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| − | = AIME 2000 II =
| + | {{AoPSWiki:Sandbox/header}} <!-- Please do not delete this line --> |
| | + | In the computer world, a '''sandbox''' is a place to test and experiment -- essentially, it's a place to play. |
| | | | |
| − | == Problem 13 ==
| + | This is the AoPSWiki Sandbox. Feel free to experiment here. |
| − | The equation <math>2000x^6+100x^5+10x^3+x-2=0</math> has exactly two real roots, one of which is <math>\frac{m+\sqrt{n}}r</math>, where <math>m</math>, <math>n</math> and <math>r</math> are integers, <math>m</math> and <math>r</math> are relatively prime, and <math>r>0</math>. Find <math>m+n+r</math>.
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| | | | |
| − | == Problem 14 ==
| + | Warning: anything you place here is subject to deletion without notice. |
| − | 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>.
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| − | == Problem 15 == | + | === Sandbox Area === |
| − | 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>
| + | <asy> |
| | + | draw((0,0)--(3,9),black); |
| | + | label("$71^\circ$",(0,0),NE); |
| | + | draw((3,9)--(6,0),black); |
| | + | label("$71^\circ$",(6,0),NW); |
| | + | draw((0,0)--(6,0),black); |
| | + | label("$38^\circ$",(3,8),S); |
| | + | dot((0,0)); |
| | + | dot((3,9)); |
| | + | dot((6,0)); |
| | + | draw((-1,7)--(2.3,5.9),black); |
| | + | draw((-1,6)--(2,6),red); |
| | + | label("$19^\circ$",(1,6),NW); |
| | + | </asy> |
| | | | |
| − | = AIME 2001 II =
| + | <math>\lim_{x\to0}\frac{a}{x}</math> |
| | | | |
| − | == Problem 13 ==
| + | The characteristic polynomial <math>f(\lambda)</math> of the |
| − | In quadrilateral <math>ABCD</math>, <math>\angle{BAD}\cong\angle{ADC}</math> and <math>\angle{ABD}\cong\angle{BCD}</math>, <math>AB = 8</math>, <math>BD = 10</math>, and <math>BC = 6</math>. The length <math>CD</math> may be written in the form <math>\frac {m}{n}</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m + n</math>.
| + | <math>3 \times 3</math> matrix |
| − | | + | <math> |
| − | == Problem 14 ==
| + | \left( |
| − | There are <math>2n</math> complex numbers that satisfy both <math>z^{28} - z^{8} - 1 = 0</math> and <math>\mid z \mid = 1</math>. These numbers have the form <math>z_{m} = \cos\theta_{m} + i\sin\theta_{m}</math>, where <math>0\leq\theta_{1} < \theta_{2} < \ldots < \theta_{2n} < 360</math> and angles are measured in degrees. Find the value of <math>\theta_{2} + \theta_{4} + \ldots + \theta_{2n}</math>.
| + | \begin{array}{ccc} |
| − | | + | a & b & c \\ |
| − | == Problem 15 ==
| + | d & e & f \\ |
| − | 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>.
| + | g & h & i \end{array} |
| − | | + | \right)</math> |
| − | = AIME 2002 II =
| + | is given by the equation |
| − | | + | <math> f(\lambda) |
| − | == Problem 13 ==
| + | = \left| |
| − | In triangle <math>ABC</math>, point <math>D</math> is on <math>\overline{BC}</math> with <math>CD=2</math> and <math>DB=5</math>, point <math>E</math> is on <math>\overline{AC}</math> with <math>CE=1</math> and <math>EA=32</math>, <math>AB=8</math>, and <math>\overline{AD}</math> and <math>\overline{BE}</math> intersect at <math>P</math>. Points <math>Q</math> and <math>R</math> lie on <math>\overline{AB}</math> so that <math>\overline{PQ}</math> is parallel to <math>\overline{CA}</math> and <math>\overline{PR}</math> is parallel to <math>\overline{CB}</math>. It is given that the ratio of the area of triangle <math>PQR</math> to the area of triangle <math>ABC</math> is <math>m/n</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m+n</math>.
| + | \begin{array}{ccc} |
| − | | + | \lambda - a & -b & -c \\ |
| − | == Problem 14 ==
| + | -d & \lambda - e & -f \\ |
| − | The perimeter of triangle <math>APM</math> is <math>152</math>, and the angle <math>PAM</math> is a right angle. A circle of radius <math>19</math> with center <math>O</math> on <math>\overline{AP}</math> is drawn so that it is tangent to <math>\overline{AM}</math> and <math>\overline{PM}</math>. Given that <math>OP=m/n</math> where <math>m</math> and <math>n</math> are relatively prime positive integers, find <math>m+n</math>.
| + | -g & -h & \lambda - i \end{array} |
| − | | + | \right|.</math> |
| − | == Problem 15 ==
| |
| − | Circles <math>\mathcal{C}_{1}</math> and <math>\mathcal{C}_{2}</math> intersect at two points, one of which is <math>(9,6)</math>, and the product of the radii is <math>68</math>. The x-axis and the line <math>y = mx</math>, where <math>m > 0</math>, are tangent to both circles. It is given that <math>m</math> can be written in the form <math>a\sqrt {b}/c</math>, where <math>a</math>, <math>b</math>, and <math>c</math> are positive integers, <math>b</math> is not divisible by the square of any prime, and <math>a</math> and <math>c</math> are relatively prime. Find <math>a + b + c</math>.
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![[asy] draw((0,0)--(3,9),black); label("$71^\circ$",(0,0),NE); draw((3,9)--(6,0),black); label("$71^\circ$",(6,0),NW); draw((0,0)--(6,0),black); label("$38^\circ$",(3,8),S); dot((0,0)); dot((3,9)); dot((6,0)); draw((-1,7)--(2.3,5.9),black); draw((-1,6)--(2,6),red); label("$19^\circ$",(1,6),NW); [/asy]](http://latex.artofproblemsolving.com/3/4/9/349e45904453e378fffc7cf94a55247ad2cc73f3.png)
of the
matrix
is given by the equation
