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Tangents drawn from the point P(3,8) to the circle x^2+y^2-6x-4y-11=0 touches the circle at the point A and B . Then equation of circumcircle of a Triangle PAB is I have a doubt:: why PO act as a diameter for Circumcenter of ...

Find all real solutions to the system of equations: x+y=\sqrt{4z-1} ...(1) y+z=\sqrt{4x-1} ...(2) z+x=\sqrt{4z-1} ...(3) ...

P is an interior point of an equilateral \Delta ABC so that *Image* , and BP and CP meet AC and AB at D and E respectively. Suppose that PB : PC = AD : AE. Find \angle BPC . ...

Find the solutions to the system of equations in reals: x+\left \lfloor y \right \rfloor+\left \left \{ z\left. \right \} \right.=1.1 \left \lfloor x \right \rfloor+\left \{ y\left. \right \} \right.+z=2.2 \left \{ x\left. \r ...

\hspace{-16}$It is known about real $\mathbf{a}$ and $\mathbf{b}$ that the inequality $\mathbf{a\; cosx +b\;cos3x >1}$\\\\ has no real solutions.then Prove that $\mathbf{\mid b\mid \leq 1}. ...

\hspace{-16}(1)\;\;\mathbf{\int \frac{1}{2x\sqrt{1-x}\sqrt{2-x+\sqrt{1-x}}}dx}$\\\\\\ $(2)\;\;\mathbf{\int\frac{1}{(x+1).\sqrt{1+x-x^2}}dx}$ ...

\hspace{-16}$If $\mathbf{\mid z^2+1 \mid =2\mid z+1\mid }$\\\\ Then find Max. value of \mathbf{z} ...

Anyone have any idea, about where usually the INPhO is held in Kolkata? Which school or Institution? Where was it held in 2010?? ...

1.Find all positive integers n, for which "2n+12n+2011n" is a perfect square. ...

1)Let a,b,c be roots of the equation x3-3x2+1=0.Find (a-2)(b-2)(c-2)? 2)Suppose f(x)=x3+px2+qx+72 is divisible by both x2+ax+b and x2+bx+a(where p,q,a,b are constants and a≠b).Find the sum of squares of the roots of the cub ...

Two mirrors at an angle θº produce 5 images of a point. The number of images produced when θ is decreased to θº – 30º is (A) 9 (B) 10 (C) 11 (D) 12 ...

Prove that \sum_{i=1}^{p}i \binom{p}{i}(n-1)^{p-i} = p n^{p-1} n is real, p is natural number ...

\hspace{-16}$Prove that If $\mathbf{n}$ is an positive Integer Then,\\\\ $\mathbf{\sum_{k=1}^{n}\cos^4\left(\frac{k\pi}{2n+1}\right)=\frac{6n-5}{16}}$ thanks guys now i have edited it ...

\hspace{-16}$Find the First Digit after the decimal point of $\mathbf{\left(2+\sqrt{5}\right)^{2010}}$\\\\ ...

\hspace{-16}$How many Integer $\mathbf{n}$ are there such that $\mathbf{n+20}$ and $\mathbf{n-20}$ are Perfect Squares\\\\ ...

1)If the range of the function f(x) = x2+ax+b/x2+2x+3 is [-5,4],a,b are natural nos,then find the value of a2 + b2 ? 2)Given,x,y belong to reals and x2 +y2 > 0.If the max and min value of the expression x2 +y2/x2+xy+4y2 ar ...

1)Number of ways 10 persons can take seats in a row of 24 fixed seats so that no two persons take consecutive seats is 2)C02 + C12 + C22 + .... + Cn2 = 3)C1+4.C2 + 7.C3 + ... + (3n-2).Cn = 4)Number of integral terms in the ex ...

1) Please do this sum without using Complex no.(Actually this sum can be solved by pure trigonometry...) 1: if theta= pi/1999 then find the value of cos(theta) x cos(2theta) x cos(3theta) x ..... cos 999(theta) ...

Evaluate: ∫01 xa-1/ln x dx ...

1)A(2,5),B(-1,3), and C(5,-1) are the vertices of a triangle. The image of the point (1,2) with respect to the median through A is 2)Equation of the circle passing through (0,0),(4,3) and (-3,4) is 3)The points (5,-4,2),(4,-3 ...

The no. of natural Numbers ≤ 2012 which are relatively prime to 2012 is : ...

If cosα + cosβ + cosγ = sinα + sinβ + sinγ, then prove that i) cos2α + cos2β + cos2γ =0 & ii) sin2α + sin2β + sin2γ =0 ...

Find the coeff. of x15 in the expansion of (1 - x)(1 - 2x)(1 - 22x).............(1 - 215x). Please help me its urgent. ...

\hspace{-16}$If $\mathbf{1\leq x<y\leq 100}$ and $\mathbf{x,y\in\mathbb{Z}}$. Then find the Probability that $\mathbf{i^x+i^y\in\mathbb{R}}$\\\\ Where $\mathbf{i=\sqrt{-1}}$ ...

if roots of the equation x2-10cx-11d=0 are a,b and those of x2-10ax-11b=0 are c,d,then find the value of (a+b+c+d)? ...

The number of distinct terms in the expansion of ( 1/xy + 1/zy + 1/xz +x+y+z)2 and ( 1/z + 1/y + 1/x +x+y+z)2 are? ...

For Ag(s) | AgCl(s) | Cl-(aq) | Br- / AgBr(s) | Ag(s) , E° cell shall be : A) - 1/38.91 log10 Ksp AgBr/Ksp AgCl A) - 1/38.91 log10 Ksp AgCl/Ksp AgBr A) - 1/38.91 loge Ksp AgBr/Ksp AgCl A) - 1/38.91 loge Ksp AgCl/Ksp AgBr My ...

Please see this question here. http://www.artofproblemsolving.com/Forum/viewtopic.php?f=296&t=454883&p=2556669#p2556669 ...

\hspace{-16}$If $\mathbf{a,b\in\mathbb{Z^{+}}}.$ Then prove that $\mathbf{\frac{a}{b}<\sqrt{2}<\frac{a+2b}{a+b}}$ ...

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