https://artofproblemsolving.com/wiki/api.php?action=feedcontributions&user=B-flat&feedformat=atom AoPS Wiki - User contributions [en] 2022-05-24T02:46:53Z User contributions MediaWiki 1.31.1 https://artofproblemsolving.com/wiki/index.php?title=2004_AMC_12A_Problems/Problem_15&diff=11185 2004 AMC 12A Problems/Problem 15 2006-11-06T02:40:37Z <p>B-flat: </p> <hr /> <div>==Problem==<br /> Brenda and Sally run in opposite directions on a circular track, starting at diametrically opposite points. They first meet after Brenda has run 100 meters. They next meet after Sally has run 150 meters past their first meeting point. Each girl runs at a constant speed. What is the length of the track in meters?<br /> <br /> &lt;math&gt; \mathrm{(A) \ } 250 \qquad \mathrm{(B) \ } 300 \qquad \mathrm{(C) \ } 350 \qquad \mathrm{(D) \ } 400\qquad \mathrm{(E) \ } 500 &lt;/math&gt;</div> B-flat https://artofproblemsolving.com/wiki/index.php?title=2004_AMC_10A_Problems/Problem_24&diff=11184 2004 AMC 10A Problems/Problem 24 2006-11-06T02:40:21Z <p>B-flat: NOT COMPLETE!!! (will finish later)</p> <hr /> <div>==Problem==<br /> Let &lt;math&gt;a_1,a_2,\cdots&lt;/math&gt;, be a sequence with the following properties.<br /> <br /> (i) &lt;math&gt;a_1=1&lt;/math&gt;, and<br /> <br /> (ii) &lt;math&gt;a_{2n}=n\cdot a_n&lt;/math&gt; for any positive integer &lt;math&gt;n&lt;/math&gt;.<br /> <br /> What is the value of &lt;math&gt;a_{2^{100}}&lt;/math&gt;?<br /> <br /> &lt;math&gt; \mathrm{(A) \ } 1 \qquad \mathrm{(B) \ } 2^{99} \qquad \mathrm{(C) \ } 2^{100} \qquad \mathrm{(D) \ } 2^{4050} \qquad \mathrm{(E) \ } 2^{9999} &lt;/math&gt;<br /> <br /> ==Solution==<br /> Note that<br /> <br /> &lt;math&gt;a_2=2a_1&lt;/math&gt;<br /> <br /> &lt;math&gt;a_{2^2}=2\cdot a_2=2\cdot1=2&lt;/math&gt;<br /> <br /> &lt;math&gt;a_{2^3}=4\cdot a_4=2^3\cdot2^{2+1}&lt;/math&gt;<br /> <br /> &lt;math&gt;a_{2^8}=8\cdot a_8=2^3\cdot&lt;/math&gt;</div> B-flat https://artofproblemsolving.com/wiki/index.php?title=2004_AMC_10A_Problems/Problem_18&diff=11157 2004 AMC 10A Problems/Problem 18 2006-11-05T23:42:15Z <p>B-flat: </p> <hr /> <div>==Problem==<br /> A sequence of three real numbers forms an arithmetic progression with a first term of 9. If 2 is added to the secon term and 20 is added to the third term, the three resulting numbers form a geometric progression. What is the smallest possible value for the third term of the geometric progression?<br /> <br /> &lt;math&gt; \mathrm{(A) \ } 1 \qquad \mathrm{(B) \ } 4 \qquad \mathrm{(C) \ } 36 \qquad \mathrm{(D) \ } 49 \qquad \mathrm{(E) \ } 81 &lt;/math&gt;<br /> <br /> ==Solution==<br /> Let d be the difference between terms in the arithmetic progression, such that first three terms are 9, 9+d, and 9+2d. The terms of the geometric progression will be 9, 11+d, and 29+2d. Because they are in a geometric progression, we can say<br /> <br /> &lt;math&gt;(11+d)^2=9(29+2d)&lt;/math&gt;, note that the terms of a geometric progression would be &lt;math&gt;\frac{x}r&lt;/math&gt;, x, and &lt;math&gt;xr&lt;/math&gt;, so &lt;math&gt;x^2=\frac{x}r\times r&lt;/math&gt;.<br /> <br /> Solve this equation.<br /> <br /> &lt;math&gt;d^2+4d-140=0&lt;/math&gt;<br /> <br /> &lt;math&gt;(d-10)(d+14)=0&lt;/math&gt;<br /> <br /> Substituting &lt;math&gt;d=10&lt;/math&gt; into the geometric progression gives the terms 9, 21, and 49; substituting &lt;math&gt;d=14&lt;/math&gt; gives 9, -3, and 1. Therefore, the smallest possible 3rd term is 1 &lt;math&gt;\Rightarrow\mathrm{(A)}&lt;/math&gt;.<br /> <br /> ==See Also==<br /> <br /> *[[2004 AMC 10A Problems]]<br /> <br /> *[[2004 AMC 10A Problems/Problem 17|Previous Problem]]<br /> <br /> *[[2004 AMC 10A Problems/Problem 19|Next Problem]]</div> B-flat https://artofproblemsolving.com/wiki/index.php?title=User:B-flat&diff=10810 User:B-flat 2006-11-04T22:33:50Z <p>B-flat: </p> <hr /> <div>== b-flat ==<br /> <br /> Here is my user page.<br /> <br /> I live in North Carolina, and I enjoy math, programming, chess, biking, and playing piano.<br /> <br /> == Contributions ==<br /> <br /> * Made a bunch of articles look nicer<br /> <br /> * Created Area of common geometric figures<br /> * Added Area of quadrilaterals<br /> <br /> * Created [[Greatest common divisor]] (GCD)<br /> <br /> * Added Approximating [[pi]] - June 26, 2006<br /> <br /> * Added [[Chinese Remainder Theorem]] Example - June 27, 2006<br /> <br /> * Added Introduction to Completing the Square - June 27, 2005<br /> <br /> * Created [[Rhombus]] Page - June 27, 2006<br /> <br /> * Created [[Parallelogram]] Page - June 27, 2006<br /> <br /> * Created [[Rectangle]] Page - June 27, 2006<br /> <br /> * Created [[Kite]] Page - June 27, 2006<br /> <br /> * Created [[Trapezoid]] Page - June 27, 2006<br /> <br /> * Created [[Isosceles trapezoid]] Page - June 27, 2006<br /> <br /> * Added Perfect Square trinomials to [[perfect square]] page - June 29, 2006<br /> <br /> * Created [[2006 AMC 10A]] Page - June 29, 2006<br /> <br /> * Added AMC 20A problems Page - July 1, 2006<br /> <br /> * Created [[constant]] Page - July 6, 2006<br /> <br /> * Created [[variable]] Page - July 6, 2006<br /> <br /> * Created [[perimeter]] Page - July 6, 2006<br /> <br /> * Created [[interval]] Page - July 11, 2006<br /> <br /> * Created [[contrapositive]] Page - July 16, 2006<br /> <br /> * Updated [[LaTeX]] Page - July 22, 2006<br /> <br /> * Created [[Proper fraction]] Page - October 19, 2006<br /> <br /> * Created [[2005 AMC 10A Problems]] Page - November 4, 2006<br /> <br /> * Created [[2005 AMC 10A Problems/Problem 20]] Page - November 4, 2006<br /> <br /> * Created [[2004 AMC 10A Problems/Problem 1]] Page - November 4, 2006<br /> <br /> * Created [[2004 AMC 10A Problems/Problem 3]] Page - November 4, 2006<br /> <br /> * Created [[2004 AMC 10A Problems/Problem 4]] Page - November 4, 2006<br /> <br /> * Created [[2004 AMC 10A Problems/Problem 6]] Page - November 4, 2006<br /> <br /> * Created [[2004 AMC 10A Problems/Problem 7]] Page - November 4, 2006<br /> <br /> * Created [[2004 AMC 10A Problems/Problem 8]] Page - November 4, 2006<br /> <br /> * Created [[2004 AMC 10A Problems/Problem 10]] Page - November 4, 2006<br /> <br /> * Created [[2004 AMC 10A Problems/Problem 11]] Page - November 4, 2006<br /> <br /> * Created [[2004 AMC 10A Problems/Problem 12]] Page - November 4, 2006<br /> <br /> * Created [[2004 AMC 10A Problems/Problem 13]] Page - November 4, 2006<br /> <br /> * Created [[2004 AMC 10A Problems/Problem 14]] Page - November 4, 2006<br /> <br /> * Created [[2004 AMC 10A Problems/Problem 15]] Page - November 4, 2006</div> B-flat https://artofproblemsolving.com/wiki/index.php?title=2004_AMC_10A_Problems/Problem_15&diff=10809 2004 AMC 10A Problems/Problem 15 2006-11-04T22:32:36Z <p>B-flat: </p> <hr /> <div>==Problem==<br /> Given that &lt;math&gt;-4\leq x\leq-2&lt;/math&gt; and &lt;math&gt;2\leq y\leq4&lt;/math&gt;, what is the largest possible value of (x+y)/x?<br /> <br /> &lt;math&gt; \mathrm{(A) \ } -1 \qquad \mathrm{(B) \ } -\frac12 \qquad \mathrm{(C) \ } 0 \qquad \mathrm{(D) \ } \frac12 {2}\qquad \mathrm{(E) \ } 1 &lt;/math&gt;<br /> <br /> ==Solution==<br /> Rewrite &lt;math&gt;\frac{(x+y)}x&lt;/math&gt; as &lt;math&gt;\frac{x}x+\frac{y}x=1+\frac{y}x&lt;/math&gt;.<br /> <br /> We also know that &lt;math&gt;\frac{y}x&lt;0&lt;/math&gt; because &lt;math&gt;x&lt;/math&gt; and &lt;math&gt;y&lt;/math&gt; are of opposite parity.<br /> <br /> Therefore, &lt;math&gt;1+\frac{y}x&lt;/math&gt; is maximized when &lt;math&gt;\frac{y}x&lt;/math&gt; is minimized, which occurs when &lt;math&gt;|x|&lt;/math&gt; is the largest and &lt;math&gt;|y|&lt;/math&gt; is the smallest.<br /> <br /> This occurs at (-4,2), so &lt;math&gt;\frac{x+y}x=1-\frac12=\frac12\Rightarrow \mathrm{(D)}&lt;/math&gt;.<br /> <br /> ==See Also==<br /> <br /> *[[2004 AMC 10A Problems]]<br /> <br /> *[[2004 AMC 10A Problems/Problem 14|Previous Problem]]<br /> <br /> *[[2004 AMC 10A Problems/Problem 16|Next Problem]]</div> B-flat https://artofproblemsolving.com/wiki/index.php?title=2004_AMC_10A_Problems/Problem_14&diff=10807 2004 AMC 10A Problems/Problem 14 2006-11-04T22:25:16Z <p>B-flat: </p> <hr /> <div>==Problem==<br /> The average value of all the pennies, nickels, dimes, and quarters in Paula's purse is 20 cents. If she had one more quarter, the average would be 21 cents. How many dimes does she have in her purse?<br /> <br /> &lt;math&gt; \mathrm{(A) \ } 0 \qquad \mathrm{(B) \ } 1 \qquad \mathrm{(C) \ } 2 \qquad \mathrm{(D) \ } 3 {2}\qquad \mathrm{(E) \ } 4 &lt;/math&gt;<br /> <br /> ==Solution==<br /> Let &lt;math&gt;n&lt;/math&gt; be the number of coins in Paula's purse. Thus, the total value of the coins in her purse is &lt;math&gt;20n&lt;/math&gt;. When 1 more quarter is added, there are &lt;math&gt;n+1&lt;/math&gt; coins, with an average of 21, or &lt;math&gt;21(n+1)&lt;/math&gt; total cents. This can also be expressed as &lt;math&gt;20n+25&lt;/math&gt;, so we set them equal and solve for &lt;math&gt;n&lt;/math&gt;.<br /> <br /> &lt;math&gt;\displaystyle21(n+1)=20n+25&lt;/math&gt;<br /> <br /> &lt;math&gt;\displaystyle n=4&lt;/math&gt;<br /> <br /> Therefore, the total value of the coins was &lt;math&gt;20\times4=80&lt;/math&gt; cents, which can only be made by using 3 quarters and 1 nickel, so there aren't any dimes &lt;math&gt;\Rightarrow\mathrm{(A)}&lt;/math&gt;.<br /> <br /> ==See Also==<br /> <br /> *[[2004 AMC 10A Problems]]<br /> <br /> *[[2004 AMC 10A Problems/Problem 13|Previous Problem]]<br /> <br /> *[[2004 AMC 10A Problems/Problem 15|Next Problem]]</div> B-flat https://artofproblemsolving.com/wiki/index.php?title=2004_AMC_10A_Problems/Problem_13&diff=10806 2004 AMC 10A Problems/Problem 13 2006-11-04T22:18:05Z <p>B-flat: </p> <hr /> <div>==Problem==<br /> At a party, each man danced with exactly three women and each woman danced with exactly two men. Twelve men attended the party. How many women attended the party?<br /> <br /> &lt;math&gt; \mathrm{(A) \ } 8 \qquad \mathrm{(B) \ } 12 \qquad \mathrm{(C) \ } 16 \qquad \mathrm{(D) \ } 18 {2}\qquad \mathrm{(E) \ } 24 &lt;/math&gt;<br /> <br /> ==Solution==<br /> If each man danced with 3 women, then there were a total of &lt;math&gt;3\times12=36&lt;/math&gt; pairs of a man and a women. However, each women only danced with 2 men, so there must have been &lt;math&gt;\frac{36}2=18&lt;/math&gt; women &lt;math&gt;\Rightarrow\mathrm{(D)}&lt;/math&gt;.<br /> <br /> ==See Also==<br /> <br /> *[[2004 AMC 10A Problems]]<br /> <br /> *[[2004 AMC 10A Problems/Problem 12|Previous Problem]]<br /> <br /> *[[2004 AMC 10A Problems/Problem 14|Next Problem]]</div> B-flat https://artofproblemsolving.com/wiki/index.php?title=2004_AMC_10A_Problems/Problem_12&diff=10805 2004 AMC 10A Problems/Problem 12 2006-11-04T22:14:55Z <p>B-flat: </p> <hr /> <div>==Problem==<br /> Henry's Hamburger Heaven offers its hamburgers with the following condiments: ketchup, mustard, mayonnaise, tomato, lettuce, pickles, cheese, and onions. A costomer can choose one, two, or three meat patties, and any collection of condiments. How many different kinds of hamburgers can be ordered?<br /> <br /> &lt;math&gt; \mathrm{(A) \ } 24 \qquad \mathrm{(B) \ } 256 \qquad \mathrm{(C) \ } 768 \qquad \mathrm{(D) \ } 40,320 {2}\qquad \mathrm{(E) \ } 120,960 &lt;/math&gt;<br /> <br /> ==Solution==<br /> A costomer can either order a condiment, or not, and there are 8 condiments. Therefore, there are &lt;math&gt;2^8=256&lt;/math&gt; ways to order the condiments.<br /> <br /> There are also 3 choices for the meat, making a total of &lt;math&gt;256\times3=768&lt;/math&gt; possible hamburgers &lt;math&gt;\Rightarrow\mathrm{(C)}&lt;/math&gt;.<br /> <br /> ==See Also==<br /> <br /> *[[2004 AMC 10A Problems]]<br /> <br /> *[[2004 AMC 10A Problems/Problem 11|Previous Problem]]<br /> <br /> *[[2004 AMC 10A Problems/Problem 13|Next Problem]]</div> B-flat https://artofproblemsolving.com/wiki/index.php?title=2004_AMC_12A_Problems/Problem_9&diff=10803 2004 AMC 12A Problems/Problem 9 2006-11-04T22:07:23Z <p>B-flat: </p> <hr /> <div>==Problem==<br /> A company sells peanut butter in cylindrical jars. Marketing research suggests that using wider jars will increase sales. If the diameter of the jars is increased by &lt;math&gt;25\%&lt;/math&gt; without altering the volume, by what percent must the height be decreased?<br /> <br /> &lt;math&gt; \mathrm{(A) \ } 10 \qquad \mathrm{(B) \ } 25 \qquad \mathrm{(C) \ } 36 \qquad \mathrm{(D) \ } 50 {2}\qquad \mathrm{(E) \ } 60 &lt;/math&gt;<br /> <br /> ==Solution==<br /> When the diameter is increased by &lt;math&gt;25\%&lt;/math&gt;, is is increased by &lt;math&gt;\frac54&lt;/math&gt;, so the area of the base is increased by &lt;math&gt;\left(\frac54\right)^2=\frac{25}{16}&lt;/math&gt;.<br /> <br /> To keep the volume the same, the height must be &lt;math&gt;\frac{1}{\frac{25}{16}}=\frac{16}{25}&lt;/math&gt; of the original height, which is a &lt;math&gt;36\%&lt;/math&gt; reduction &lt;math&gt;\Rightarrow\mathrm{(C)}&lt;/math&gt;.<br /> <br /> ==See Also==<br /> <br /> *[[2004 AMC 10A Problems]]<br /> <br /> *[[2004 AMC 10A Problems/Problem 10|Previous Problem]]<br /> <br /> *[[2004 AMC 10A Problems/Problem 12|Next Problem]]</div> B-flat https://artofproblemsolving.com/wiki/index.php?title=2004_AMC_10A_Problems/Problem_10&diff=10801 2004 AMC 10A Problems/Problem 10 2006-11-04T21:56:08Z <p>B-flat: </p> <hr /> <div>==Problem==<br /> Coin &lt;math&gt;A&lt;/math&gt; is flipped three times and coin &lt;math&gt;B&lt;/math&gt; is flipped four times. What is the probability that the number of heads obtained from flipping the two fair foins is the same?<br /> <br /> &lt;math&gt; \mathrm{(A) \ } \frac{29}{128} \qquad \mathrm{(B) \ } \frac{23}{128} \qquad \mathrm{(C) \ } \frac14 \qquad \mathrm{(D) \ } \frac{35}{128} {2}\qquad \mathrm{(E) \ } \frac12 &lt;/math&gt;<br /> <br /> ==Solution==<br /> There are 4 ways that the same number of heads will be obtained; 0, 1, 2, or 3 heads.<br /> <br /> The probability of both getting 0 heads is &lt;math&gt;\left(\frac12\right)^3{3\choose0}\left(\frac12\right)^4{4\choose0}=\frac1{128}&lt;/math&gt;.<br /> <br /> The probability of both getting 1 head is<br /> &lt;math&gt;\left(\frac12\right)^3{3\choose1}\left(\frac12\right)^4{4\choose1}=\frac{12}{128}&lt;/math&gt;<br /> <br /> The probability of both getting 2 heads is<br /> &lt;math&gt;\left(\frac12\right)^3{3\choose2}\left(\frac12\right)^4{4\choose2}=\frac{18}{128}&lt;/math&gt;<br /> <br /> The probability of both getting 3 heads is<br /> &lt;math&gt;\left(\frac12\right)^3{3\choose3}\left(\frac12\right)^4{4\choose3}=\frac{4}{128}&lt;/math&gt;<br /> <br /> Therefore, the probabiliy of flipping the same number of heads is:<br /> &lt;math&gt;\frac{1+12+18+4}{128}=\frac{35}{128}\Rightarrow\mathrm{(D)}&lt;/math&gt;<br /> <br /> ==See Also==<br /> <br /> *[[2004 AMC 10A Problems]]<br /> <br /> *[[2004 AMC 10A Problems/Problem 9|Previous Problem]]<br /> <br /> *[[2004 AMC 10A Problems/Problem 11|Next Problem]]</div> B-flat https://artofproblemsolving.com/wiki/index.php?title=User:B-flat&diff=10795 User:B-flat 2006-11-04T21:45:26Z <p>B-flat: </p> <hr /> <div>== b-flat ==<br /> <br /> Here is my user page.<br /> <br /> I live in North Carolina, and I enjoy math, programming, chess, biking, and playing piano.<br /> <br /> == Contributions ==<br /> <br /> * Made a bunch of articles look nicer<br /> <br /> * Created Area of common geometric figures<br /> * Added Area of quadrilaterals<br /> <br /> * Created [[Greatest common divisor]] (GCD)<br /> <br /> * Added Approximating [[pi]] - June 26, 2006<br /> <br /> * Added [[Chinese Remainder Theorem]] Example - June 27, 2006<br /> <br /> * Added Introduction to Completing the Square - June 27, 2005<br /> <br /> * Created [[Rhombus]] Page - June 27, 2006<br /> <br /> * Created [[Parallelogram]] Page - June 27, 2006<br /> <br /> * Created [[Rectangle]] Page - June 27, 2006<br /> <br /> * Created [[Kite]] Page - June 27, 2006<br /> <br /> * Created [[Trapezoid]] Page - June 27, 2006<br /> <br /> * Created [[Isosceles trapezoid]] Page - June 27, 2006<br /> <br /> * Added Perfect Square trinomials to [[perfect square]] page - June 29, 2006<br /> <br /> * Created [[2006 AMC 10A]] Page - June 29, 2006<br /> <br /> * Added AMC 20A problems Page - July 1, 2006<br /> <br /> * Created [[constant]] Page - July 6, 2006<br /> <br /> * Created [[variable]] Page - July 6, 2006<br /> <br /> * Created [[perimeter]] Page - July 6, 2006<br /> <br /> * Created [[interval]] Page - July 11, 2006<br /> <br /> * Created [[contrapositive]] Page - July 16, 2006<br /> <br /> * Updated [[LaTeX]] Page - July 22, 2006<br /> <br /> * Created [[Proper fraction]] Page - October 19, 2006<br /> <br /> * Created [[2005 AMC 10A Problems]] Page - November 4, 2006<br /> <br /> * Created [[2005 AMC 10A Problems/Problem 20]] Page - November 4, 2006<br /> <br /> * Created [[2004 AMC 10A Problems/Problem 1]] Page - November 4, 2006<br /> <br /> * Created [[2004 AMC 10A Problems/Problem 3]] Page - November 4, 2006<br /> <br /> * Created [[2004 AMC 10A Problems/Problem 4]] Page - November 4, 2006<br /> <br /> * Created [[2004 AMC 10A Problems/Problem 6]] Page - November 4, 2006<br /> <br /> * Created [[2004 AMC 10A Problems/Problem 7]] Page - November 4, 2006<br /> <br /> * Created [[2004 AMC 10A Problems/Problem 8]] Page - November 4, 2006</div> B-flat https://artofproblemsolving.com/wiki/index.php?title=2004_AMC_12A_Problems/Problem_7&diff=10794 2004 AMC 12A Problems/Problem 7 2006-11-04T21:43:44Z <p>B-flat: </p> <hr /> <div>==Problem==<br /> A game is played with tokens according to the following rule. In each round, the player with the most tokens gives one token to each of the other players and also places one token in the discard pile. The game ends when some player runs out of tokens. Players &lt;math&gt;A&lt;/math&gt;, &lt;math&gt;B&lt;/math&gt;, and &lt;math&gt;C&lt;/math&gt; start with 15, 14, and 13 tokens, respectively. How many rounds will there be in the game?<br /> <br /> &lt;math&gt; \mathrm{(A) \ } 36 \qquad \mathrm{(B) \ } 37 \qquad \mathrm{(C) \ } 38 \qquad \mathrm{(D) \ } 39 {2}\qquad \mathrm{(E) \ } 40 &lt;/math&gt;<br /> <br /> ==Solution==<br /> Look at a set of 3 rounds, where the players have &lt;math&gt;x+1&lt;/math&gt;, &lt;math&gt;x&lt;/math&gt;, and &lt;math&gt;x-1&lt;/math&gt; tokens. Each of the players will gain two tokens from the others and give away 3 tokens, so overall, each player will lose 1 token.<br /> <br /> Therefore, after 12 sets of 3 rounds, or 36 rounds, the players will have 3, 2, and 1 tokens, repectively. After 1 more round, player &lt;math&gt;A&lt;/math&gt; will give away his last 3 tokens and the game will stop &lt;math&gt;\Rightarrow\mathrm{(B)}&lt;/math&gt;.<br /> <br /> ==See Also==<br /> <br /> *[[2004 AMC 10A Problems]]<br /> <br /> *[[2004 AMC 10A Problems/Problem 7|Previous Problem]]<br /> <br /> *[[2004 AMC 10A Problems/Problem 9|Next Problem]]</div> B-flat https://artofproblemsolving.com/wiki/index.php?title=User:B-flat&diff=10791 User:B-flat 2006-11-04T21:35:41Z <p>B-flat: </p> <hr /> <div>== b-flat ==<br /> <br /> Here is my user page.<br /> <br /> I live in North Carolina, and I enjoy math, programming, chess, biking, and playing piano.<br /> <br /> == Contributions ==<br /> <br /> * Made a bunch of articles look nicer<br /> <br /> * Created Area of common geometric figures<br /> * Added Area of quadrilaterals<br /> <br /> * Created [[Greatest common divisor]] (GCD)<br /> <br /> * Added Approximating [[pi]] - June 26, 2006<br /> <br /> * Added [[Chinese Remainder Theorem]] Example - June 27, 2006<br /> <br /> * Added Introduction to Completing the Square - June 27, 2005<br /> <br /> * Created [[Rhombus]] Page - June 27, 2006<br /> <br /> * Created [[Parallelogram]] Page - June 27, 2006<br /> <br /> * Created [[Rectangle]] Page - June 27, 2006<br /> <br /> * Created [[Kite]] Page - June 27, 2006<br /> <br /> * Created [[Trapezoid]] Page - June 27, 2006<br /> <br /> * Created [[Isosceles trapezoid]] Page - June 27, 2006<br /> <br /> * Added Perfect Square trinomials to [[perfect square]] page - June 29, 2006<br /> <br /> * Created [[2006 AMC 10A]] Page - June 29, 2006<br /> <br /> * Added AMC 20A problems Page - July 1, 2006<br /> <br /> * Created [[constant]] Page - July 6, 2006<br /> <br /> * Created [[variable]] Page - July 6, 2006<br /> <br /> * Created [[perimeter]] Page - July 6, 2006<br /> <br /> * Created [[interval]] Page - July 11, 2006<br /> <br /> * Created [[contrapositive]] Page - July 16, 2006<br /> <br /> * Updated [[LaTeX]] Page - July 22, 2006<br /> <br /> * Created [[Proper fraction]] Page - October 19, 2006<br /> <br /> * Created [[2005 AMC 10A Problems]] Page - November 4, 2006<br /> <br /> * Created [[2005 AMC 10A Problems/Problem 20]] Page - November 4, 2006<br /> <br /> * Created [[2004 AMC 10A Problems/Problem 1]] Page - November 4, 2006<br /> <br /> * Created [[2004 AMC 10A Problems/Problem 3]] Page - November 4, 2006<br /> <br /> * Created [[2004 AMC 10A Problems/Problem 4]] Page - November 4, 2006<br /> <br /> * Created [[2004 AMC 10A Problems/Problem 6]] Page - November 4, 2006<br /> <br /> * Created [[2004 AMC 10A Problems/Problem 7]] Page - November 4, 2006</div> B-flat https://artofproblemsolving.com/wiki/index.php?title=2004_AMC_10A_Problems/Problem_7&diff=10790 2004 AMC 10A Problems/Problem 7 2006-11-04T21:34:22Z <p>B-flat: </p> <hr /> <div>==Problem==<br /> A grocer stacks oranges in a pyramid-like stack whose rectangular base is 5 oranges by 8 oranges. Each orange above the first level rests in a pocket formed by four oranges below. The stack is completed by a single row of oranges. How many oranges are in the stack?<br /> <br /> &lt;math&gt; \mathrm{(A) \ } 96 \qquad \mathrm{(B) \ } 98 \qquad \mathrm{(C) \ } 100 \qquad \mathrm{(D) \ } 101 {2}\qquad \mathrm{(E) \ } 134 &lt;/math&gt;<br /> <br /> ==Solution==<br /> There are &lt;math&gt;5\times8=40&lt;/math&gt; oranges on the 1st layer of the stack. When the 2nd layer is added on top of the first, it will be a layer of &lt;math&gt;4\times7=28&lt;/math&gt; oranges. When the third layer is added on top of the 2nd, it will be a layer of &lt;math&gt;3\times6=18&lt;/math&gt; oranges, etc.<br /> <br /> Therefore, there are &lt;math&gt;5\times8+4\times7+3\times6+2\times5+1\times4=40+28+18+10+4=100&lt;/math&gt; oranges in the stack &lt;math&gt;\Rightarrow\mathrm{(C)}&lt;/math&gt;.<br /> <br /> ==See Also==<br /> <br /> *[[2004 AMC 10A Problems]]<br /> <br /> *[[2004 AMC 10A Problems/Problem 6|Previous Problem]]<br /> <br /> *[[2004 AMC 10A Problems/Problem 8|Next Problem]]</div> B-flat https://artofproblemsolving.com/wiki/index.php?title=2004_AMC_12A_Problems/Problem_4&diff=10789 2004 AMC 12A Problems/Problem 4 2006-11-04T21:28:38Z <p>B-flat: </p> <hr /> <div>==Problem==<br /> Bertha has 6 daughters and no sons. Some of her daughters have 6 daughters, and the rest have none. Bertha has a total of 30 daughters and granddaughters, and no great-granddaughters. How many of Bertha's daughters and grand-daughters have no daughters?<br /> <br /> &lt;math&gt; \mathrm{(A) \ } 22 \qquad \mathrm{(B) \ } 23 \qquad \mathrm{(C) \ } 24 \qquad \mathrm{(D) \ } 25 {2}\qquad \mathrm{(E) \ } 26 &lt;/math&gt;<br /> <br /> ==Solution==<br /> Since Bertha has 6 daughters, Bertha has &lt;math&gt;30-6=24&lt;/math&gt; granddaughters, of which none have daughters. Of Bertha's daughters, &lt;math&gt;\frac{24}6=4&lt;/math&gt; have daughters, so &lt;math&gt;6-4=2&lt;/math&gt; do not have daughters.<br /> <br /> Therefore, of Bertha's daughters and granddaughters, &lt;math&gt;24+2=26&lt;/math&gt; do not have daughters &lt;math&gt;\Rightarrow\mathrm{(E)}&lt;/math&gt;.<br /> <br /> ==See Also==<br /> <br /> *[[2004 AMC 10A Problems]]<br /> <br /> *[[2004 AMC 10A Problems/Problem 5|Previous Problem]]<br /> <br /> *[[2004 AMC 10A Problems/Problem 7|Next Problem]]</div> B-flat https://artofproblemsolving.com/wiki/index.php?title=2004_AMC_10A_Problems/Problem_4&diff=10787 2004 AMC 10A Problems/Problem 4 2006-11-04T21:20:18Z <p>B-flat: </p> <hr /> <div>==Problem==<br /> What is the value of &lt;math&gt;x&lt;/math&gt; if &lt;math&gt;|x-1|=|x-2|&lt;/math&gt;?<br /> <br /> &lt;math&gt; \mathrm{(A) \ } -\frac12 \qquad \mathrm{(B) \ } \frac12 \qquad \mathrm{(C) \ } 1 \qquad \mathrm{(D) \ } \frac32 {2}\qquad \mathrm{(E) \ } 2 &lt;/math&gt;<br /> <br /> ==Solution==<br /> &lt;math&gt;|x-1|&lt;/math&gt; is the same thing as saying the distance between &lt;math&gt;x&lt;/math&gt; and &lt;math&gt;1&lt;/math&gt;; &lt;math&gt;|x-2|&lt;/math&gt; is the same thing as saying the distance between &lt;math&gt;x&lt;/math&gt; and &lt;math&gt;2&lt;/math&gt;.<br /> <br /> Therefore, &lt;math&gt;x&lt;/math&gt; is the same distance from &lt;math&gt;1&lt;/math&gt; and &lt;math&gt;2&lt;/math&gt;, so &lt;math&gt;x=\frac{1+2}2=\frac32\Rightarrow\mathrm{(D)}&lt;/math&gt;.<br /> <br /> ==See Also==<br /> <br /> *[[2004 AMC 10A Problems]]<br /> <br /> *[[2004 AMC 10A Problems/Problem 3|Previous Problem]]<br /> <br /> *[[2004 AMC 10A Problems/Problem 5|Next Problem]]</div> B-flat https://artofproblemsolving.com/wiki/index.php?title=2004_AMC_12A_Problems/Problem_1&diff=10786 2004 AMC 12A Problems/Problem 1 2006-11-04T21:15:26Z <p>B-flat: </p> <hr /> <div>==Problem==<br /> Alicia earns 20 dollars per hour, of which &lt;math&gt;1.45\%&lt;/math&gt; is deducted to pay local taxes. How many cents per hour of Alicia's wages are used to pay local taxes?<br /> <br /> &lt;math&gt; \mathrm{(A) \ } 0.0029 \qquad \mathrm{(B) \ } 0.029 \qquad \mathrm{(C) \ } 0.29 \qquad \mathrm{(D) \ } 2.9 {2}\qquad \mathrm{(E) \ } 29 &lt;/math&gt;<br /> <br /> ==Solution==<br /> 20 dollars is the same as 2000 cents, and &lt;math&gt;1.45\%&lt;/math&gt; of 2000 is &lt;math&gt;0.0145\times2000=29&lt;/math&gt; cents &lt;math&gt;\Rightarrow\mathrm{(E)}&lt;/math&gt;.<br /> <br /> ==See Also==<br /> <br /> *[[2004 AMC 10A Problems]]<br /> <br /> *[[2004 AMC 10A Problems/Problem 2|Previous Problem]]<br /> <br /> *[[2004 AMC 10A Problems/Problem 4|Next Problem]]</div> B-flat https://artofproblemsolving.com/wiki/index.php?title=User:B-flat&diff=10785 User:B-flat 2006-11-04T21:08:40Z <p>B-flat: </p> <hr /> <div>== b-flat ==<br /> <br /> Here is my user page.<br /> <br /> I live in North Carolina, and I enjoy math, programming, chess, biking, and playing piano.<br /> <br /> == Contributions ==<br /> <br /> * Made a bunch of articles look nicer<br /> <br /> * Created Area of common geometric figures<br /> * Added Area of quadrilaterals<br /> <br /> * Created [[Greatest common divisor]] (GCD)<br /> <br /> * Added Approximating [[pi]] - June 26, 2006<br /> <br /> * Added [[Chinese Remainder Theorem]] Example - June 27, 2006<br /> <br /> * Added Introduction to Completing the Square - June 27, 2005<br /> <br /> * Created [[Rhombus]] Page - June 27, 2006<br /> <br /> * Created [[Parallelogram]] Page - June 27, 2006<br /> <br /> * Created [[Rectangle]] Page - June 27, 2006<br /> <br /> * Created [[Kite]] Page - June 27, 2006<br /> <br /> * Created [[Trapezoid]] Page - June 27, 2006<br /> <br /> * Created [[Isosceles trapezoid]] Page - June 27, 2006<br /> <br /> * Added Perfect Square trinomials to [[perfect square]] page - June 29, 2006<br /> <br /> * Created [[2006 AMC 10A]] Page - June 29, 2006<br /> <br /> * Added AMC 20A problems Page - July 1, 2006<br /> <br /> * Created [[constant]] Page - July 6, 2006<br /> <br /> * Created [[variable]] Page - July 6, 2006<br /> <br /> * Created [[perimeter]] Page - July 6, 2006<br /> <br /> * Created [[interval]] Page - July 11, 2006<br /> <br /> * Created [[contrapositive]] Page - July 16, 2006<br /> <br /> * Updated [[LaTeX]] Page - July 22, 2006<br /> <br /> * Created [[Proper fraction]] Page - October 19, 2006<br /> <br /> * Created [[2005 AMC 10A Problems]] Page - November 4, 2006<br /> <br /> * Created [[2005 AMC 10A Problems/Problem 20]] Page - November 4, 2006<br /> <br /> * Created [[2004 AMC 10A Problems/Problem 1]] Page - November 4, 2006</div> B-flat https://artofproblemsolving.com/wiki/index.php?title=2004_AMC_10A_Problems/Problem_1&diff=10784 2004 AMC 10A Problems/Problem 1 2006-11-04T21:07:04Z <p>B-flat: </p> <hr /> <div>==Problem==<br /> You and five friends need to raise &lt;math&gt;1500&lt;/math&gt; dollars in donations for a charity, dividing the fundraising equally. How many dollars will each of you need to raise?<br /> <br /> &lt;math&gt; \mathrm{(A) \ } 250\qquad \mathrm{(B) \ } 300 \qquad \mathrm{(C) \ } 1500 \qquad \mathrm{(D) \ } 7500 \qquad \mathrm{(E) \ } 9000 &lt;/math&gt;<br /> <br /> ==Solution==<br /> There are 6 people to split the &lt;math&gt;1500&lt;/math&gt; dollars among, so each person must raise &lt;math&gt;\frac{1500}6=250&lt;/math&gt; dollars. &lt;math&gt;\Rightarrow\mathrm{(C)}&lt;/math&gt;<br /> <br /> ==See Also==<br /> *[[2004 AMC 10A Problems]]<br /> <br /> *[[2004 AMC 10A Problems/Problem 2|Next Problem]]</div> B-flat https://artofproblemsolving.com/wiki/index.php?title=User:B-flat&diff=10779 User:B-flat 2006-11-04T20:44:43Z <p>B-flat: /* Contributions */</p> <hr /> <div>== b-flat ==<br /> <br /> Here is my user page.<br /> <br /> I live in North Carolina, and I enjoy math, programming, chess, biking, and playing piano.<br /> <br /> == Contributions ==<br /> <br /> * Made a bunch of articles look nicer<br /> <br /> * Created Area of common geometric figures<br /> * Added Area of quadrilaterals<br /> <br /> * Created [[Greatest common divisor]] (GCD)<br /> <br /> * Added Approximating [[pi]] - June 26, 2006<br /> <br /> * Added [[Chinese Remainder Theorem]] Example - June 27, 2006<br /> <br /> * Added Introduction to Completing the Square - June 27, 2005<br /> <br /> * Created [[Rhombus]] Page - June 27, 2006<br /> <br /> * Created [[Parallelogram]] Page - June 27, 2006<br /> <br /> * Created [[Rectangle]] Page - June 27, 2006<br /> <br /> * Created [[Kite]] Page - June 27, 2006<br /> <br /> * Created [[Trapezoid]] Page - June 27, 2006<br /> <br /> * Created [[Isosceles trapezoid]] Page - June 27, 2006<br /> <br /> * Added Perfect Square trinomials to [[perfect square]] page - June 29, 2006<br /> <br /> * Created [[2006 AMC 10A]] Page - June 29, 2006<br /> <br /> * Added AMC 20A problems Page - July 1, 2006<br /> <br /> * Created [[constant]] Page - July 6, 2006<br /> <br /> * Created [[variable]] Page - July 6, 2006<br /> <br /> * Created [[perimeter]] Page - July 6, 2006<br /> <br /> * Created [[interval]] Page - July 11, 2006<br /> <br /> * Created [[contrapositive]] Page - July 16, 2006<br /> <br /> * Updated [[LaTeX]] Page - July 22, 2006<br /> <br /> * Created [[Proper fraction]] Page - October 19, 2006<br /> <br /> * Created [[2005 AMC 10A Problems]] Page - November 4, 2006</div> B-flat https://artofproblemsolving.com/wiki/index.php?title=2005_AMC_10A_Problems/Problem_20&diff=10777 2005 AMC 10A Problems/Problem 20 2006-11-04T20:43:09Z <p>B-flat: </p> <hr /> <div>==Problem==<br /> An equiangular octagon has four sides of length 1 and four sides of length &lt;math&gt;\sqrt2/2&lt;/math&gt;, arranged so that no two consecutive sides have the same length. What is the area of the octagon?<br /> <br /> &lt;math&gt; \mathrm{(A) \ } \frac72\qquad \mathrm{(B) \ } \frac{7\sqrt2}{2}\qquad \mathrm{(C) \ } \frac{5+4\sqrt2}{2}\qquad \mathrm{(D) \ } \frac{4+5\sqrt2}{2}\qquad \mathrm{(E) \ } 7 &lt;/math&gt;<br /> <br /> ==Solution==<br /> The area of the octagon can be divided up into 5 squares with side &lt;math&gt;\frac{\sqrt2}2&lt;/math&gt; and 4 right triangles, which are half the area of each of the squares.<br /> <br /> Therefore, the area of the octagon is equal to the area of &lt;math&gt;5+4\left(\frac12\right)=7&lt;/math&gt; squares.<br /> <br /> The area of each square is &lt;math&gt;\left(\frac{\sqrt2}2\right)^2=\frac12&lt;/math&gt;, so the area of 7 squares is &lt;math&gt;\frac72\Rightarrow\mathrm{(C)}&lt;/math&gt;.<br /> <br /> ==See Also==<br /> *[[2005 AMC 10A Problems]]<br /> <br /> *[[2005 AMC 10A Problems/Problem 19|Previous Problem]]<br /> <br /> *[[2005 AMC 10A Problems/Problem 21|Next Problem]]<br /> <br /> [[Category:Introductory Geometry Problems]]</div> B-flat https://artofproblemsolving.com/wiki/index.php?title=2005_AMC_10A_Problems&diff=10775 2005 AMC 10A Problems 2006-11-04T20:37:27Z <p>B-flat: </p> <hr /> <div>[[2005 AMC 10A Problems/Problem 1|#1]]<br /> <br /> While eating out, Mike and Joe each tipped their server &lt;math&gt;\&lt;math&gt;2&lt;/math&gt;. Mike tipped &lt;math&gt;10\%&lt;/math&gt; of his bll and Joe tipped &lt;math&gt;20%&lt;/math&gt; of his bill. What was the difference, in dollars between their bills? <br /> <br /> &lt;math&gt; \mathrm{(A) \ } 2\qquad \mathrm{(B) \ } 4\qquad \mathrm{(C) \ } 5\qquad \mathrm{(D) \ } 10\qquad \mathrm{(E) \ } 20 &lt;/math&gt;<br /> <br /> ----<br /> <br /> [[2005 AMC 10A Problems/Problem 2|#2]]<br /> <br /> For each pair of real numbers &lt;math&gt;a&lt;/math&gt;&lt;math&gt;\neq&lt;/math&gt;&lt;math&gt;b&lt;/math&gt;, define the [[operation]] &lt;math&gt;\star&lt;/math&gt; as<br /> <br /> &lt;math&gt; (a \star b) = \frac{a+b}{a-b} &lt;/math&gt;.<br /> <br /> What is the value of &lt;math&gt; ((1 \star 2) \star 3)&lt;/math&gt;?<br /> <br /> &lt;math&gt; \mathrm{(A) \ } -\frac{2}{3}\qquad \mathrm{(B) \ } -\frac{1}{5}\qquad \mathrm{(C) \ } 0\qquad \mathrm{(D) \ } \frac{1}{2}\qquad \mathrm{(E) \ } \textrm{This\, value\, is\, not\, defined.} &lt;/math&gt;<br /> <br /> ----<br /> <br /> [[2005 AMC 10A Problems/Problem 3|#3]]<br /> <br /> <br /> The equations &lt;math&gt; 2x + 7 = 3 &lt;/math&gt; and &lt;math&gt; bx - 10 = -2 &lt;/math&gt; have the same solution &lt;math&gt;x&lt;/math&gt;. What is the value of &lt;math&gt;b&lt;/math&gt;? <br /> <br /> &lt;math&gt; \mathrm{(A) \ } -8\qquad \mathrm{(B) \ } -4\qquad \mathrm{(C) \ } -2\qquad \mathrm{(D) \ } 4\qquad \mathrm{(E) \ } 8 &lt;/math&gt;<br /> <br /> ----<br /> <br /> [[2005 AMC 10A Problems/Problem 4|#4]]<br /> <br /> <br /> A rectangle with a [[diagonal]] of length &lt;math&gt;x&lt;/math&gt; is twice as long as it is wide. What is the area of the rectangle? <br /> <br /> &lt;math&gt; \mathrm{(A) \ } \frac{1}{4}x^2\qquad \mathrm{(B) \ } \frac{2}{5}x^2\qquad \mathrm{(C) \ } \frac{1}{2}x^2\qquad \mathrm{(D) \ } x^2\qquad \mathrm{(E) \ } \frac{3}{2}x^2 &lt;/math&gt;<br /> <br /> ----<br /> <br /> [[2005 AMC 10A Problems/Problem 5|#5]]<br /> <br /> <br /> A store normally sells windows at &lt;math&gt;&lt;/math&gt;100&lt;/math&gt; each. This week the store is offering one free window for each purchase of four. Dave needs seven windows and Doug needs eight windows. How many dollars will they save if they purchase the windows together rather than separately?<br /> <br /> &lt;math&gt; \mathrm{(A) \ } 100\qquad \mathrm{(B) \ } 200\qquad \mathrm{(C) \ } 300\qquad \mathrm{(D) \ } 400\qquad \mathrm{(E) \ } 500 &lt;/math&gt;<br /> <br /> ----<br /> <br /> [[2005 AMC 10A Problems/Problem 6|#6]]<br /> <br /> <br /> The average (mean) of &lt;math&gt;20&lt;/math&gt; numbers is &lt;math&gt;30&lt;/math&gt;, and the average of &lt;math&gt;30&lt;/math&gt; other numbers is &lt;math&gt;20&lt;/math&gt;. What is the average of all &lt;math&gt;50&lt;/math&gt; numbers?<br /> <br /> &lt;math&gt; \mathrm{(A) \ } 23\qquad \mathrm{(B) \ } 24\qquad \mathrm{(C) \ } 25\qquad \mathrm{(D) \ } 26\qquad \mathrm{(E) \ } 27 &lt;/math&gt;<br /> <br /> ----<br /> <br /> [[2005 AMC 10A Problems/Problem 7|#7]]<br /> <br /> <br /> Josh and Mike live &lt;math&gt;13&lt;/math&gt; miles apart. Yesterday Josh started to ride his bicycle toward Mike's house. A little later Mike started to ride his bicycle toward Josh's house. When they met, Josh had ridden for twice the length of time as Mike and at four-fifths of Mike's rate. How many miles had Mike ridden when they met? <br /> <br /> &lt;math&gt; \mathrm{(A) \ } 4\qquad \mathrm{(B) \ } 5\qquad \mathrm{(C) \ } 6\qquad \mathrm{(D) \ } 7\qquad \mathrm{(E) \ } 8 &lt;/math&gt;<br /> <br /> ----<br /> <br /> [[2005 AMC 10A Problems/Problem 9|#9]]<br /> <br /> <br /> Three tiles are marked &lt;math&gt;X&lt;/math&gt; and two other tiles are marked &lt;math&gt;O&lt;/math&gt;. The five tiles are randomly arranged in a row. What is the probability that the arrangement reads &lt;math&gt;XOXOX&lt;/math&gt;?<br /> <br /> &lt;math&gt; \mathrm{(A) \ } \frac{1}{12}\qquad \mathrm{(B) \ } \frac{1}{10}\qquad \mathrm{(C) \ } \frac{1}{6}\qquad \mathrm{(D) \ } \frac{1}{4}\qquad \mathrm{(E) \ } \frac{1}{3} &lt;/math&gt;<br /> <br /> ----<br /> <br /> [[2005 AMC 10A Problems/Problem 10|#10]]<br /> <br /> <br /> There are two values of &lt;math&gt;a&lt;/math&gt; for which the equation &lt;math&gt; 4x^2 + ax + 8x + 9 = 0 &lt;/math&gt; has only one solution for &lt;math&gt;x&lt;/math&gt;. What is the sum of those values of &lt;math&gt;a&lt;/math&gt;?<br /> <br /> &lt;math&gt; \mathrm{(A) \ } -16\qquad \mathrm{(B) \ } -8\qquad \mathrm{(C) \ } 0\qquad \mathrm{(D) \ } 8\qquad \mathrm{(E) \ } 20 &lt;/math&gt;<br /> <br /> ----<br /> <br /> [[2005 AMC 10A Problems/Problem 11|#11]]<br /> <br /> <br /> A wooden cube &lt;math&gt;n&lt;/math&gt; units on a side is painted red on all six faces and then cut into &lt;math&gt;n^3&lt;/math&gt; unit cubes. Exactly one-fourth of the total number of faces of the unit cubes are red. What is &lt;math&gt;n&lt;/math&gt;?<br /> <br /> &lt;math&gt; \mathrm{(A) \ } 3\qquad \mathrm{(B) \ } 4\qquad \mathrm{(C) \ } 5\qquad \mathrm{(D) \ } 6\qquad \mathrm{(E) \ } 7 &lt;/math&gt;<br /> <br /> ----<br /> <br /> [[2005 AMC 10A Problems/Problem 12|#12]]<br /> <br /> <br /> The figure shown is called a ''trefoil'' and is constructed by drawing circular sectors about the sides of the congruent equilateral triangles. What is the area of a trefoil whose horizontal base has length &lt;math&gt;2&lt;/math&gt;?<br /> <br /> [[Image:2005amc10a12.gif]]<br /> <br /> &lt;math&gt; \mathrm{(A) \ } \frac{1}{3}\pi+\frac{\sqrt{3}}{2}\qquad \mathrm{(B) \ } \frac{2}{3}\pi\qquad \mathrm{(C) \ } \frac{2}{3}\pi+\frac{\sqrt{3}}{4}\qquad \mathrm{(D) \ } \frac{2}{3}\pi+\frac{\sqrt{3}}{3}\qquad \mathrm{(E) \ } \frac{2}{3}\pi+\frac{\sqrt{3}}{2} &lt;/math&gt;<br /> <br /> ----<br /> <br /> [[2005 AMC 10A Problems/Problem 13|#13]]<br /> <br /> <br /> How many positive integers &lt;math&gt;n&lt;/math&gt; satisfy the following condition:<br /> <br /> &lt;math&gt; (130n)^{50} &gt; n^{100} &gt; 2^{200} &lt;/math&gt;?<br /> <br /> &lt;math&gt; \mathrm{(A) \ } 0\qquad \mathrm{(B) \ } 7\qquad \mathrm{(C) \ } 12\qquad \mathrm{(D) \ } 65\qquad \mathrm{(E) \ } 125 &lt;/math&gt;<br /> <br /> ----<br /> <br /> [[2005 AMC 10A Problems/Problem 14|#14]]<br /> <br /> <br /> How many three-digit numbers satisfy the property that the middle digit is the average of the first and the last digits? <br /> <br /> &lt;math&gt; \mathrm{(A) \ } 41\qquad \mathrm{(B) \ } 42\qquad \mathrm{(C) \ } 43\qquad \mathrm{(D) \ } 44\qquad \mathrm{(E) \ } 45 &lt;/math&gt;<br /> <br /> ----<br /> <br /> [[2005 AMC 10A Problems/Problem 15|#15]]<br /> <br /> <br /> How many positive cubes divide &lt;math&gt; 3! \cdot 5! \cdot 7! &lt;/math&gt; ?<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 /> ----<br /> <br /> [[2005 AMC 10A Problems/Problem 16|#16]]<br /> <br /> <br /> The sum of the digits of a two-digit number is subtracted from the number. The units digit of the result is &lt;math&gt;6&lt;/math&gt;. How many two-digit numbers have this property? <br /> <br /> &lt;math&gt; \mathrm{(A) \ } 5\qquad \mathrm{(B) \ } 7\qquad \mathrm{(C) \ } 9\qquad \mathrm{(D) \ } 10\qquad \mathrm{(E) \ } 19 &lt;/math&gt;<br /> <br /> ----<br /> <br /> [[2005 AMC 10A Problems/Problem 17|#17]]<br /> <br /> <br /> In the five-sided star shown, the letters &lt;math&gt;A&lt;/math&gt;, &lt;math&gt;B&lt;/math&gt;, &lt;math&gt;C&lt;/math&gt;, &lt;math&gt;D&lt;/math&gt;, and &lt;math&gt;E&lt;/math&gt; are replaced by the numbers &lt;math&gt;3&lt;/math&gt;, &lt;math&gt;5&lt;/math&gt;, &lt;math&gt;6&lt;/math&gt;, &lt;math&gt;7&lt;/math&gt;, and &lt;math&gt;9&lt;/math&gt;, although not necessarily in this order. The sums of the numbers at the ends of the line segments &lt;math&gt;AB&lt;/math&gt;, &lt;math&gt;BC&lt;/math&gt;, &lt;math&gt;CD&lt;/math&gt;, &lt;math&gt;DE&lt;/math&gt;, and &lt;math&gt;EA&lt;/math&gt; form an arithmetic sequence, although not necessarily in this order. What is the middle term of the sequence? <br /> <br /> [[Image:2005amc10a17.gif]]<br /> <br /> &lt;math&gt; \mathrm{(A) \ } 9\qquad \mathrm{(B) \ } 10\qquad \mathrm{(C) \ } 11\qquad \mathrm{(D) \ } 12\qquad \mathrm{(E) \ } 13 &lt;/math&gt;<br /> <br /> ----<br /> <br /> [[2005 AMC 10A Problems/Problem 18|#18]]<br /> <br /> <br /> Team A and team B play a series. The first team to win three games wins the series. Each team is equally likely to win each game, there are no ties, and the outcomes of the individual games are independent. If team B wins the second game and team A wins the series, what is the probability that team B wins the first game? <br /> <br /> &lt;math&gt; \mathrm{(A) \ } \frac{1}{5}\qquad \mathrm{(B) \ } \frac{1}{4}\qquad \mathrm{(C) \ } \frac{1}{3}\qquad \mathrm{(D) \ } \frac{1}{2}\qquad \mathrm{(E) \ } \frac{2}{3} &lt;/math&gt;<br /> <br /> ----<br /> <br /> [[2005 AMC 10A Problems/Problem 19|#19]]<br /> <br /> <br /> ----<br /> <br /> [[2005 AMC 10A Problems/Problem 20|#20]]<br /> <br /> <br /> An equiangular octagon has four sides of length 1 and four sides of length &lt;math&gt;\sqrt2/2&lt;/math&gt;, arranged so that no two consecutive sides have the same length. What is the area of the octagon?<br /> <br /> &lt;math&gt; \mathrm{(A) \ } \frac72\qquad \mathrm{(B) \ } \frac{7\sqrt2}{2}\qquad \mathrm{(C) \ } \frac{5+4\sqrt2}{2}\qquad \mathrm{(D) \ } \frac{4+5\sqrt2}{2}\qquad \mathrm{(E) \ } 7 &lt;/math&gt;<br /> <br /> ----<br /> <br /> [[2005 AMC 10A Problems/Problem 21|#21]]<br /> <br /> <br /> For how many positive integers &lt;math&gt;n&lt;/math&gt; does &lt;math&gt; 1+2+...+n &lt;/math&gt; evenly divide &lt;math&gt;6n&lt;/math&gt;? <br /> <br /> &lt;math&gt; \mathrm{(A) \ } 3\qquad \mathrm{(B) \ } 5\qquad \mathrm{(C) \ } 7\qquad \mathrm{(D) \ } 9\qquad \mathrm{(E) \ } 11 &lt;/math&gt;<br /> <br /> ----<br /> <br /> [[2005 AMC 10A Problems/Problem 22|#22]]<br /> <br /> <br /> Let &lt;math&gt;S&lt;/math&gt; be the [[set]] of the &lt;math&gt;2005&lt;/math&gt; smallest positive multiples of &lt;math&gt;4&lt;/math&gt;, and let &lt;math&gt;T&lt;/math&gt; be the set of the &lt;math&gt;2005&lt;/math&gt; smallest positive multiples of &lt;math&gt;6&lt;/math&gt;. How many elements are common to &lt;math&gt;S&lt;/math&gt; and &lt;math&gt;T&lt;/math&gt;?<br /> <br /> &lt;math&gt; \mathrm{(A) \ } 166\qquad \mathrm{(B) \ } 333\qquad \mathrm{(C) \ } 500\qquad \mathrm{(D) \ } 668\qquad \mathrm{(E) \ } 1001 &lt;/math&gt;<br /> <br /> ----<br /> <br /> [[2005 AMC 10A Problems/Problem 23|#23]]<br /> <br /> <br /> ----<br /> <br /> [[2005 AMC 10A Problems/Problem 24|#24]]<br /> <br /> <br /> For each positive integer &lt;math&gt; m &gt; 1 &lt;/math&gt;, let &lt;math&gt;P(m)&lt;/math&gt; denote the greatest prime factor of &lt;math&gt;m&lt;/math&gt;. For how many positive integers &lt;math&gt;n&lt;/math&gt; is it true that both &lt;math&gt; P(n) = \sqrt{n} &lt;/math&gt; and &lt;math&gt; P(n+48) = \sqrt{n+48} &lt;/math&gt;?<br /> <br /> &lt;math&gt; \mathrm{(A) \ } 0\qquad \mathrm{(B) \ } 1\qquad \mathrm{(C) \ } 3\qquad \mathrm{(D) \ } 4\qquad \mathrm{(E) \ } 5 &lt;/math&gt;<br /> <br /> ----<br /> <br /> [[2005 AMC 10A Problems/Problem 25|#25]]<br /> <br /> <br /> In &lt;math&gt;ABC&lt;/math&gt; we have &lt;math&gt; AB = 25 &lt;/math&gt;, &lt;math&gt; BC = 39 &lt;/math&gt;, and &lt;math&gt;AC=42&lt;/math&gt;. Points &lt;math&gt;D&lt;/math&gt; and &lt;math&gt;E&lt;/math&gt; are on &lt;math&gt;AB&lt;/math&gt; and &lt;math&gt;AC&lt;/math&gt; respectively, with &lt;math&gt; AD = 19 &lt;/math&gt; and &lt;math&gt; AE = 14 &lt;/math&gt;. What is the [[ratio]] of the area of triangle &lt;math&gt;ADE&lt;/math&gt; to the area of the [[quadrilateral]] &lt;math&gt;BCED&lt;/math&gt;?<br /> <br /> &lt;math&gt; \mathrm{(A) \ } \frac{266}{1521}\qquad \mathrm{(B) \ } \frac{19}{75}\qquad \mathrm{(C) \ } \frac{1}{3}\qquad \mathrm{(D) \ } \frac{19}{56}\qquad \mathrm{(E) \ } 1 &lt;/math&gt;</div> B-flat https://artofproblemsolving.com/wiki/index.php?title=2005_AMC_10A_Problems/Problem_20&diff=10772 2005 AMC 10A Problems/Problem 20 2006-11-04T20:30:36Z <p>B-flat: </p> <hr /> <div>==Problem==<br /> An equiangular octagon has four sides of length 1 and four sides of length &lt;math&gt;\sqrt2/2&lt;/math&gt;, arranged so that no two consecutive sides have the same length. What is the area of the octagon?<br /> <br /> &lt;math&gt; \mathrm{(A) \ } \frac72\qquad \mathrm{(B) \ } \frac{7\sqrt2}{2}\qquad \mathrm{(C) \ } \frac{5+4\sqrt2}{2}\qquad \mathrm{(D) \ } \frac{4+5\sqrt2}{2}\qquad \mathrm{(E) \ } 7 &lt;/math&gt;<br /> <br /> ==See Also==<br /> *[[2005 AMC 10A Problems]]<br /> <br /> *[[2005 AMC 10A Problems/Problem 19|Previous Problem]]<br /> <br /> *[[2005 AMC 10A Problems/Problem 21|Next Problem]]<br /> <br /> [[Category:Introductory Geometry Problems]]</div> B-flat https://artofproblemsolving.com/wiki/index.php?title=2005_AMC_10A_Problems/Problem_20&diff=10771 2005 AMC 10A Problems/Problem 20 2006-11-04T20:28:30Z <p>B-flat: </p> <hr /> <div>==Problem==<br /> An equiangular octagon has four sides of length 1 and four sides of length &lt;math&gt;\sqrt2/2&lt;/math&gt;, arranged so that no two consecutive sides have the same length. What is the area of the octagon?<br /> <br /> &lt;math&gt; \mathrm{(A) \ } \frac72\qquad \mathrm{(B) \ } \frac{7\sqrt2}{2}\qquad \mathrm{(C) \ } \frac{5+4\sqrt2}{2}\qquad \mathrm{(D) \ } \frac{4+5\sqrt2}{2}\qquad \mathrm{(E) \ } 7 &lt;/math&gt;</div> B-flat https://artofproblemsolving.com/wiki/index.php?title=2005_AMC_10A_Problems&diff=10735 2005 AMC 10A Problems 2006-11-04T16:50:22Z <p>B-flat: Started - will finish later (b-flat)</p> <hr /> <div>[[2005 AMC 10A Problems/Problem 1|#1]]<br /> <br /> While eating out, Mike and Joe each tipped their server &lt;math&gt;\$2&lt;/math&gt;. Mike tipped &lt;math&gt;10%&lt;/math&gt; of his bll and Joe tipped &lt;math&gt;20%&lt;/math&gt; of his bill. What was the difference, in dollars between their bills? <br /> <br /> &lt;math&gt; \mathrm{(A) \ } 2\qquad \mathrm{(B) \ } 4\qquad \mathrm{(C) \ } 5\qquad \mathrm{(D) \ } 10\qquad \mathrm{(E) \ } 20 &lt;/math&gt;<br /> <br /> ----<br /> <br /> [[2005 AMC 10A Problems/Problem 2|#2]]<br /> <br /> For each pair of real numbers &lt;math&gt;a&lt;/math&gt;&lt;math&gt;\neq&lt;/math&gt;&lt;math&gt;b&lt;/math&gt;, define the [[operation]] &lt;math&gt;\star&lt;/math&gt; as<br /> <br /> &lt;math&gt; (a \star b) = \frac{a+b}{a-b} &lt;/math&gt;.<br /> <br /> What is the value of &lt;math&gt; ((1 \star 2) \star 3)&lt;/math&gt;?<br /> <br /> &lt;math&gt; \mathrm{(A) \ } -\frac{2}{3}\qquad \mathrm{(B) \ } -\frac{1}{5}\qquad \mathrm{(C) \ } 0\qquad \mathrm{(D) \ } \frac{1}{2}\qquad \mathrm{(E) \ } \textrm{This\, value\, is\, not\, defined.} &lt;/math&gt;<br /> <br /> ----<br /> <br /> [[2005 AMC 10A Problems/Problem 3|#3]]<br /> <br /> <br /> The equations &lt;math&gt; 2x + 7 = 3 &lt;/math&gt; and &lt;math&gt; bx - 10 = -2 &lt;/math&gt; have the same solution &lt;math&gt;x&lt;/math&gt;. What is the value of &lt;math&gt;b&lt;/math&gt;? <br /> <br /> &lt;math&gt; \mathrm{(A) \ } -8\qquad \mathrm{(B) \ } -4\qquad \mathrm{(C) \ } -2\qquad \mathrm{(D) \ } 4\qquad \mathrm{(E) \ } 8 &lt;/math&gt;<br /> <br /> ----<br /> <br /> [[2005 AMC 10A Problems/Problem 4|#4]]<br /> <br /> <br /> A rectangle with a [[diagonal]] of length &lt;math&gt;x&lt;/math&gt; is twice as long as it is wide. What is the area of the rectangle? <br /> <br /> &lt;math&gt; \mathrm{(A) \ } \frac{1}{4}x^2\qquad \mathrm{(B) \ } \frac{2}{5}x^2\qquad \mathrm{(C) \ } \frac{1}{2}x^2\qquad \mathrm{(D) \ } x^2\qquad \mathrm{(E) \ } \frac{3}{2}x^2 &lt;/math&gt;</div> B-flat https://artofproblemsolving.com/wiki/index.php?title=Quadratic_reciprocity&diff=10390 Quadratic reciprocity 2006-10-18T23:46:22Z <p>B-flat: </p> <hr /> <div>Let &lt;math&gt;p&lt;/math&gt; be a [[prime number|prime]], and let &lt;math&gt;a&lt;/math&gt; be any integer not divisible by &lt;math&gt;p&lt;/math&gt;. Then we can define the [[Legendre symbol]] &lt;math&gt;\left(\frac{a}{p}\right)=\begin{cases} 1 &amp; a\mathrm{\ is\ a\ quadratic\ residue\ modulo\ } p, \\ -1 &amp; \mathrm{otherwise}.\end{cases}&lt;/math&gt; <br /> <br /> We say that &lt;math&gt;a&lt;/math&gt; is a '''quadratic residue''' modulo &lt;math&gt;p&lt;/math&gt; if there exists an integer &lt;math&gt;n&lt;/math&gt; so that &lt;math&gt;n^2\equiv a\pmod p&lt;/math&gt;. We can then define &lt;math&gt;\left(\frac{a}{p}\right)=0&lt;/math&gt; if &lt;math&gt;a&lt;/math&gt; is divisible by &lt;math&gt;p&lt;/math&gt;.<br /> <br /> == Quadratic Reciprocity Theorem ==<br /> <br /> There are three parts. Let &lt;math&gt;p&lt;/math&gt; and &lt;math&gt;q&lt;/math&gt; be distinct [[odd integer | odd]] primes. Then the following hold:<br /> <br /> * &lt;math&gt;\left(\frac{-1}{p}\right)=(-1)^{(p-1)/4}&lt;/math&gt;.<br /> * &lt;math&gt;\left(\frac{2}{p}\right)=(-1)^{(p^2-1)/8}&lt;/math&gt;.<br /> * &lt;math&gt;\left(\frac{p}{q}\right)\left(\frac{q}{p}\right)=(-1)^{(p-1)/4\ (q-1)/4}&lt;/math&gt;.<br /> <br /> This theorem can help us evaluate Legendre symbols, since the following laws also apply:<br /> <br /> * If &lt;math&gt;a\equiv b\pmod{p}&lt;/math&gt;, then &lt;math&gt;\left(\frac{a}{p}\right)=\left(\frac{b}{p}\right)&lt;/math&gt;.<br /> * &lt;math&gt;\left(\frac{ab}{p}\right)=\left(\frac{a}{p}\right)\left(\frac{b}{p}\right)&lt;/math&gt;.<br /> <br /> There also exist quadratic reciprocity laws in other [[ring of integers|rings of integers]]. (I'll put that here later if I remember.)</div> B-flat https://artofproblemsolving.com/wiki/index.php?title=User:B-flat&diff=10389 User:B-flat 2006-10-18T23:45:10Z <p>B-flat: </p> <hr /> <div>== b-flat ==<br /> <br /> Here is my user page.<br /> <br /> I live in North Carolina, and I enjoy math, programming, chess, biking, and playing piano.<br /> <br /> == Contributions ==<br /> <br /> * Made a bunch of articles look nicer<br /> <br /> * Created Area of common geometric figures<br /> * Added Area of quadrilaterals<br /> <br /> * Created [[Greatest common divisor]] (GCD)<br /> <br /> * Added Approximating [[pi]] - June 26, 2006<br /> <br /> * Added [[Chinese Remainder Theorem]] Example - June 27, 2006<br /> <br /> * Added Introduction to Completing the Square - June 27, 2005<br /> <br /> * Created [[Rhombus]] Page - June 27, 2006<br /> <br /> * Created [[Parallelogram]] Page - June 27, 2006<br /> <br /> * Created [[Rectangle]] Page - June 27, 2006<br /> <br /> * Created [[Kite]] Page - June 27, 2006<br /> <br /> * Created [[Trapezoid]] Page - June 27, 2006<br /> <br /> * Created [[Isosceles trapezoid]] Page - June 27, 2006<br /> <br /> * Added Perfect Square trinomials to [[perfect square]] page - June 29, 2006<br /> <br /> * Created [[2006 AMC 10A]] Page - June 29, 2006<br /> <br /> * Added AMC 20A problems Page - July 1, 2006<br /> <br /> * Created [[constant]] Page - July 6, 2006<br /> <br /> * Created [[variable]] Page - July 6, 2006<br /> <br /> * Created [[perimeter]] Page - July 6, 2006<br /> <br /> * Created [[interval]] Page - July 11, 2006<br /> <br /> * Created [[contrapositive]] Page - July 16, 2006<br /> <br /> * Updated [[LaTeX]] Page - July 22, 2006<br /> <br /> * Created [[Proper fraction]] Page - October 19, 2006</div> B-flat https://artofproblemsolving.com/wiki/index.php?title=Proper_fraction&diff=10388 Proper fraction 2006-10-18T23:44:16Z <p>B-flat: </p> <hr /> <div>A '''proper fraction''' is a fraction such that the absolute value is less than 1.<br /> <br /> ===Examples===<br /> &lt;math&gt;\frac34&lt;/math&gt;<br /> <br /> &lt;math&gt;\frac{\pi}{6}&lt;/math&gt;<br /> <br /> ===See Also===<br /> [[Improper fraction]]</div> B-flat https://artofproblemsolving.com/wiki/index.php?title=User:B-flat&diff=8130 User:B-flat 2006-07-23T01:34:52Z <p>B-flat: /* Contributions */</p> <hr /> <div>== b-flat ==<br /> <br /> Here is my user page.<br /> <br /> I live in North Carolina, and I enjoy math, programming, chess, biking, and playing piano.<br /> <br /> == Contributions ==<br /> <br /> * Made a bunch of articles look nicer<br /> <br /> * Created Area of common geometric figures<br /> * Added Area of quadrilaterals<br /> <br /> * Created [[Greatest common divisor]] (GCD)<br /> <br /> * Added Approximating [[pi]] - June 26, 2006<br /> <br /> * Added [[Chinese Remainder Theorem]] Example - June 27, 2006<br /> <br /> * Added Introduction to Completing the Square - June 27, 2005<br /> <br /> * Created [[Rhombus]] Page - June 27, 2006<br /> <br /> * Created [[Parallelogram]] Page - June 27, 2006<br /> <br /> * Created [[Rectangle]] Page - June 27, 2006<br /> <br /> * Created [[Kite]] Page - June 27, 2006<br /> <br /> * Created [[Trapezoid]] Page - June 27, 2006<br /> <br /> * Created [[Isosceles trapezoid]] Page - June 27, 2006<br /> <br /> * Added Perfect Square trinomials to [[perfect square]] page - June 29, 2006<br /> <br /> * Created [[2006 AMC 10A]] Page - June 29, 2006<br /> <br /> * Added AMC 20A problems Page - July 1, 2006<br /> <br /> * Created [[constant]] Page - July 6, 2006<br /> <br /> * Created [[variable]] Page - July 6, 2006<br /> <br /> * Created [[perimeter]] Page - July 6, 2006<br /> <br /> * Created [[interval]] Page - July 11, 2006<br /> <br /> * Created [[contrapositive]] Page - July 16, 2006<br /> <br /> * Updated [[LaTeX]] Page - July 22, 2006</div> B-flat https://artofproblemsolving.com/wiki/index.php?title=LaTeX&diff=8129 LaTeX 2006-07-23T01:34:10Z <p>B-flat: updated</p> <hr /> <div>'''LaTeX''' is a typesetting language used primarily to type mathematical expressions in an elegant fashion. For example, without LaTeX, &lt;math&gt;\frac{35}{137}&lt;/math&gt; would have to be written as 35/137. To use LaTeX in the forums, enclose your LaTeX code with dollar signs: &lt;math&gt;your codes here&lt;/math&gt;. To use LaTeX on AoPSWiki, enclose your code with math tags instead of dollar signs, like so: &lt;nowiki&gt;&lt;math&gt;your codes here&lt;/math&gt;&lt;/nowiki&gt;<br /> <br /> CHANGE: Dollar signs can now be used to use LaTeX.<br /> <br /> ==Useful Codes==<br /> <br /> '''\boxed{Answer}''' Produces a box around your Answer. Cannot be used in Wiki<br /> <br /> '''\frac{a}{b}''' Produces a common fraction with [[numerator]] ''a'' and [[denominator]] ''b''. &lt;math&gt;\frac{a}{b}&lt;/math&gt;<br /> <br /> '''^\circ''' Produces the degrees symbol. &lt;math&gt;a^{\circ}&lt;/math&gt;<br /> <br /> '''\text{Your Text Here}''' Produces text within LaTeX. &lt;math&gt;\mbox{Your Text Here}&lt;/math&gt;<br /> <br /> '''\sqrt{x}''' Produces the square root of ''x''. &lt;math&gt;\sqrt{x}&lt;/math&gt;<br /> <br /> '''a\equiv b \mod{c}''' Produces a is equivalent to b, mod c. &lt;math&gt;a\equiv b \mod{c}&lt;/math&gt;<br /> <br /> '''\binom{9}{3}''' Produces 9 choose 3. Cannot be used in Wiki.<br /> <br /> '''x^{y}''' Produces x to the power of y. &lt;math&gt;x^y&lt;/math&gt;<br /> <br /> '''x_{y}''' Produces x with y in subscript. &lt;math&gt;x_y&lt;/math&gt;<br /> <br /> '''\rightarrow''' Produces an arrow to the right. &lt;math&gt;\rightarrow&lt;/math&gt;<br /> <br /> Note that on AoPSWiki, many codes that work on the AoPS forums do not work. Also, a helpful tip is that if LaTeX fails to render within AoPSWiki, try adding the code \displaystyle to the beginning of the string of LaTeX. This often fixes minor rendering problems.<br /> <br /> <br /> Also note that you do not have to use braces, &quot;{&quot; and &quot;}&quot;, when you only want one character in the operation.<br /> ===Example===<br /> x^y is the same as x^{y}<br /> <br /> ==Fonts==<br /> <br /> === Font families ===<br /> <br /> * Roman (default): \textrm{...}<br /> * Sans-serif: \textsf{...}<br /> * Monospace (typewriter): \texttt{...}<br /> <br /> === Font sizes ===<br /> <br /> To activate a font size, write '{\tiny{This text is tiny}}', for example.<br /> <br /> * \tiny (5 pt.)<br /> * \scriptsize (7 pt.)<br /> * \footnotesize (8 pt.)<br /> * \small (9 pt.)<br /> * \normalsize (10 pt.)<br /> * \large (12 pt.)<br /> * \Large (14 pt.)<br /> * \LARGE (18 pt.)<br /> * \huge (20 pt.)<br /> * \Huge (24 pt.)<br /> <br /> === Font styles ===<br /> <br /> * Bold \textbf{...}<br /> * Italics \textit{...}<br /> * Slanted \textsl{...}<br /> * Small capitals \textsc{...}<br /> * Sans-serif \textsf{...}<br /> * Monospace \texttt{...}<br /> * Emphasis \emph{...}<br /> <br /> ==Tutorials &amp; Tools ==<br /> <br /> * [http://www.artofproblemsolving.com/LaTeX/AoPS_L_About.php AoPS LaTeX Guide]<br /> *[http://en.wikipedia.org/wiki/LaTeX Wikipedia Article]<br /> *[http://sciencesoft.at/index.jsp?link=latex&amp;lang=en&amp;wiki=1 This] is a useful site that will change LaTeX input into a PNG image.<br /> <br /> <br /> <br /> {{tutorial}}</div> B-flat https://artofproblemsolving.com/wiki/index.php?title=LaTeX&diff=8128 LaTeX 2006-07-23T01:32:24Z <p>B-flat: /* Useful Codes */</p> <hr /> <div>'''LaTeX''' is a typesetting language used primarily to type mathematical expressions in an elegant fashion. For example, without LaTeX, &lt;math&gt;\frac{35}{137}&lt;/math&gt; would have to be written as 35/137. To use LaTeX in the forums, enclose your LaTeX code with dollar signs: &lt;math&gt;your codes here&lt;/math&gt;. To use LaTeX on AoPSWiki, enclose your code with math tags instead of dollar signs, like so: &lt;nowiki&gt;&lt;math&gt;your codes here&lt;/math&gt;&lt;/nowiki&gt;<br /> <br /> ==Useful Codes==<br /> <br /> '''\boxed{Answer}''' Produces a box around your Answer. Cannot be used in Wiki<br /> <br /> '''\frac{a}{b}''' Produces a common fraction with [[numerator]] ''a'' and [[denominator]] ''b''. &lt;math&gt;\frac{a}{b}&lt;/math&gt;<br /> <br /> '''^\circ''' Produces the degrees symbol. &lt;math&gt;a^{\circ}&lt;/math&gt;<br /> <br /> '''\text{Your Text Here}''' Produces text within LaTeX. &lt;math&gt;\mbox{Your Text Here}&lt;/math&gt;<br /> <br /> '''\sqrt{x}''' Produces the square root of ''x''. &lt;math&gt;\sqrt{x}&lt;/math&gt;<br /> <br /> '''a\equiv b \mod{c}''' Produces a is equivalent to b, mod c. &lt;math&gt;a\equiv b \mod{c}&lt;/math&gt;<br /> <br /> '''\binom{9}{3}''' Produces 9 choose 3. Cannot be used in Wiki.<br /> <br /> '''x^{y}''' Produces x to the power of y. &lt;math&gt;x^y&lt;/math&gt;<br /> <br /> '''x_{y}''' Produces x with y in subscript. &lt;math&gt;x_y&lt;/math&gt;<br /> <br /> '''\rightarrow''' Produces an arrow to the right. &lt;math&gt;\rightarrow&lt;/math&gt;<br /> <br /> Note that on AoPSWiki, many codes that work on the AoPS forums do not work. Also, a helpful tip is that if LaTeX fails to render within AoPSWiki, try adding the code \displaystyle to the beginning of the string of LaTeX. This often fixes minor rendering problems.<br /> <br /> <br /> Also note that you do not have to use braces, &quot;{&quot; and &quot;}&quot;, when you only want one character in the operation.<br /> ===Example===<br /> x^y is the same as x^{y}<br /> <br /> ==Fonts==<br /> <br /> === Font families ===<br /> <br /> * Roman (default): \textrm{...}<br /> * Sans-serif: \textsf{...}<br /> * Monospace (typewriter): \texttt{...}<br /> <br /> === Font sizes ===<br /> <br /> To activate a font size, write '{\tiny{This text is tiny}}', for example.<br /> <br /> * \tiny (5 pt.)<br /> * \scriptsize (7 pt.)<br /> * \footnotesize (8 pt.)<br /> * \small (9 pt.)<br /> * \normalsize (10 pt.)<br /> * \large (12 pt.)<br /> * \Large (14 pt.)<br /> * \LARGE (18 pt.)<br /> * \huge (20 pt.)<br /> * \Huge (24 pt.)<br /> <br /> === Font styles ===<br /> <br /> * Bold \textbf{...}<br /> * Italics \textit{...}<br /> * Slanted \textsl{...}<br /> * Small capitals \textsc{...}<br /> * Sans-serif \textsf{...}<br /> * Monospace \texttt{...}<br /> * Emphasis \emph{...}<br /> <br /> ==Tutorials &amp; Tools ==<br /> <br /> * [http://www.artofproblemsolving.com/LaTeX/AoPS_L_About.php AoPS LaTeX Guide]<br /> *[http://en.wikipedia.org/wiki/LaTeX Wikipedia Article]<br /> *[http://sciencesoft.at/index.jsp?link=latex&amp;lang=en&amp;wiki=1 This] is a useful site that will change LaTeX input into a PNG image.<br /> <br /> <br /> <br /> {{tutorial}}</div> B-flat https://artofproblemsolving.com/wiki/index.php?title=2006_AMC_10A_Problems/Problem_18&diff=8127 2006 AMC 10A Problems/Problem 18 2006-07-23T01:12:58Z <p>B-flat: /* Solution */</p> <hr /> <div>== Problem ==<br /> A license plate in a certain state consists of 4 digits, not necessarily distinct, and 2 letters, also not necessarily distinct. These six characters may appear in any order, except that the two letters must appear next to each other. How many distinct license plates are possible? <br /> <br /> &lt;math&gt;\mathrm{(A) \ } 10^4\times 26^2\qquad\mathrm{(B) \ } 10^3\times 26^3\qquad\mathrm{(C) \ } 5\times 10^4\times 26^2\qquad\mathrm{(D) \ } 10^2\times 26^4\qquad\mathrm{(E) \ } 5\times 10^3\times 26^3\qquad&lt;/math&gt;<br /> == Solution ==<br /> There are &lt;math&gt;10\cdot10\cdot10\cdot10 = 10^4&lt;/math&gt; ways to choose 4 digits.<br /> <br /> There are &lt;math&gt;26 \cdot 26 = 26^2&lt;/math&gt; ways to choose the 2 letters. <br /> <br /> For the letters to be next to each other, they can be the 1st and 2nd, 2nd and 3rd, 3rd and 4th, 4th and 5th, or the 5th and 6th characters. <br /> So, there are &lt;math&gt;5&lt;/math&gt; choices for position.<br /> <br /> Therefore there are &lt;math&gt; 5\times 10^4\times 26^2 &lt;/math&gt; distinct license plates &lt;math&gt; \Rightarrow (C) &lt;/math&gt;<br /> <br /> == See Also ==<br /> *[[2006 AMC 10A Problems]]</div> B-flat https://artofproblemsolving.com/wiki/index.php?title=User:B-flat&diff=7820 User:B-flat 2006-07-17T01:08:30Z <p>B-flat: /* Contributions */</p> <hr /> <div>== b-flat ==<br /> <br /> Here is my user page.<br /> <br /> I live in North Carolina, and I enjoy math, programming, chess, biking, and playing piano.<br /> <br /> == Contributions ==<br /> <br /> * Made a bunch of articles look nicer<br /> <br /> * Created Area of common geometric figures<br /> * Added Area of quadrilaterals<br /> <br /> * Created [[Greatest common divisor]] (GCD)<br /> <br /> * Added Approximating [[pi]] - June 26, 2006<br /> <br /> * Added [[Chinese Remainder Theorem]] Example - June 27, 2006<br /> <br /> * Added Introduction to Completing the Square - June 27, 2005<br /> <br /> * Created [[Rhombus]] Page - June 27, 2006<br /> <br /> * Created [[Parallelogram]] Page - June 27, 2006<br /> <br /> * Created [[Rectangle]] Page - June 27, 2006<br /> <br /> * Created [[Kite]] Page - June 27, 2006<br /> <br /> * Created [[Trapezoid]] Page - June 27, 2006<br /> <br /> * Created [[Isosceles trapezoid]] Page - June 27, 2006<br /> <br /> * Added Perfect Square trinomials to [[perfect square]] page - June 29, 2006<br /> <br /> * Created [[2006 AMC 10A]] Page - June 29, 2006<br /> <br /> * Added AMC 20A problems Page - July 1, 2006<br /> <br /> * Created [[constant]] Page - July 6, 2006<br /> <br /> * Created [[variable]] Page - July 6, 2006<br /> <br /> * Created [[perimeter]] Page - July 6, 2006<br /> <br /> * Created [[interval]] Page - July 11, 2006<br /> <br /> * Created [[contrapositive]] Page - July 16, 2006</div> B-flat https://artofproblemsolving.com/wiki/index.php?title=Contrapositive&diff=7819 Contrapositive 2006-07-17T01:07:41Z <p>B-flat: </p> <hr /> <div>A '''contrapositive''' of a statement is always true, assuming that the conditional statement is true. However, if the conditional statement is false, then the contrapositive is also false.<br /> <br /> A [[conditional]] statement is usually expressed as<br /> <br /> If '''P''', then '''Q'''<br /> <br /> The contrapositive statement is usually expressed as<br /> <br /> If not '''Q''', then not '''P'''<br /> <br /> where '''P''' denotes a condition and '''Q''' denotes another condition.<br /> <br /> == Examples ==<br /> <br /> Given the conditional statement &quot;If a polygon has 3 sides, then it is a triangle&quot;, the contrapositive is &quot;If a polygon is not a triangle, then it does not have 3 sides&quot;.</div> B-flat https://artofproblemsolving.com/wiki/index.php?title=User_talk:JBL&diff=7467 User talk:JBL 2006-07-12T17:22:42Z <p>B-flat: </p> <hr /> <div>I do that so that I have a more organized list, not a mess like on the my contributions page. This way, it is easier to reference.<br /> <br /> b-flat</div> B-flat https://artofproblemsolving.com/wiki/index.php?title=User:B-flat&diff=7306 User:B-flat 2006-07-11T17:11:44Z <p>B-flat: /* Contributions */</p> <hr /> <div>== b-flat ==<br /> <br /> Here is my user page.<br /> <br /> I live in North Carolina, and I enjoy math, programming, chess, biking, and playing piano.<br /> <br /> == Contributions ==<br /> <br /> * Made a bunch of articles look nicer<br /> <br /> * Created Area of common geometric figures<br /> * Added Area of quadrilaterals<br /> <br /> * Created [[Greatest common divisor]] (GCD)<br /> <br /> * Added Approximating [[pi]] - June 26, 2006<br /> <br /> * Added [[Chinese Remainder Theorem]] Example - June 27, 2006<br /> <br /> * Added Introduction to Completing the Square - June 27, 2005<br /> <br /> * Created [[Rhombus]] Page - June 27, 2006<br /> <br /> * Created [[Parallelogram]] Page - June 27, 2006<br /> <br /> * Created [[Rectangle]] Page - June 27, 2006<br /> <br /> * Created [[Kite]] Page - June 27, 2006<br /> <br /> * Created [[Trapezoid]] Page - June 27, 2006<br /> <br /> * Created [[Isosceles trapezoid]] Page - June 27, 2006<br /> <br /> * Added Perfect Square trinomials to [[perfect square]] page - June 29, 2006<br /> <br /> * Created [[2006 AMC 10A]] Page - June 29, 2006<br /> <br /> * Added AMC 20A problems Page - July 1, 2006<br /> <br /> * Created [[constant]] Page - July 6, 2006<br /> <br /> * Created [[variable]] Page - July 6, 2006<br /> <br /> * Created [[perimeter]] Page - July 6, 2006<br /> <br /> * Created [[interval]] Page - July 11, 2006</div> B-flat https://artofproblemsolving.com/wiki/index.php?title=Interval&diff=7305 Interval 2006-07-11T17:11:18Z <p>B-flat: </p> <hr /> <div>== Definition ==<br /> <br /> An '''interval''' is a range of values. The most common uses of an interval are for [[domain]] and [[range]].<br /> <br /> == Symbols ==<br /> <br /> If an interval has either ( or ) on it, the values at the end are NOT included.<br /> <br /> If an interval has either [ or ] on it, the values at the end ARE included.<br /> <br /> Note: &lt;math&gt;-\infty&lt;/math&gt; and &lt;math&gt;\infty&lt;/math&gt; are never included as endpoints<br /> <br /> == Examples ==<br /> <br /> * (2,3) means all real values between 2 and 3, but not including 2 and 3<br /> <br /> * [-2,0) means all real values between -2 and 0, but does not include 0</div> B-flat https://artofproblemsolving.com/wiki/index.php?title=2006_AMC_10A_Problems/Problem_2&diff=6176 2006 AMC 10A Problems/Problem 2 2006-07-06T18:05:32Z <p>B-flat: /* Solution */</p> <hr /> <div>== Problem ==<br /> Define &lt;math&gt;x\otimes y=x^3-y&lt;/math&gt;. What is &lt;math&gt;h\otimes (h\otimes h)&lt;/math&gt;?<br /> <br /> &lt;math&gt; \mathrm{(A) \ } -h\qquad \mathrm{(B) \ } 0\qquad \mathrm{(C) \ } h\qquad \mathrm{(D) \ } 2h\qquad \mathrm{(E) \ } h^3 &lt;/math&gt;<br /> == Solution ==<br /> Plugging our values in the function, we have <br /> <br /> &lt;center&gt;&lt;math&gt;\displaystyle h^3-h&lt;/math&gt;&lt;/center&gt;<br /> <br /> Plugging in the function once more, we have<br /> <br /> &lt;center&gt;&lt;math&gt;\displaystyle h^3-(h^3-h)=h, <br /> (C)&lt;/math&gt;&lt;/center&gt;<br /> <br /> == See Also ==<br /> *[[2006 AMC 10A Problems]]</div> B-flat https://artofproblemsolving.com/wiki/index.php?title=User:B-flat&diff=6175 User:B-flat 2006-07-06T18:02:42Z <p>B-flat: /* Contributions */</p> <hr /> <div>== b-flat ==<br /> <br /> Here is my user page.<br /> <br /> I live in North Carolina, and I enjoy math, programming, chess, biking, and playing piano.<br /> <br /> == Contributions ==<br /> <br /> * Made a bunch of articles look nicer<br /> <br /> * Created Area of common geometric figures<br /> * Added Area of quadrilaterals<br /> <br /> * Created [[Greatest common divisor]] (GCD)<br /> <br /> * Added Approximating [[pi]] - June 26, 2006<br /> <br /> * Added [[Chinese Remainder Theorem]] Example - June 27, 2006<br /> <br /> * Added Introduction to Completing the Square - June 27, 2005<br /> <br /> * Created [[Rhombus]] Page - June 27, 2006<br /> <br /> * Created [[Parallelogram]] Page - June 27, 2006<br /> <br /> * Created [[Rectangle]] Page - June 27, 2006<br /> <br /> * Created [[Kite]] Page - June 27, 2006<br /> <br /> * Created [[Trapezoid]] Page - June 27, 2006<br /> <br /> * Created [[Isosceles trapezoid]] Page - June 27, 2006<br /> <br /> * Added Perfect Square trinomials to [[perfect square]] page - June 29, 2006<br /> <br /> * Created [[2006 AMC 10A]] Page - June 29, 2006<br /> <br /> * Added AMC 20A problems Page - July 1, 2006<br /> <br /> * Created [[constant]] Page - July 6, 2006<br /> <br /> * Created [[variable]] Page - July 6, 2006<br /> <br /> * Created [[perimeter]] Page - July 6, 2006</div> B-flat https://artofproblemsolving.com/wiki/index.php?title=Perimeter&diff=6174 Perimeter 2006-07-06T18:01:47Z <p>B-flat: </p> <hr /> <div>== Definition ==<br /> <br /> The '''perimeter''' of a geometric figure is the distance around the edge of the figure. '''Perimeter''' is often denoted by P. The '''perimeter''' of a circle is called its circumference.<br /> <br /> == Formulas ==<br /> <br /> * Square - 4s, where s is the side length<br /> <br /> * Rectangle - 2(l+w), where l is the length and w is the width<br /> <br /> * Circle - &lt;math&gt;2\pi r&lt;/math&gt;, where r is the radius<br /> <br /> * Regular geometric figure with n sides - ns, where s is the length of the side</div> B-flat https://artofproblemsolving.com/wiki/index.php?title=Contest_strategies&diff=6173 Contest strategies 2006-07-06T17:56:29Z <p>B-flat: /* Educated Guessing */</p> <hr /> <div>Although all [[math competitions]] require knowledge of math to do well in, there are certain ways to do slightly better than normally using some contest strategies.<br /> <br /> ==General Strategies==<br /> <br /> ===Skipping Problems===<br /> <br /> Instead of spending twenty minutes on a problem, some people choose to spend thirty seconds reading the problem, then skip it. The time saved skipping the problem can help you check and solve other problems, and you can always return to the problem that you skipped. This concept helps double where you are rewarded for a blank answer, such as the [[SAT]], the [[AMC 10]], and the [[AMC 12]].<br /> <br /> In the MathCounts Sprint competition of 2006, many of the harder problems were presented earlier on in the test, putting people who did not skip problems at a large disadvantage - Daesun Yim, the 5th place written finisher, skipped five out of thirty problems on the Sprint round, and he received a 23/30 on the Sprint round.<br /> <br /> ===Educated Guessing===<br /> <br /> Educated guessing is the art of taking a, well, educated guess. An educated guess takes less time than completely solving the problem, especially when the problem stumps you, and if you can narrow the problem down to a couple of choices, you may be able to take a good guess, saving you time and giving you a chance of getting the answer correct. This tactic helps the most in competitions with multiple choice answers. It also helps where a wrong answer and an answer left blank are counted as the same amount of points, such as the [[AMC 8]] and [[MathCounts]].<br /> <br /> == See also ==<br /> * [[Mathematics competition resources]]<br /> * [[Mathematical problem solving]]</div> B-flat https://artofproblemsolving.com/wiki/index.php?title=User:B-flat&diff=6164 User:B-flat 2006-07-06T17:28:23Z <p>B-flat: /* Contributions */</p> <hr /> <div>== b-flat ==<br /> <br /> Here is my user page.<br /> <br /> I live in North Carolina, and I enjoy math, programming, chess, biking, and playing piano.<br /> <br /> == Contributions ==<br /> <br /> * Made a bunch of articles look nicer<br /> <br /> * Created Area of common geometric figures<br /> * Added Area of quadrilaterals<br /> <br /> * Created [[Greatest common divisor]] (GCD)<br /> <br /> * Added Approximating [[pi]] - June 26, 2006<br /> <br /> * Added [[Chinese Remainder Theorem]] Example - June 27, 2006<br /> <br /> * Added Introduction to Completing the Square - June 27, 2005<br /> <br /> * Created [[Rhombus]] Page - June 27, 2006<br /> <br /> * Created [[Parallelogram]] Page - June 27, 2006<br /> <br /> * Created [[Rectangle]] Page - June 27, 2006<br /> <br /> * Created [[Kite]] Page - June 27, 2006<br /> <br /> * Created [[Trapezoid]] Page - June 27, 2006<br /> <br /> * Created [[Isosceles trapezoid]] Page - June 27, 2006<br /> <br /> * Added Perfect Square trinomials to [[perfect square]] page - June 29, 2006<br /> <br /> * Created [[2006 AMC 10A]] Page - June 29, 2006<br /> <br /> * Added AMC 20A problems Page - July 1, 2006<br /> <br /> * Created [[constant]] Page - July 6, 2006<br /> <br /> * Created [[variable]] Page - July 6, 2006</div> B-flat https://artofproblemsolving.com/wiki/index.php?title=Variable&diff=6163 Variable 2006-07-06T17:25:28Z <p>B-flat: </p> <hr /> <div>== Definition ==<br /> <br /> A '''variable''' does not have a set value. Its value can change, and a '''variable''' is used to represent unknowns in equations.<br /> <br /> Any symbol may be used as a variable, but the most common variables are x, y, z, n, i, a, b, and c.<br /> <br /> == Examples ==<br /> <br /> What plus 7 is 10?<br /> <br /> Write an equation that uses a '''variable''' to represent the part that is unknown.<br /> <br /> &lt;math&gt;\displaystyle7+n=10&lt;/math&gt;</div> B-flat https://artofproblemsolving.com/wiki/index.php?title=User:B-flat&diff=6161 User:B-flat 2006-07-06T17:11:17Z <p>B-flat: /* Contributions */</p> <hr /> <div>== b-flat ==<br /> <br /> Here is my user page.<br /> <br /> I live in North Carolina, and I enjoy math, programming, chess, biking, and playing piano.<br /> <br /> == Contributions ==<br /> <br /> * Made a bunch of articles look nicer<br /> <br /> * Created Area of common geometric figures<br /> * Added Area of quadrilaterals<br /> <br /> * Created [[Greatest common divisor]] (GCD)<br /> <br /> * Added Approximating [[pi]] - June 26, 2006<br /> <br /> * Added [[Chinese Remainder Theorem]] Example - June 27, 2006<br /> <br /> * Added Introduction to Completing the Square - June 27, 2005<br /> <br /> * Created [[Rhombus]] Page - June 27, 2006<br /> <br /> * Created [[Parallelogram]] Page - June 27, 2006<br /> <br /> * Created [[Rectangle]] Page - June 27, 2006<br /> <br /> * Created [[Kite]] Page - June 27, 2006<br /> <br /> * Created [[Trapezoid]] Page - June 27, 2006<br /> <br /> * Created [[Isosceles trapezoid]] Page - June 27, 2006<br /> <br /> * Added Perfect Square trinomials to [[perfect square]] page - June 29, 2006<br /> <br /> * Created [[2006 AMC 10A]] Page - June 29, 2006<br /> <br /> * Added AMC 20A problems Page - July 1, 2006<br /> <br /> * Created [[constant]] Page - July 6, 2006</div> B-flat https://artofproblemsolving.com/wiki/index.php?title=Constant&diff=6160 Constant 2006-07-06T17:10:26Z <p>B-flat: </p> <hr /> <div>== Definition ==<br /> <br /> A '''constant''' is a number that remains the same, unlike a [[variable]], that can change.<br /> <br /> == Examples ==<br /> <br /> &lt;math&gt;\displaystyle\pi&lt;/math&gt;, or [[pi]] is a famous constant that is approximately 3.14159.<br /> <br /> &lt;math&gt;\displaystyle e&lt;/math&gt;, or [[e]] is another famous constant that is approximately 2.71828.<br /> <br /> All numbers are constants because they do not change their value; they remain the same forever.</div> B-flat https://artofproblemsolving.com/wiki/index.php?title=User:B-flat&diff=5767 User:B-flat 2006-07-01T16:37:49Z <p>B-flat: /* Contributions */</p> <hr /> <div>== b-flat ==<br /> <br /> Here is my user page.<br /> <br /> I live in North Carolina, and I enjoy math, programming, chess, biking, and playing piano.<br /> <br /> == Contributions ==<br /> <br /> * Made a bunch of articles look nicer<br /> <br /> * Created Area of common geometric figures<br /> * Added Area of quadrilaterals<br /> <br /> * Created [[Greatest common divisor]] (GCD)<br /> <br /> * Added Approximating [[pi]] - June 26, 2006<br /> <br /> * Added [[Chinese Remainder Theorem]] Example - June 27, 2006<br /> <br /> * Added Introduction to Completing the Square - June 27, 2005<br /> <br /> * Created [[Rhombus]] Page - June 27, 2006<br /> <br /> * Created [[Parallelogram]] Page - June 27, 2006<br /> <br /> * Created [[Rectangle]] Page - June 27, 2006<br /> <br /> * Created [[Kite]] Page - June 27, 2006<br /> <br /> * Created [[Trapezoid]] Page - June 27, 2006<br /> <br /> * Created [[Isosceles trapezoid]] Page - June 27, 2006<br /> <br /> * Added Perfect Square trinomials to [[perfect square]] page - June 29, 2006<br /> <br /> * Created [[2006 AMC 10A]] Page - June 29, 2006<br /> <br /> * Added AMC 20A problems Page - July 1, 2006</div> B-flat https://artofproblemsolving.com/wiki/index.php?title=2006_AMC_10A_Problems&diff=5766 2006 AMC 10A Problems 2006-07-01T16:37:05Z <p>B-flat: </p> <hr /> <div>==Problem 1==<br /> <br /> Sandwiches at Joe's Fast Food cost$3 each and sodas cost $2 each. How many dollars will it cost to purchase 5 sandwiches and 8 sodas?<br /> <br /> &lt;center&gt;&lt;math&gt; \mathrm{(A) \ } 31\qquad \mathrm{(B) \ } 32\qquad \mathrm{(C) \ } 33\qquad \mathrm{(D) \ } 34\qquad \mathrm{(E) \ } 35 &lt;/math&gt;&lt;/center&gt;<br /> <br /> [[2006 AMC 10A Problem 1|Solution]]<br /> <br /> == Problem 2 ==<br /> <br /> Define &lt;math&gt;x\otimes y=x^3-y&lt;/math&gt;. What is &lt;math&gt;h\otimes (h\otimes h)&lt;/math&gt;?<br /> <br /> &lt;center&gt;&lt;math&gt; \mathrm{(A) \ } -h\qquad \mathrm{(B) \ } 0\qquad \mathrm{(C) \ } h\qquad \mathrm{(D) \ } 2h\qquad \mathrm{(E) \ } h^3 &lt;/math&gt;&lt;/center&gt;<br /> <br /> [[2006 AMC 10A Problem 2|Solution]]<br /> <br /> == Problem 3 ==<br /> <br /> The ratio of Mary's age to Alice's age is 3:5. Alice is 30 years old. How many years old is Mary?<br /> <br /> &lt;center&gt;&lt;math&gt; \mathrm{(A) \ } 15\qquad \mathrm{(B) \ } 18\qquad \mathrm{(C) \ } 20\qquad \mathrm{(D) \ } 24\qquad \mathrm{(E) \ } 50 &lt;/math&gt;&lt;/center&gt;<br /> <br /> [[2006 AMC 10A Problem 3|Solution]]<br /> <br /> == Problem 4 ==<br /> <br /> A digital watch displays hours and minutes with AM and PM. What is the largest possible sum of the digits in the display?<br /> <br /> &lt;center&gt;&lt;math&gt; \mathrm{(A) \ } 17\qquad \mathrm{(B) \ } 19\qquad \mathrm{(C) \ } 21\qquad \mathrm{(D) \ } 22\qquad \mathrm{(E) \ } 23 &lt;/math&gt;&lt;/center&gt;<br /> <br /> [[2006 AMC 10A Problem 4|Solution]]<br /> <br /> == Problem 5 ==<br /> <br /> Doug and Dave shared a pizza with 8 equally-sized slices. Doug wanted a plain pizza, but Dave wanted anchovies on half of the pizza. The cost of a plain pizza was$8, and there was an additional cost of \$2 for putting anchovies on one half. Dave at all of the slices of anchovy piaaz and one plain slice. Doug ate the remainder. Each then paid for what he had eaten. How many more dollars did Dave pay than Doug?<br /> <br /> &lt;center&gt;&lt;math&gt; \mathrm{(A) \ } 1\qquad \mathrm{(B) \ } 2\qquad \mathrm{(C) \ } 3\qquad \mathrm{(D) \ } 4\qquad \mathrm{(E) \ } 5 &lt;/math&gt;&lt;/center&gt;<br /> <br /> [[2006 AMC 10A Problem 5|Solution]]<br /> <br /> == Problem 6 ==<br /> <br /> What non-zero real value for &lt;math&gt;\displaystyle x&lt;/math&gt; satisfies &lt;math&gt;\displaystyle(7x)^{14}=(14x)^7&lt;/math&gt;?<br /> <br /> &lt;center&gt;&lt;math&gt; \mathrm{(A) \ } \frac17\qquad \mathrm{(B) \ } \frac27\qquad \mathrm{(C) \ } 1\qquad \mathrm{(D) \ } 7\qquad \mathrm{(E) \ } 14 &lt;/math&gt;&lt;/center&gt;<br /> <br /> [[2006 AMC 10A Problem 6|Solution]]<br /> <br /> == Problem 7 ==<br /> <br /> Missing diagram<br /> <br /> The &lt;math&gt;8x18&lt;/math&gt; rectangle &lt;math&gt;ABCD&lt;/math&gt; is cut into two congruent hexagons, as shown, in such a way that the two hexagons can be repositioned without overlap to form a square. What is &lt;math&gt;y&lt;/math&gt;?<br /> <br /> &lt;center&gt;&lt;math&gt; \mathrm{(A) \ } 6\qquad \mathrm{(B) \ } 7\qquad \mathrm{(C) \ } 8\qquad \mathrm{(D) \ } 9\qquad \mathrm{(E) \ } 10 &lt;/math&gt;&lt;/center&gt;<br /> <br /> [[2006 AMC 10A Problem 7|Solution]]<br /> <br /> == Problem 8 ==<br /> <br /> A parabola with equation &lt;math&gt;\displaystyle y=x^2+bx+c&lt;/math&gt; passes through the points (2,3) and (4,3). What is &lt;math&gt;\displaystyle c&lt;/math&gt;?<br /> <br /> &lt;center&gt;&lt;math&gt; \mathrm{(A) \ } 2\qquad \mathrm{(B) \ } 5\qquad \mathrm{(C) \ } 7\qquad \mathrm{(D) \ } 10\qquad \mathrm{(E) \ } 11 &lt;/math&gt;&lt;/center&gt;<br /> <br /> [[2006 AMC 10A Problem 8|Solution]]<br /> <br /> == Problem 9 ==<br /> <br /> How many sets of two or more consecutive positive integers have a sum of 15?<br /> <br /> &lt;center&gt;&lt;math&gt; \mathrm{(A) \ } 1\qquad \mathrm{(B) \ } 2\qquad \mathrm{(C) \ } 3\qquad \mathrm{(D) \ } 4\qquad \mathrm{(E) \ } 5 &lt;/math&gt;&lt;/center&gt;<br /> <br /> [[2006 AMC 10A Problem 9|Solution]]<br /> <br /> == Problem 10 ==<br /> <br /> For how many real values of &lt;math&gt;\displaystyle x&lt;/math&gt; is &lt;math&gt;\sqrt{120-\sqrt{x}}&lt;/math&gt; an integer?<br /> <br /> &lt;center&gt;&lt;math&gt; \mathrm{(A) \ } 3\qquad \mathrm{(B) \ } 6\qquad \mathrm{(C) \ } 9\qquad \mathrm{(D) \ } 10\qquad \mathrm{(E) \ } 11 &lt;/math&gt;&lt;/center&gt;<br /> <br /> [[2006 AMC 10A Problem 10|Solution]]<br /> <br /> == Problem 11 ==<br /> <br /> Which of the following describes the graph of the equation &lt;math&gt;\displaystyle(x+y)^2=x^2+y^2&lt;/math&gt;?<br /> <br /> &lt;center&gt;&lt;math&gt; \mathrm{(A) \ } the empty set\qquad \mathrm{(B) \ } one point\qquad \mathrm{(C) \ } two lines\qquad \mathrm{(D) \ } a circle\qquad \mathrm{(E) \ } the entire plane &lt;/math&gt;&lt;/center&gt;<br /> <br /> [[2006 AMC 10A Problem 11|Solution]]<br /> <br /> == Problem 12 ==<br /> <br /> Missing diagram<br /> <br /> Rolly wishes to secure his dog with an 8-foot rope to a square shed that is 16 feet on each side. His preliminary drawings are shown.<br /> <br /> Which of these arrangements give the dog the greater area to roam, and by how many square feet?<br /> <br /> &lt;center&gt;&lt;math&gt; \mathrm{(A) \ } I, by 8\pi\qquad \mathrm{(B) \ } I, by 6\pi\qquad \mathrm{(C) \ } II, by 4\pi\qquad \mathrm{(D) \ } II, by 8\pi\qquad \mathrm{(E) \ } II, by 10\pi &lt;/math&gt;&lt;/center&gt;<br /> <br /> [[2006 AMC 10A Problem 12|Solution]]</div> B-flat https://artofproblemsolving.com/wiki/index.php?title=User:B-flat&diff=5373 User:B-flat 2006-06-29T21:33:20Z <p>B-flat: /* Contributions */</p> <hr /> <div>== b-flat ==<br /> <br /> Here is my user page.<br /> <br /> I live in North Carolina, and I enjoy math, programming, chess, biking, and playing piano.<br /> <br /> == Contributions ==<br /> <br /> * Made a bunch of articles look nicer<br /> <br /> * Created Area of common geometric figures<br /> * Added Area of quadrilaterals<br /> <br /> * Created [[Greatest common divisor]] (GCD)<br /> <br /> * Added Approximating [[pi]] - June 26, 2006<br /> <br /> * Added [[Chinese Remainder Theorem]] Example - June 27, 2006<br /> <br /> * Added Introduction to Completing the Square - June 27, 2005<br /> <br /> * Created [[Rhombus]] Page - June 27, 2006<br /> <br /> * Created [[Parallelogram]] Page - June 27, 2006<br /> <br /> * Created [[Rectangle]] Page - June 27, 2006<br /> <br /> * Created [[Kite]] Page - June 27, 2006<br /> <br /> * Created [[Trapezoid]] Page - June 27, 2006<br /> <br /> * Created [[Isosceles trapezoid]] Page - June 27, 2006<br /> <br /> * Added Perfect Square trinomials to [[perfect square]] page - June 29, 2006<br /> <br /> * Created [[2006 AMC 10A]] Page - June 29, 2006</div> B-flat https://artofproblemsolving.com/wiki/index.php?title=2006_AMC_10A&diff=5372 2006 AMC 10A 2006-06-29T21:30:57Z <p>B-flat: </p> <hr /> <div>2006 AMC 10A problems and solutions. The first link contains the full set of test problems. The rest contain each individual problem and its solution.<br /> <br /> * [[2006 AMC 10A Problems]]<br /> * [[2006 AMC 10A Problem 1]]<br /> * [[2006 AMC 10A Problem 2]]<br /> * [[2006 AMC 10A Problem 3]]<br /> * [[2006 AMC 10A Problem 4]]<br /> * [[2006 AMC 10A Problem 5]]<br /> * [[2006 AMC 10A Problem 6]]<br /> * [[2006 AMC 10A Problem 7]]<br /> * [[2006 AMC 10A Problem 8]]<br /> * [[2006 AMC 10A Problem 9]]<br /> * [[2006 AMC 10A Problem 10]]<br /> * [[2006 AMC 10A Problem 11]]<br /> * [[2006 AMC 10A Problem 12]]<br /> * [[2006 AMC 10A Problem 13]]<br /> * [[2006 AMC 10A Problem 14]]<br /> * [[2006 AMC 10A Problem 15]]<br /> * [[2006 AMC 10A Problem 16]]<br /> * [[2006 AMC 10A Problem 17]]<br /> * [[2006 AMC 10A Problem 18]]<br /> * [[2006 AMC 10A Problem 19]]<br /> * [[2006 AMC 10A Problem 20]]<br /> * [[2006 AMC 10A Problem 21]]<br /> * [[2006 AMC 10A Problem 22]]<br /> * [[2006 AMC 10A Problem 23]]<br /> * [[2006 AMC 10A Problem 24]]<br /> * [[2006 AMC 10A Problem 25]]<br /> <br /> == See also ==</div> B-flat https://artofproblemsolving.com/wiki/index.php?title=User:B-flat&diff=5280 User:B-flat 2006-06-29T16:29:21Z <p>B-flat: /* Contributions */</p> <hr /> <div>== b-flat ==<br /> <br /> Here is my user page.<br /> <br /> I live in North Carolina, and I enjoy math, programming, chess, biking, and playing piano.<br /> <br /> == Contributions ==<br /> <br /> * Made a bunch of articles look nicer<br /> <br /> * Created Area of common geometric figures<br /> * Added Area of quadrilaterals<br /> <br /> * Created [[Greatest common divisor]] (GCD)<br /> <br /> * Added Approximating [[pi]] - June 26, 2006<br /> <br /> * Added [[Chinese Remainder Theorem]] Example - June 27, 2006<br /> <br /> * Added Introduction to Completing the Square - June 27, 2005<br /> <br /> * Created [[Rhombus]] Page - June 27, 2006<br /> <br /> * Created [[Parallelogram]] Page - June 27, 2006<br /> <br /> * Created [[Rectangle]] Page - June 27, 2006<br /> <br /> * Created [[Kite]] Page - June 27, 2006<br /> <br /> * Created [[Trapezoid]] Page - June 27, 2006<br /> <br /> * Created [[Isosceles trapezoid]] Page - June 27, 2006<br /> <br /> * Added Perfect Square trinomials to [[perfect square]] page</div> B-flat