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Let " x " and " y " be positive integers such that " xy " divides " x 2 + y 2 + 1 " . Prove that - x 2 + y 2 + 1/xy = 3 . I will add some more if I see that people are taking interest in this . ...
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1 . What is a " convex " function ? 2 . Prove that " y = x 2 " is a convex function . 3 . Out of 15 persons , 6 are girls while 9 are boys . We want the boys to form a line with increasing heights , and also the girls , but t ...
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1 . Solve - x 3 - 3 x = x + 2 ...
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Prove that for any " x , y > 0 " , x x + y y ≥ x y + y x ...
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3 . *Image* " a r , b r and c r" are three sequences of real numbers . ...
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Prove the following two inequalities : - 1 . *Image* Please note that " a r and b r " are two sequences of real numbers . 2 . *Image* Also infer when the two equalities hold . Note : - The first problem has got many solutions ...
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Suppose we construct a plane such that it only contains points whose ordinates as well the abscissae are integers . Each point of this plane is labeled by a positive integer . Each of these numbers is - 1 > The arithmetic ...
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The mass density per unit area of each circle is " σ " , and all the surfaces are frictionless . It is also given that the radius of each circle is " R " . Find the force required to keep the two bottommost circles together ...
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Evaluate the following expression : - *Image* ...
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Calculate the value of " L " as depicted below - *Image* Hint : - It is not as big as looks . ...
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A helicopter can hover when the power output of its engine is " P " . A second helicopter is an exact copy of the first one, but its linear dimensions are half those of the original. What power output is needed to enable this ...
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1 . *Image* A particle of mass " m " rests on a straight groove along which it is constrained to move . The mass is attached to a vertical rubber band suspended froma rigid support . Initially , the band is of its natural len ...
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1 . *Image* A plane monochromatic light wave of intesity I 0 falls normally on the surfaces of the opaque screens in the figure here . Find the intensity of light at the behind the blackened points of the screens . ...
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Last question of old NCERT : Chapter - EMI A line charge " λ " per unit length is lodged uniformly onto the rim of a wheel of mass " M " and radius " R " . The wheel has light non - conducting spokes and is free to rotate wi ...
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1 > Prove that if " an " is a sequence of real numbers such that , " lim ( n → ∞ ) [ an ] = x " then , " lim ( n → ∞ ) [ a1 + a2 + ...........an/n ] = x " also . Source - Michael Spivak ...
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Let " a " and " b " be real numbers in the interval [ 0 , Î / 2 ] . Then for how many integral valued pairs of " a " and " b " must be there who satisfy .......... sin 6 ( a ) + 3 sin 2 ( a ) cos 2 ( b ) + cos 6 ( b ) = 1 Not ...
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This is a part of my initiative - Not A Doubt : A snowball is thrown against a wall . Where does its momentum go ? Where does its energy go ? Explain clearly . ...
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Since our exams are approaching fast , I have taken an initiative to post and solve problems that I have found to be quite intriguing . These are all from IIT - JEE syllabus , and I hope to create a field of mutual participat ...
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Since I am posting after a long time , I am starting with a very elemental problem which is in fact incorrectly solved in Arihant ............... But I loved solving it ........... Find the number of 2 numbers " x " and " y " ...
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These are not doubts - I am just looking for simpler ways to solve these :- 1 > Given two plain - parallel electrodes , at a distance " d " , and kept at voltages " 0 " and " V0 " , find the magnetic field per unit length ...
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1 > Solve the following equation for " x " - 16 x 6 - 24 x 4 + 10 x 2 - 1 = 1/8 x 2 > The positive reals " x " , " y " , " z " satisfy the following equation - x 2 + x y + y 2/3 = 25 y 2/3 + z 2 = 9 z 2 + z x + x 2 = 16 ...
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A system consists of a cylindrical anode of radius B and a coaxial cylindrical cathode of radius A . A filament located along the axis of the system carries a heating current I . Find the least potential difference between th ...
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Start with the set S - { 3 , 4 , 12 } . In each step , you may chose two numbers " a " and " b " from S and replace " ( . 6 a - . 8 b) " by " ( . 8 a + . 6 b ) " . Can you form a set P - { 4 , 6 , 12 } by finitely many steps ...
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1 . Does there exist a function " f " defined on the set of positive real numbers which takes only positive real values , and satisfies , i > f ( x f ( y ) ) = y f ( x ) ii > f ( x ) tends to zero as " x " approaches in ...
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