https://artofproblemsolving.com/wiki/api.php?action=feedcontributions&user=Cj13609517288&feedformat=atom AoPS Wiki - User contributions [en] 2021-09-21T03:13:44Z User contributions MediaWiki 1.31.1 https://artofproblemsolving.com/wiki/index.php?title=2011_AMC_10B_Problems/Problem_18&diff=110608 2011 AMC 10B Problems/Problem 18 2019-10-26T16:04:11Z <p>Cj13609517288: /* Solution 2 (with trig) */</p> <hr /> <div>== Problem==<br /> <br /> Rectangle &lt;math&gt;ABCD&lt;/math&gt; has &lt;math&gt;AB = 6&lt;/math&gt; and &lt;math&gt;BC = 3&lt;/math&gt;. Point &lt;math&gt;M&lt;/math&gt; is chosen on side &lt;math&gt;AB&lt;/math&gt; so that &lt;math&gt;\angle AMD = \angle CMD&lt;/math&gt;. What is the degree measure of &lt;math&gt;\angle AMD&lt;/math&gt;?<br /> <br /> &lt;math&gt; \textbf{(A)}\ 15 \qquad\textbf{(B)}\ 30 \qquad\textbf{(C)}\ 45 \qquad\textbf{(D)}\ 60 \qquad\textbf{(E)}\ 75&lt;/math&gt;<br /> [[Category: Introductory Geometry Problems]]<br /> <br /> ==Solution 1==<br /> <br /> &lt;center&gt;&lt;asy&gt;<br /> unitsize(10mm);<br /> defaultpen(linewidth(.5pt)+fontsize(10pt));<br /> dotfactor=3;<br /> <br /> pair A=(0,3), B=(6,3), C=(6,0), D=(0,0);<br /> pair M=(0.80385,3);<br /> <br /> draw(A--B--C--D--cycle);<br /> draw(M--C);<br /> draw(M--D);<br /> draw(anglemark(A,M,D));<br /> draw(anglemark(D,M,C));<br /> draw(anglemark(C,D,M));<br /> <br /> pair[] ps={A,B,C,D,M};<br /> dot(ps);<br /> label(&quot;$A$&quot;,A,NW);<br /> label(&quot;$B$&quot;,B,NE);<br /> label(&quot;$C$&quot;,C,SE);<br /> label(&quot;$D$&quot;,D,SW);<br /> label(&quot;$M$&quot;,M,N);<br /> label(&quot;$6$&quot;,midpoint(C--M),SW);<br /> label(&quot;$3$&quot;,midpoint(B--C),E);<br /> label(&quot;$6$&quot;,midpoint(C--D),S);<br /> <br /> &lt;/asy&gt;<br /> &lt;/center&gt;<br /> <br /> It is given that &lt;math&gt;\angle AMD \sim \angle CMD&lt;/math&gt;. Since &lt;math&gt;\angle AMD&lt;/math&gt; and &lt;math&gt;\angle CDM&lt;/math&gt; are alternate interior angles and &lt;math&gt;\overline{AB} \parallel \overline{DC}&lt;/math&gt;, &lt;math&gt;\angle AMD \cong \angle CDM \longrightarrow \angle CMD \cong \angle CDM&lt;/math&gt;. Use the [[Base Angle Theorem]] to show &lt;math&gt;\overline{DC} \cong \overline{MC}&lt;/math&gt;. We know that &lt;math&gt;ABCD&lt;/math&gt; is a [[rectangle]], so it follows that &lt;math&gt;\overline{MC} = 6&lt;/math&gt;. We notice that &lt;math&gt;\triangle BMC&lt;/math&gt; is a &lt;math&gt;30-60-90&lt;/math&gt; triangle, and &lt;math&gt;\angle BMC = 30^{\circ}&lt;/math&gt;. If we let &lt;math&gt;x&lt;/math&gt; be the measure of &lt;math&gt;\angle AMD,&lt;/math&gt; then<br /> &lt;cmath&gt;\begin{align*}<br /> 2x + 30 &amp;= 180\\<br /> 2x &amp;= 150\\<br /> x &amp;= \boxed{\textbf{(E)} 75}<br /> \end{align*}&lt;/cmath&gt;<br /> <br /> ==Solution 2 (with trig)==<br /> <br /> Let &lt;math&gt;\angle{DMC} = \angle{AMD} = \theta&lt;/math&gt;. If we let &lt;math&gt;AM = x&lt;/math&gt;, we have that &lt;math&gt;MD = \sqrt{x^2 + 9}&lt;/math&gt;, by the Pythagorean Theorem, and similarily, &lt;math&gt;MC = \sqrt{x^2 - 12x + 45}&lt;/math&gt;. Applying LOC, we see that <br /> &lt;cmath&gt;2x^2 + 54 - 12x - 2 \sqrt{x^4 - 12x^3 + 54x^2 - 108x + 405} \cdot \cos (\theta) = 36&lt;/cmath&gt; and &lt;cmath&gt;\tan (\theta) = \frac{3}{x}&lt;/cmath&gt; YAY!!! We have two equations for two variables... that are terribly ugly. Well, we'll try to solve it. First of all, note that &lt;math&gt;\sin (\theta) = \frac{3}{\sqrt{x^2+9}}&lt;/math&gt;, so solving for &lt;math&gt;\cos (\theta)&lt;/math&gt; in terms of &lt;math&gt;x&lt;/math&gt;, we get that &lt;math&gt;\cos (\theta) = \frac{x \cdot \sqrt{x^2 + 9}}{x^2 + 9}&lt;/math&gt;. The equation now becomes <br /> <br /> &lt;cmath&gt;2x^2 + 54 - 12x - 2 \sqrt{x^4 - 12x^3 + 54x^2 - 108x + 405} \cdot \frac{x \cdot \sqrt{x^2 + 9}}{x^2 + 9} = 36&lt;/cmath&gt;<br /> Simplifying, we get<br /> <br /> &lt;cmath&gt;4x^4 - 48x^3 + 216x^2 - 432x + 324&lt;/cmath&gt;<br /> <br /> Now, we apply the quartic formula to get <br /> <br /> &lt;cmath&gt;x = 6 \pm 3 \sqrt{3}&lt;/cmath&gt;<br /> <br /> We can easily see that &lt;math&gt;x = 6 + 3 \sqrt{3}&lt;/math&gt; is an invalid solution. Thus, &lt;math&gt;x = 6 - 3 \sqrt{3}&lt;/math&gt;. <br /> <br /> Finally, since &lt;math&gt;\tan (\theta) = \frac{3}{6 - 3 \sqrt{3}} = 2 + \sqrt{3}&lt;/math&gt;, &lt;math&gt;\theta = \frac{5 + 12n}{12} \pi&lt;/math&gt;, where &lt;math&gt;n&lt;/math&gt; is any integer. Converting to degrees, we have that &lt;math&gt;\theta = 75 + 180n&lt;/math&gt;. Since &lt;math&gt;0 &lt; \theta &lt; 90&lt;/math&gt;, we have that &lt;math&gt;\theta = \boxed{75}&lt;/math&gt;. &lt;math&gt;\square&lt;/math&gt; <br /> <br /> ~ilovepi3.14<br /> <br /> == See Also==<br /> <br /> {{AMC10 box|year=2011|ab=B|num-b=17|num-a=19}}<br /> {{MAA Notice}}</div> Cj13609517288 https://artofproblemsolving.com/wiki/index.php?title=2013_AMC_10A_Problems/Problem_24&diff=110139 2013 AMC 10A Problems/Problem 24 2019-10-05T21:43:29Z <p>Cj13609517288: /* Solution 2 */</p> <hr /> <div>Central High School is competing against Northern High School in a backgammon match. Each school has three players, and the contest rules require that each player play two games against each of the other school's players. The match takes place in six rounds, with three games played simultaneously in each round. In how many different ways can the match be scheduled?<br /> <br /> &lt;math&gt; \textbf{(A)}\ 540\qquad\textbf{(B)}\ 600\qquad\textbf{(C)}\ 720\qquad\textbf{(D)}\ 810\qquad\textbf{(E)}\ 900&lt;/math&gt;<br /> <br /> ==Solution 1==<br /> <br /> Let us label the players of the first team &lt;math&gt;A&lt;/math&gt;, &lt;math&gt;B&lt;/math&gt;, and &lt;math&gt;C&lt;/math&gt;, and those of the second team, &lt;math&gt;X&lt;/math&gt;, &lt;math&gt;Y&lt;/math&gt;, and &lt;math&gt;Z&lt;/math&gt;. <br /> <br /> 1. One way of scheduling all six distinct rounds could be:<br /> <br /> Round 1----&gt;&lt;math&gt;AX&lt;/math&gt; &lt;math&gt;BY&lt;/math&gt; &lt;math&gt;CZ&lt;/math&gt;<br /> Round 2----&gt;&lt;math&gt;AX&lt;/math&gt; &lt;math&gt;BZ&lt;/math&gt; &lt;math&gt;CY&lt;/math&gt;<br /> Round 3----&gt;&lt;math&gt;AY&lt;/math&gt; &lt;math&gt;BX&lt;/math&gt; &lt;math&gt;CZ&lt;/math&gt;<br /> Round 4----&gt;&lt;math&gt;AY&lt;/math&gt; &lt;math&gt;BZ&lt;/math&gt; &lt;math&gt;CX&lt;/math&gt;<br /> Round 5----&gt;&lt;math&gt;AZ&lt;/math&gt; &lt;math&gt;BX&lt;/math&gt; &lt;math&gt;CY&lt;/math&gt;<br /> Round 6----&gt;&lt;math&gt;AZ&lt;/math&gt; &lt;math&gt;BY&lt;/math&gt; &lt;math&gt;CX&lt;/math&gt;<br /> <br /> The above mentioned schedule ensures that each player of one team plays twice with each player from another team. Now you can generate a completely new schedule by permutating those &lt;math&gt;6&lt;/math&gt; rounds and that can be done in &lt;math&gt;6!&lt;/math&gt;=&lt;math&gt;720&lt;/math&gt; ways.<br /> <br /> 2. One can also make the schedule in such a way that two rounds are repeated. <br /> <br /> (a)<br /> Round 1----&gt;&lt;math&gt;AX&lt;/math&gt; &lt;math&gt;BZ&lt;/math&gt; &lt;math&gt;CY&lt;/math&gt;<br /> Round 2----&gt;&lt;math&gt;AX&lt;/math&gt; &lt;math&gt;BZ&lt;/math&gt; &lt;math&gt;CY&lt;/math&gt;<br /> Round 3----&gt;&lt;math&gt;AY&lt;/math&gt; &lt;math&gt;BX&lt;/math&gt; &lt;math&gt;CZ&lt;/math&gt;<br /> Round 4----&gt;&lt;math&gt;AY&lt;/math&gt; &lt;math&gt;BX&lt;/math&gt; &lt;math&gt;CZ&lt;/math&gt;<br /> Round 5----&gt;&lt;math&gt;AZ&lt;/math&gt; &lt;math&gt;BY&lt;/math&gt; &lt;math&gt;CX&lt;/math&gt;<br /> Round 6----&gt;&lt;math&gt;AZ&lt;/math&gt; &lt;math&gt;BY&lt;/math&gt; &lt;math&gt;CX&lt;/math&gt;<br /> <br /> (b)<br /> Round 1----&gt;&lt;math&gt;AX&lt;/math&gt; &lt;math&gt;BY&lt;/math&gt; &lt;math&gt;CZ&lt;/math&gt;<br /> Round 2----&gt;&lt;math&gt;AX&lt;/math&gt; &lt;math&gt;BY&lt;/math&gt; &lt;math&gt;CZ&lt;/math&gt;<br /> Round 3----&gt;&lt;math&gt;AY&lt;/math&gt; &lt;math&gt;BZ&lt;/math&gt; &lt;math&gt;CX&lt;/math&gt;<br /> Round 4----&gt;&lt;math&gt;AY&lt;/math&gt; &lt;math&gt;BZ&lt;/math&gt; &lt;math&gt;CX&lt;/math&gt;<br /> Round 5----&gt;&lt;math&gt;AZ&lt;/math&gt; &lt;math&gt;BX&lt;/math&gt; &lt;math&gt;CY&lt;/math&gt;<br /> Round 6----&gt;&lt;math&gt;AZ&lt;/math&gt; &lt;math&gt;BX&lt;/math&gt; &lt;math&gt;CY&lt;/math&gt;<br /> <br /> As mentioned earlier any permutation of (a) and (b) will also give us a new schedule. For both (a) and (b) the number of permutations are<br /> &lt;math&gt;\frac{6!}{2!2!2!}&lt;/math&gt; = &lt;math&gt;90&lt;/math&gt;<br /> <br /> <br /> So the total number of schedules is &lt;math&gt;720+90+90&lt;/math&gt; =&lt;math&gt;\boxed{\textbf{(E)} 900}&lt;/math&gt;.<br /> <br /> ==Solution 2==<br /> Label the players of the first team &lt;math&gt;A&lt;/math&gt;, &lt;math&gt;B&lt;/math&gt;, and &lt;math&gt;C&lt;/math&gt;, and those of the second team, &lt;math&gt;X&lt;/math&gt;, &lt;math&gt;Y&lt;/math&gt;, and &lt;math&gt;Z&lt;/math&gt;. We can start by assigning an opponent to person &lt;math&gt;A&lt;/math&gt; for all &lt;math&gt;6&lt;/math&gt; games. Since &lt;math&gt;A&lt;/math&gt; has to play each of &lt;math&gt;X&lt;/math&gt;, &lt;math&gt;Y&lt;/math&gt;, and &lt;math&gt;Z&lt;/math&gt; twice, there are &lt;math&gt;\frac{6!}{2!2!2!} = 90&lt;/math&gt; ways to do this. We can assume that the opponents for &lt;math&gt;A&lt;/math&gt; in the &lt;math&gt;6&lt;/math&gt; rounds are &lt;math&gt;X&lt;/math&gt;, &lt;math&gt;X&lt;/math&gt;, &lt;math&gt;Y&lt;/math&gt;, &lt;math&gt;Y&lt;/math&gt;, &lt;math&gt;Z&lt;/math&gt;, &lt;math&gt;Z&lt;/math&gt; and multiply by &lt;math&gt;90&lt;/math&gt; afterwards.<br /> <br /> <br /> Notice that for every valid assignment of the opponents of &lt;math&gt;A&lt;/math&gt; and &lt;math&gt;B&lt;/math&gt;, there is only &lt;math&gt;1&lt;/math&gt; valid assignment of opponents for &lt;math&gt;C&lt;/math&gt;. More specifically, the opponents for &lt;math&gt;C&lt;/math&gt; are the leftover opponents after the opponents for &lt;math&gt;A&lt;/math&gt; and &lt;math&gt;B&lt;/math&gt; are chosen in each round. Therefore, all we have to do is assign the opponents for &lt;math&gt;B&lt;/math&gt;. This is the same as finding the number of permutations of &lt;math&gt;X&lt;/math&gt;, &lt;math&gt;Y&lt;/math&gt;, and &lt;math&gt;Z&lt;/math&gt; that do not have a &lt;math&gt;X&lt;/math&gt; in the first two spots, an &lt;math&gt;Y&lt;/math&gt; in the next two spots, and a &lt;math&gt;Z&lt;/math&gt; in the final two spots.<br /> <br /> <br /> <br /> We can use casework to find this by using the fact that after we put down the &lt;math&gt;X&lt;/math&gt;'s and &lt;math&gt;Y&lt;/math&gt;'s first there is &lt;math&gt;1&lt;/math&gt; way to put down the &lt;math&gt;Z&lt;/math&gt;'s (the two remaining spots).<br /> <br /> If &lt;math&gt;X&lt;/math&gt;'s are put in the middle 2 spots, then there is &lt;math&gt;1&lt;/math&gt; way to assign spots to &lt;math&gt;Y&lt;/math&gt;, namely the last &lt;math&gt;2&lt;/math&gt; spots. (If one of the last two spots are left empty, there will have to be a &lt;math&gt;Z&lt;/math&gt; there, which which not valid).<br /> <br /> If &lt;math&gt;X&lt;/math&gt;'s are put in the last 2 spots, then there is &lt;math&gt;1&lt;/math&gt; way to assign spots to &lt;math&gt;Y&lt;/math&gt;.<br /> <br /> Finally, if one &lt;math&gt;X&lt;/math&gt; was put in on of the middle two spots and one &lt;math&gt;X&lt;/math&gt; was put in one of the last two spots, there are &lt;math&gt;2\cdot2&lt;/math&gt; ways to assign spots to &lt;math&gt;X&lt;/math&gt; and &lt;math&gt;2\cdot1&lt;/math&gt; ways to assign spots to &lt;math&gt;Y&lt;/math&gt; (one of the first two spots and the remaining spot in the last 2).<br /> <br /> <br /> <br /> There are &lt;math&gt;1+1+2\cdot2\cdot2\cdot1 = 10&lt;/math&gt; ways to assign opponents to &lt;math&gt;B&lt;/math&gt; and &lt;math&gt;90\cdot10 = 900&lt;/math&gt; ways to order the games. &lt;math&gt;\boxed{\textbf{(E)} 900}&lt;/math&gt;<br /> <br /> ==See Also==<br /> <br /> {{AMC10 box|year=2013|ab=A|num-b=23|num-a=25}}<br /> <br /> [[Category:Introductory Combinatorics Problems]]<br /> {{MAA Notice}}</div> Cj13609517288