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Let [x] denote the greatest integer less than or equal to x and {x} = x-[x] (commonly known as fractional part of x). Find all continuous functions f such that {f(x+y)} ={f(x)}+{f(y)} ...
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Is it possible to find a continuous function defined over all real numbers such that its graph intersects any non-vertical line in infinitely many points? ...
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P(x) is a quadratic such that (1) its leading coefficient is 1 (2) P(x) and P(P(P(x))) share a root. Then prove that P(0)*P(1)=0 ...
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Dear friends The contest session of IMO 2011 starts tomorrow Lets root for our team. You can check them out http://official.imo2011.nl/year_reg_team.aspx?year=2011&code=IND. Its great to see a girl member, Mrudul Thatte on th ...
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f is a real-valued differentiable function defined on [0,1]. If f(0)=0 and f(1)=1, then prove that there exist distinct points x1,x2 in [0,1] satisfying 1/f'(x1) + 1/f'(x2) = 2 ...
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Let S = (z_1,z_2,...,z_{2010}) We construct for each z_i \in S the set S_i = (z_iz_1,z_i,z_2,...,z_iz_{2010}) If it is true that S_i = S for 1 \le i \le 2010 then prove that (1) |z_i| =1 (2) z \in S \Rightarrow \overline{z} \ ...
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Find all functions f: \mathbb{R}^+ \rightarrow \mathbb{R} , such that whenever x,y \in \mathbb{R}^+ , f(x+y) = f(x^2+y^2) [Note, no conditions of continuity, differentiability are given] ...
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Prove that \sqrt [44] {\tan 1^{\circ} \tan 2^{\circ}...\tan 44^{\circ}} < \sqrt 2 -1 <\frac{\tan 1^{\circ} + \tan 2^{\circ}+...+\tan 44^{\circ}}{44} ...
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Some days ago one student there posted a prob to be solved http://www.goiit.com/posts/list/differential-calculus-inequality-1042884.htm I (Hari Shankar there) foolishly got tempted to post a solution, perhaps out of some misp ...
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George Bernard Dantzig (November 8, 1914 – May 13, 2005) was an American mathematician, and the Professor Emeritus of Transportation Sciences and Professor of Operations Research and of Computer Science at Stanford. Dantzig ...
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IMO 2010 is currently underway at Kazakhstan. The actual competition will be on 8th-9th as far as I can make out. http://michaelnielsen.org/polymath1/index.php?title=Imo_2010 will feature a joint effort at solving the toughes ...
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A fun result. Its more suited for 11th students A sequence is as follows: 1,2,3,4,5,10,20,40,80,160,... i.e. the 1st 5 nos. in AP and then onwards a GP. Prove that every natural number can be obtained as sums of distinct term ...
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Let a_0 = a_1 = 1 and a_{n+1} = 1 + \frac{a_1^2}{a_0} + \frac{a_2^2}{a_1} + ...+ \frac{a_n^2}{a_{n-1}} Find a closed formula for an (i.e. express an as a function of n) ...
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Mr and Mrs. Bajaj invite three other couples to a party and as usual some people shake hands. Spouses do not shake hands with each other. At the end of introductions, Mr. Bajaj asks all others including Mrs. Bajaj how many ti ...
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Saw this in some coaching institute material. Its easy, but their method seemed tedious. Lets see how targetiitians fare: x,y \in \mathbb{R} and x+iy = \frac{3}{\cos \theta + i \sin \theta+2} Evaluate 4x-x^2-y^2 ...
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http://www.openthemagazine.com/article/nation/where-are-they-now is a nice article that tracks what many JEE AIR 1's have gone on to do after BTech at the IITs ...
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Denote by Sk the sum to k terms of an AP. If for some two indices m and n it is known that Sm=Sn, for what index i, does Si attain a real extremum. ...
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A recurrence satisfies a_{n+1} = a_n + \sqrt{a_n+a_{n+1}} a1 =1 Obtain an explicit formula for an i.e. find a function f(n) such that an = f(n) edited: first term provided now, sry. ...
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8 pieces are placed on a chessboard such that no two pieces lie on the same row or column. Prove that an even number of pieces lie on black squares ...
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This is fun to work out. Let \tau(m) represent the number of divisors of the natural number m. Then prove that \tau(1)+\tau(1)+ ...+\tau(n) = \left[\frac{n}{1} \right] + \left[\frac{n}{2} \right]+...+\left[\frac{n}{n} \right] ...
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Find all roots of the polynomial 4x^6 - 6x^2+2\sqrt 2 = 0 ...
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Consider the polynomials f(x) =a_1 +a_2x+a_3x^2+a_4x^4 and g(x) =b_1 +b_2x+b_3x^2+b_4x^4 where all coefficients are real It is known that \forall \ x \in \mathbb{R} \ \ [f(x)] = [g(x)] Is it necessary that f(x) = g(x)? ...
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Could anyone post the questions please? I am told it was held today. ...
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If xy+yz+zx = 3, prove that \sqrt{1+x^4} + \sqrt {1+y^4} + \sqrt {1+z^4} \ge 3 \sqrt 2 ...
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Prove that (\sin \gamma + a \cos \gamma)(\sin \gamma + b \cos \gamma) \le 1 + \left( \frac{a+b}{2} \right)^2 for reals a,b ...
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The set of integers is divided into n infinite APs. If the common differences of the APs are denoted by di, find \sum_{i=1}^n \frac{1}{d_i} ...
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Find the number of roots of the equation z^7+4z^2+11=0 satisfying 1<|z|<2 ...
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Good prob flicked from another forum: Given complex numbers a,b,c,d such that \frac{a-d}{b-c} and \frac{b-d}{c-a} are purely imaginary, prove that \frac{c-d}{a-b} is also purely imaginary ...
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f is a continuous function that maps the closed unit interval I = [0,1] into itself. Prove that if f(f(x)) = x for all x ε I, then f is monotonic ...
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Prove that among the +ve numbers a,2a,3a,...,(n-1)a, there is one that differs by an integer by at most 1/n ...
