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  1. <?xml version='1.0' encoding='UTF-8'?><?xml-stylesheet href="http://www.blogger.com/styles/atom.css" type="text/css"?><feed xmlns='http://www.w3.org/2005/Atom' xmlns:openSearch='http://a9.com/-/spec/opensearchrss/1.0/' xmlns:blogger='http://schemas.google.com/blogger/2008' xmlns:georss='http://www.georss.org/georss' xmlns:gd="http://schemas.google.com/g/2005" xmlns:thr='http://purl.org/syndication/thread/1.0'><id>tag:blogger.com,1999:blog-32747608</id><updated>2024-03-05T22:29:35.822+05:30</updated><title type='text'>Vidyamanohar</title><subtitle type='html'>A place I would like to share some concepts, ideas, puzzles that simulate the minds of those interested in Math - for competitions or fun.</subtitle><link rel='http://schemas.google.com/g/2005#feed' type='application/atom+xml' href='http://vidyamanohar.blogspot.com/feeds/posts/default'/><link rel='self' type='application/atom+xml' href='http://www.blogger.com/feeds/32747608/posts/default'/><link rel='alternate' type='text/html' href='http://vidyamanohar.blogspot.com/'/><link rel='hub' href='http://pubsubhubbub.appspot.com/'/><link rel='next' type='application/atom+xml' href='http://www.blogger.com/feeds/32747608/posts/default?start-index=26&amp;max-results=25'/><author><name>Vidyamanohar</name><uri>http://www.blogger.com/profile/14537585081760370998</uri><email>noreply@blogger.com</email><gd:image rel='http://schemas.google.com/g/2005#thumbnail' width='16' height='16' src='https://img1.blogblog.com/img/b16-rounded.gif'/></author><generator version='7.00' uri='http://www.blogger.com'>Blogger</generator><openSearch:totalResults>50</openSearch:totalResults><openSearch:startIndex>1</openSearch:startIndex><openSearch:itemsPerPage>25</openSearch:itemsPerPage><entry><id>tag:blogger.com,1999:blog-32747608.post-3668451493563642334</id><published>2014-06-06T21:58:00.000+05:30</published><updated>2014-06-06T21:58:26.321+05:30</updated><title type='text'>Puzzle 50 - Integration</title><content type='html'>&lt;div dir=&quot;ltr&quot; style=&quot;text-align: left;&quot; trbidi=&quot;on&quot;&gt;Evaluate&lt;br /&gt;
  2. &lt;a href=&quot;http://www.codecogs.com/eqnedit.php?latex=\int&amp;space;{\frac{{1&amp;space;-&amp;space;7{{\cos&amp;space;}^2}x}}{{{{\sin&amp;space;}^7}x{{\cos&amp;space;}^2}x}}dx}&quot; target=&quot;_blank&quot;&gt;&lt;img src=&quot;http://latex.codecogs.com/gif.latex?\int&amp;space;{\frac{{1&amp;space;-&amp;space;7{{\cos&amp;space;}^2}x}}{{{{\sin&amp;space;}^7}x{{\cos&amp;space;}^2}x}}dx}&quot; title=&quot;\int {\frac{{1 - 7{{\cos }^2}x}}{{{{\sin }^7}x{{\cos }^2}x}}dx}&quot; /&gt;&lt;/a&gt;&lt;br /&gt;
  3. &lt;/div&gt;</content><link rel='replies' type='application/atom+xml' href='http://vidyamanohar.blogspot.com/feeds/3668451493563642334/comments/default' title='Post Comments'/><link rel='replies' type='text/html' href='http://www.blogger.com/comment.g?blogID=32747608&amp;postID=3668451493563642334' title='0 Comments'/><link rel='edit' type='application/atom+xml' href='http://www.blogger.com/feeds/32747608/posts/default/3668451493563642334'/><link rel='self' type='application/atom+xml' href='http://www.blogger.com/feeds/32747608/posts/default/3668451493563642334'/><link rel='alternate' type='text/html' href='http://vidyamanohar.blogspot.com/2014/06/puzzle-50-integration.html' title='Puzzle 50 - Integration'/><author><name>Vidyamanohar</name><uri>http://www.blogger.com/profile/14537585081760370998</uri><email>noreply@blogger.com</email><gd:image rel='http://schemas.google.com/g/2005#thumbnail' width='16' height='16' src='https://img1.blogblog.com/img/b16-rounded.gif'/></author><thr:total>0</thr:total></entry><entry><id>tag:blogger.com,1999:blog-32747608.post-8296761361643329022</id><published>2014-05-26T16:10:00.000+05:30</published><updated>2014-05-26T16:10:52.279+05:30</updated><title type='text'>Puzzle 49</title><content type='html'>&lt;div dir=&quot;ltr&quot; style=&quot;text-align: left;&quot; trbidi=&quot;on&quot;&gt;A rectangular sheet of paper ABCD is with width AD=1 and length AB more than the width but not more than double of it.  The paper is folded through vertex A so that the edge along AD falls onto AB.  Without unfolding, the paper is folded again along a line through B so that CB now lies on AB.  Now the result is a triangular piece of paper.  There is a region on this triangular sheet that has four layers of paper.  Find the area of that region in terms of the length x, of the rectangle.&lt;br /&gt;
  4. &lt;/div&gt;</content><link rel='replies' type='application/atom+xml' href='http://vidyamanohar.blogspot.com/feeds/8296761361643329022/comments/default' title='Post Comments'/><link rel='replies' type='text/html' href='http://www.blogger.com/comment.g?blogID=32747608&amp;postID=8296761361643329022' title='1 Comments'/><link rel='edit' type='application/atom+xml' href='http://www.blogger.com/feeds/32747608/posts/default/8296761361643329022'/><link rel='self' type='application/atom+xml' href='http://www.blogger.com/feeds/32747608/posts/default/8296761361643329022'/><link rel='alternate' type='text/html' href='http://vidyamanohar.blogspot.com/2014/05/puzzle-49.html' title='Puzzle 49'/><author><name>Vidyamanohar</name><uri>http://www.blogger.com/profile/14537585081760370998</uri><email>noreply@blogger.com</email><gd:image rel='http://schemas.google.com/g/2005#thumbnail' width='16' height='16' src='https://img1.blogblog.com/img/b16-rounded.gif'/></author><thr:total>1</thr:total></entry><entry><id>tag:blogger.com,1999:blog-32747608.post-8903530171073968732</id><published>2014-05-24T16:56:00.001+05:30</published><updated>2014-05-25T08:16:43.038+05:30</updated><title type='text'>Puzzle 48 - Roots of a cubic</title><content type='html'>&lt;div dir=&quot;ltr&quot; style=&quot;text-align: left;&quot; trbidi=&quot;on&quot;&gt;From Rolle’s theorem, we can conclude that between every pair of distinct roots of a polynomial P(x), there lies a root of its derivative P’(x).  If P’(x) keeps its sign unchanged between any two distinct real numbers, then it is possible that P(x) does not have a root between them and hence may keep the same sign between those real numbers.  Further if P(x) has two equal roots, then it will have a common root with its derivative. Consider &lt;a href=&quot;http://www.codecogs.com/eqnedit.php?latex=P(x)&amp;space;=&amp;space;8{x^3}&amp;space;-&amp;space;12(k&amp;space;&amp;plus;&amp;space;3){x^2}&amp;space;&amp;plus;&amp;space;72kx&amp;space;-&amp;space;32&quot; target=&quot;_blank&quot;&gt;&lt;img src=&quot;http://latex.codecogs.com/gif.latex?P(x)&amp;space;=&amp;space;8{x^3}&amp;space;-&amp;space;12(k&amp;space;&amp;plus;&amp;space;3){x^2}&amp;space;&amp;plus;&amp;space;72kx&amp;space;-&amp;space;32&quot; title=&quot;P(x) = 8{x^3} - 12(k + 3){x^2} + 72kx - 32&quot; /&gt;&lt;/a&gt; with &lt;a href=&quot;http://www.codecogs.com/eqnedit.php?latex=k&quot; target=&quot;_blank&quot;&gt;&lt;img src=&quot;http://latex.codecogs.com/gif.latex?k&quot; title=&quot;k&quot; /&gt;&lt;/a&gt; being an integer between &lt;a href=&quot;http://www.codecogs.com/eqnedit.php?latex=-2&quot; target=&quot;_blank&quot;&gt;&lt;img src=&quot;http://latex.codecogs.com/gif.latex?-2&quot; title=&quot;-2&quot; /&gt;&lt;/a&gt; and &lt;a href=&quot;http://www.codecogs.com/eqnedit.php?latex=8&quot; target=&quot;_blank&quot;&gt;&lt;img src=&quot;http://latex.codecogs.com/gif.latex?8&quot; title=&quot;8&quot; /&gt;&lt;/a&gt; both inclusive, then find the number of values of &lt;a href=&quot;http://www.codecogs.com/eqnedit.php?latex=k&quot; target=&quot;_blank&quot;&gt;&lt;img src=&quot;http://latex.codecogs.com/gif.latex?k&quot; title=&quot;k&quot; /&gt;&lt;/a&gt; if &lt;a href=&quot;http://www.codecogs.com/eqnedit.php?latex=P(x)&amp;space;=&amp;space;0&quot; target=&quot;_blank&quot;&gt;&lt;img src=&quot;http://latex.codecogs.com/gif.latex?P(x)&amp;space;=&amp;space;0&quot; title=&quot;P(x) = 0&quot; /&gt;&lt;/a&gt; has to have &lt;br /&gt;
  5. (i) one real root and two complex roots, (ii) only two equal roots&lt;br /&gt;
  6. &lt;br /&gt;
  7. &lt;br /&gt;
  8. &lt;/div&gt;</content><link rel='replies' type='application/atom+xml' href='http://vidyamanohar.blogspot.com/feeds/8903530171073968732/comments/default' title='Post Comments'/><link rel='replies' type='text/html' href='http://www.blogger.com/comment.g?blogID=32747608&amp;postID=8903530171073968732' title='1 Comments'/><link rel='edit' type='application/atom+xml' href='http://www.blogger.com/feeds/32747608/posts/default/8903530171073968732'/><link rel='self' type='application/atom+xml' href='http://www.blogger.com/feeds/32747608/posts/default/8903530171073968732'/><link rel='alternate' type='text/html' href='http://vidyamanohar.blogspot.com/2014/05/roots-of-cubic.html' title='Puzzle 48 - Roots of a cubic'/><author><name>Vidyamanohar</name><uri>http://www.blogger.com/profile/14537585081760370998</uri><email>noreply@blogger.com</email><gd:image rel='http://schemas.google.com/g/2005#thumbnail' width='16' height='16' src='https://img1.blogblog.com/img/b16-rounded.gif'/></author><thr:total>1</thr:total></entry><entry><id>tag:blogger.com,1999:blog-32747608.post-4972611518954337005</id><published>2013-05-02T19:47:00.000+05:30</published><updated>2013-05-02T19:47:17.591+05:30</updated><title type='text'>Puzzle 47</title><content type='html'>&lt;div dir=&quot;ltr&quot; style=&quot;text-align: left;&quot; trbidi=&quot;on&quot;&gt;Semi circles are drawn on each side of a square.  An elastic band is stretched tightly around the shape formed (see the figure).  Find the length of the elastic band in terms of the length x, of the side of the square.&lt;br /&gt;
  9. &lt;br /&gt;
  10. &lt;a href=&quot;http://3.bp.blogspot.com/-LUdalBMjafc/UYJ0p26IDHI/AAAAAAAAAgc/81h77PVYQcs/s1600/Puzzle+47+Image.jpg&quot; imageanchor=&quot;1&quot; &gt;&lt;img border=&quot;0&quot; src=&quot;http://3.bp.blogspot.com/-LUdalBMjafc/UYJ0p26IDHI/AAAAAAAAAgc/81h77PVYQcs/s320/Puzzle+47+Image.jpg&quot; /&gt;&lt;/a&gt;&lt;br /&gt;
  11. &lt;/div&gt;</content><link rel='replies' type='application/atom+xml' href='http://vidyamanohar.blogspot.com/feeds/4972611518954337005/comments/default' title='Post Comments'/><link rel='replies' type='text/html' href='http://www.blogger.com/comment.g?blogID=32747608&amp;postID=4972611518954337005' title='4 Comments'/><link rel='edit' type='application/atom+xml' href='http://www.blogger.com/feeds/32747608/posts/default/4972611518954337005'/><link rel='self' type='application/atom+xml' href='http://www.blogger.com/feeds/32747608/posts/default/4972611518954337005'/><link rel='alternate' type='text/html' href='http://vidyamanohar.blogspot.com/2013/05/puzzle-47.html' title='Puzzle 47'/><author><name>Vidyamanohar</name><uri>http://www.blogger.com/profile/14537585081760370998</uri><email>noreply@blogger.com</email><gd:image rel='http://schemas.google.com/g/2005#thumbnail' width='16' height='16' src='https://img1.blogblog.com/img/b16-rounded.gif'/></author><media:thumbnail xmlns:media="http://search.yahoo.com/mrss/" url="http://3.bp.blogspot.com/-LUdalBMjafc/UYJ0p26IDHI/AAAAAAAAAgc/81h77PVYQcs/s72-c/Puzzle+47+Image.jpg" height="72" width="72"/><thr:total>4</thr:total></entry><entry><id>tag:blogger.com,1999:blog-32747608.post-5232959216065850977</id><published>2013-04-25T09:47:00.001+05:30</published><updated>2013-04-25T09:47:39.419+05:30</updated><title type='text'>Puzzle 46 - Squares</title><content type='html'>&lt;div dir=&quot;ltr&quot; style=&quot;text-align: left;&quot; trbidi=&quot;on&quot;&gt;Square ABCD, of side length 26, is centered at O.  With D as center, draw another square EFGH with side length 22, none of its sides parallel to those of ABCD, so that OE is kept perpendicular to DA.  Find the area of the square ABCD that is not inside the square EFGH.&lt;br /&gt;
  12. &lt;/div&gt;</content><link rel='replies' type='application/atom+xml' href='http://vidyamanohar.blogspot.com/feeds/5232959216065850977/comments/default' title='Post Comments'/><link rel='replies' type='text/html' href='http://www.blogger.com/comment.g?blogID=32747608&amp;postID=5232959216065850977' title='3 Comments'/><link rel='edit' type='application/atom+xml' href='http://www.blogger.com/feeds/32747608/posts/default/5232959216065850977'/><link rel='self' type='application/atom+xml' href='http://www.blogger.com/feeds/32747608/posts/default/5232959216065850977'/><link rel='alternate' type='text/html' href='http://vidyamanohar.blogspot.com/2013/04/puzzle-46-squares.html' title='Puzzle 46 - Squares'/><author><name>Vidyamanohar</name><uri>http://www.blogger.com/profile/14537585081760370998</uri><email>noreply@blogger.com</email><gd:image rel='http://schemas.google.com/g/2005#thumbnail' width='16' height='16' src='https://img1.blogblog.com/img/b16-rounded.gif'/></author><thr:total>3</thr:total></entry><entry><id>tag:blogger.com,1999:blog-32747608.post-8109598347257317395</id><published>2013-04-23T16:48:00.001+05:30</published><updated>2013-04-23T16:48:46.814+05:30</updated><title type='text'>Puzzle 45 - Similar Triangles </title><content type='html'>&lt;div dir=&quot;ltr&quot; style=&quot;text-align: left;&quot; trbidi=&quot;on&quot;&gt;Let A is the set of all triangles in a plane. Define a function &lt;img src=&quot;http://latex.codecogs.com/gif.latex?f:A \times A \to {R^ + }&quot; title=&quot;f:A \times A \to {R^ + }&quot; /&gt; as &lt;img src=&quot;http://latex.codecogs.com/gif.latex?f\left( {{T_1},\,{T_2}} \right) = a&quot; title=&quot;f\left( {{T_1},\,{T_2}} \right) = a&quot; /&gt; implies that triangle &lt;img src=&quot;http://latex.codecogs.com/gif.latex?{T_2}&quot; title=&quot;{T_2}&quot; /&gt; is similar to &lt;img src=&quot;http://latex.codecogs.com/gif.latex?{T_1}&quot; title=&quot;{T_1}&quot; /&gt; with the dilation constant or scale factor ‘a’.  Find all the possible triangles &lt;img src=&quot;http://latex.codecogs.com/gif.latex?{T_1}&quot; title=&quot;{T_1}&quot; /&gt; and &lt;img src=&quot;http://latex.codecogs.com/gif.latex?{T_2}&quot; title=&quot;{T_2}&quot; /&gt; such that&lt;br /&gt;
  13. &lt;br /&gt;
  14. a. Lengths of the sides of &lt;img src=&quot;http://latex.codecogs.com/gif.latex?{T_1}&quot; title=&quot;{T_1}&quot; /&gt; and &lt;img src=&quot;http://latex.codecogs.com/gif.latex?{T_2}&quot; title=&quot;{T_2}&quot; /&gt; are integers&lt;br /&gt;
  15. b. The perimeter and area of  &lt;img src=&quot;http://latex.codecogs.com/gif.latex?{T_2}&quot; title=&quot;{T_2}&quot; /&gt; are same.&lt;br /&gt;
  16. &lt;/div&gt;</content><link rel='replies' type='application/atom+xml' href='http://vidyamanohar.blogspot.com/feeds/8109598347257317395/comments/default' title='Post Comments'/><link rel='replies' type='text/html' href='http://www.blogger.com/comment.g?blogID=32747608&amp;postID=8109598347257317395' title='0 Comments'/><link rel='edit' type='application/atom+xml' href='http://www.blogger.com/feeds/32747608/posts/default/8109598347257317395'/><link rel='self' type='application/atom+xml' href='http://www.blogger.com/feeds/32747608/posts/default/8109598347257317395'/><link rel='alternate' type='text/html' href='http://vidyamanohar.blogspot.com/2013/04/puzzle-45-similar-triangles.html' title='Puzzle 45 - Similar Triangles '/><author><name>Vidyamanohar</name><uri>http://www.blogger.com/profile/14537585081760370998</uri><email>noreply@blogger.com</email><gd:image rel='http://schemas.google.com/g/2005#thumbnail' width='16' height='16' src='https://img1.blogblog.com/img/b16-rounded.gif'/></author><thr:total>0</thr:total></entry><entry><id>tag:blogger.com,1999:blog-32747608.post-3864401593060678904</id><published>2013-04-22T18:41:00.001+05:30</published><updated>2013-04-22T18:53:43.572+05:30</updated><title type='text'>Puzzle 44</title><content type='html'>&lt;div dir=&quot;ltr&quot; style=&quot;text-align: left;&quot; trbidi=&quot;on&quot;&gt;W(a, b) is the centre of the circle. TE is a tangent to the circle at E, which lies on the y-axis. The circle touches the x-axis at D. The equation of TE is&lt;img src=&quot;http://latex.codecogs.com/gif.latex?y = \frac{3}{4}x - 9&quot; title=&quot;y = \frac{3}{4}x - 9&quot; /&gt;.  Calculate the values of a and b.&lt;br /&gt;
  17. &lt;br /&gt;
  18. &lt;/div&gt;</content><link rel='replies' type='application/atom+xml' href='http://vidyamanohar.blogspot.com/feeds/3864401593060678904/comments/default' title='Post Comments'/><link rel='replies' type='text/html' href='http://www.blogger.com/comment.g?blogID=32747608&amp;postID=3864401593060678904' title='6 Comments'/><link rel='edit' type='application/atom+xml' href='http://www.blogger.com/feeds/32747608/posts/default/3864401593060678904'/><link rel='self' type='application/atom+xml' href='http://www.blogger.com/feeds/32747608/posts/default/3864401593060678904'/><link rel='alternate' type='text/html' href='http://vidyamanohar.blogspot.com/2013/04/puzzle-44.html' title='Puzzle 44'/><author><name>Vidyamanohar</name><uri>http://www.blogger.com/profile/14537585081760370998</uri><email>noreply@blogger.com</email><gd:image rel='http://schemas.google.com/g/2005#thumbnail' width='16' height='16' src='https://img1.blogblog.com/img/b16-rounded.gif'/></author><thr:total>6</thr:total></entry><entry><id>tag:blogger.com,1999:blog-32747608.post-3175884381641940615</id><published>2012-03-28T20:14:00.002+05:30</published><updated>2012-03-28T20:14:48.595+05:30</updated><title type='text'>Puzzle 43 - Locus of midpoints</title><content type='html'>&lt;div dir=&quot;ltr&quot; style=&quot;text-align: left;&quot; trbidi=&quot;on&quot;&gt;
  19. Two circles &lt;img src=&quot;http://latex.codecogs.com/gif.latex?%7BC_1%7D,%5C,%7BC_2%7D&quot; title=&quot;{C_1},\,{C_2}&quot; /&gt; with different radii are given in the plane touching each other externally at &lt;img src=&quot;http://latex.codecogs.com/gif.latex?T&quot; title=&quot;T&quot; /&gt;. Consider points &lt;img src=&quot;http://latex.codecogs.com/gif.latex?A&quot; title=&quot;A&quot; /&gt; on &lt;img src=&quot;http://latex.codecogs.com/gif.latex?%7BC_1%7D&quot; title=&quot;{C_1}&quot; /&gt; and  &lt;img src=&quot;http://latex.codecogs.com/gif.latex?B&quot; title=&quot;B&quot; /&gt; on &lt;img src=&quot;http://latex.codecogs.com/gif.latex?%7BC_2%7D&quot; title=&quot;{C_1}&quot; /&gt;, such that &lt;img src=&quot;http://latex.codecogs.com/gif.latex?AB&quot; title=&quot;AB&quot; /&gt;  subtends right angle at &lt;img src=&quot;http://latex.codecogs.com/gif.latex?T&quot; title=&quot;T&quot; /&gt;. Find the locus of midpoints of all such segments  &lt;img src=&quot;http://latex.codecogs.com/gif.latex?AB&quot; title=&quot;AB&quot; /&gt;.&lt;/div&gt;</content><link rel='replies' type='application/atom+xml' href='http://vidyamanohar.blogspot.com/feeds/3175884381641940615/comments/default' title='Post Comments'/><link rel='replies' type='text/html' href='http://www.blogger.com/comment.g?blogID=32747608&amp;postID=3175884381641940615' title='6 Comments'/><link rel='edit' type='application/atom+xml' href='http://www.blogger.com/feeds/32747608/posts/default/3175884381641940615'/><link rel='self' type='application/atom+xml' href='http://www.blogger.com/feeds/32747608/posts/default/3175884381641940615'/><link rel='alternate' type='text/html' href='http://vidyamanohar.blogspot.com/2012/03/puzzle-43-locus-of-midpoints.html' title='Puzzle 43 - Locus of midpoints'/><author><name>Vidyamanohar</name><uri>http://www.blogger.com/profile/14537585081760370998</uri><email>noreply@blogger.com</email><gd:image rel='http://schemas.google.com/g/2005#thumbnail' width='16' height='16' src='https://img1.blogblog.com/img/b16-rounded.gif'/></author><thr:total>6</thr:total></entry><entry><id>tag:blogger.com,1999:blog-32747608.post-2886629115265288338</id><published>2012-03-27T08:44:00.003+05:30</published><updated>2012-03-27T08:44:21.988+05:30</updated><title type='text'>Puzzle 42</title><content type='html'>&lt;div dir=&quot;ltr&quot; style=&quot;text-align: left;&quot; trbidi=&quot;on&quot;&gt;
  20. &lt;div dir=&quot;ltr&quot; style=&quot;text-align: left;&quot; trbidi=&quot;on&quot;&gt;
  21. &lt;br /&gt;
  22. Let &lt;img src=&quot;http://latex.codecogs.com/gif.latex?%5Cinline%20f:R%20%5Cto%20R&quot; title=&quot;\inline f:R \to R&quot; /&gt; is a function such that &lt;img src=&quot;http://latex.codecogs.com/gif.latex?%5Cinline%20f%28f%28x%29%29%20=%20%7Bx%5E2%7D%20-%20x%20+%201&quot; title=&quot;\inline f(f(x)) = {x^2} - x + 1&quot; /&gt;.  Find the value of f(0).&lt;/div&gt;
  23. &lt;/div&gt;
  24. </content><link rel='replies' type='application/atom+xml' href='http://vidyamanohar.blogspot.com/feeds/2886629115265288338/comments/default' title='Post Comments'/><link rel='replies' type='text/html' href='http://www.blogger.com/comment.g?blogID=32747608&amp;postID=2886629115265288338' title='2 Comments'/><link rel='edit' type='application/atom+xml' href='http://www.blogger.com/feeds/32747608/posts/default/2886629115265288338'/><link rel='self' type='application/atom+xml' href='http://www.blogger.com/feeds/32747608/posts/default/2886629115265288338'/><link rel='alternate' type='text/html' href='http://vidyamanohar.blogspot.com/2012/03/puzzle-42.html' title='Puzzle 42'/><author><name>Vidyamanohar</name><uri>http://www.blogger.com/profile/14537585081760370998</uri><email>noreply@blogger.com</email><gd:image rel='http://schemas.google.com/g/2005#thumbnail' width='16' height='16' src='https://img1.blogblog.com/img/b16-rounded.gif'/></author><thr:total>2</thr:total></entry><entry><id>tag:blogger.com,1999:blog-32747608.post-859914485379857291</id><published>2012-03-15T14:07:00.000+05:30</published><updated>2012-03-15T14:07:18.600+05:30</updated><title type='text'>Puzzlee 41 - Polynomials + Trigonometry</title><content type='html'>&lt;div dir=&quot;ltr&quot; style=&quot;text-align: left;&quot; trbidi=&quot;on&quot;&gt;
  25. If &lt;a href=&quot;http://www.codecogs.com/eqnedit.php?latex=%5Calpha%20%3C%20%5Cbeta%20%3C%20%5Cgamma&quot; target=&quot;_blank&quot;&gt;&lt;img src=&quot;http://latex.codecogs.com/gif.latex?%5Calpha%20%3C%20%5Cbeta%20%3C%20%5Cgamma&quot; title=&quot;\alpha &amp;lt; \beta &amp;lt; \gamma&quot; /&gt;&lt;/a&gt; are three real roots of the equation &lt;a href=&quot;http://www.codecogs.com/eqnedit.php?latex=%7Bx%5E3%7D%20-%203x%20-%201%20=%200&quot; target=&quot;_blank&quot;&gt;&lt;img src=&quot;http://latex.codecogs.com/gif.latex?%7Bx%5E3%7D%20-%203x%20-%201%20=%200&quot; title=&quot;{x^3} - 3x - 1 = 0&quot; /&gt;&lt;/a&gt;, show that &lt;a href=&quot;http://www.codecogs.com/eqnedit.php?latex=%7B%5Cgamma%20%5E2%7D%20-%20%5Cgamma%20=%20%7B%5Cbeta%20%5E2%7D%20-%20%5Calpha&quot; target=&quot;_blank&quot;&gt;&lt;img src=&quot;http://latex.codecogs.com/gif.latex?%7B%5Cgamma%20%5E2%7D%20-%20%5Cgamma%20=%20%7B%5Cbeta%20%5E2%7D%20-%20%5Calpha&quot; title=&quot;{\gamma ^2} - \gamma = {\beta ^2} - \alpha&quot; /&gt;&lt;/a&gt;.&lt;/div&gt;
  26. </content><link rel='replies' type='application/atom+xml' href='http://vidyamanohar.blogspot.com/feeds/859914485379857291/comments/default' title='Post Comments'/><link rel='replies' type='text/html' href='http://www.blogger.com/comment.g?blogID=32747608&amp;postID=859914485379857291' title='1 Comments'/><link rel='edit' type='application/atom+xml' href='http://www.blogger.com/feeds/32747608/posts/default/859914485379857291'/><link rel='self' type='application/atom+xml' href='http://www.blogger.com/feeds/32747608/posts/default/859914485379857291'/><link rel='alternate' type='text/html' href='http://vidyamanohar.blogspot.com/2012/03/puzzlee-41-polynomials-trigonometry.html' title='Puzzlee 41 - Polynomials + Trigonometry'/><author><name>Vidyamanohar</name><uri>http://www.blogger.com/profile/14537585081760370998</uri><email>noreply@blogger.com</email><gd:image rel='http://schemas.google.com/g/2005#thumbnail' width='16' height='16' src='https://img1.blogblog.com/img/b16-rounded.gif'/></author><thr:total>1</thr:total></entry><entry><id>tag:blogger.com,1999:blog-32747608.post-9160217841798511245</id><published>2012-03-14T11:24:00.000+05:30</published><updated>2012-03-14T11:24:22.834+05:30</updated><title type='text'>Puzzle 40 - Matrices</title><content type='html'>&lt;div dir=&quot;ltr&quot; style=&quot;text-align: left;&quot; trbidi=&quot;on&quot;&gt;
  27. Let &lt;a href=&quot;http://www.codecogs.com/eqnedit.php?latex=A&quot; target=&quot;_blank&quot;&gt;&lt;img src=&quot;http://latex.codecogs.com/gif.latex?A&quot; title=&quot;A&quot; /&gt;&lt;/a&gt;  and &lt;a href=&quot;http://www.codecogs.com/eqnedit.php?latex=B&quot; target=&quot;_blank&quot;&gt;&lt;img src=&quot;http://latex.codecogs.com/gif.latex?B&quot; title=&quot;B&quot; /&gt;&lt;/a&gt; are two different &lt;a href=&quot;http://www.codecogs.com/eqnedit.php?latex=3%20%5Ctimes%203&quot; target=&quot;_blank&quot;&gt;&lt;img src=&quot;http://latex.codecogs.com/gif.latex?3%20%5Ctimes%203&quot; title=&quot;3 \times 3&quot; /&gt;&lt;/a&gt; matrices such that &lt;a href=&quot;http://www.codecogs.com/eqnedit.php?latex=%7BA%5E3%7D%20=%20%7BB%5E3%7D&quot; target=&quot;_blank&quot;&gt;&lt;img src=&quot;http://latex.codecogs.com/gif.latex?%7BA%5E3%7D%20=%20%7BB%5E3%7D&quot; title=&quot;{A^3} = {B^3}&quot; /&gt;&lt;/a&gt; and &lt;a href=&quot;http://www.codecogs.com/eqnedit.php?latex=%7BA%5E2%7DB%20=%20%7BB%5E2%7DA&quot; target=&quot;_blank&quot;&gt;&lt;img src=&quot;http://latex.codecogs.com/gif.latex?%7BA%5E2%7DB%20=%20%7BB%5E2%7DA&quot; title=&quot;{A^2}B = {B^2}A&quot; /&gt;&lt;/a&gt;.  Does the matrix  &lt;a href=&quot;http://www.codecogs.com/eqnedit.php?latex=%7BA%5E2%7D%20@plus;%20%7BB%5E2%7D&quot; target=&quot;_blank&quot;&gt;&lt;img src=&quot;http://latex.codecogs.com/gif.latex?%7BA%5E2%7D%20+%20%7BB%5E2%7D&quot; title=&quot;{A^2} + {B^2}&quot; /&gt;&lt;/a&gt; have inverse?  &lt;/div&gt;</content><link rel='replies' type='application/atom+xml' href='http://vidyamanohar.blogspot.com/feeds/9160217841798511245/comments/default' title='Post Comments'/><link rel='replies' type='text/html' href='http://www.blogger.com/comment.g?blogID=32747608&amp;postID=9160217841798511245' title='1 Comments'/><link rel='edit' type='application/atom+xml' href='http://www.blogger.com/feeds/32747608/posts/default/9160217841798511245'/><link rel='self' type='application/atom+xml' href='http://www.blogger.com/feeds/32747608/posts/default/9160217841798511245'/><link rel='alternate' type='text/html' href='http://vidyamanohar.blogspot.com/2012/03/puzzle-40-matrices.html' title='Puzzle 40 - Matrices'/><author><name>Vidyamanohar</name><uri>http://www.blogger.com/profile/14537585081760370998</uri><email>noreply@blogger.com</email><gd:image rel='http://schemas.google.com/g/2005#thumbnail' width='16' height='16' src='https://img1.blogblog.com/img/b16-rounded.gif'/></author><thr:total>1</thr:total></entry><entry><id>tag:blogger.com,1999:blog-32747608.post-4765496346833026343</id><published>2012-03-13T08:07:00.003+05:30</published><updated>2012-03-13T08:08:15.057+05:30</updated><title type='text'>Puzzle 39 - 2012 Integration</title><content type='html'>&lt;div dir=&quot;ltr&quot; style=&quot;text-align: left;&quot; trbidi=&quot;on&quot;&gt;
  28. Find the value of &lt;a href=&quot;http://www.codecogs.com/eqnedit.php?latex=%5Cfrac%7B%7B6539%7D%7D%7B%7B101%7D%7D%5Cint%5Climits_%7B%20-%201%7D%5E1%20%7B%5Cfrac%7B%7B%7Bt%5E%7B12%7D%7D%20@plus;%2031%7D%7D%7B%7B1%20@plus;%20%7B%7B2011%7D%5Et%7D%7D%7Ddt%7D&quot; target=&quot;_blank&quot;&gt;&lt;img src=&quot;http://latex.codecogs.com/gif.latex?%5Cfrac%7B%7B6539%7D%7D%7B%7B101%7D%7D%5Cint%5Climits_%7B%20-%201%7D%5E1%20%7B%5Cfrac%7B%7B%7Bt%5E%7B12%7D%7D%20+%2031%7D%7D%7B%7B1%20+%20%7B%7B2011%7D%5Et%7D%7D%7Ddt%7D&quot; title=&quot;\frac{{6539}}{{101}}\int\limits_{ - 1}^1 {\frac{{{t^{12}} + 31}}{{1 + {{2011}^t}}}dt}&quot; /&gt;&lt;/a&gt;&lt;/div&gt;</content><link rel='replies' type='application/atom+xml' href='http://vidyamanohar.blogspot.com/feeds/4765496346833026343/comments/default' title='Post Comments'/><link rel='replies' type='text/html' href='http://www.blogger.com/comment.g?blogID=32747608&amp;postID=4765496346833026343' title='3 Comments'/><link rel='edit' type='application/atom+xml' href='http://www.blogger.com/feeds/32747608/posts/default/4765496346833026343'/><link rel='self' type='application/atom+xml' href='http://www.blogger.com/feeds/32747608/posts/default/4765496346833026343'/><link rel='alternate' type='text/html' href='http://vidyamanohar.blogspot.com/2012/03/puzzle-39-2012-integration.html' title='Puzzle 39 - 2012 Integration'/><author><name>Vidyamanohar</name><uri>http://www.blogger.com/profile/14537585081760370998</uri><email>noreply@blogger.com</email><gd:image rel='http://schemas.google.com/g/2005#thumbnail' width='16' height='16' src='https://img1.blogblog.com/img/b16-rounded.gif'/></author><thr:total>3</thr:total></entry><entry><id>tag:blogger.com,1999:blog-32747608.post-2965977441802031738</id><published>2012-03-08T20:15:00.001+05:30</published><updated>2012-03-08T20:16:11.629+05:30</updated><title type='text'>Puzzle 38 - Point inside the Circle</title><content type='html'>&lt;div dir=&quot;ltr&quot; style=&quot;text-align: left;&quot; trbidi=&quot;on&quot;&gt;
  29. Find the values of &lt;a href=&quot;http://www.codecogs.com/eqnedit.php?latex=%5Calpha&quot; target=&quot;_blank&quot;&gt;&lt;img src=&quot;http://latex.codecogs.com/gif.latex?%5Calpha&quot; title=&quot;\alpha&quot; /&gt;&lt;/a&gt;  for which the point &lt;a href=&quot;http://www.codecogs.com/eqnedit.php?latex=%28%5Calpha%20-%201,%5C;%5Calpha%20@plus;%201%29&quot; target=&quot;_blank&quot;&gt;&lt;img src=&quot;http://latex.codecogs.com/gif.latex?%28%5Calpha%20-%201,%5C;%5Calpha%20+%201%29&quot; title=&quot;(\alpha - 1,\;\alpha + 1)&quot; /&gt;&lt;/a&gt; lies in the larger segment of the circle &lt;a href=&quot;http://www.codecogs.com/eqnedit.php?latex=%7Bx%5E2%7D%20@plus;%20%7By%5E2%7D%20-%20x%20-%20y%20-%206%20=%200&quot; target=&quot;_blank&quot;&gt;&lt;img src=&quot;http://latex.codecogs.com/gif.latex?%7Bx%5E2%7D%20+%20%7By%5E2%7D%20-%20x%20-%20y%20-%206%20=%200&quot; title=&quot;{x^2} + {y^2} - x - y - 6 = 0&quot; /&gt;&lt;/a&gt; made by the chord whose equation is x + y – 2 = 0  &lt;/div&gt;</content><link rel='replies' type='application/atom+xml' href='http://vidyamanohar.blogspot.com/feeds/2965977441802031738/comments/default' title='Post Comments'/><link rel='replies' type='text/html' href='http://www.blogger.com/comment.g?blogID=32747608&amp;postID=2965977441802031738' title='3 Comments'/><link rel='edit' type='application/atom+xml' href='http://www.blogger.com/feeds/32747608/posts/default/2965977441802031738'/><link rel='self' type='application/atom+xml' href='http://www.blogger.com/feeds/32747608/posts/default/2965977441802031738'/><link rel='alternate' type='text/html' href='http://vidyamanohar.blogspot.com/2012/03/puzzle-38-point-inside-circle.html' title='Puzzle 38 - Point inside the Circle'/><author><name>Vidyamanohar</name><uri>http://www.blogger.com/profile/14537585081760370998</uri><email>noreply@blogger.com</email><gd:image rel='http://schemas.google.com/g/2005#thumbnail' width='16' height='16' src='https://img1.blogblog.com/img/b16-rounded.gif'/></author><thr:total>3</thr:total></entry><entry><id>tag:blogger.com,1999:blog-32747608.post-4841013005833874827</id><published>2012-03-06T10:30:00.001+05:30</published><updated>2012-03-06T10:31:47.309+05:30</updated><title type='text'>Puzzle 37 - Extreme Values of Integral</title><content type='html'>&lt;div dir=&quot;ltr&quot; style=&quot;text-align: left;&quot; trbidi=&quot;on&quot;&gt;
  30. Define the function &lt;a href=&quot;http://www.codecogs.com/eqnedit.php?latex=f%28x%29%20=%20%5Cint%5Climits_0%5E%7B%5Cfrac%7B%5Cpi%20%7D%7B2%7D%7D%20%7B%5Cfrac%7B%7B%5Ccos%20%7Ct%20-%20x%7C%7D%7D%7B%7B1%20@plus;%20%5Csin%20%7Ct%20-%20x%7C%7D%7D%7D%20dt;%5C,0%20%5Cle%20x%20%5Cle%20%5Cpi&quot; target=&quot;_blank&quot;&gt;&lt;img src=&quot;http://latex.codecogs.com/gif.latex?f%28x%29%20=%20%5Cint%5Climits_0%5E%7B%5Cfrac%7B%5Cpi%20%7D%7B2%7D%7D%20%7B%5Cfrac%7B%7B%5Ccos%20%7Ct%20-%20x%7C%7D%7D%7B%7B1%20+%20%5Csin%20%7Ct%20-%20x%7C%7D%7D%7D%20dt;%5C,0%20%5Cle%20x%20%5Cle%20%5Cpi&quot; title=&quot;f(x) = \int\limits_0^{\frac{\pi }{2}} {\frac{{\cos |t - x|}}{{1 + \sin |t - x|}}} dt;\,0 \le x \le \pi&quot; /&gt;&lt;/a&gt;, find the maximum and minimum values of f(x) in the interval &lt;a href=&quot;http://www.codecogs.com/eqnedit.php?latex=[0,%5C,%5Cpi%20]&quot; target=&quot;_blank&quot;&gt;&lt;img src=&quot;http://latex.codecogs.com/gif.latex?[0,%5C,%5Cpi%20]&quot; title=&quot;[0,\,\pi ]&quot; /&gt;&lt;/a&gt;&lt;/div&gt;</content><link rel='replies' type='application/atom+xml' href='http://vidyamanohar.blogspot.com/feeds/4841013005833874827/comments/default' title='Post Comments'/><link rel='replies' type='text/html' href='http://www.blogger.com/comment.g?blogID=32747608&amp;postID=4841013005833874827' title='0 Comments'/><link rel='edit' type='application/atom+xml' href='http://www.blogger.com/feeds/32747608/posts/default/4841013005833874827'/><link rel='self' type='application/atom+xml' href='http://www.blogger.com/feeds/32747608/posts/default/4841013005833874827'/><link rel='alternate' type='text/html' href='http://vidyamanohar.blogspot.com/2012/03/puzzle-37-extreme-values-of-integral.html' title='Puzzle 37 - Extreme Values of Integral'/><author><name>Vidyamanohar</name><uri>http://www.blogger.com/profile/14537585081760370998</uri><email>noreply@blogger.com</email><gd:image rel='http://schemas.google.com/g/2005#thumbnail' width='16' height='16' src='https://img1.blogblog.com/img/b16-rounded.gif'/></author><thr:total>0</thr:total></entry><entry><id>tag:blogger.com,1999:blog-32747608.post-3989585422273742661</id><published>2012-02-25T10:22:00.001+05:30</published><updated>2012-02-25T10:23:06.146+05:30</updated><title type='text'>Puzzle 36 - Functional Equation</title><content type='html'>&lt;div dir=&quot;ltr&quot; style=&quot;text-align: left;&quot; trbidi=&quot;on&quot;&gt;
  31. Find the function f(x) if f(x) is a non constant and satisfies the equation &lt;a href=&quot;http://www.codecogs.com/eqnedit.php?latex=f%28x%29%20=%20%7Bx%5E2%7D%20-%20%5Cint%5Climits_0%5E1%20%7B%7B%7B%28f%28t%29%20@plus;%20x%29%7D%5E2%7D%7D%20dt&quot; target=&quot;_blank&quot;&gt;&lt;img src=&quot;http://latex.codecogs.com/gif.latex?f%28x%29%20=%20%7Bx%5E2%7D%20-%20%5Cint%5Climits_0%5E1%20%7B%7B%7B%28f%28t%29%20+%20x%29%7D%5E2%7D%7D%20dt&quot; title=&quot;f(x) = {x^2} - \int\limits_0^1 {{{(f(t) + x)}^2}} dt&quot; /&gt;&lt;/a&gt;&lt;/div&gt;</content><link rel='replies' type='application/atom+xml' href='http://vidyamanohar.blogspot.com/feeds/3989585422273742661/comments/default' title='Post Comments'/><link rel='replies' type='text/html' href='http://www.blogger.com/comment.g?blogID=32747608&amp;postID=3989585422273742661' title='1 Comments'/><link rel='edit' type='application/atom+xml' href='http://www.blogger.com/feeds/32747608/posts/default/3989585422273742661'/><link rel='self' type='application/atom+xml' href='http://www.blogger.com/feeds/32747608/posts/default/3989585422273742661'/><link rel='alternate' type='text/html' href='http://vidyamanohar.blogspot.com/2012/02/puzzle-36-functional-equation.html' title='Puzzle 36 - Functional Equation'/><author><name>Vidyamanohar</name><uri>http://www.blogger.com/profile/14537585081760370998</uri><email>noreply@blogger.com</email><gd:image rel='http://schemas.google.com/g/2005#thumbnail' width='16' height='16' src='https://img1.blogblog.com/img/b16-rounded.gif'/></author><thr:total>1</thr:total></entry><entry><id>tag:blogger.com,1999:blog-32747608.post-6192190097607246570</id><published>2012-02-25T08:23:00.002+05:30</published><updated>2012-02-25T08:23:50.474+05:30</updated><title type='text'>Puzzle 35 - Complex system of equations</title><content type='html'>&lt;div dir=&quot;ltr&quot; style=&quot;text-align: left;&quot; trbidi=&quot;on&quot;&gt;
  32. Find the complex roots of the system of equations &lt;br /&gt;
  33. &lt;br /&gt;
  34. &lt;a href=&quot;http://www.codecogs.com/eqnedit.php?latex=%28x%20@plus;%20y%29%28%7Bx%5E2%7D%20-%20%7By%5E2%7D%29%20=%201176&quot; target=&quot;_blank&quot;&gt;&lt;img src=&quot;http://latex.codecogs.com/gif.latex?%28x%20+%20y%29%28%7Bx%5E2%7D%20-%20%7By%5E2%7D%29%20=%201176&quot; title=&quot;(x + y)({x^2} - {y^2}) = 1176&quot; /&gt;&lt;/a&gt;&lt;br /&gt;
  35. &lt;br /&gt;
  36. &lt;a href=&quot;http://www.codecogs.com/eqnedit.php?latex=%28x%20-%20y%29%28%7Bx%5E2%7D%20@plus;%20%7By%5E2%7D%29%20=%20696&quot; target=&quot;_blank&quot;&gt;&lt;img src=&quot;http://latex.codecogs.com/gif.latex?%28x%20-%20y%29%28%7Bx%5E2%7D%20+%20%7By%5E2%7D%29%20=%20696&quot; title=&quot;(x - y)({x^2} + {y^2}) = 696&quot; /&gt;&lt;/a&gt;&lt;/div&gt;</content><link rel='replies' type='application/atom+xml' href='http://vidyamanohar.blogspot.com/feeds/6192190097607246570/comments/default' title='Post Comments'/><link rel='replies' type='text/html' href='http://www.blogger.com/comment.g?blogID=32747608&amp;postID=6192190097607246570' title='1 Comments'/><link rel='edit' type='application/atom+xml' href='http://www.blogger.com/feeds/32747608/posts/default/6192190097607246570'/><link rel='self' type='application/atom+xml' href='http://www.blogger.com/feeds/32747608/posts/default/6192190097607246570'/><link rel='alternate' type='text/html' href='http://vidyamanohar.blogspot.com/2012/02/puzzle-35-complex-system-of-equations.html' title='Puzzle 35 - Complex system of equations'/><author><name>Vidyamanohar</name><uri>http://www.blogger.com/profile/14537585081760370998</uri><email>noreply@blogger.com</email><gd:image rel='http://schemas.google.com/g/2005#thumbnail' width='16' height='16' src='https://img1.blogblog.com/img/b16-rounded.gif'/></author><thr:total>1</thr:total></entry><entry><id>tag:blogger.com,1999:blog-32747608.post-8871149758317773586</id><published>2012-02-23T22:20:00.002+05:30</published><updated>2012-02-23T22:21:25.731+05:30</updated><title type='text'>Puzzle 34 - Analytical Geometry</title><content type='html'>Vertex A of the triangle ABC is (14, 8).  If the orthocenter of the triangle is H(8, 1) and the midpoint of the side BC is (3, -2), find the coordinates of the points B and C.</content><link rel='replies' type='application/atom+xml' href='http://vidyamanohar.blogspot.com/feeds/8871149758317773586/comments/default' title='Post Comments'/><link rel='replies' type='text/html' href='http://www.blogger.com/comment.g?blogID=32747608&amp;postID=8871149758317773586' title='1 Comments'/><link rel='edit' type='application/atom+xml' href='http://www.blogger.com/feeds/32747608/posts/default/8871149758317773586'/><link rel='self' type='application/atom+xml' href='http://www.blogger.com/feeds/32747608/posts/default/8871149758317773586'/><link rel='alternate' type='text/html' href='http://vidyamanohar.blogspot.com/2012/02/puzzle-34-analytical-geometry.html' title='Puzzle 34 - Analytical Geometry'/><author><name>Vidyamanohar</name><uri>http://www.blogger.com/profile/14537585081760370998</uri><email>noreply@blogger.com</email><gd:image rel='http://schemas.google.com/g/2005#thumbnail' width='16' height='16' src='https://img1.blogblog.com/img/b16-rounded.gif'/></author><thr:total>1</thr:total></entry><entry><id>tag:blogger.com,1999:blog-32747608.post-9016614885748251892</id><published>2012-02-23T11:06:00.000+05:30</published><updated>2012-02-23T11:08:21.395+05:30</updated><title type='text'>Puzzle 33 - Number Grid</title><content type='html'>&lt;div dir=&quot;ltr&quot; style=&quot;text-align: left;&quot; trbidi=&quot;on&quot;&gt;
  37. &lt;br /&gt;
  38. &lt;div class=&quot;separator&quot; style=&quot;clear: both; text-align: left;&quot;&gt;
  39. &lt;span style=&quot;line-height: 115%;&quot;&gt;In
  40. the grid shown, a number is to be placed in each smaller square so that the
  41. product of all the three numbers in any row, column, or diagonal is the same
  42. positive number.&amp;nbsp; Find the sum of x and
  43. y.&lt;/span&gt;&lt;/div&gt;
  44. &lt;div class=&quot;separator&quot; style=&quot;clear: both; text-align: center;&quot;&gt;
  45. &lt;a href=&quot;http://1.bp.blogspot.com/-f0urkAS-K3A/T0XQC5jUtnI/AAAAAAAAAYA/oVQORlIK0kI/s1600/Puzzle+33+Image.jpg&quot; imageanchor=&quot;1&quot; style=&quot;margin-left: 1em; margin-right: 1em;&quot;&gt;&lt;img border=&quot;0&quot; src=&quot;http://1.bp.blogspot.com/-f0urkAS-K3A/T0XQC5jUtnI/AAAAAAAAAYA/oVQORlIK0kI/s1600/Puzzle+33+Image.jpg&quot; /&gt;&lt;/a&gt;&lt;/div&gt;
  46. &lt;div class=&quot;MsoNormal&quot;&gt;
  47. &lt;span style=&quot;line-height: 115%;&quot;&gt;&lt;br /&gt;&lt;/span&gt;&lt;/div&gt;
  48. &lt;/div&gt;</content><link rel='replies' type='application/atom+xml' href='http://vidyamanohar.blogspot.com/feeds/9016614885748251892/comments/default' title='Post Comments'/><link rel='replies' type='text/html' href='http://www.blogger.com/comment.g?blogID=32747608&amp;postID=9016614885748251892' title='3 Comments'/><link rel='edit' type='application/atom+xml' href='http://www.blogger.com/feeds/32747608/posts/default/9016614885748251892'/><link rel='self' type='application/atom+xml' href='http://www.blogger.com/feeds/32747608/posts/default/9016614885748251892'/><link rel='alternate' type='text/html' href='http://vidyamanohar.blogspot.com/2012/02/puzzle-33-number-grid.html' title='Puzzle 33 - Number Grid'/><author><name>Vidyamanohar</name><uri>http://www.blogger.com/profile/14537585081760370998</uri><email>noreply@blogger.com</email><gd:image rel='http://schemas.google.com/g/2005#thumbnail' width='16' height='16' src='https://img1.blogblog.com/img/b16-rounded.gif'/></author><media:thumbnail xmlns:media="http://search.yahoo.com/mrss/" url="http://1.bp.blogspot.com/-f0urkAS-K3A/T0XQC5jUtnI/AAAAAAAAAYA/oVQORlIK0kI/s72-c/Puzzle+33+Image.jpg" height="72" width="72"/><thr:total>3</thr:total></entry><entry><id>tag:blogger.com,1999:blog-32747608.post-8926144919893060053</id><published>2012-02-19T08:46:00.003+05:30</published><updated>2012-02-19T08:47:02.799+05:30</updated><title type='text'>Puzzle 32 - Trigonometry</title><content type='html'>&lt;div dir=&quot;ltr&quot; style=&quot;text-align: left;&quot; trbidi=&quot;on&quot;&gt;
  49. If &lt;a href=&quot;http://www.codecogs.com/eqnedit.php?latex=%5Calpha%20@plus;%20%5Cbeta%20@plus;%20%5Cgamma%20=%20%5Cpi&quot; target=&quot;_blank&quot;&gt;&lt;img src=&quot;http://latex.codecogs.com/gif.latex?%5Calpha%20+%20%5Cbeta%20+%20%5Cgamma%20=%20%5Cpi&quot; title=&quot;\alpha + \beta + \gamma = \pi&quot; /&gt;&lt;/a&gt; and &lt;a href=&quot;http://www.codecogs.com/eqnedit.php?latex=%5Cprod%20%7B%5Ctan%20%5Cleft%28%20%7B%5Cfrac%7B%7B%5Calpha%20@plus;%20%5Cbeta%20-%20%5Cgamma%20%7D%7D%7B4%7D%7D%20%5Cright%29%7D%20=%201&quot; target=&quot;_blank&quot;&gt;&lt;img src=&quot;http://latex.codecogs.com/gif.latex?%5Cprod%20%7B%5Ctan%20%5Cleft%28%20%7B%5Cfrac%7B%7B%5Calpha%20+%20%5Cbeta%20-%20%5Cgamma%20%7D%7D%7B4%7D%7D%20%5Cright%29%7D%20=%201&quot; title=&quot;\prod {\tan \left( {\frac{{\alpha + \beta - \gamma }}{4}} \right)} = 1&quot; /&gt;&lt;/a&gt; find the value of &lt;a href=&quot;http://www.codecogs.com/eqnedit.php?latex=%5Csum%20%7B%5Csin%20%5Calpha%20%7D%20@plus;%20%5Cprod%20%7B%5Csin%20%5Calpha%20%7D&quot; target=&quot;_blank&quot;&gt;&lt;img src=&quot;http://latex.codecogs.com/gif.latex?%5Csum%20%7B%5Csin%20%5Calpha%20%7D%20+%20%5Cprod%20%7B%5Csin%20%5Calpha%20%7D&quot; title=&quot;\sum {\sin \alpha } + \prod {\sin \alpha }&quot; /&gt;&lt;/a&gt;&lt;/div&gt;</content><link rel='replies' type='application/atom+xml' href='http://vidyamanohar.blogspot.com/feeds/8926144919893060053/comments/default' title='Post Comments'/><link rel='replies' type='text/html' href='http://www.blogger.com/comment.g?blogID=32747608&amp;postID=8926144919893060053' title='1 Comments'/><link rel='edit' type='application/atom+xml' href='http://www.blogger.com/feeds/32747608/posts/default/8926144919893060053'/><link rel='self' type='application/atom+xml' href='http://www.blogger.com/feeds/32747608/posts/default/8926144919893060053'/><link rel='alternate' type='text/html' href='http://vidyamanohar.blogspot.com/2012/02/puzzle-32-trigonometry.html' title='Puzzle 32 - Trigonometry'/><author><name>Vidyamanohar</name><uri>http://www.blogger.com/profile/14537585081760370998</uri><email>noreply@blogger.com</email><gd:image rel='http://schemas.google.com/g/2005#thumbnail' width='16' height='16' src='https://img1.blogblog.com/img/b16-rounded.gif'/></author><thr:total>1</thr:total></entry><entry><id>tag:blogger.com,1999:blog-32747608.post-133920125933046921</id><published>2012-02-18T15:03:00.001+05:30</published><updated>2012-02-18T15:04:05.456+05:30</updated><title type='text'>Puzzle 31 - Limit</title><content type='html'>&lt;div dir=&quot;ltr&quot; style=&quot;text-align: left;&quot; trbidi=&quot;on&quot;&gt;Evaluate &lt;a href=&quot;http://www.codecogs.com/eqnedit.php?latex=\mathop {\lim }\limits_{n \to \infty } \left( {\frac{1}{{\sqrt 2 }}\sqrt {\frac{1}{2} @plus; \frac{1}{2}\frac{1}{{\sqrt 2 }}} \sqrt {\frac{1}{2} @plus; \frac{1}{2}\sqrt {\frac{1}{2} @plus; \frac{1}{2}\frac{1}{{\sqrt 2 }}} } ...n\,terms} \right)&quot; target=&quot;_blank&quot;&gt;&lt;img src=&quot;http://latex.codecogs.com/gif.latex?\mathop {\lim }\limits_{n \to \infty } \left( {\frac{1}{{\sqrt 2 }}\sqrt {\frac{1}{2} + \frac{1}{2}\frac{1}{{\sqrt 2 }}} \sqrt {\frac{1}{2} + \frac{1}{2}\sqrt {\frac{1}{2} + \frac{1}{2}\frac{1}{{\sqrt 2 }}} } ...n\,terms} \right)&quot; title=&quot;\mathop {\lim }\limits_{n \to \infty } \left( {\frac{1}{{\sqrt 2 }}\sqrt {\frac{1}{2} + \frac{1}{2}\frac{1}{{\sqrt 2 }}} \sqrt {\frac{1}{2} + \frac{1}{2}\sqrt {\frac{1}{2} + \frac{1}{2}\frac{1}{{\sqrt 2 }}} } ...n\,terms} \right)&quot; /&gt;&lt;/a&gt;&lt;br /&gt;
  50. &lt;/div&gt;</content><link rel='replies' type='application/atom+xml' href='http://vidyamanohar.blogspot.com/feeds/133920125933046921/comments/default' title='Post Comments'/><link rel='replies' type='text/html' href='http://www.blogger.com/comment.g?blogID=32747608&amp;postID=133920125933046921' title='3 Comments'/><link rel='edit' type='application/atom+xml' href='http://www.blogger.com/feeds/32747608/posts/default/133920125933046921'/><link rel='self' type='application/atom+xml' href='http://www.blogger.com/feeds/32747608/posts/default/133920125933046921'/><link rel='alternate' type='text/html' href='http://vidyamanohar.blogspot.com/2012/02/puzzle-31-limit.html' title='Puzzle 31 - Limit'/><author><name>Vidyamanohar</name><uri>http://www.blogger.com/profile/14537585081760370998</uri><email>noreply@blogger.com</email><gd:image rel='http://schemas.google.com/g/2005#thumbnail' width='16' height='16' src='https://img1.blogblog.com/img/b16-rounded.gif'/></author><thr:total>3</thr:total></entry><entry><id>tag:blogger.com,1999:blog-32747608.post-5592265595906877484</id><published>2012-02-17T08:37:00.002+05:30</published><updated>2012-02-23T17:18:21.571+05:30</updated><title type='text'>Puzzle 30 - Parabola and Circle</title><content type='html'>&lt;div dir=&quot;ltr&quot; style=&quot;text-align: left;&quot; trbidi=&quot;on&quot;&gt;
  51. The tangent and normal at the point &lt;a href=&quot;http://www.codecogs.com/eqnedit.php?latex=P%28at%5E2,%202at%29&quot; target=&quot;_blank&quot;&gt;&lt;img src=&quot;http://latex.codecogs.com/gif.latex?P%28at%5E2,%202at%29&quot; title=&quot;P(at^2, 2at)&quot; /&gt;&lt;/a&gt; to the curve &lt;a href=&quot;http://www.codecogs.com/eqnedit.php?latex=y%5E2%20=%204ax&quot; target=&quot;_blank&quot;&gt;&lt;img src=&quot;http://latex.codecogs.com/gif.latex?y%5E2%20=%204ax&quot; title=&quot;y^2 = 4ax&quot; /&gt;&lt;/a&gt; meet the x-axis and y-axis in T and G respectively, then find the angle at which the tangent at P to the curve is inclined to the tangent at P to the circle through the points T, P, G.&lt;/div&gt;</content><link rel='replies' type='application/atom+xml' href='http://vidyamanohar.blogspot.com/feeds/5592265595906877484/comments/default' title='Post Comments'/><link rel='replies' type='text/html' href='http://www.blogger.com/comment.g?blogID=32747608&amp;postID=5592265595906877484' title='1 Comments'/><link rel='edit' type='application/atom+xml' href='http://www.blogger.com/feeds/32747608/posts/default/5592265595906877484'/><link rel='self' type='application/atom+xml' href='http://www.blogger.com/feeds/32747608/posts/default/5592265595906877484'/><link rel='alternate' type='text/html' href='http://vidyamanohar.blogspot.com/2012/02/puzzle-30-parabola-and-circle.html' title='Puzzle 30 - Parabola and Circle'/><author><name>Vidyamanohar</name><uri>http://www.blogger.com/profile/14537585081760370998</uri><email>noreply@blogger.com</email><gd:image rel='http://schemas.google.com/g/2005#thumbnail' width='16' height='16' src='https://img1.blogblog.com/img/b16-rounded.gif'/></author><thr:total>1</thr:total></entry><entry><id>tag:blogger.com,1999:blog-32747608.post-4902654930476283835</id><published>2012-02-14T17:35:00.000+05:30</published><updated>2012-02-14T17:35:59.599+05:30</updated><title type='text'>Puzzle 29 - Integration</title><content type='html'>&lt;div dir=&quot;ltr&quot; style=&quot;text-align: left;&quot; trbidi=&quot;on&quot;&gt;
  52. If &lt;a href=&quot;http://www.codecogs.com/eqnedit.php?latex=%5Cint%20%7B%5Cfrac%7B%7B1%20-%207%7B%7B%5Ccos%20%7D%5E2%7Dx%7D%7D%7B%7B%7B%7B%5Csin%20%7D%5E7%7Dx%7B%7B%5Ccos%20%7D%5E2%7Dx%7D%7Ddx%7D%20=%20%5Cfrac%7B%7Bf%28x%29%7D%7D%7B%7B%7B%7B%5Csin%20%7D%5E7%7Dx%7D%7D%20@plus;%20C&quot; target=&quot;_blank&quot;&gt;&lt;img src=&quot;http://latex.codecogs.com/gif.latex?%5Cint%20%7B%5Cfrac%7B%7B1%20-%207%7B%7B%5Ccos%20%7D%5E2%7Dx%7D%7D%7B%7B%7B%7B%5Csin%20%7D%5E7%7Dx%7B%7B%5Ccos%20%7D%5E2%7Dx%7D%7Ddx%7D%20=%20%5Cfrac%7B%7Bf%28x%29%7D%7D%7B%7B%7B%7B%5Csin%20%7D%5E7%7Dx%7D%7D%20+%20C&quot; title=&quot;\int {\frac{{1 - 7{{\cos }^2}x}}{{{{\sin }^7}x{{\cos }^2}x}}dx} = \frac{{f(x)}}{{{{\sin }^7}x}} + C&quot; /&gt;&lt;/a&gt;, then find the value of &lt;a href=&quot;http://www.codecogs.com/eqnedit.php?latex=f%27%280%29%20@plus;%20f%27%27%28%5Cfrac%7B%5Cpi%20%7D%7B4%7D%29&quot; target=&quot;_blank&quot;&gt;&lt;img src=&quot;http://latex.codecogs.com/gif.latex?f%27%280%29%20+%20f%27%27%28%5Cfrac%7B%5Cpi%20%7D%7B4%7D%29&quot; title=&quot;f&#39;(0) + f&#39;&#39;(\frac{\pi }{4})&quot; /&gt;&lt;/a&gt;&lt;/div&gt;</content><link rel='replies' type='application/atom+xml' href='http://vidyamanohar.blogspot.com/feeds/4902654930476283835/comments/default' title='Post Comments'/><link rel='replies' type='text/html' href='http://www.blogger.com/comment.g?blogID=32747608&amp;postID=4902654930476283835' title='1 Comments'/><link rel='edit' type='application/atom+xml' href='http://www.blogger.com/feeds/32747608/posts/default/4902654930476283835'/><link rel='self' type='application/atom+xml' href='http://www.blogger.com/feeds/32747608/posts/default/4902654930476283835'/><link rel='alternate' type='text/html' href='http://vidyamanohar.blogspot.com/2012/02/puzzle-29-integration.html' title='Puzzle 29 - Integration'/><author><name>Vidyamanohar</name><uri>http://www.blogger.com/profile/14537585081760370998</uri><email>noreply@blogger.com</email><gd:image rel='http://schemas.google.com/g/2005#thumbnail' width='16' height='16' src='https://img1.blogblog.com/img/b16-rounded.gif'/></author><thr:total>1</thr:total></entry><entry><id>tag:blogger.com,1999:blog-32747608.post-4186725529071106915</id><published>2012-02-13T12:34:00.001+05:30</published><updated>2012-03-09T06:31:16.646+05:30</updated><title type='text'>Puzzle 28 - Integration</title><content type='html'>&lt;div dir=&quot;ltr&quot; style=&quot;text-align: left;&quot; trbidi=&quot;on&quot;&gt;Let &lt;a href=&quot;http://www.codecogs.com/eqnedit.php?latex=%7B%5Crm%7Bf%7D%7D%5Cleft%28%20%7B%5Crm%7Bn%7D%7D%20%5Cright%29&quot; target=&quot;_blank&quot;&gt;&lt;img src=&quot;http://latex.codecogs.com/gif.latex?%7B%5Crm%7Bf%7D%7D%5Cleft%28%20%7B%5Crm%7Bn%7D%7D%20%5Cright%29&quot; title=&quot;{\rm{f}}\left( {\rm{n}} \right)&quot; /&gt;&lt;/a&gt; denotes the coefficient of &lt;a href=&quot;http://www.codecogs.com/eqnedit.php?latex=%7B%7B%5Crm%7Bx%7D%7D%5E%7B%7B%5Crm%7B2%7D%7D0%7B%5Crm%7B12%7D%7D%7D%7D&quot; target=&quot;_blank&quot;&gt;&lt;img src=&quot;http://latex.codecogs.com/gif.latex?%7B%7B%5Crm%7Bx%7D%7D%5E%7B%7B%5Crm%7B2%7D%7D0%7B%5Crm%7B12%7D%7D%7D%7D&quot; title=&quot;{{\rm{x}}^{{\rm{2}}0{\rm{12}}}}&quot; /&gt;&lt;/a&gt; in the expansion of &lt;a href=&quot;http://www.codecogs.com/eqnedit.php?latex=%7B%5Cleft%28%20%7B%7B%5Crm%7B1%7D%7D%20@plus;%20%7B%5Crm%7Bx%7D%7D%7D%20%5Cright%29%5E%7B%5Crm%7Bn%7D%7D%7D&quot; target=&quot;_blank&quot;&gt;&lt;img src=&quot;http://latex.codecogs.com/gif.latex?%7B%5Cleft%28%20%7B%7B%5Crm%7B1%7D%7D%20+%20%7B%5Crm%7Bx%7D%7D%7D%20%5Cright%29%5E%7B%5Crm%7Bn%7D%7D%7D&quot; title=&quot;{\left( {{\rm{1}} + {\rm{x}}} \right)^{\rm{n}}}&quot; /&gt;&lt;/a&gt;.  Then evaluate &lt;a href=&quot;http://www.codecogs.com/eqnedit.php?latex=%5Cint%5Climits_0%5E1%20%7Bf%28%20-%20t%20-%201%29%5Cleft[%20%7B%5Cfrac%7B1%7D%7B%7Bt%20@plus;%201%7D%7D%20@plus;%20%5Cfrac%7B1%7D%7B%7Bt%20@plus;%202%7D%7D%20@plus;%20%5Cfrac%7B1%7D%7B%7Bt%20@plus;%203%7D%7D...%5Cfrac%7B1%7D%7B%7Bt%20@plus;%202012%7D%7D%7D%20%5Cright]%7D%20dt&quot; target=&quot;_blank&quot;&gt;&lt;img src=&quot;http://latex.codecogs.com/gif.latex?%5Cint%5Climits_0%5E1%20%7Bf%28%20-%20t%20-%201%29%5Cleft[%20%7B%5Cfrac%7B1%7D%7B%7Bt%20+%201%7D%7D%20+%20%5Cfrac%7B1%7D%7B%7Bt%20+%202%7D%7D%20+%20%5Cfrac%7B1%7D%7B%7Bt%20+%203%7D%7D...%5Cfrac%7B1%7D%7B%7Bt%20+%202012%7D%7D%7D%20%5Cright]%7D%20dt&quot; title=&quot;\int\limits_0^1 {f( - t - 1)\left[ {\frac{1}{{t + 1}} + \frac{1}{{t + 2}} + \frac{1}{{t + 3}}...\frac{1}{{t + 2012}}} \right]} dt&quot; /&gt;&lt;/a&gt;&lt;/div&gt;</content><link rel='replies' type='application/atom+xml' href='http://vidyamanohar.blogspot.com/feeds/4186725529071106915/comments/default' title='Post Comments'/><link rel='replies' type='text/html' href='http://www.blogger.com/comment.g?blogID=32747608&amp;postID=4186725529071106915' title='1 Comments'/><link rel='edit' type='application/atom+xml' href='http://www.blogger.com/feeds/32747608/posts/default/4186725529071106915'/><link rel='self' type='application/atom+xml' href='http://www.blogger.com/feeds/32747608/posts/default/4186725529071106915'/><link rel='alternate' type='text/html' href='http://vidyamanohar.blogspot.com/2012/02/let-denotes-coefficient-of-in-expansion.html' title='Puzzle 28 - Integration'/><author><name>Vidyamanohar</name><uri>http://www.blogger.com/profile/14537585081760370998</uri><email>noreply@blogger.com</email><gd:image rel='http://schemas.google.com/g/2005#thumbnail' width='16' height='16' src='https://img1.blogblog.com/img/b16-rounded.gif'/></author><thr:total>1</thr:total></entry><entry><id>tag:blogger.com,1999:blog-32747608.post-4835372386132050831</id><published>2012-02-13T08:58:00.000+05:30</published><updated>2012-02-13T09:01:41.880+05:30</updated><title type='text'>Puzzle 27 - Determinant</title><content type='html'>&lt;div dir=&quot;ltr&quot; style=&quot;text-align: left;&quot; trbidi=&quot;on&quot;&gt;
  53. Define the function &lt;a href=&quot;http://www.codecogs.com/eqnedit.php?latex=f%28x%29%20=%20%28%7Bp_1%7D%20-%20x%29%28%7Bp_2%7D%20-%20x%29%28%7Bp_3%7D%20-%20x%29...%28%7Bp_n%7D%20-%20x%29;&quot; target=&quot;_blank&quot;&gt;&lt;img src=&quot;http://latex.codecogs.com/gif.latex?f%28x%29%20=%20%28%7Bp_1%7D%20-%20x%29%28%7Bp_2%7D%20-%20x%29%28%7Bp_3%7D%20-%20x%29...%28%7Bp_n%7D%20-%20x%29;&quot; title=&quot;f(x) = ({p_1} - x)({p_2} - x)({p_3} - x)...({p_n} - x);&quot; /&gt;&lt;/a&gt;&lt;a href=&quot;http://www.codecogs.com/eqnedit.php?latex=%7Bp_i%7D%20%5Cin%20R&quot; target=&quot;_blank&quot;&gt;&lt;img src=&quot;http://latex.codecogs.com/gif.latex?%7Bp_i%7D%20%5Cin%20R&quot; title=&quot;{p_i} \in R&quot; /&gt;&lt;/a&gt;.  Let a and b are two unequal real numbers.  Let M is an n×n matrix with all a’s above the diagonal and b’s below the diagonal and &lt;a href=&quot;http://www.codecogs.com/eqnedit.php?latex=%7Bp_1%7D,%5C,%5C,%7Bp_2%7D,%5C,%5C,%7Bp_3%7D,%5C,...%7Bp_n%7D&quot; target=&quot;_blank&quot;&gt;&lt;img src=&quot;http://latex.codecogs.com/gif.latex?%7Bp_1%7D,%5C,%5C,%7Bp_2%7D,%5C,%5C,%7Bp_3%7D,%5C,...%7Bp_n%7D&quot; title=&quot;{p_1},\,\,{p_2},\,\,{p_3},\,...{p_n}&quot; /&gt;&lt;/a&gt; along the diagonal.&lt;br /&gt;
  54. Show that the determinant of M is given by &lt;a href=&quot;http://www.codecogs.com/eqnedit.php?latex=%5Cfrac%7B%7Bbf%28a%29%20-%20af%28b%29%7D%7D%7B%7Bb%20-%20a%7D%7D&quot; target=&quot;_blank&quot;&gt;&lt;img src=&quot;http://latex.codecogs.com/gif.latex?%5Cfrac%7B%7Bbf%28a%29%20-%20af%28b%29%7D%7D%7B%7Bb%20-%20a%7D%7D&quot; title=&quot;\frac{{bf(a) - af(b)}}{{b - a}}&quot; /&gt;&lt;/a&gt; &lt;/div&gt;</content><link rel='replies' type='application/atom+xml' href='http://vidyamanohar.blogspot.com/feeds/4835372386132050831/comments/default' title='Post Comments'/><link rel='replies' type='text/html' href='http://www.blogger.com/comment.g?blogID=32747608&amp;postID=4835372386132050831' title='2 Comments'/><link rel='edit' type='application/atom+xml' href='http://www.blogger.com/feeds/32747608/posts/default/4835372386132050831'/><link rel='self' type='application/atom+xml' href='http://www.blogger.com/feeds/32747608/posts/default/4835372386132050831'/><link rel='alternate' type='text/html' href='http://vidyamanohar.blogspot.com/2012/02/puzzle-27-determinant.html' title='Puzzle 27 - Determinant'/><author><name>Vidyamanohar</name><uri>http://www.blogger.com/profile/14537585081760370998</uri><email>noreply@blogger.com</email><gd:image rel='http://schemas.google.com/g/2005#thumbnail' width='16' height='16' src='https://img1.blogblog.com/img/b16-rounded.gif'/></author><thr:total>2</thr:total></entry><entry><id>tag:blogger.com,1999:blog-32747608.post-6169461715881874760</id><published>2012-01-23T12:00:00.000+05:30</published><updated>2012-01-23T12:00:53.687+05:30</updated><title type='text'>Puzzle 26</title><content type='html'>&lt;div dir=&quot;ltr&quot; style=&quot;text-align: left;&quot; trbidi=&quot;on&quot;&gt;
  55. &lt;br /&gt;
  56. &lt;div class=&quot;MsoNormal&quot;&gt;
  57. &lt;span style=&quot;line-height: 115%; mso-bidi-font-size: 12.0pt; mso-no-proof: yes;&quot;&gt;&lt;span style=&quot;font-family: Verdana, sans-serif;&quot;&gt;Consider the three digit numbers 301, 431, 602, 715, 842,
  58. 856, 973, 986&lt;o:p&gt;&lt;/o:p&gt;&lt;/span&gt;&lt;/span&gt;&lt;/div&gt;
  59. &lt;div class=&quot;MsoNormal&quot;&gt;
  60. &lt;span style=&quot;line-height: 115%; mso-bidi-font-size: 12.0pt; mso-no-proof: yes;&quot;&gt;&lt;span style=&quot;font-family: Verdana, sans-serif;&quot;&gt;They ALL satisfy the following properties:&lt;o:p&gt;&lt;/o:p&gt;&lt;/span&gt;&lt;/span&gt;&lt;/div&gt;
  61. &lt;div class=&quot;MsoNormal&quot;&gt;
  62. &lt;span style=&quot;line-height: 115%; mso-bidi-font-size: 12.0pt; mso-no-proof: yes;&quot;&gt;&lt;span style=&quot;font-family: Verdana, sans-serif;&quot;&gt;1)&amp;nbsp; All &amp;nbsp;the digits are different&lt;o:p&gt;&lt;/o:p&gt;&lt;/span&gt;&lt;/span&gt;&lt;/div&gt;
  63. &lt;div class=&quot;MsoNormal&quot;&gt;
  64. &lt;span style=&quot;line-height: 115%; mso-bidi-font-size: 12.0pt; mso-no-proof: yes;&quot;&gt;&lt;span style=&quot;font-family: Verdana, sans-serif;&quot;&gt;2)&amp;nbsp; When the digits are
  65. placed along a cirlce, each digit is three times the previous digit, or one
  66. unit more, or two units more (while doing this, keep in mind that if this
  67. number comes out to be more than 9, only the units digit is taken).&amp;nbsp; Hence 7 is followed by 1, 2 or 3.&amp;nbsp; 3 is followed by 9, 0 or 1.&amp;nbsp; If the number ends with the digit 8, then the
  68. first digit can be 4, 5 or 6.&lt;o:p&gt;&lt;/o:p&gt;&lt;/span&gt;&lt;/span&gt;&lt;/div&gt;
  69. &lt;div class=&quot;MsoNormal&quot;&gt;
  70. &lt;span style=&quot;line-height: 115%; mso-bidi-font-size: 12.0pt; mso-no-proof: yes;&quot;&gt;&lt;span style=&quot;font-family: Verdana, sans-serif;&quot;&gt;Not only the above listed numbers, their cyclic permutations
  71. also satisfy the above properties.&lt;o:p&gt;&lt;/o:p&gt;&lt;/span&gt;&lt;/span&gt;&lt;/div&gt;
  72. &lt;div class=&quot;MsoNormal&quot;&gt;
  73. &lt;span style=&quot;line-height: 115%; mso-bidi-font-size: 12.0pt; mso-no-proof: yes;&quot;&gt;&lt;span style=&quot;font-family: Verdana, sans-serif;&quot;&gt;Hence the above list gives 22 three digit numbers.&amp;nbsp; We know, no number starts with a zero.&lt;o:p&gt;&lt;/o:p&gt;&lt;/span&gt;&lt;/span&gt;&lt;/div&gt;
  74. &lt;div class=&quot;MsoNormal&quot;&gt;
  75. &lt;span style=&quot;line-height: 115%; mso-bidi-font-size: 12.0pt; mso-no-proof: yes;&quot;&gt;&lt;span style=&quot;font-family: Verdana, sans-serif;&quot;&gt;Now, try for four digited numbers that satisfy the above
  76. properties.&amp;nbsp; How many such numbers are
  77. there, if we consider their cyclic permutations also as different number?&lt;/span&gt;&lt;o:p&gt;&lt;/o:p&gt;&lt;/span&gt;&lt;/div&gt;
  78. &lt;/div&gt;</content><link rel='replies' type='application/atom+xml' href='http://vidyamanohar.blogspot.com/feeds/6169461715881874760/comments/default' title='Post Comments'/><link rel='replies' type='text/html' href='http://www.blogger.com/comment.g?blogID=32747608&amp;postID=6169461715881874760' title='2 Comments'/><link rel='edit' type='application/atom+xml' href='http://www.blogger.com/feeds/32747608/posts/default/6169461715881874760'/><link rel='self' type='application/atom+xml' href='http://www.blogger.com/feeds/32747608/posts/default/6169461715881874760'/><link rel='alternate' type='text/html' href='http://vidyamanohar.blogspot.com/2012/01/puzzle-26.html' title='Puzzle 26'/><author><name>Vidyamanohar</name><uri>http://www.blogger.com/profile/14537585081760370998</uri><email>noreply@blogger.com</email><gd:image rel='http://schemas.google.com/g/2005#thumbnail' width='16' height='16' src='https://img1.blogblog.com/img/b16-rounded.gif'/></author><thr:total>2</thr:total></entry></feed>

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