{"id":159,"date":"2007-05-05T14:22:43","date_gmt":"2007-05-05T14:22:43","guid":{"rendered":"http:\/\/www.sixthform.info\/maths\/?p=159"},"modified":"2020-03-14T19:29:30","modified_gmt":"2020-03-14T19:29:30","slug":"problems-problems","status":"publish","type":"post","link":"https:\/\/www.sixthform.info\/maths\/?p=159","title":{"rendered":"Problems, Problems"},"content":{"rendered":"<p>The <a target=\"_blank\" href=\"http:\/\/www.artofproblemsolving.com\/\" title=\"Art of Problem Solving (AopS)\" rel=\"noopener noreferrer\">Art of Problem Solving (AopS)<\/a> site encourages, teaches and promotes mathematics competitions from small local ones right up to the <a target=\"_blank\" href=\"http:\/\/imo.math.ca\/\" title=\"International Mathematical Olympiad (IMO)\" rel=\"noopener noreferrer\">International Mathematical Olympiad (IMO)<\/a>. As it says on its front page:<\/p>\n<blockquote><p><em>Is math class too easy for you? Looking for a greater challenge?<br \/>\nYou&#8217;ve come to the right place.<\/em><\/p><\/blockquote>\n<p>It has an excellent introduction to <img src='\/maths\/latexrender\/pictures\/c51d7e23458ca0e7373a8ed6ab56b2b9.gif' title='\\LaTeX' alt='\\LaTeX' align=absmiddle> <a target=\"_blank\" href=\"http:\/\/www.artofproblemsolving.com\/LaTeX\/AoPS_L_About.php\" title=\"Introduction to LaTeX\" rel=\"noopener noreferrer\">site<\/a>\u00c2\u00a0and a forum to discuss problems. The forum has an <a target=\"_blank\" href=\"http:\/\/www.artofproblemsolving.com\/Forum\/feed.php\" title=\"AoPS RSS feed\" rel=\"noopener noreferrer\">RSS feed<\/a> where students pose new problems every day. So if you&#8217;re &#8220;looking for a greater challenge&#8221; subscribe to this feed. Here is a random sample of some of the problems posed recently, which range from very easy to extremely difficult. Click on the problem number to go to the discussion on it.<\/p>\n<ul><a target=\"_blank\" href=\"http:\/\/www.artofproblemsolving.com\/Forum\/viewtopic.php?t=145301\" title=\"Problem 1\" rel=\"noopener noreferrer\">1<\/a>. We define addition in a different way\u00c2\u00a0to usual; an addition statement is true only if the letters in the addends is a rearrangement of the letters in the sum. For example,<br \/>\n10 + 6 = 16?<br \/>\nTEN + SIX = TENSIX = SIXTEN, but to be 16 it would need another E.<br \/>\nFind a &#8220;true&#8221; addition a + b = c + d.<\/p>\n<p><a target=\"_blank\" href=\"http:\/\/www.artofproblemsolving.com\/Forum\/viewtopic.php?t=147261\" title=\"Problem 2\" rel=\"noopener noreferrer\">2<\/a>. Prove that <img src='\/maths\/latexrender\/pictures\/a13c6e45ad4c8be110e5eaba68218688.gif' title='1+\\frac{1}{1!}+\\frac{1}{2!}+\\frac{1}{3!}+\\dots&lt;2.8' alt='1+\\frac{1}{1!}+\\frac{1}{2!}+\\frac{1}{3!}+\\dots&lt;2.8' align=absmiddle>.\n\n<a target=\"_blank\" href=\"http:\/\/www.artofproblemsolving.com\/Forum\/viewtopic.php?t=146864\" title=\"Problem 3\" rel=\"noopener noreferrer\">3<\/a>. Let <img src='\/maths\/latexrender\/pictures\/a44c56c8177e32d3613988f4dba7962e.gif' title='a,b,c' alt='a,b,c' align=absmiddle> be nonzero real numbers. Find all ordered pairs <img src='\/maths\/latexrender\/pictures\/61387437566c024c911e4099fb69c76a.gif' title='(a,b,c)' alt='(a,b,c)' align=absmiddle> such that <img src='\/maths\/latexrender\/pictures\/a821fa58619a4430dc941c38509e71b9.gif' title='\\displaystyle\\frac{2(a-b-c)}{a^{2}}=\\frac{4b-a-2c}{b^{2}}=\\frac{4c-a-2b}{c^{2}}' alt='\\displaystyle\\frac{2(a-b-c)}{a^{2}}=\\frac{4b-a-2c}{b^{2}}=\\frac{4c-a-2b}{c^{2}}' align=absmiddle>.<\/p>\n<p><a target=\"_blank\" href=\"http:\/\/www.artofproblemsolving.com\/Forum\/viewtopic.php?t=146687\" title=\"Problem 4\" rel=\"noopener noreferrer\">4<\/a>. <img src='\/maths\/latexrender\/pictures\/8fa14cdd754f91cc6554c9e71929cce7.gif' title='f' alt='f' align=absmiddle> is a continuous complex-valued function satisfying:<br \/>\ni) <img src='\/maths\/latexrender\/pictures\/86afccc3394981e3f2ad82449c641939.gif' title='|f(z)| = |z|' alt='|f(z)| = |z|' align=absmiddle><br \/>\nii) <img src='\/maths\/latexrender\/pictures\/881a4e6f204e1a065fd54b0d062ab5db.gif' title='|f(z)-z| = |z|' alt='|f(z)-z| = |z|' align=absmiddle><br \/>\nFind <img src='\/maths\/latexrender\/pictures\/8e16290a303e8aee57c62de73b4da98a.gif' title='f(f(f(2007)))' alt='f(f(f(2007)))' align=absmiddle><\/p>\n<p><a target=\"_blank\" href=\"http:\/\/www.artofproblemsolving.com\/Forum\/viewtopic.php?t=146815\" title=\"Problem 5\" rel=\"noopener noreferrer\">5<\/a>. If <img src='\/maths\/latexrender\/pictures\/0cc175b9c0f1b6a831c399e269772661.gif' title='a' alt='a' align=absmiddle> and <img src='\/maths\/latexrender\/pictures\/92eb5ffee6ae2fec3ad71c777531578f.gif' title='b' alt='b' align=absmiddle> are relatively coprime, find all possible values of <img src='\/maths\/latexrender\/pictures\/363452dd0a07bad8601339ab3893c08c.gif' title='\\gcd(a+b,a^{2}+b^{2})' alt='\\gcd(a+b,a^{2}+b^{2})' align=absmiddle>.<\/p>\n<p><a target=\"_blank\" href=\"http:\/\/www.artofproblemsolving.com\/Forum\/viewtopic.php?t=147254\" title=\"Problem 6\" rel=\"noopener noreferrer\">6<\/a>. Let <img src='\/maths\/latexrender\/pictures\/ce04be1226e56f48da55b6c130d45b94.gif' title='A,B,C' alt='A,B,C' align=absmiddle> be three angles of\u00c2\u00a0 <img src='\/maths\/latexrender\/pictures\/635759e23aaf6e02541e3b72d65268d0.gif' title='\\triangle ABC' alt='\\triangle ABC' align=absmiddle>. Prove that <img src='\/maths\/latexrender\/pictures\/746472843d3d6fe8bd4ac00e29adadce.gif' title='(1-\\cos A)(1-\\cos B)(1-\\cos C)\\ge\\cos A\\cos B\\cos C' alt='(1-\\cos A)(1-\\cos B)(1-\\cos C)\\ge\\cos A\\cos B\\cos C' align=absmiddle>.<\/p>\n<p><a target=\"_blank\" href=\"http:\/\/www.artofproblemsolving.com\/Forum\/viewtopic.php?t=147243\" title=\"Problem 7\" rel=\"noopener noreferrer\">7<\/a>. For each function <img src='\/maths\/latexrender\/pictures\/8fa14cdd754f91cc6554c9e71929cce7.gif' title='f' alt='f' align=absmiddle> which is defined for all real numbers and satisfies <img src='\/maths\/latexrender\/pictures\/0ed9a7118e1b926963505eb81c50400c.gif' title='f(xy)=xf(y)+yf(x)' alt='f(xy)=xf(y)+yf(x)' align=absmiddle> and <img src='\/maths\/latexrender\/pictures\/1b47c3c8f26b266ffa587ef78151fbf6.gif' title='f(x+y)=f(x^{1993})+f(y^{1993})' alt='f(x+y)=f(x^{1993})+f(y^{1993})' align=absmiddle> determine the value of <img src='\/maths\/latexrender\/pictures\/079388fa9feeeff7a7c1e68cd7f4adc3.gif' title='f(\\sqrt{5753})' alt='f(\\sqrt{5753})' align=absmiddle>.<\/p>\n<p><a target=\"_blank\" href=\"http:\/\/www.artofproblemsolving.com\/Forum\/viewtopic.php?t=147239\" title=\"Problem 8\" rel=\"noopener noreferrer\">8<\/a>. Let <img src='\/maths\/latexrender\/pictures\/a44c56c8177e32d3613988f4dba7962e.gif' title='a,b,c' alt='a,b,c' align=absmiddle> be positive reals such that <img src='\/maths\/latexrender\/pictures\/bed7e19e687f07025333c93d456a42aa.gif' title='a+b+c=1' alt='a+b+c=1' align=absmiddle>. Prove that <img src='\/maths\/latexrender\/pictures\/53b8c90b2dbaf274ae66e5c3f9215922.gif' title='7(ab+bc+ca) \\le 9abc+2' alt='7(ab+bc+ca) \\le 9abc+2' align=absmiddle>.<\/p>\n<p><a target=\"_blank\" href=\"http:\/\/www.artofproblemsolving.com\/Forum\/viewtopic.php?t=146416\" title=\"Problem 9\" rel=\"noopener noreferrer\">9<\/a>. For <img src='\/maths\/latexrender\/pictures\/995446dc2c5760f8d1f414b94ef8ac47.gif' title='0 \\leq d \\leq 9' alt='0 \\leq d \\leq 9' align=absmiddle>, we define the numbers <img src='\/maths\/latexrender\/pictures\/66fb1a5703aee95f7d5b919ee76dca47.gif' title='S_{d}=1+d+d^{2}+\\cdots+d^{2006}' alt='S_{d}=1+d+d^{2}+\\cdots+d^{2006}' align=absmiddle>. Find the last digit of the number <img src='\/maths\/latexrender\/pictures\/5c8a12fbbe5c1da91d82fc79fc83c844.gif' title='S_{0}+S_{1}+\\cdots+S_{9}' alt='S_{0}+S_{1}+\\cdots+S_{9}' align=absmiddle>.<\/p>\n<p><a target=\"_blank\" href=\"http:\/\/www.artofproblemsolving.com\/Forum\/viewtopic.php?t=146418\" title=\"Problem 10\" rel=\"noopener noreferrer\">10<\/a>. The product of several distinct positive integers is divisible by <img src='\/maths\/latexrender\/pictures\/29845498f936cd33ed7b23755524f2be.gif' title='2006^2' alt='2006^2' align=absmiddle>. Determine the minimum value the sum of such numbers can take.<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>The Art of Problem Solving (AopS) site encourages, teaches and promotes mathematics competitions from small local ones right up to the International Mathematical Olympiad (IMO). As it says on its front page: Is math class too easy for you? Looking for a greater challenge? You&#8217;ve come to the right place. It has an excellent introduction [&hellip;]<\/p>\n","protected":false},"author":3,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[],"class_list":["post-159","post","type-post","status-publish","format-standard","hentry","category-articles"],"_links":{"self":[{"href":"https:\/\/www.sixthform.info\/maths\/index.php?rest_route=\/wp\/v2\/posts\/159","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.sixthform.info\/maths\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.sixthform.info\/maths\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.sixthform.info\/maths\/index.php?rest_route=\/wp\/v2\/users\/3"}],"replies":[{"embeddable":true,"href":"https:\/\/www.sixthform.info\/maths\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=159"}],"version-history":[{"count":1,"href":"https:\/\/www.sixthform.info\/maths\/index.php?rest_route=\/wp\/v2\/posts\/159\/revisions"}],"predecessor-version":[{"id":287,"href":"https:\/\/www.sixthform.info\/maths\/index.php?rest_route=\/wp\/v2\/posts\/159\/revisions\/287"}],"wp:attachment":[{"href":"https:\/\/www.sixthform.info\/maths\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=159"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.sixthform.info\/maths\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=159"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.sixthform.info\/maths\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=159"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}