Prove that \(\dfrac{n}{\sqrt{n-1}}\ge2\) with \(n>1\)

Prove that P\(=\dfrac{1}{2^2}+\dfrac{1}{3^2}+.......+\dfrac{1}{2013^2}\) isn't a integer

Prove that A=(x+y)(x+2y)(x+3y)(x+4y)+y⁴ is a square of a number

Give S=\(\dfrac{1}{101}+\dfrac{1}{102}+......+\dfrac{1}{200}\).Prove that:\(S>\dfrac{7}{12}\)

Find the maximum of expression \(Q=\dfrac{3\left(x+1\right)}{x^3+x^2+x+1}\)

Solve the equation:\(\left\{{}\begin{matrix}x^3-6x^2+2y^2+15x+5y=21\\x^2+y^2+xy-7x-6y+14=0\end{matrix}\right.\)

Give a,b,c\(\ge0\) and ab+bc+ca=1.Find the minimum of expression P=\(10a^2+10b^2+c^2\)

Solve the equation:\(\sqrt{x+7}-\sqrt{x-82}=x-2017\)

Let ABC be a right with hypotenuse BC.An altitude from A is drawn,intersecting the opposite side BC at H such that AC=8 cm,BH=\(\dfrac{18}{5}\) cm.Find the area of ABC

Find x is a integer for A=x³+9x²+23x+15 \(⋮\)16

Give a,b>0 for a+b\(\le\dfrac{4}{5}\).Prove that:\(a+b+\dfrac{a+b}{ab}\ge\dfrac{29}{5}\)

Find x,y,z when know \(\dfrac{x^2}{2}+\dfrac{y^2}{3}+\dfrac{z^2}{4}=\dfrac{x^2+y^2+z^2}{5}\)

Give ABC triangle has \(\angle A=2\angle B=4\angle C\).Prove that:\(\dfrac{1}{BC}+\dfrac{1}{AC}=\dfrac{1}{AB}\)

Solve the equation:\(\dfrac{x-a}{bc}+\dfrac{x-b}{ca}+\dfrac{x-c}{ab}=2\left(\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}\right)\) with a,b,c are parameters

Let a,b,c\(\in\left[-1;4\right]\) for a+2b+3c\(\le4\).Prove that:\(a^2+2b^2+3c^2\le36\)

Let a,b,c>0 .Prove that:\(\dfrac{a}{3a^2+2b^2+c^2}+\dfrac{b}{3b^2+2c^2+a^2}+\dfrac{c}{3c^2+2a^2+b^2}\le\dfrac{1}{6}.\left(\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}\right)\)

Find x,y are integer fỏ 2x²+3y²+4x=19

Find a,b for \(a^2+2b^2=2011\)

Prove that:  \(a^3+b^3⋮6\) when \(a+b⋮6\)

Prove that:1³+2³+..............+2016³=n² with n\(\in N\)*.Then find n