https://artofproblemsolving.com/wiki/api.php?action=feedcontributions&user=Ohmcfifth&feedformat=atomAoPS Wiki - User contributions [en]2024-03-28T10:45:08ZUser contributionsMediaWiki 1.31.1https://artofproblemsolving.com/wiki/index.php?title=2016_AIME_I_Problems/Problem_4&diff=773122016 AIME I Problems/Problem 42016-03-04T22:04:56Z<p>Ohmcfifth: Created page with "==Problem== A right prism with height <math>h</math> has bases that are regular hexagons with sides of length 12. A vertex <math>A</math> of the prism and its three adjacent v..."</p>
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<div>==Problem==<br />
A right prism with height <math>h</math> has bases that are regular hexagons with sides of length 12. A vertex <math>A</math> of the prism and its three adjacent vertices are the vertices of a triangular pyramid. The dihedral angle (the angle between the two planes) formed by the face of the pyramid that lies in a base of the prism and the face of the pyramid that does not contain <math>A</math> measures <math>60</math> degrees. Find <math>h^2</math>.</div>Ohmcfifthhttps://artofproblemsolving.com/wiki/index.php?title=2016_AIME_I_Problems/Problem_7&diff=772952016 AIME I Problems/Problem 72016-03-04T21:53:09Z<p>Ohmcfifth: Created page with "==Problem== For integers <math>a</math> and <math>b</math> consider the complex number <cmath>\frac{\sqrt{ab+2016}}{ab+100}-({\frac{\sqrt{|a+b|}}{ab+100}})i</cmath> Find the ..."</p>
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<div>==Problem==<br />
For integers <math>a</math> and <math>b</math> consider the complex number<br />
<cmath>\frac{\sqrt{ab+2016}}{ab+100}-({\frac{\sqrt{|a+b|}}{ab+100}})i</cmath><br />
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Find the number of ordered pairs of integers <math>(a,b)</math> such that this complex number is a real number.</div>Ohmcfifthhttps://artofproblemsolving.com/wiki/index.php?title=2016_AIME_I_Problems/Problem_2&diff=772862016 AIME I Problems/Problem 22016-03-04T21:48:20Z<p>Ohmcfifth: /* Solution */</p>
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<div>==Problem 2==<br />
Two dice appear to be normal dice with their faces numbered from <math>1</math> to <math>6</math>, but each die is weighted so that the probability of rolling the number <math>k</math> is directly proportional to <math>k</math>. The probability of rolling a <math>7</math> with this pair of dice is <math>\frac{m}{n}</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m+n</math>.<br />
==Solution==<br />
It is easier to think of the dice as 21 sided dice with 6 sixes, 5 fives, etc. Then there are 21^2=441 possible roles. There are 2*(1*6+2*5+3*4)=56 roles that will result in a seven. The odds are therefore <math>56/441=8/63</math>. The answer is <math>8+63=071</math></div>Ohmcfifth