https://artofproblemsolving.com/wiki/api.php?action=feedcontributions&user=Stormbreaker7984&feedformat=atom AoPS Wiki - User contributions [en] 2021-04-23T09:41:21Z User contributions MediaWiki 1.31.1 https://artofproblemsolving.com/wiki/index.php?title=Simon%27s_Favorite_Factoring_Trick&diff=129725 Simon's Favorite Factoring Trick 2020-07-29T16:19:33Z <p>Stormbreaker7984: /* The General Statement */</p> <hr /> <div><br /> ==The General Statement==<br /> You have been hacked.<br /> <br /> == Applications ==<br /> This factorization frequently shows up on contest problems, especially those heavy on algebraic manipulation. Usually &lt;math&gt;x&lt;/math&gt; and &lt;math&gt;y&lt;/math&gt; are variables and &lt;math&gt;j,k&lt;/math&gt; are known constants. Also, it is typically necessary to add the &lt;math&gt;jk&lt;/math&gt; term to both sides to perform the factorization.<br /> <br /> == Fun Practice Problems ==<br /> ===Introductory===<br /> *Two different [[prime number]]s between &lt;math&gt;4&lt;/math&gt; and &lt;math&gt;18&lt;/math&gt; are chosen. When their sum is subtracted from their product, which of the following numbers could be obtained?<br /> <br /> &lt;math&gt; \mathrm{(A) \ 22 } \qquad \mathrm{(B) \ 60 } \qquad \mathrm{(C) \ 119 } \qquad \mathrm{(D) \ 180 } \qquad \mathrm{(E) \ 231 } &lt;/math&gt;<br /> <br /> ([[2000 AMC 12/Problem 6|Source]])<br /> <br /> ===Intermediate===<br /> *&lt;math&gt;m, n&lt;/math&gt; are integers such that &lt;math&gt;m^2 + 3m^2n^2 = 30n^2 + 517&lt;/math&gt;. Find &lt;math&gt;3m^2n^2&lt;/math&gt;.<br /> <br /> ([[1987 AIME Problems/Problem 5|Source]])<br /> <br /> ===Olympiad===<br /> <br /> *The integer &lt;math&gt;N&lt;/math&gt; is positive. There are exactly 2005 ordered pairs &lt;math&gt;(x, y)&lt;/math&gt; of positive integers satisfying:<br /> <br /> &lt;cmath&gt;\frac 1x +\frac 1y = \frac 1N&lt;/cmath&gt;<br /> <br /> Prove that &lt;math&gt;N&lt;/math&gt; is a perfect square. <br /> <br /> Source: (British Mathematical Olympiad Round 3, 2005)<br /> <br /> == See More==<br /> * [[Algebra]]<br /> * [[Factoring]]<br /> <br /> [[Category:Elementary algebra]]<br /> [[Category:Theorems]]</div> Stormbreaker7984 https://artofproblemsolving.com/wiki/index.php?title=How_many_times_does_the_digit_9_appear_in_the_list_of_all_integers_from_1_to_500%3F_(The_number_$_99_$,_for_example,_is_counted_twice,_because_$9$_appears_two_times_in_it.)&diff=123925 How many times does the digit 9 appear in the list of all integers from 1 to 500? (The number $99$, for example, is counted twice, because $9$ appears two times in it.) 2020-06-05T20:57:23Z <p>Stormbreaker7984: The answer is... 100</p> <hr /> <div>How many times does the digit 9 appear in the list of all integers from 1 to 500? (The number &lt;math&gt; 99 &lt;/math&gt;, for example, is counted twice, because &lt;math&gt;9&lt;/math&gt; appears two times in it.)<br /> The answer is.....<br /> <br /> <br /> The easiest approach is to consider how many times 9 can appear in the units place, how many times in the tens place, and how many times in the hundreds place. If we put a 9 in the units place, there are 10 choices for the tens place and 5 choices for the hundreds digit (including 0), for a total of 50 times. Likewise, if we put a 9 in the tens place, there are 10 choices for the units place and 5 choices for the hundreds digit, for a total of 50 times. Since 9 cannot appear in the hundreds digit, there are &lt;math&gt;50+50=\boxed{100}&lt;/math&gt; appearances of the digit 9.</div> Stormbreaker7984