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 3 ==
+
== Problem 2 ==
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>?
+
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>.
  
[[1990 AIME Problems/Problem 3|Solution]]
+
[[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?
  
== Problem 5 ==
+
[[1992 AIME Problems/Problem 6|Solution]]
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?
+
== Problem 6 ==
 
+
The graphs of the equations
[[1991 AIME Problems/Problem 5|Solution]]
+
<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 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 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]]
 
  
 +
[[1994 AIME Problems/Problem 6|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>.
+
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>.
 
 
[[1992 AIME Problems/Problem 8|Solution]]
 
  
 +
[[1993 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 boxIf <math>p\,</math> is written as a fraction in lowest terms, what is the sum of the numerator and denominator?
+
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 integersHow many such ordered pairs <math>(a,b)\,</math> are there?
  
[[1993 AIME Problems/Problem 7|Solution]]
+
[[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 ==
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>.
+
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>?
 
 
[[Image:AIME_1989_Problem_12.png]]
 
  
[[1989 AIME Problems/Problem 12|Solution]]
+
[[1993 AIME Problems/Problem 12|Solution]]
 
== Problem 11 ==
 
== 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 belowThe 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>.
+
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 tallEach 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?
  
<asy>
+
[[1994 AIME Problems/Problem 11|Solution]]
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 ==
 
== 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>.
+
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>
  
[[1991 AIME Problems/Problem 12|Solution]]
+
[[1995 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>?
+
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>.
  
[[1993 AIME Problems/Problem 10|Solution]]
+
[[1996 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?
+
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>
  
[[1989 AIME Problems/Problem 13|Solution]]
+
[[1995 AIME Problems/Problem 13|Solution]]
 
== Problem 14 ==
 
== 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
+
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>
<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
+
[[1995 AIME Problems/Problem 14|Solution]]
<center><math>r+si=(a_ma_{m-1}\ldots a_1a_0)_{-n+i}</math></center>
+
== Problem 13 ==
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
+
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
<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]]
+
<cmath>\dfrac{\text{Area}(\triangle ADB)}{\text{Area}(\triangle ABC)}</cmath>
== 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]]
+
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>.
  
[[1990 AIME Problems/Problem 14|Solution]]
+
[[1996 AIME Problems/Problem 13|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?
+
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>.
 +
 
  
[[1992 AIME Problems/Problem 15|Solution]]
+
[[1996 AIME Problems/Problem 15|Solution]]
== Problem 14 ==
+
== Problem 13 ==
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>.
+
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>
  
[[1993 AIME Problems/Problem 14|Solution]]
+
[[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:

Problem 2

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

Solution

Problem 6

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

Solution

Problem 6

The graphs of the equations

$y=k, \qquad y=\sqrt{3}x+2k, \qquad y=-\sqrt{3}x+2k,$

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

Solution

Problem 8

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

Solution

Problem 7

For certain ordered pairs $(a,b)\,$ of real numbers, the system of equations

$ax+by=1\,$
$x^2+y^2=50\,$

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

Solution

Problem 8

The points $(0,0)\,$, $(a,11)\,$, and $(b,37)\,$ are the vertices of an equilateral triangle. Find the value of $ab\,$.

Solution

Problem 12

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

Solution

Problem 11

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

Solution

Problem 12

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

Solution

Problem 10

Find the smallest positive integer solution to $\tan{19x^{\circ}}=\dfrac{\cos{96^{\circ}}+\sin{96^{\circ}}}{\cos{96^{\circ}}-\sin{96^{\circ}}}$.

Solution

Problem 13

Let $f(n)$ be the integer closest to $\sqrt[4]{n}.$ Find $\sum_{k=1}^{1995}\frac 1{f(k)}.$

Solution

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 $m\pi-n\sqrt{d},$ where $m, n,$ and $d_{}$ are positive integers and $d_{}$ is not divisible by the square of any prime number. Find $m+n+d.$

Solution

Problem 13

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

\[\dfrac{\text{Area}(\triangle ADB)}{\text{Area}(\triangle ABC)}\]

can be written in the form $\dfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

Solution

Problem 15

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


Solution

Problem 13

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

Solution

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