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Difference between revisions of "AoPS Wiki:Sandbox"
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= AIME 2000 II =
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=== AIME 2000 II ===
 
  
 
== Problem 13 ==
 
== Problem 13 ==
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Find the least positive integer <math>n</math> such that <center><math>\frac 1{\sin 45^\circ\sin 46^\circ}+\frac 1{\sin 47^\circ\sin 48^\circ}+\cdots+\frac 1{\sin 133^\circ\sin 134^\circ}=\frac 1{\sin n^\circ}.</math></center>
 
Find the least positive integer <math>n</math> such that <center><math>\frac 1{\sin 45^\circ\sin 46^\circ}+\frac 1{\sin 47^\circ\sin 48^\circ}+\cdots+\frac 1{\sin 133^\circ\sin 134^\circ}=\frac 1{\sin n^\circ}.</math></center>
  
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=== AIME 2001 II ===
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= AIME 2001 II =
  
 
== Problem 13 ==
 
== Problem 13 ==
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Let <math>EFGH</math>, <math>EFDC</math>, and <math>EHBC</math> be three adjacent square faces of a cube, for which <math>EC = 8</math>, and let <math>A</math> be the eighth vertex of the cube. Let <math>I</math>, <math>J</math>, and <math>K</math>, be the points on <math>\overline{EF}</math>, <math>\overline{EH}</math>, and <math>\overline{EC}</math>, respectively, so that <math>EI = EJ = EK = 2</math>. A solid <math>S</math> is obtained by drilling a tunnel through the cube. The sides of the tunnel are planes parallel to <math>\overline{AE}</math>, and containing the edges, <math>\overline{IJ}</math>, <math>\overline{JK}</math>, and <math>\overline{KI}</math>. The surface area of <math>S</math>, including the walls of the tunnel, is <math>m + n\sqrt {p}</math>, where <math>m</math>, <math>n</math>, and <math>p</math> are positive integers and <math>p</math> is not divisible by the square of any prime. Find <math>m + n + p</math>.
 
Let <math>EFGH</math>, <math>EFDC</math>, and <math>EHBC</math> be three adjacent square faces of a cube, for which <math>EC = 8</math>, and let <math>A</math> be the eighth vertex of the cube. Let <math>I</math>, <math>J</math>, and <math>K</math>, be the points on <math>\overline{EF}</math>, <math>\overline{EH}</math>, and <math>\overline{EC}</math>, respectively, so that <math>EI = EJ = EK = 2</math>. A solid <math>S</math> is obtained by drilling a tunnel through the cube. The sides of the tunnel are planes parallel to <math>\overline{AE}</math>, and containing the edges, <math>\overline{IJ}</math>, <math>\overline{JK}</math>, and <math>\overline{KI}</math>. The surface area of <math>S</math>, including the walls of the tunnel, is <math>m + n\sqrt {p}</math>, where <math>m</math>, <math>n</math>, and <math>p</math> are positive integers and <math>p</math> is not divisible by the square of any prime. Find <math>m + n + p</math>.
  
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=== AIME 2002 II ===
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= AIME 2002 II =
  
 
== Problem 13 ==
 
== Problem 13 ==

Revision as of 08:27, 12 April 2010

AIME 2000 II

Problem 13

The equation $2000x^6+100x^5+10x^3+x-2=0$ has exactly two real roots, one of which is $\frac{m+\sqrt{n}}r$, where $m$, $n$ and $r$ are integers, $m$ and $r$ are relatively prime, and $r>0$. Find $m+n+r$.

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 $f_1-f_2+f_3-f_4+\cdots+(-1)^{j+1}f_j$.

Problem 15

Find the least positive integer $n$ such that

$\frac 1{\sin 45^\circ\sin 46^\circ}+\frac 1{\sin 47^\circ\sin 48^\circ}+\cdots+\frac 1{\sin 133^\circ\sin 134^\circ}=\frac 1{\sin n^\circ}.$

AIME 2001 II

Problem 13

In quadrilateral $ABCD$, $\angle{BAD}\cong\angle{ADC}$ and $\angle{ABD}\cong\angle{BCD}$, $AB = 8$, $BD = 10$, and $BC = 6$. The length $CD$ may be written in the form $\frac {m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

Problem 14

There are $2n$ complex numbers that satisfy both $z^{28} - z^{8} - 1 = 0$ and $\mid z \mid = 1$. These numbers have the form $z_{m} = \cos\theta_{m} + i\sin\theta_{m}$, where $0\leq\theta_{1} < \theta_{2} < \ldots < \theta_{2n} < 360$ and angles are measured in degrees. Find the value of $\theta_{2} + \theta_{4} + \ldots + \theta_{2n}$.

Problem 15

Let $EFGH$, $EFDC$, and $EHBC$ be three adjacent square faces of a cube, for which $EC = 8$, and let $A$ be the eighth vertex of the cube. Let $I$, $J$, and $K$, be the points on $\overline{EF}$, $\overline{EH}$, and $\overline{EC}$, respectively, so that $EI = EJ = EK = 2$. A solid $S$ is obtained by drilling a tunnel through the cube. The sides of the tunnel are planes parallel to $\overline{AE}$, and containing the edges, $\overline{IJ}$, $\overline{JK}$, and $\overline{KI}$. The surface area of $S$, including the walls of the tunnel, is $m + n\sqrt {p}$, where $m$, $n$, and $p$ are positive integers and $p$ is not divisible by the square of any prime. Find $m + n + p$.

AIME 2002 II

Problem 13

In triangle $ABC$, point $D$ is on $\overline{BC}$ with $CD=2$ and $DB=5$, point $E$ is on $\overline{AC}$ with $CE=1$ and $EA=32$, $AB=8$, and $\overline{AD}$ and $\overline{BE}$ intersect at $P$. Points $Q$ and $R$ lie on $\overline{AB}$ so that $\overline{PQ}$ is parallel to $\overline{CA}$ and $\overline{PR}$ is parallel to $\overline{CB}$. It is given that the ratio of the area of triangle $PQR$ to the area of triangle $ABC$ is $m/n$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

Problem 14

The perimeter of triangle $APM$ is $152$, and the angle $PAM$ is a right angle. A circle of radius $19$ with center $O$ on $\overline{AP}$ is drawn so that it is tangent to $\overline{AM}$ and $\overline{PM}$. Given that $OP=m/n$ where $m$ and $n$ are relatively prime positive integers, find $m+n$.

Problem 15

Circles $\mathcal{C}_{1}$ and $\mathcal{C}_{2}$ intersect at two points, one of which is $(9,6)$, and the product of the radii is $68$. The x-axis and the line $y = mx$, where $m > 0$, are tangent to both circles. It is given that $m$ can be written in the form $a\sqrt {b}/c$, where $a$, $b$, and $c$ are positive integers, $b$ is not divisible by the square of any prime, and $a$ and $c$ are relatively prime. Find $a + b + c$.

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