https://artofproblemsolving.com/wiki/api.php?action=feedcontributions&user=Liopleurodon&feedformat=atom AoPS Wiki - User contributions [en] 2022-10-03T17:31:15Z User contributions MediaWiki 1.31.1 https://artofproblemsolving.com/wiki/index.php?title=2017_USAJMO_Problems&diff=85328 2017 USAJMO Problems 2017-04-20T23:18:23Z <p>Liopleurodon: /* Problem 6 */</p> <hr /> <div>==Day 1==<br /> <br /> Note: For any geometry problem whose statement begins with an asterisk (&lt;math&gt;*&lt;/math&gt;), the first page of the solution must be a large, in-scale, clearly labeled diagram. Failure to meet this requirement will result in an automatic 1-point deduction. <br /> <br /> ===Problem 1===<br /> Prove that there are infinitely many distinct pairs &lt;math&gt;(a,b)&lt;/math&gt; of relatively prime positive integers &lt;math&gt;a &gt; 1&lt;/math&gt; and &lt;math&gt;b &gt; 1&lt;/math&gt; such that &lt;math&gt;a^b + b^a&lt;/math&gt; is divisible by &lt;math&gt;a + b.&lt;/math&gt; <br /> <br /> [[2017 USAJMO Problems/Problem 1|Solution]]<br /> <br /> ===Problem 2===<br /> Consider the equation <br /> &lt;cmath&gt;\left(3x^3 + xy^2 \right) \left(x^2y + 3y^3 \right) = (x-y)^7.&lt;/cmath&gt;<br /> <br /> (a) Prove that there are infinitely many pairs &lt;math&gt;(x,y)&lt;/math&gt; of positive integers satisfying the equation. <br /> <br /> (b) Describe all pairs &lt;math&gt;(x,y)&lt;/math&gt; of positive integers satisfying the equation. <br /> <br /> [[2017 USAJMO Problems/Problem 2|Solution]]<br /> ===Problem 3===<br /> (&lt;math&gt;*&lt;/math&gt;) Let &lt;math&gt;ABC&lt;/math&gt; be an equilateral triangle and let &lt;math&gt;P&lt;/math&gt; be a point on its circumcircle. Let lines &lt;math&gt;PA&lt;/math&gt; and &lt;math&gt;BC&lt;/math&gt; intersect at &lt;math&gt;D&lt;/math&gt;; let lines &lt;math&gt;PB&lt;/math&gt; and &lt;math&gt;CA&lt;/math&gt; intersect at &lt;math&gt;E&lt;/math&gt;; and let lines &lt;math&gt;PC&lt;/math&gt; and &lt;math&gt;AB&lt;/math&gt; intersect at &lt;math&gt;F&lt;/math&gt;. Prove that the area of triangle &lt;math&gt;DEF&lt;/math&gt; is twice the area of triangle &lt;math&gt;ABC&lt;/math&gt;.<br /> <br /> [[2017 USAJMO Problems/Problem 3|Solution]]<br /> <br /> ==Day 2==<br /> ===Problem 4===<br /> <br /> ===Problem 5===<br /> <br /> ===Problem 6===<br /> {{MAA Notice}}<br /> <br /> {{USAJMO newbox|year= 2017 |before=[[2016 USAJMO]]|after=[[2018 USAJMO]]}}</div> Liopleurodon https://artofproblemsolving.com/wiki/index.php?title=2017_USAJMO_Problems&diff=85327 2017 USAJMO Problems 2017-04-20T23:18:06Z <p>Liopleurodon: /* Problem 6 */</p> <hr /> <div>==Day 1==<br /> <br /> Note: For any geometry problem whose statement begins with an asterisk (&lt;math&gt;*&lt;/math&gt;), the first page of the solution must be a large, in-scale, clearly labeled diagram. Failure to meet this requirement will result in an automatic 1-point deduction. <br /> <br /> ===Problem 1===<br /> Prove that there are infinitely many distinct pairs &lt;math&gt;(a,b)&lt;/math&gt; of relatively prime positive integers &lt;math&gt;a &gt; 1&lt;/math&gt; and &lt;math&gt;b &gt; 1&lt;/math&gt; such that &lt;math&gt;a^b + b^a&lt;/math&gt; is divisible by &lt;math&gt;a + b.&lt;/math&gt; <br /> <br /> [[2017 USAJMO Problems/Problem 1|Solution]]<br /> <br /> ===Problem 2===<br /> Consider the equation <br /> &lt;cmath&gt;\left(3x^3 + xy^2 \right) \left(x^2y + 3y^3 \right) = (x-y)^7.&lt;/cmath&gt;<br /> <br /> (a) Prove that there are infinitely many pairs &lt;math&gt;(x,y)&lt;/math&gt; of positive integers satisfying the equation. <br /> <br /> (b) Describe all pairs &lt;math&gt;(x,y)&lt;/math&gt; of positive integers satisfying the equation. <br /> <br /> [[2017 USAJMO Problems/Problem 2|Solution]]<br /> ===Problem 3===<br /> (&lt;math&gt;*&lt;/math&gt;) Let &lt;math&gt;ABC&lt;/math&gt; be an equilateral triangle and let &lt;math&gt;P&lt;/math&gt; be a point on its circumcircle. Let lines &lt;math&gt;PA&lt;/math&gt; and &lt;math&gt;BC&lt;/math&gt; intersect at &lt;math&gt;D&lt;/math&gt;; let lines &lt;math&gt;PB&lt;/math&gt; and &lt;math&gt;CA&lt;/math&gt; intersect at &lt;math&gt;E&lt;/math&gt;; and let lines &lt;math&gt;PC&lt;/math&gt; and &lt;math&gt;AB&lt;/math&gt; intersect at &lt;math&gt;F&lt;/math&gt;. Prove that the area of triangle &lt;math&gt;DEF&lt;/math&gt; is twice the area of triangle &lt;math&gt;ABC&lt;/math&gt;.<br /> <br /> [[2017 USAJMO Problems/Problem 3|Solution]]<br /> <br /> ==Day 2==<br /> ===Problem 4===<br /> <br /> ===Problem 5===<br /> <br /> ===Problem 6===<br /> Let &lt;math&gt;P_1,....,P_{2n}&lt;/math&gt; be &lt;math&gt;2n&lt;/math&gt; distinct points on the unit circle &lt;math&gt;x^2+y^2=1&lt;/math&gt; other than &lt;math&gt;(1,0)&lt;/math&gt;. Each point is colored either red or blue, with exactly &lt;math&gt;n&lt;/math&gt; of them red and &lt;math&gt;n&lt;/math&gt; of them blue. Let &lt;math&gt;R_1,...,R_{n}&lt;/math&gt; be any ordering of the red points. Let &lt;math&gt;B_1&lt;/math&gt; be the nearest blue point to &lt;math&gt;R_1&lt;/math&gt; traveling counterclockwise around the circle starting from &lt;math&gt;R_1&lt;/math&gt;. Then let &lt;math&gt;B_2&lt;/math&gt; be the nearest of the remaining blue points to &lt;math&gt;R_2&lt;/math&gt; traveling counterclockwise around the circle from &lt;math&gt;R_2&lt;/math&gt;, and so on, until we have labeled all of the blue points &lt;math&gt;B_1,...,B_{n}&lt;/math&gt;. Show that the number of counterclockwise arcs of the form &lt;math&gt;R_{i} \rightarrow B_{i}&lt;/math&gt; that contain the point &lt;math&gt;(1,0)&lt;/math&gt; is independent of the way we chose the ordering &lt;math&gt;R_1,...,R_{n}&lt;/math&gt; of the red points.<br /> <br /> {{MAA Notice}}<br /> <br /> {{USAJMO newbox|year= 2017 |before=[[2016 USAJMO]]|after=[[2018 USAJMO]]}}</div> Liopleurodon