# 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 2 == |

− | + | A circle with diameter <math>\overline{PQ}\,</math> of length 10 is internally tangent at <math>P^{}_{}</math> to a circle of radius 20. Square <math>ABCD\,</math> is constructed with <math>A\,</math> and <math>B\,</math> on the larger circle, <math>\overline{CD}\,</math> tangent at <math>Q\,</math> to the smaller circle, and the smaller circle outside <math>ABCD\,</math>. The length of <math>\overline{AB}\,</math> can be written in the form <math>m + \sqrt{n}\,</math>, where <math>m\,</math> and <math>n\,</math> are integers. Find <math>m + n\,</math>. | |

− | [[ | + | [[1994 AIME Problems/Problem 2|Solution]] |

+ | == Problem 6 == | ||

+ | For how many pairs of consecutive integers in <math>\{1000,1001,1002^{}_{},\ldots,2000\}</math> is no carrying required when the two integers are added? | ||

− | + | [[1992 AIME Problems/Problem 6|Solution]] | |

− | + | == Problem 6 == | |

− | + | The graphs of the equations | |

− | [[ | + | <center><math>y=k, \qquad y=\sqrt{3}x+2k, \qquad y=-\sqrt{3}x+2k,</math></center> |

− | + | are drawn in the coordinate plane for <math>k=-10,-9,-8,\ldots,9,10.\,</math> These 63 lines cut part of the plane into equilateral triangles of side <math>2/\sqrt{3}</math>. How many such triangles are formed? | |

− | == Problem | ||

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+ | [[1994 AIME Problems/Problem 6|Solution]] | ||

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

− | + | Let <math>S\,</math> be a set with six elements. In how many different ways can one select two not necessarily distinct subsets of <math>S\,</math> so that the union of the two subsets is <math>S\,</math>? The order of selection does not matter; for example, the pair of subsets <math>\{a, c\}\,</math>, <math>\{b, c, d, e, f\}\,</math> represents the same selection as the pair <math>\{b, c, d, e, f\}\,</math>, <math>\{a, c\}\,</math>. | |

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+ | [[1993 AIME Problems/Problem 8|Solution]] | ||

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

− | + | For certain ordered pairs <math>(a,b)\,</math> of real numbers, the system of equations | |

+ | <center><math>ax+by=1\,</math></center> | ||

+ | <center><math>x^2+y^2=50\,</math></center> | ||

+ | has at least one solution, and each solution is an ordered pair <math>(x,y)\,</math> of integers. How many such ordered pairs <math>(a,b)\,</math> are there? | ||

− | [[ | + | [[1994 AIME Problems/Problem 7|Solution]] |

+ | == Problem 8 == | ||

+ | The points <math>(0,0)\,</math>, <math>(a,11)\,</math>, and <math>(b,37)\,</math> are the vertices of an equilateral triangle. Find the value of <math>ab\,</math>. | ||

+ | [[1994 AIME Problems/Problem 8|Solution]] | ||

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

− | + | The vertices of <math>\triangle ABC</math> are <math>A = (0,0)\,</math>, <math>B = (0,420)\,</math>, and <math>C = (560,0)\,</math>. The six faces of a die are labeled with two <math>A\,</math>'s, two <math>B\,</math>'s, and two <math>C\,</math>'s. Point <math>P_1 = (k,m)\,</math> is chosen in the interior of <math>\triangle ABC</math>, and points <math>P_2\,</math>, <math>P_3\,</math>, <math>P_4, \dots</math> are generated by rolling the die repeatedly and applying the rule: If the die shows label <math>L\,</math>, where <math>L \in \{A, B, C\}</math>, and <math>P_n\,</math> is the most recently obtained point, then <math>P_{n + 1}^{}</math> is the midpoint of <math>\overline{P_n L}</math>. Given that <math>P_7 = (14,92)\,</math>, what is <math>k + m\,</math>? | |

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− | [[ | + | [[1993 AIME Problems/Problem 12|Solution]] |

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

− | + | Ninety-four bricks, each measuring <math>4''\times10''\times19'',</math> are to be stacked one on top of another to form a tower 94 bricks tall. Each brick can be oriented so it contributes <math>4''\,</math> or <math>10''\,</math> or <math>19''\,</math> to the total height of the tower. How many different tower heights can be achieved using all 94 of the bricks? | |

− | + | [[1994 AIME Problems/Problem 11|Solution]] | |

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

− | + | Pyramid <math>OABCD</math> has square base <math>ABCD,</math> congruent edges <math>\overline{OA}, \overline{OB}, \overline{OC},</math> and <math>\overline{OD},</math> and <math>\angle AOB=45^\circ.</math> Let <math>\theta</math> be the measure of the dihedral angle formed by faces <math>OAB</math> and <math>OBC.</math> Given that <math>\cos \theta=m+\sqrt{n},</math> where <math>m_{}</math> and <math>n_{}</math> are integers, find <math>m+n.</math> | |

− | [[ | + | [[1995 AIME Problems/Problem 12|Solution]] |

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

− | + | Find the smallest positive integer solution to <math>\tan{19x^{\circ}}=\dfrac{\cos{96^{\circ}}+\sin{96^{\circ}}}{\cos{96^{\circ}}-\sin{96^{\circ}}}</math>. | |

− | [[ | + | [[1996 AIME Problems/Problem 10|Solution]] |

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

− | Let <math> | + | Let <math>f(n)</math> be the integer closest to <math>\sqrt[4]{n}.</math> Find <math>\sum_{k=1}^{1995}\frac 1{f(k)}.</math> |

− | [[ | + | [[1995 AIME Problems/Problem 13|Solution]] |

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

− | + | In a circle of radius 42, two chords of length 78 intersect at a point whose distance from the center is 18. The two chords divide the interior of the circle into four regions. Two of these regions are bordered by segments of unequal lengths, and the area of either of them can be expressed uniquely in the form <math>m\pi-n\sqrt{d},</math> where <math>m, n,</math> and <math>d_{}</math> are positive integers and <math>d_{}</math> is not divisible by the square of any prime number. Find <math>m+n+d.</math> | |

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− | + | [[1995 AIME Problems/Problem 14|Solution]] | |

− | + | == Problem 13 == | |

− | + | In triangle <math>ABC</math>, <math>AB=\sqrt{30}</math>, <math>AC=\sqrt{6}</math>, and <math>BC=\sqrt{15}</math>. There is a point <math>D</math> for which <math>\overline{AD}</math> bisects <math>\overline{BC}</math>, and <math>\angle ADB</math> is a right angle. The ratio | |

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− | + | <cmath>\dfrac{\text{Area}(\triangle ADB)}{\text{Area}(\triangle ABC)}</cmath> | |

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− | + | 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>. | |

− | [[ | + | [[1996 AIME Problems/Problem 13|Solution]] |

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

− | + | In parallelogram <math>ABCD,</math> let <math>O</math> be the intersection of diagonals <math>\overline{AC}</math> and <math>\overline{BD}</math>. Angles <math>CAB</math> and <math>DBC</math> are each twice as large as angle <math>DBA,</math> and angle <math>ACB</math> is <math>r</math> times as large as angle <math>AOB</math>. Find the greatest integer that does not exceed <math>1000r</math>. | |

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− | [[ | + | [[1996 AIME Problems/Problem 15|Solution]] |

− | == Problem | + | == Problem 13 == |

− | + | If <math>\{a_1,a_2,a_3,\ldots,a_n\}</math> is a [[set]] of [[real numbers]], indexed so that <math>a_1 < a_2 < a_3 < \cdots < a_n,</math> its complex power sum is defined to be <math>a_1i + a_2i^2+ a_3i^3 + \cdots + a_ni^n,</math> where <math>i^2 = - 1.</math> Let <math>S_n</math> be the sum of the complex power sums of all nonempty [[subset]]s of <math>\{1,2,\ldots,n\}.</math> Given that <math>S_8 = - 176 - 64i</math> and <math> S_9 = p + qi,</math> where <math>p</math> and <math>q</math> are integers, find <math>|p| + |q|.</math> | |

− | [[ | + | [[1998 AIME Problems/Problem 13|Solution]] |

## Revision as of 13:10, 25 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 2

A circle with diameter of length 10 is internally tangent at to a circle of radius 20. Square is constructed with and on the larger circle, tangent at to the smaller circle, and the smaller circle outside . The length of can be written in the form , where and are integers. Find .

## Problem 6

For how many pairs of consecutive integers in is no carrying required when the two integers are added?

## Problem 6

The graphs of the equations

are drawn in the coordinate plane for These 63 lines cut part of the plane into equilateral triangles of side . How many such triangles are formed?

## Problem 8

Let be a set with six elements. In how many different ways can one select two not necessarily distinct subsets of so that the union of the two subsets is ? The order of selection does not matter; for example, the pair of subsets , represents the same selection as the pair , .

## Problem 7

For certain ordered pairs of real numbers, the system of equations

has at least one solution, and each solution is an ordered pair of integers. How many such ordered pairs are there?

## Problem 8

The points , , and are the vertices of an equilateral triangle. Find the value of .

## Problem 12

The vertices of are , , and . The six faces of a die are labeled with two 's, two 's, and two 's. Point is chosen in the interior of , and points , , are generated by rolling the die repeatedly and applying the rule: If the die shows label , where , and is the most recently obtained point, then is the midpoint of . Given that , what is ?

## Problem 11

Ninety-four bricks, each measuring are to be stacked one on top of another to form a tower 94 bricks tall. Each brick can be oriented so it contributes or or to the total height of the tower. How many different tower heights can be achieved using all 94 of the bricks?

## Problem 12

Pyramid has square base congruent edges and and Let be the measure of the dihedral angle formed by faces and Given that where and are integers, find

## Problem 10

Find the smallest positive integer solution to .

## Problem 13

Let be the integer closest to Find

## Problem 14

In a circle of radius 42, two chords of length 78 intersect at a point whose distance from the center is 18. The two chords divide the interior of the circle into four regions. Two of these regions are bordered by segments of unequal lengths, and the area of either of them can be expressed uniquely in the form where and are positive integers and is not divisible by the square of any prime number. Find

## Problem 13

In triangle , , , and . There is a point for which bisects , and is a right angle. The ratio

can be written in the form , where and are relatively prime positive integers. Find .

## Problem 15

In parallelogram let be the intersection of diagonals and . Angles and are each twice as large as angle and angle is times as large as angle . Find the greatest integer that does not exceed .

## Problem 13

If is a set of real numbers, indexed so that its complex power sum is defined to be where Let be the sum of the complex power sums of all nonempty subsets of Given that and where and are integers, find