https://artofproblemsolving.com/wiki/api.php?action=feedcontributions&user=Minirafa&feedformat=atom AoPS Wiki - User contributions [en] 2021-04-15T05:09:36Z User contributions MediaWiki 1.31.1 https://artofproblemsolving.com/wiki/index.php?title=2011_AMC_10A_Problems/Problem_23&diff=59658 2011 AMC 10A Problems/Problem 23 2014-02-15T15:40:44Z <p>Minirafa: /* Solution */</p> <hr /> <div>== Problem ==<br /> Seven students count from 1 to 1000 as follows:<br /> <br /> •Alice says all the numbers, except she skips the middle number in each consecutive group of three numbers. That is, Alice says 1, 3, 4, 6, 7, 9, . . ., 997, 999, 1000.<br /> <br /> •Barbara says all of the numbers that Alice doesn't say, except she also skips the middle number in each consecutive group of three numbers.<br /> <br /> •Candice says all of the numbers that neither Alice nor Barbara says, except she also skips the middle number in each consecutive group of three numbers.<br /> <br /> •Debbie, Eliza, and Fatima say all of the numbers that none of the students with the first names beginning before theirs in the alphabet say, except each also skips the middle number in each of her consecutive groups of three numbers.<br /> <br /> •Finally, George says the only number that no one else says.<br /> <br /> What number does George say?<br /> <br /> &lt;math&gt; \textbf{(A)}\ 37\qquad\textbf{(B)}\ 242\qquad\textbf{(C)}\ 365\qquad\textbf{(D)}\ 728\qquad\textbf{(E)}\ 998 &lt;/math&gt;<br /> <br /> == Solution ==<br /> First look at the numbers Alice says. &lt;math&gt;1, 3, 4, 6, 7, 9 \cdots&lt;/math&gt; skipping every number that is congruent to &lt;math&gt;2 \pmod 3&lt;/math&gt;. Thus, Barbara says those numbers EXCEPT every second - being &lt;math&gt;2 + 3^1 \equiv 5 \pmod{3^2=9}&lt;/math&gt;. So Barbara skips every number congruent to &lt;math&gt;5 \pmod 9&lt;/math&gt;. We continue and see: <br /> <br /> Alice skips &lt;math&gt;2 \pmod 3&lt;/math&gt;, Barbara skips &lt;math&gt;5 \pmod 9&lt;/math&gt;, Candice skips &lt;math&gt;14 \pmod {27}&lt;/math&gt;, Debbie skips &lt;math&gt;41 \pmod {81}&lt;/math&gt;, Eliza skips &lt;math&gt;122 \pmod {243}&lt;/math&gt;, and Fatima skips &lt;math&gt;365 \pmod {729}&lt;/math&gt;.<br /> <br /> Since the only number congruent to &lt;math&gt;365 \pmod {729}&lt;/math&gt; and less than &lt;math&gt;1,000&lt;/math&gt; is &lt;math&gt;365&lt;/math&gt;, the correct answer is &lt;math&gt; \boxed{365\ \mathbf{(C)}} &lt;/math&gt;.<br /> <br /> == See Also ==<br /> <br /> <br /> {{AMC10 box|year=2011|ab=A|num-b=22|num-a=24}}<br /> {{MAA Notice}}</div> Minirafa https://artofproblemsolving.com/wiki/index.php?title=2005_AMC_12B_Problems&diff=40366 2005 AMC 12B Problems 2011-07-11T21:53:56Z <p>Minirafa: /* Problem 15 */</p> <hr /> <div>== Problem 1 ==<br /> A scout troop buys &lt;math&gt;1000&lt;/math&gt; candy bars at a price of five for &lt;math&gt;2&lt;/math&gt; dollars. They sell all the candy bars at the price of two for &lt;math&gt;1&lt;/math&gt; dollar. What was their profit, in dollars?<br /> <br /> &lt;math&gt;<br /> \mathrm{(A)}\ 100 \qquad<br /> \mathrm{(B)}\ 200 \qquad<br /> \mathrm{(C)}\ 300 \qquad<br /> \mathrm{(D)}\ 400 \qquad<br /> \mathrm{(E)}\ 500<br /> &lt;/math&gt;<br /> <br /> [[2005 AMC 12B Problems/Problem 1|Solution]]<br /> <br /> == Problem 2 ==<br /> A positive number &lt;math&gt;x&lt;/math&gt; has the property that &lt;math&gt;x\%&lt;/math&gt; of &lt;math&gt;x&lt;/math&gt; is &lt;math&gt;4&lt;/math&gt;. What is &lt;math&gt;x&lt;/math&gt;?<br /> <br /> &lt;math&gt;<br /> \mathrm{(A)}\ 2 \qquad<br /> \mathrm{(B)}\ 4 \qquad<br /> \mathrm{(C)}\ 10 \qquad<br /> \mathrm{(D)}\ 20 \qquad<br /> \mathrm{(E)}\ 40<br /> &lt;/math&gt;<br /> <br /> [[2005 AMC 12B Problems/Problem 2|Solution]]<br /> <br /> == Problem 3 ==<br /> Brianna is using part of the money she earned on her weekend job to buy several equally-priced CDs. She used one ﬁfth of her money to buy one third of the CDs. What fraction of her money will she have left after she buys all the CDs? <br /> <br /> &lt;math&gt;<br /> \mathrm{(A)}\ \frac15 \qquad<br /> \mathrm{(B)}\ \frac13 \qquad<br /> \mathrm{(C)}\ \frac25 \qquad<br /> \mathrm{(D)}\ \frac23 \qquad<br /> \mathrm{(E)}\ \frac45<br /> &lt;/math&gt;<br /> <br /> [[2005 AMC 12B Problems/Problem 3|Solution]]<br /> <br /> == Problem 4 ==<br /> At the beginning of the school year, Lisa's goal was to earn an A on at least &lt;math&gt;80\%&lt;/math&gt; of her &lt;math&gt;50&lt;/math&gt; quizzes for the year. She earned an A on &lt;math&gt;22&lt;/math&gt; of the first &lt;math&gt;30&lt;/math&gt; quizzes. If she is to achieve her goal, on at most how many of the remaining quizzes can she earn a grade lower than an A?<br /> <br /> &lt;math&gt;<br /> \mathrm{(A)}\ 1 \qquad<br /> \mathrm{(B)}\ 2 \qquad<br /> \mathrm{(C)}\ 3 \qquad<br /> \mathrm{(D)}\ 4 \qquad<br /> \mathrm{(E)}\ 5<br /> &lt;/math&gt;<br /> <br /> [[2005 AMC 12B Problems/Problem 4|Solution]]<br /> <br /> == Problem 5 ==<br /> An &lt;math&gt;8&lt;/math&gt;-foot by &lt;math&gt;10&lt;/math&gt;-foot floor is tiles with square tiles of size &lt;math&gt;1&lt;/math&gt; foot by &lt;math&gt;1&lt;/math&gt; foot. Each tile has a pattern consisting of four white quarter circles of radius &lt;math&gt;1/2&lt;/math&gt; foot centered at each corner of the tile. The remaining portion of the tile is shaded. How many square feet of the floor are shaded?<br /> <br /> &lt;asy&gt;<br /> unitsize(2cm);<br /> defaultpen(linewidth(.8pt));<br /> fill(unitsquare,gray);<br /> filldraw(Arc((0,0),.5,0,90)--(0,0)--cycle,white,black);<br /> filldraw(Arc((1,0),.5,90,180)--(1,0)--cycle,white,black);<br /> filldraw(Arc((1,1),.5,180,270)--(1,1)--cycle,white,black);<br /> filldraw(Arc((0,1),.5,270,360)--(0,1)--cycle,white,black);<br /> &lt;/asy&gt;<br /> <br /> &lt;math&gt;<br /> \mathrm{(A)}\ 80-20\pi \qquad<br /> \mathrm{(B)}\ 60-10\pi \qquad<br /> \mathrm{(C)}\ 80-10\pi \qquad<br /> \mathrm{(D)}\ 60+10\pi \qquad<br /> \mathrm{(E)}\ 80+10\pi<br /> &lt;/math&gt;<br /> <br /> [[2005 AMC 12B Problems/Problem 5|Solution]]<br /> <br /> == Problem 6 ==<br /> In &lt;math&gt;\triangle ABC&lt;/math&gt;, we have &lt;math&gt;AC=BC=7&lt;/math&gt; and &lt;math&gt;AB=2&lt;/math&gt;. Suppose that &lt;math&gt;D&lt;/math&gt; is a point on line &lt;math&gt;AB&lt;/math&gt; such that &lt;math&gt;B&lt;/math&gt; lies between &lt;math&gt;A&lt;/math&gt; and &lt;math&gt;D&lt;/math&gt; and &lt;math&gt;CD=8&lt;/math&gt;. What is &lt;math&gt;BD&lt;/math&gt;?<br /> <br /> &lt;math&gt;<br /> \mathrm{(A)}\ 3 \qquad<br /> \mathrm{(B)}\ 2\sqrt{3} \qquad<br /> \mathrm{(C)}\ 4 \qquad<br /> \mathrm{(D)}\ 5 \qquad<br /> \mathrm{(E)}\ 4\sqrt{2}<br /> &lt;/math&gt;<br /> <br /> [[2005 AMC 12B Problems/Problem 6|Solution]]<br /> <br /> == Problem 7 ==<br /> What is the area enclosed by the graph of &lt;math&gt;|3x|+|4y|=12&lt;/math&gt;?<br /> <br /> &lt;math&gt;<br /> \mathrm{(A)}\ 6 \qquad<br /> \mathrm{(B)}\ 12 \qquad<br /> \mathrm{(C)}\ 16 \qquad<br /> \mathrm{(D)}\ 24 \qquad<br /> \mathrm{(E)}\ 25<br /> &lt;/math&gt;<br /> <br /> [[2005 AMC 12B Problems/Problem 7|Solution]]<br /> <br /> == Problem 8 ==<br /> For how many values of &lt;math&gt;a&lt;/math&gt; is it true that the line &lt;math&gt;y = x + a&lt;/math&gt; passes through the<br /> vertex of the parabola &lt;math&gt;y = x^2 + a^2&lt;/math&gt; ?<br /> <br /> &lt;math&gt;<br /> \mathrm{(A)}\ 0 \qquad<br /> \mathrm{(B)}\ 1 \qquad<br /> \mathrm{(C)}\ 2 \qquad<br /> \mathrm{(D)}\ 10 \qquad<br /> \mathrm{(E)}\ \text{infinitely many}<br /> &lt;/math&gt;<br /> <br /> [[2005 AMC 12B Problems/Problem 8|Solution]]<br /> <br /> == Problem 9 ==<br /> On a certain math exam, &lt;math&gt;10&lt;/math&gt;% of the students got &lt;math&gt;70&lt;/math&gt; points, &lt;math&gt;25&lt;/math&gt;% got &lt;math&gt;80&lt;/math&gt; points, &lt;math&gt;20&lt;/math&gt;% got &lt;math&gt;85&lt;/math&gt; points, &lt;math&gt;15&lt;/math&gt;% got &lt;math&gt;90&lt;/math&gt; points, and the rest got &lt;math&gt;95&lt;/math&gt; points. What is the difference between the mean and the median score on this exam?<br /> <br /> &lt;math&gt;\mathrm{(A)}\ {{{0}}} \qquad \mathrm{(B)}\ {{{1}}} \qquad \mathrm{(C)}\ {{{2}}} \qquad \mathrm{(D)}\ {{{4}}} \qquad \mathrm{(E)}\ {{{5}}}&lt;/math&gt;<br /> <br /> [[2005 AMC 12B Problems/Problem 9|Solution]]<br /> <br /> == Problem 10 ==<br /> The first term of a sequence is &lt;math&gt;2005&lt;/math&gt;. Each succeeding term is the sum of the cubes of the digits of the previous terms. What is the &lt;math&gt;2005^{\text{th}}&lt;/math&gt; term of the sequence?<br /> <br /> &lt;math&gt;\mathrm{(A)}\ {{{29}}} \qquad \mathrm{(B)}\ {{{55}}} \qquad \mathrm{(C)}\ {{{85}}} \qquad \mathrm{(D)}\ {{{133}}} \qquad \mathrm{(E)}\ {{{250}}}&lt;/math&gt;<br /> <br /> [[2005 AMC 12B Problems/Problem 10|Solution]]<br /> <br /> == Problem 11 ==<br /> An envelope contains eight bills: &lt;math&gt;2&lt;/math&gt; ones, &lt;math&gt;2&lt;/math&gt; fives, &lt;math&gt;2&lt;/math&gt; tens, and &lt;math&gt;2&lt;/math&gt; twenties. Two bills are drawn at random without replacement. What is the probability that their sum is &lt;math&gt;20&lt;/math&gt; or more?<br /> <br /> &lt;math&gt;\mathrm{(A)}\ {{{\frac{1}{4}}}} \qquad \mathrm{(B)}\ {{{\frac{2}{7}}}} \qquad \mathrm{(C)}\ {{{\frac{3}{7}}}} \qquad \mathrm{(D)}\ {{{\frac{1}{2}}}} \qquad \mathrm{(E)}\ {{{\frac{2}{3}}}}&lt;/math&gt;<br /> <br /> [[2005 AMC 12B Problems/Problem 11|Solution]]<br /> <br /> == Problem 12 ==<br /> The [[quadratic equation]] &lt;math&gt;x^2+mx+n&lt;/math&gt; has roots twice those of &lt;math&gt;x^2+px+m&lt;/math&gt;, and none of &lt;math&gt;m,n,&lt;/math&gt; and &lt;math&gt;p&lt;/math&gt; is zero. What is the value of &lt;math&gt;n/p&lt;/math&gt;?<br /> <br /> &lt;math&gt;\mathrm{(A)}\ {{{1}}} \qquad \mathrm{(B)}\ {{{2}}} \qquad \mathrm{(C)}\ {{{4}}} \qquad \mathrm{(D)}\ {{{8}}} \qquad \mathrm{(E)}\ {{{16}}}&lt;/math&gt;<br /> <br /> [[2005 AMC 12B Problems/Problem 12|Solution]]<br /> <br /> == Problem 13 ==<br /> Suppose that &lt;math&gt;4^{x_1}=5&lt;/math&gt;, &lt;math&gt;5^{x_2}=6&lt;/math&gt;, &lt;math&gt;6^{x_3}=7&lt;/math&gt;, ... , &lt;math&gt;127^{x_{124}}=128&lt;/math&gt;. What is &lt;math&gt;x_1x_2...x_{124}&lt;/math&gt;?<br /> <br /> &lt;math&gt;\mathrm{(A)}\ {{{2}}} \qquad \mathrm{(B)}\ {{{\frac{5}{2}}}} \qquad \mathrm{(C)}\ {{{3}}} \qquad \mathrm{(D)}\ {{{\frac{7}{2}}}} \qquad \mathrm{(E)}\ {{{4}}}&lt;/math&gt;<br /> <br /> [[2005 AMC 12B Problems/Problem 13|Solution]]<br /> <br /> == Problem 14 ==<br /> <br /> A circle having center &lt;math&gt;(0,k)&lt;/math&gt;, with &lt;math&gt;k&gt;6&lt;/math&gt;,is tangent to the lines &lt;math&gt;y=x&lt;/math&gt;, &lt;math&gt;y=-x&lt;/math&gt; and &lt;math&gt;y=6&lt;/math&gt;. What is the radius of this circle?<br /> <br /> &lt;math&gt;<br /> \mathrm{(A)}\ 6\sqrt{2}-6 \qquad<br /> \mathrm{(B)}\ 6 \qquad<br /> \mathrm{(C)}\ 6\sqrt{2} \qquad<br /> \mathrm{(D)}\ 12 \qquad<br /> \mathrm{(E)}\ 6+6\sqrt{2}<br /> &lt;/math&gt;<br /> <br /> [[2005 AMC 12B Problems/Problem 14|Solution]]<br /> <br /> == Problem 15 ==<br /> <br /> The sum of four two-digit numbers is &lt;math&gt;221&lt;/math&gt;. None of the eight digits is &lt;math&gt;0&lt;/math&gt; and no two of them are the same. Which of the following is '''not''' included among the eight digits?<br /> <br /> &lt;math&gt;<br /> \mathrm{(A)}\ 1 \qquad<br /> \mathrm{(B)}\ 2 \qquad<br /> \mathrm{(C)}\ 3 \qquad<br /> \mathrm{(D)}\ 4 \qquad<br /> \mathrm{(E)}\ 5<br /> &lt;/math&gt;<br /> <br /> [[2005 AMC 12B Problems/Problem 15|Solution]]<br /> <br /> == Problem 16 ==<br /> Eight spheres of radius 1, one per octant, are each tangent to the coordinate planes. What is the radius of the smallest sphere, centered at the origin, that contains these eight spheres?<br /> <br /> &lt;math&gt;<br /> \mathrm (A)\ \sqrt{2} \qquad<br /> \mathrm (B)\ \sqrt{3} \qquad<br /> \mathrm (C)\ 1+\sqrt{2}\qquad<br /> \mathrm (D)\ 1+\sqrt{3}\qquad<br /> \mathrm (E)\ 3<br /> &lt;/math&gt;<br /> <br /> [[2005 AMC 12B Problems/Problem 16|Solution]]<br /> <br /> == Problem 17 ==<br /> <br /> How many distinct four-tuples &lt;math&gt;(a, b, c, d)&lt;/math&gt; of rational numbers are there with<br /> <br /> &lt;math&gt;a \cdot \log_{10} 2+b \cdot \log_{10} 3 +c \cdot \log_{10} 5 + d \cdot \log_{10} 7 = 2005&lt;/math&gt;?<br /> <br /> &lt;math&gt;<br /> \mathrm{(A)}\ 0 \qquad<br /> \mathrm{(B)}\ 1 \qquad<br /> \mathrm{(C)}\ 17 \qquad<br /> \mathrm{(D)}\ 2004 \qquad<br /> \mathrm{(E)}\ \text{infinitely many}<br /> &lt;/math&gt;<br /> <br /> [[2005 AMC 12B Problems/Problem 17|Solution]]<br /> <br /> == Problem 18 ==<br /> <br /> Let &lt;math&gt;A(2,2)&lt;/math&gt; and &lt;math&gt;B(7,7)&lt;/math&gt; be points in the plane. Define &lt;math&gt;R&lt;/math&gt; as the region in the first quadrant consisting of those points &lt;math&gt;C&lt;/math&gt; such that &lt;math&gt;\triangle ABC&lt;/math&gt; is an acute triangle. What is the closest integer to the area of the region &lt;math&gt;R&lt;/math&gt;?<br /> <br /> &lt;math&gt;<br /> \mathrm{(A)}\ 25 \qquad<br /> \mathrm{(B)}\ 39 \qquad<br /> \mathrm{(C)}\ 51 \qquad<br /> \mathrm{(D)}\ 60 \qquad<br /> \mathrm{(E)}\ 80 \qquad<br /> &lt;/math&gt;<br /> <br /> [[2005 AMC 12B Problems/Problem 18|Solution]]<br /> <br /> == Problem 19 ==<br /> <br /> Let &lt;math&gt;x&lt;/math&gt; and &lt;math&gt;y&lt;/math&gt; be two-digit integers such that &lt;math&gt;y&lt;/math&gt; is obtained by reversing the digits of &lt;math&gt;x&lt;/math&gt;. The integers &lt;math&gt;x&lt;/math&gt; and &lt;math&gt;y&lt;/math&gt; satisfy &lt;math&gt;x^{2}-y^{2}=m^{2}&lt;/math&gt; for some positive integer &lt;math&gt;m&lt;/math&gt;. What is &lt;math&gt;x+y+m&lt;/math&gt;?<br /> <br /> &lt;math&gt;<br /> \mathrm{(A)}\ 88 \qquad<br /> \mathrm{(B)}\ 112 \qquad<br /> \mathrm{(C)}\ 116 \qquad<br /> \mathrm{(D)}\ 144 \qquad<br /> \mathrm{(E)}\ 154 \qquad<br /> &lt;/math&gt;<br /> <br /> [[2005 AMC 12B Problems/Problem 19|Solution]]<br /> <br /> == Problem 20 ==<br /> <br /> Let &lt;math&gt;a,b,c,d,e,f,g&lt;/math&gt; and &lt;math&gt;h&lt;/math&gt; be distinct elements in the set<br /> <br /> &lt;cmath&gt;\{-7,-5,-3,-2,2,4,6,13\}.&lt;/cmath&gt;<br /> <br /> What is the minimum possible value of<br /> <br /> &lt;cmath&gt;(a+b+c+d)^{2}+(e+f+g+h)^{2}?&lt;/cmath&gt;<br /> <br /> &lt;math&gt;<br /> \mathrm{(A)}\ 30 \qquad<br /> \mathrm{(B)}\ 32 \qquad<br /> \mathrm{(C)}\ 34 \qquad<br /> \mathrm{(D)}\ 40 \qquad<br /> \mathrm{(E)}\ 50<br /> &lt;/math&gt;<br /> <br /> [[2005 AMC 12B Problems/Problem 20|Solution]]<br /> <br /> == Problem 21 ==<br /> A positive integer &lt;math&gt;n&lt;/math&gt; has &lt;math&gt;60&lt;/math&gt; divisors and &lt;math&gt;7n&lt;/math&gt; has &lt;math&gt;80&lt;/math&gt; divisors. What is the greatest integer &lt;math&gt;k&lt;/math&gt; such that &lt;math&gt;7^k&lt;/math&gt; divides &lt;math&gt;n&lt;/math&gt;?<br /> <br /> &lt;math&gt;\mathrm{(A)}\ {{{0}}} \qquad \mathrm{(B)}\ {{{1}}} \qquad \mathrm{(C)}\ {{{2}}} \qquad \mathrm{(D)}\ {{{3}}} \qquad \mathrm{(E)}\ {{{4}}}&lt;/math&gt;<br /> <br /> [[2005 AMC 12B Problems/Problem 21|Solution]]<br /> <br /> == Problem 22 ==<br /> <br /> A sequence of complex numbers &lt;math&gt;z_{0}, z_{1}, z_{2}, ...&lt;/math&gt; is defined by the rule<br /> <br /> &lt;cmath&gt;z_{n+1} = \frac {iz_{n}}{\overline {z_{n}}},&lt;/cmath&gt;<br /> <br /> where &lt;math&gt;\overline {z_{n}}&lt;/math&gt; is the [[complex conjugate]] of &lt;math&gt;z_{n}&lt;/math&gt; and &lt;math&gt;i^{2}=-1&lt;/math&gt;. Suppose that &lt;math&gt;|z_{0}|=1&lt;/math&gt; and &lt;math&gt;z_{2005}=1&lt;/math&gt;. How many possible values are there for &lt;math&gt;z_{0}&lt;/math&gt;?<br /> <br /> &lt;math&gt;<br /> \textbf{(A)}\ 1 \qquad <br /> \textbf{(B)}\ 2 \qquad <br /> \textbf{(C)}\ 4 \qquad <br /> \textbf{(D)}\ 2005 \qquad <br /> \textbf{(E)}\ 2^{2005}<br /> &lt;/math&gt;<br /> <br /> [[2005 AMC 12B Problems/Problem 22|Solution]]<br /> <br /> == Problem 23 ==<br /> <br /> Let &lt;math&gt;S&lt;/math&gt; be the set of ordered triples &lt;math&gt;(x,y,z)&lt;/math&gt; of real numbers for which<br /> <br /> &lt;cmath&gt;\log_{10}(x+y) = z \text{ and } \log_{10}(x^{2}+y^{2}) = z+1.&lt;/cmath&gt;<br /> There are real numbers &lt;math&gt;a&lt;/math&gt; and &lt;math&gt;b&lt;/math&gt; such that for all ordered triples &lt;math&gt;(x,y.z)&lt;/math&gt; in &lt;math&gt;S&lt;/math&gt; we have &lt;math&gt;x^{3}+y^{3}=a \cdot 10^{3z} + b \cdot 10^{2z}.&lt;/math&gt; What is the value of &lt;math&gt;a+b?&lt;/math&gt;<br /> <br /> &lt;math&gt;<br /> \textbf{(A)}\ \frac {15}{2} \qquad <br /> \textbf{(B)}\ \frac {29}{2} \qquad <br /> \textbf{(C)}\ 15 \qquad <br /> \textbf{(D)}\ \frac {39}{2} \qquad <br /> \textbf{(E)}\ 24<br /> &lt;/math&gt;<br /> <br /> [[2005 AMC 12B Problems/Problem 23|Solution]]<br /> <br /> == Problem 24 ==<br /> All three vertices of an equilateral triangle are on the parabola &lt;math&gt;y=x^2&lt;/math&gt;, and one of its sides has a slope of &lt;math&gt;2&lt;/math&gt;. The &lt;math&gt;x&lt;/math&gt;-coordinates of the three vertices have a sum of &lt;math&gt;m/n&lt;/math&gt;, where &lt;math&gt;m&lt;/math&gt; and &lt;math&gt;n&lt;/math&gt; are relatively prime positive integers. What is the value of &lt;math&gt;m+n&lt;/math&gt;?<br /> <br /> &lt;math&gt;\mathrm{(A)}\ {{{14}}} \qquad \mathrm{(B)}\ {{{15}}} \qquad \mathrm{(C)}\ {{{16}}} \qquad \mathrm{(D)}\ {{{17}}} \qquad \mathrm{(E)}\ {{{18}}}&lt;/math&gt;<br /> <br /> [[2005 AMC 12B Problems/Problem 24|Solution]]<br /> <br /> == Problem 25 ==<br /> <br /> Six ants simultaneously stand on the six [[vertex|vertices]] of a regular [[octahedron]], with each ant at a different vertex. Simultaneously and independently, each ant moves from its vertex to one of the four adjacent vertices, each with equal [[probability]]. What is the probability that no two ants arrive at the same vertex?<br /> <br /> &lt;math&gt;\mathrm{(A)}\ \frac {5}{256}<br /> \qquad\mathrm{(B)}\ \frac {21}{1024}<br /> \qquad\mathrm{(C)}\ \frac {11}{512}<br /> \qquad\mathrm{(D)}\ \frac {23}{1024}<br /> \qquad\mathrm{(E)}\ \frac {3}{128}&lt;/math&gt;<br /> <br /> [[2005 AMC 12B Problems/Problem 25|Solution]]<br /> <br /> == See also ==<br /> * [[AMC 12]]<br /> * [[AMC 12 Problems and Solutions]]<br /> * [[2005 AMC 12B]]<br /> * [http://www.artofproblemsolving.com/Community/AoPS_Y_MJ_Transcripts.php?mj_id=49 2005 AMC B Math Jam Transcript]<br /> * [[Mathematics competition resources]]</div> Minirafa https://artofproblemsolving.com/wiki/index.php?title=2003_AMC_12A_Problems/Problem_13&diff=40196 2003 AMC 12A Problems/Problem 13 2011-07-04T15:57:47Z <p>Minirafa: /* Problem */</p> <hr /> <div>== Problem ==<br /> The [[polygon]] enclosed by the solid lines in the figure consists of 4 [[congruent]] [[square (geometry) | squares]] joined [[edge]]-to-edge. One more congruent square is attached to an edge at one of the nine positions indicated. How many of the nine resulting polygons can be folded to form a [[cube (geometry) | cube]] with one face missing? <br /> <br /> [[Image:2003amc10a10.gif]]<br /> <br /> &lt;math&gt; \mathrm{(A) \ } 2\qquad \mathrm{(B) \ } 3\qquad \mathrm{(C) \ } 4\qquad \mathrm{(D) \ } 5\qquad \mathrm{(E) \ } 6 &lt;/math&gt;<br /> <br /> == Solution ==<br /> [[Image:2003amc10a10.gif]]<br /> <br /> Let the squares be labeled &lt;math&gt;A&lt;/math&gt;, &lt;math&gt;B&lt;/math&gt;, &lt;math&gt;C&lt;/math&gt;, and &lt;math&gt;D&lt;/math&gt;.<br /> <br /> When the polygon is folded, the &quot;right&quot; edge of square &lt;math&gt;A&lt;/math&gt; becomes adjacent to the &quot;bottom edge&quot; of square &lt;math&gt;C&lt;/math&gt;, and the &quot;bottom&quot; edge of square &lt;math&gt;A&lt;/math&gt; becomes adjacent to the &quot;bottom&quot; edge of square &lt;math&gt;D&lt;/math&gt;. <br /> <br /> So, any &quot;new&quot; square that is attatched to those edges will prevent the polygon from becoming a cube with one face missing. <br /> <br /> Therefore, squares &lt;math&gt;1&lt;/math&gt;, &lt;math&gt;2&lt;/math&gt;, and &lt;math&gt;3&lt;/math&gt; will prevent the polygon from becoming a cube with one face missing.<br /> <br /> Squares &lt;math&gt;4&lt;/math&gt;, &lt;math&gt;5&lt;/math&gt;, &lt;math&gt;6&lt;/math&gt;, &lt;math&gt;7&lt;/math&gt;, &lt;math&gt;8&lt;/math&gt;, and &lt;math&gt;9&lt;/math&gt; will allow the polygon to become a cube with one face missing when folded. <br /> <br /> Thus the answer is &lt;math&gt;6 \Rightarrow E&lt;/math&gt;.<br /> <br /> == See Also ==<br /> *[[2003 AMC 12A Problems]]<br /> {{AMC12 box|year=2003|ab=A|num-b=12|num-a=14}}<br /> <br /> [[Category:Introductory Geometry Problems]]</div> Minirafa https://artofproblemsolving.com/wiki/index.php?title=2011_AMC_10A_Problems/Problem_23&diff=37090 2011 AMC 10A Problems/Problem 23 2011-02-23T02:55:26Z <p>Minirafa: /* Problem */</p> <hr /> <div>== Problem ==<br /> Seven students count from 1 to 1000 as follows:<br /> <br /> •Alice says all the numbers, except she skips the middle number in each consecutive group of three numbers. That is, Alice says 1, 3, 4, 6, 7, 9, . . ., 997, 999, 1000.<br /> <br /> •Barbara says all of the numbers that Alice doesn't say, except she also skips the middle number in each consecutive group of three numbers.<br /> <br /> •Candice says all of the numbers that neither Alice nor Barbara says, except she also skips the middle number in each consecutive group of three numbers.<br /> <br /> •Debbie, Eliza, and Fatima say all of the numbers that none of the students with the first names beginning before theirs in the alphabet say, except each also skips the middle number in each of her consecutive groups of three numbers.<br /> <br /> •Finally, George says the only number that no one else says.<br /> <br /> What number does George say?<br /> <br /> &lt;math&gt; \textbf{(A)}\ 37\qquad\textbf{(B)}\ 242\qquad\textbf{(C)}\ 365\qquad\textbf{(D)}\ 728\qquad\textbf{(E)}\ 998 &lt;/math&gt;<br /> <br /> == Solution ==<br /> First look at the numbers Alice says. 1, 3, 4, 6, 7, 9 ... skipping every number that is congruent to 2 mod 3. Thus, Barbara says those numbers EXCEPT every second - being &lt;math&gt;2 + 3^1 = 5&lt;/math&gt; mod &lt;math&gt;3^2=9&lt;/math&gt;. So Barbara skips every number congruent to 5 mod 9. We continue on and see: (see this for yourself)<br /> <br /> Alice skips 2 mod 3, Barbara skips 5 mod 9, Candice skips 14 mod 27, Debbie skips 41 mod 81, Eliza skips 122 mod 423, and Fatima skips 365 mod 729.<br /> <br /> Since the only number congruent to 365 mod 729 and less than 1,000 is 365, the correct answer is '''(C)'''.</div> Minirafa