https://artofproblemsolving.com/wiki/index.php?title=1968_IMO_Problems/Problem_3&feed=atom&action=history 1968 IMO Problems/Problem 3 - Revision history 2022-05-22T10:59:41Z Revision history for this page on the wiki MediaWiki 1.31.1 https://artofproblemsolving.com/wiki/index.php?title=1968_IMO_Problems/Problem_3&diff=45664&oldid=prev 1=2: Created page with "==Problem== Consider the system of equations <cmath>ax_1^2 + bx_1 + c = x_2</cmath> <cmath>ax_2^2 + bx_2 + c = x_3</cmath> <cmath> \cdots </cmath> <cmath>ax_{n-1}^2 + bx_{n-1} +..." 2012-03-20T18:19:28Z <p>Created page with &quot;==Problem== Consider the system of equations &lt;cmath&gt;ax_1^2 + bx_1 + c = x_2&lt;/cmath&gt; &lt;cmath&gt;ax_2^2 + bx_2 + c = x_3&lt;/cmath&gt; &lt;cmath&gt; \cdots &lt;/cmath&gt; &lt;cmath&gt;ax_{n-1}^2 + bx_{n-1} +...&quot;</p> <p><b>New page</b></p><div>==Problem==<br /> <br /> Consider the system of equations<br /> &lt;cmath&gt;ax_1^2 + bx_1 + c = x_2&lt;/cmath&gt;<br /> &lt;cmath&gt;ax_2^2 + bx_2 + c = x_3&lt;/cmath&gt;<br /> &lt;cmath&gt; \cdots &lt;/cmath&gt;<br /> &lt;cmath&gt;ax_{n-1}^2 + bx_{n-1} + c = x_n&lt;/cmath&gt;<br /> &lt;cmath&gt;ax_n^2 + bx_n + c = x_1&lt;/cmath&gt;<br /> with unknowns &lt;math&gt;x_1, x_2, \cdots, x_n&lt;/math&gt; where &lt;math&gt;a, b, c&lt;/math&gt; are real and &lt;math&gt;a \neq 0&lt;/math&gt;. Let &lt;math&gt;\Delta = (b - 1)^2 - 4ac&lt;/math&gt;. Prove that for this system<br /> <br /> (a) if &lt;math&gt;\Delta &lt; 0&lt;/math&gt;, there is no solution,<br /> <br /> (b) if &lt;math&gt;\Delta = 0&lt;/math&gt;, there is exactly one solution,<br /> <br /> (c) if &lt;math&gt;\Delta &gt; 0&lt;/math&gt;, there is more than one solution.<br /> <br /> ==Solution==<br /> <br /> Adding the &lt;math&gt;n&lt;/math&gt; equations together yields<br /> <br /> &lt;cmath&gt;\sum_{i=1}^{n} (ax_i^2+(b-1)x_i+c)=0&lt;/cmath&gt;<br /> <br /> Let &lt;math&gt;s_i=ax_i^2+(b-1)x_i+c&lt;/math&gt;.<br /> <br /> (a) If &lt;math&gt;\Delta&lt;0&lt;/math&gt;, then there is no solution to the quadratic equation &lt;math&gt;ax^2+(b-1)x+c&lt;/math&gt;, as the determinant is negative. This implies that either &lt;math&gt;s_i&gt;0&lt;/math&gt; for all &lt;math&gt;i&lt;/math&gt;, or &lt;math&gt;s_i&lt;0&lt;/math&gt; for all &lt;math&gt;i&lt;/math&gt;. In either case the above summation cannot be 0, which implies that there are no solutions to the given system of equations. &lt;math&gt;\blacksquare&lt;/math&gt;<br /> <br /> (b) If &lt;math&gt;\Delta=0&lt;/math&gt;, then there is exactly one solution to the quadratic equation &lt;math&gt;ax^2+(b-1)x+c&lt;/math&gt; (let it be &lt;math&gt;r&lt;/math&gt;), and either &lt;math&gt;s_i\geq 0&lt;/math&gt; for all &lt;math&gt;i&lt;/math&gt;, or &lt;math&gt;s_i\leq 0&lt;/math&gt; for all &lt;math&gt;i&lt;/math&gt;. The only way that the above summation is 0 is if &lt;math&gt;s_i=0&lt;/math&gt; for all &lt;math&gt;i&lt;/math&gt;. As there is exactly one &lt;math&gt;x_i&lt;/math&gt; that makes &lt;math&gt;s_i=0&lt;/math&gt; (namely &lt;math&gt;x_i=r&lt;/math&gt;), then the only possible solution to the system of equations is &lt;math&gt;(x_1,x_2,\dots , x_n)=(r, r, \dots , r)&lt;/math&gt;. It's not hard to show that this works, so when &lt;math&gt;\Delta=0&lt;/math&gt; the system of equations has exactly one solution. &lt;math&gt;\blacksquare&lt;/math&gt;<br /> <br /> (c) If &lt;math&gt;\Delta&gt;1&lt;/math&gt;, then there are exactly two solutions to the quadratic equation &lt;math&gt;ax^2+(b-1)x+c&lt;/math&gt;. Let the roots be &lt;math&gt;r_1&lt;/math&gt; and &lt;math&gt;r_2&lt;/math&gt;. If &lt;math&gt;x_i=r_1&lt;/math&gt; for all &lt;math&gt;i&lt;/math&gt;, then &lt;math&gt;ax_i^2+bx_i+c=ar_1^2+br_1+c=r_1=x_{i+1}&lt;/math&gt;. This shows that &lt;math&gt;(x_1,x_2,\dots ,x_n)=(r_1,r_1,\dots, r_1)&lt;/math&gt; is a solution. We can show that &lt;math&gt;(x_1,x_2,\dots ,x_n)=(r_2,r_2,\dots, r_2)&lt;/math&gt; is another solution using the same reasoning, which shows that the equation has more than one solution. &lt;math&gt;\blacksquare&lt;/math&gt;<br /> <br /> ==See Also==<br /> <br /> {{IMO box|year=1968|num-b=2|num-a=4}}<br /> * [http://www.artofproblemsolving.com/Forum/viewtopic.php?p=361675&amp;sid=6798c42a2ab57f3ca82ffba974ed589c#p361675 Discussion on AoPS/MathLinks]<br /> <br /> [[Category:Olympiad Algebra Problems]]</div> 1=2