{}</math>, <math>AC=7^{}_{}</math>, <math>AD=18^{}_{}</math>, <math>BC=36^{}_{}</math>, <math>BD=27^{}_{}</math>, and <math>CD=13^{}_{}</math>, as shown in the figure. Let <math>d^{}_{}</math> be the distance between the midpoints of edges <math>AB^{}_{}</math> and <math>CD^{}_{}</math>. Find <math>d^{2}_{}</math>. | |

− | [[ | + | [[Image:AIME_1989_Problem_12.png]] |

− | == Problem | + | [[1989 AIME Problems/Problem 12|Solution]] |

− | + | == Problem 11 == | |

+ | Twelve congruent disks are placed on a circle <math>C^{}_{}</math> of radius 1 in such a way that the twelve disks cover <math>C^{}_{}</math>, no two of the disks overlap, and so that each of the twelve disks is tangent to its two neighbors. The resulting arrangement of disks is shown in the figure below. The sum of the areas of the twelve disks can be written in the form <math>\pi(a-b\sqrt{c})</math>, where <math>a,b,c^{}_{}</math> are positive integers and <math>c^{}_{}</math> is not divisible by the square of any prime. Find <math>a+b+c^{}_{}</math>. | ||

− | + | <asy> | |

− | + | unitsize(100); | |

+ | draw(Circle((0,0),1)); | ||

+ | dot((0,0)); | ||

+ | draw((0,0)--(1,0)); | ||

+ | label("$1$", (0.5,0), S); | ||

− | + | for (int i=0; i<12; ++i) | |

+ | { | ||

+ | dot((cos(i*pi/6), sin(i*pi/6))); | ||

+ | } | ||

− | + | for (int a=1; a<24; a+=2) | |

+ | { | ||

+ | dot(((1/cos(pi/12))*cos(a*pi/12), (1/cos(pi/12))*sin(a*pi/12))); | ||

+ | draw(((1/cos(pi/12))*cos(a*pi/12), (1/cos(pi/12))*sin(a*pi/12))--((1/cos(pi/12))*cos((a+2)*pi/12), (1/cos(pi/12))*sin((a+2)*pi/12))); | ||

+ | draw(Circle(((1/cos(pi/12))*cos(a*pi/12), (1/cos(pi/12))*sin(a*pi/12)), tan(pi/12))); | ||

+ | } | ||

+ | </asy> | ||

+ | [[1991 AIME Problems/Problem 11|Solution]] | ||

+ | == Problem 12 == | ||

+ | Rhombus <math>PQRS^{}_{}</math> is inscribed in rectangle <math>ABCD^{}_{}</math> so that vertices <math>P^{}_{}</math>, <math>Q^{}_{}</math>, <math>R^{}_{}</math>, and <math>S^{}_{}</math> are interior points on sides <math>\overline{AB}</math>, <math>\overline{BC}</math>, <math>\overline{CD}</math>, and <math>\overline{DA}</math>, respectively. It is given that <math>PB^{}_{}=15</math>, <math>BQ^{}_{}=20</math>, <math>PR^{}_{}=30</math>, and <math>QS^{}_{}=40</math>. Let <math>m/n^{}_{}</math>, in lowest terms, denote the perimeter of <math>ABCD^{}_{}</math>. Find <math>m+n^{}_{}</math>. | ||

+ | |||

+ | [[1991 AIME Problems/Problem 12|Solution]] | ||

== Problem 10 == | == Problem 10 == | ||

− | + | Euler's formula states that for a convex polyhedron with <math>V\,</math> vertices, <math>E\,</math> edges, and <math>F\,</math> faces, <math>V-E+F=2\,</math>. A particular convex polyhedron has 32 faces, each of which is either a triangle or a pentagon. At each of its <math>V\,</math> vertices, <math>T\,</math> triangular faces and <math>P^{}_{}</math> pentagonal faces meet. What is the value of <math>100P+10T+V\,</math>? | |

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− | |||

+ | [[1993 AIME Problems/Problem 10|Solution]] | ||

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

− | + | Let <math>S^{}_{}</math> be a subset of <math>\{1,2,3^{}_{},\ldots,1989\}</math> such that no two members of <math>S^{}_{}</math> differ by <math>4^{}_{}</math> or <math>7^{}_{}</math>. What is the largest number of elements <math>S^{}_{}</math> can have? | |

− | |||

− | |||

− | == Problem | + | [[1989 AIME Problems/Problem 13|Solution]] |

− | + | == Problem 14 == | |

+ | Given a positive integer <math>n^{}_{}</math>, it can be shown that every complex number of the form <math>r+si^{}_{}</math>, where <math>r^{}_{}</math> and <math>s^{}_{}</math> are integers, can be uniquely expressed in the base <math>-n+i^{}_{}</math> using the integers <math>1,2^{}_{},\ldots,n^2</math> as digits. That is, the equation | ||

+ | <center><math>r+si=a_m(-n+i)^m+a_{m-1}(-n+i)^{m-1}+\cdots +a_1(-n+i)+a_0</math></center> | ||

+ | is true for a unique choice of non-negative integer <math>m^{}_{}</math> and digits <math>a_0,a_1^{},\ldots,a_m</math> chosen from the set <math>\{0^{}_{},1,2,\ldots,n^2\}</math>, with <math>a_m\ne 0^{}){}</math>. We write | ||

+ | <center><math>r+si=(a_ma_{m-1}\ldots a_1a_0)_{-n+i}</math></center> | ||

+ | to denote the base <math>-n+i^{}_{}</math> expansion of <math>r+si^{}_{}</math>. There are only finitely many integers <math>k+0i^{}_{}</math> that have four-digit expansions | ||

+ | <center><math>k=(a_3a_2a_1a_0)_{-3+i^{}_{}}~~~~a_3\ne 0.</math></center> | ||

+ | Find the sum of all such <math>k^{}_{}</math>. | ||

− | + | [[1989 AIME Problems/Problem 14|Solution]] | |

− | + | == Problem 14 == | |

− | < | + | The rectangle <math>ABCD^{}_{}</math> below has dimensions <math>AB^{}_{} = 12 \sqrt{3}</math> and <math>BC^{}_{} = 13 \sqrt{3}</math>. Diagonals <math>\overline{AC}</math> and <math>\overline{BD}</math> intersect at <math>P^{}_{}</math>. If triangle <math>ABP^{}_{}</math> is cut out and removed, edges <math>\overline{AP}</math> and <math>\overline{BP}</math> are joined, and the figure is then creased along segments <math>\overline{CP}</math> and <math>\overline{DP}</math>, we obtain a triangular pyramid, all four of whose faces are isosceles triangles. Find the volume of this pyramid. |

− | < | ||

− | [[ | + | [[Image:AIME_1990_Problem_14.png]] |

+ | [[1990 AIME Problems/Problem 14|Solution]] | ||

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

− | + | Define a positive integer <math>n^{}_{}</math> to be a factorial tail if there is some positive integer <math>m^{}_{}</math> such that the decimal representation of <math>m!</math> ends with exactly <math>n</math> zeroes. How many positive integers less than <math>1992</math> are not factorial tails? | |

− | [[ | + | [[1992 AIME Problems/Problem 15|Solution]] |

+ | == Problem 14 == | ||

+ | A rectangle that is inscribed in a larger rectangle (with one vertex on each side) is called ''unstuck'' if it is possible to rotate (however slightly) the smaller rectangle about its center within the confines of the larger. Of all the rectangles that can be inscribed unstuck in a 6 by 8 rectangle, the smallest perimeter has the form <math>\sqrt{N}\,</math>, for a positive integer <math>N\,</math>. Find <math>N\,</math>. | ||

− | [[ | + | [[1993 AIME Problems/Problem 14|Solution]] |

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## Revision as of 17:43, 24 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

Let be a regular and be a regular such that each interior angle of is as large as each interior angle of . What's the largest possible value of ?

## Problem 5

Given a rational number, write it as a fraction in lowest terms and calculate the product of the resulting numerator and denominator. For how many rational numbers between 0 and 1 will be the resulting product?

## Problem 4

In Pascal's Triangle, each entry is the sum of the two entries above it. In which row of Pascal's Triangle do three consecutive entries occur that are in the ratio ?

## Problem 9

Suppose that and that where is in lowest terms. Find

## Problem 8

For any sequence of real numbers , define to be the sequence , whose term is . Suppose that all of the terms of the sequence are , and that . Find .

## Problem 7

Three numbers, , , , are drawn randomly and without replacement from the set . Three other numbers, , , , are then drawn randomly and without replacement from the remaining set of 997 numbers. Let be the probability that, after a suitable rotation, a brick of dimensions can be enclosed in a box of dimensions , with the sides of the brick parallel to the sides of the box. If is written as a fraction in lowest terms, what is the sum of the numerator and denominator?

## Problem 12

Let be a tetrahedron with , , , , , and , as shown in the figure. Let be the distance between the midpoints of edges and . Find .

## Problem 11

Twelve congruent disks are placed on a circle of radius 1 in such a way that the twelve disks cover , no two of the disks overlap, and so that each of the twelve disks is tangent to its two neighbors. The resulting arrangement of disks is shown in the figure below. The sum of the areas of the twelve disks can be written in the form , where are positive integers and is not divisible by the square of any prime. Find .

## Problem 12

Rhombus is inscribed in rectangle so that vertices , , , and are interior points on sides , , , and , respectively. It is given that , , , and . Let , in lowest terms, denote the perimeter of . Find .

## Problem 10

Euler's formula states that for a convex polyhedron with vertices, edges, and faces, . A particular convex polyhedron has 32 faces, each of which is either a triangle or a pentagon. At each of its vertices, triangular faces and pentagonal faces meet. What is the value of ?

## Problem 13

Let be a subset of such that no two members of differ by or . What is the largest number of elements can have?

## Problem 14

Given a positive integer , it can be shown that every complex number of the form , where and are integers, can be uniquely expressed in the base using the integers as digits. That is, the equation

is true for a unique choice of non-negative integer and digits chosen from the set , with . We write

to denote the base expansion of . There are only finitely many integers that have four-digit expansions

Find the sum of all such .

## Problem 14

The rectangle below has dimensions and . Diagonals and intersect at . If triangle is cut out and removed, edges and are joined, and the figure is then creased along segments and , we obtain a triangular pyramid, all four of whose faces are isosceles triangles. Find the volume of this pyramid.

## Problem 15

Define a positive integer to be a factorial tail if there is some positive integer such that the decimal representation of ends with exactly zeroes. How many positive integers less than are not factorial tails?

## Problem 14

A rectangle that is inscribed in a larger rectangle (with one vertex on each side) is called *unstuck* if it is possible to rotate (however slightly) the smaller rectangle about its center within the confines of the larger. Of all the rectangles that can be inscribed unstuck in a 6 by 8 rectangle, the smallest perimeter has the form , for a positive integer . Find .