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 2 ==
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== Problem 3 ==
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>.
+
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?
  
[[1994 AIME Problems/Problem 2|Solution]]
+
[[1997 AIME Problems/Problem 3|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 5 ==
== Problem 6 ==
+
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 graphs of the equations
+
 
<center><math>y=k, \qquad y=\sqrt{3}x+2k, \qquad y=-\sqrt{3}x+2k,</math></center>
+
[[1995 AIME Problems/Problem 5|Solution]]
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 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>.
 +
 
 +
[[2001 AIME I Problems/Problem 4|Solution]]
 +
 
 +
== 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>
 +
 
 +
[[1994 AIME Problems/Problem 9|Solution]]
  
[[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>.
+
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?
 +
 
 +
[[1995 AIME Problems/Problem 8|Solution]]
  
[[1993 AIME Problems/Problem 8|Solution]]
+
== Problem 9 ==
== Problem 7 ==
+
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>
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]]
+
[[Image:AIME_1995_Problem_9.png]]
== 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]]
+
[[1995 AIME Problems/Problem 9|Solution]]
== 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>?
 
  
[[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?
+
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>.
 +
 
 +
[[1996 AIME Problems/Problem 11|Solution]]
 +
 
 +
== 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]]
  
[[1994 AIME Problems/Problem 11|Solution]]
 
 
== 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>
+
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>?
  
[[1995 AIME Problems/Problem 12|Solution]]
+
[[1998 AIME Problems/Problem 12|Solution]]
== 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]]
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== Problem 11 ==
== Problem 13 ==
+
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>
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]]
+
[[1999 AIME Problems/Problem 11|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>
+
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>
  
[[1995 AIME Problems/Problem 14|Solution]]
+
[[1999 AIME Problems/Problem 14|Solution]]
== Problem 13 ==
+
== Problem 14 ==
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
+
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>.
 
 
<cmath>\dfrac{\text{Area}(\triangle ADB)}{\text{Area}(\triangle ABC)}</cmath>
 
 
 
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]]
+
[[2000 AIME I Problems/Problem 14|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>.
+
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>.
  
[[1996 AIME Problems/Problem 15|Solution]]
+
[[2000 AIME II Problems/Problem 14|Solution]]
 
== Problem 13 ==
 
== 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>
+
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>
  
[[1998 AIME Problems/Problem 13|Solution]]
+
[[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:

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?

Solution

Problem 5

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

Solution

Problem 4

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

Solution

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 $p/q,\,$ where $p\,$ and $q\,$ are relatively prime positive integers. Find $p+q.\,$

Solution

Problem 8

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

Solution

Problem 9

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

AIME 1995 Problem 9.png

Solution

Problem 11

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

Solution

Problem 13

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

$\Big|\big| |x|-2\big|-1\Big|+\Big|\big| |y|-2\big|-1\Big|=1.$

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

Solution

Problem 12

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

Solution

Problem 11

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

Solution

Problem 14

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

Solution

Problem 14

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

Solution

Problem 15

A stack of $2000$ cards is labelled with the integers from $1$ to $2000,$ 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: $1,2,3,\ldots,1999,2000.$ In the original stack of cards, how many cards were above the card labeled $1999$?

Solution

Problem 14

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

Solution

Problem 13

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

Solution

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