https://artofproblemsolving.com/wiki/api.php?action=feedcontributions&user=Mathsweat+notreally&feedformat=atom AoPS Wiki - User contributions [en] 2022-05-25T15:22:51Z User contributions MediaWiki 1.31.1 https://artofproblemsolving.com/wiki/index.php?title=Chittur_Gopalakrishnavishwanathasrinivasaiyer_Lemma&diff=162526 Chittur Gopalakrishnavishwanathasrinivasaiyer Lemma 2021-09-21T00:59:10Z <p>Mathsweat notreally: </p> <hr /> <div> We know that an &lt;math&gt;x&lt;/math&gt; exists that equal to &lt;math&gt;42\, \cdot&lt;/math&gt; &lt;math&gt;\text{mod} \sqrt{4761}.&lt;/math&gt; This &lt;math&gt;x&lt;/math&gt; is very powerful in competition math problems. Usually coming up on JMO and AMO geo problems. The Euler Line intersects the radical axis at &lt;math&gt;(x^n, n^x)&lt;/math&gt; where &lt;math&gt;n&lt;/math&gt; is the number of composite factors the radius has. This theorem is also used in Newton's Sums, as the &lt;math&gt;n&lt;/math&gt;th root unity is the same thing as &lt;math&gt;x^n&lt;/math&gt; &lt;math&gt;\text{mod}&lt;/math&gt; &lt;math&gt;(42*10\cdot(70-1)^n).&lt;/math&gt; Finally, you'll se it in combo! The number ways you can shuffle &lt;math&gt;n&lt;/math&gt; things into &lt;math&gt;n^2 + nk + 1&lt;/math&gt; items where &lt;math&gt;k&lt;/math&gt; is the number of partitions in an item is the &lt;math&gt;x^{23\cdot3}.&lt;/math&gt; My coaches Iyer Sir and Barnes approved this nice lemma.</div> Mathsweat notreally https://artofproblemsolving.com/wiki/index.php?title=BestieTheorem&diff=147662 BestieTheorem 2021-02-21T19:43:44Z <p>Mathsweat notreally: </p> <hr /> <div>&lt;math&gt;\text{\LARGE{The Bestie Theorem}}&lt;/math&gt;<br /> The bestie theorem is a very easy theorem to learn....<br /> <br /> <br /> All it states is that a bestie can keep on growing and growing as you feed him food to eat.<br /> It can solve any problem and this is the formula:<br /> <br /> <br /> <br /> &lt;math&gt; \text{Bestie Weight} = \text{Bestie Weight 10 minutes ago}^\text{Bestie Weight 5 minutes ago x Amount of food fed}&lt;/math&gt;<br /> <br /> <br /> Simple!!<br /> <br /> This is one of the most important formulas in all of mathematics!!</div> Mathsweat notreally https://artofproblemsolving.com/wiki/index.php?title=BestieTheorem&diff=147660 BestieTheorem 2021-02-21T19:42:48Z <p>Mathsweat notreally: </p> <hr /> <div>&lt;math&gt;\text{\LARGE{The Bestie Theorem}}&lt;/math&gt;<br /> The bestie theorem is a very easy theorem to learn....<br /> <br /> <br /> All it states is that a bestie can keep on growing and growing as you feed him food to eat.<br /> It can solve any problem and this is the formula:<br /> <br /> <br /> <br /> &lt;math&gt; \text{Bestie Weight} = \text{Bestie Weight 10 minutes ago}^\text{Bestie Weight 5 minutes ago x Amount of food fed}&lt;/math&gt;<br /> <br /> <br /> Simple!!<br /> <br /> This is one of the most important formulas in all of mathematics!!<br /> <br /> This is from a user<br /> <br /> OMG THIS THROEM IS SOO USEFUL. THIS HELPED ME WITH ATLEAST 3 PROBLEMS ON THE AIME. THANK YOU THANK YOU THANK YOU SOOO MUCH.</div> Mathsweat notreally https://artofproblemsolving.com/wiki/index.php?title=1981_IMO_Problems/Problem_1&diff=144489 1981 IMO Problems/Problem 1 2021-02-02T00:08:13Z <p>Mathsweat notreally: </p> <hr /> <div>== Problem ==<br /> <br /> &lt;math&gt;P&lt;/math&gt; is a point inside a given triangle &lt;math&gt;ABC&lt;/math&gt;. &lt;math&gt;D, E, F&lt;/math&gt; are the feet of the perpendiculars from &lt;math&gt;P&lt;/math&gt; to the lines &lt;math&gt;BC, CA, AB&lt;/math&gt;, respectively. Find all &lt;math&gt;P&lt;/math&gt; for which<br /> <br /> &lt;center&gt;<br /> &lt;math&gt;<br /> \frac{BC}{PD} + \frac{CA}{PE} + \frac{AB}{PF}<br /> &lt;/math&gt;<br /> &lt;/center&gt;<br /> <br /> is least.<br /> <br /> == Solution ==<br /> <br /> We note that &lt;math&gt;BC \cdot PD + CA \cdot PE + AB \cdot PF&lt;/math&gt; is twice the triangle's area, i.e., constant. By the [[Cauchy-Schwarz Inequality]],<br /> <br /> &lt;center&gt;<br /> &lt;math&gt;<br /> {(BC \cdot PD + CA \cdot PE + AB \cdot PF) \left(\frac{BC}{PD} + \frac{CA}{PE} + \frac{AB}{PF} \right) \ge ( BC + CA + AB )^2}<br /> &lt;/math&gt;,<br /> &lt;/center&gt;<br /> <br /> with equality exactly when &lt;math&gt;PD = PE = PF &lt;/math&gt;, which occurs when &lt;math&gt;P &lt;/math&gt; is the triangle's incenter, Q.E.D.<br /> <br /> {{alternate solutions}}<br /> <br /> {{IMO box|before=First question|num-a=2|year=1981}}<br /> <br /> [[Category:Olympiad Geometry Problems]]</div> Mathsweat notreally https://artofproblemsolving.com/wiki/index.php?title=User:V_v&diff=143419 User:V v 2021-01-27T14:09:50Z <p>Mathsweat notreally: </p> <hr /> <div>cool kid<br /> <br /> <br /> bestie theorem</div> Mathsweat notreally