Level 3

Let $P\left(x\right)=a_0+a_{1}x+a_2{x}^2+\dots +a_{\mathit{n}}{x}^{\mathit{n}}$ be a polynomial in which $a_{\mathit{i}}$ is a non-negative integer for each $i\in \left\{0,1,2,3,\dots ,n\right\}$. If $P\left(1\right)=4$ and $P\left(5\right)=136$, what is the value of $P\left(3\right)$?

Let $a$ and $b$ be positive real numbers such that $\frac{1}{a}-\frac{1}{b}-\frac{1}{a+b}=0$. Find the value of ${\left(\frac{b}{a}+\frac{a}{b}\right)}^2$ .

Find the sum of all the real numbers $x$ that satisfy the equation ${\left(3^x-27\right)}^2+{\left(5^x-625\right)}^2={\left(3^x+5^x-652\right)}^2$.

Let $x,y$ and $z$ be three real numbers such that $xy+yz+xz=4$. Find the least possible value of $x^2+y^2+z^2$.

If $x, y,$ and $z$ are positive numbers satisfying $x+\frac{1}{y}=4, y+\frac{1}{z}=1$ and $z+\frac{1}{x}=\frac{7}{3}$, then find $xyz?$

If $x$ is any real number, then $\frac{x^2}{1+x^4}$ belongs to which one of the following intervals?

If $\alpha , \beta$ are the roots of $x^2+px+q=0$ and $x^{2n}+p^n{x}^n+q^n=0$ and if $\left(\frac{\alpha }{\beta }\right),\left(\frac{\beta }{\alpha }\right)$ are the roots of $x^n+1+{\left(x+1\right)}^n=0,$ then $n\left(\in N\right)$

Let $r, s$ and $t$ be the roots of the equation, $8x^3+1001x+2008=0.$ The value of ${\left(r+s\right)}^3+{\left(s+t\right)}^3+{\left(t+r\right)}^3$ is

If $\alpha$ and $\beta$ are the roots of the equation $x^2-ax+b=0$ and $A_n={\alpha }^n+{\beta }^n,$ then which of the following is true?

If $\alpha$ and $\beta , \alpha$ and $\gamma , \alpha$ and $\delta$ are the roots of the equations $a{x}^2+2bx+c=0, 2b{x}^2+cx+a=0$ and $c{x}^2+ax+2b=0,$ respectively, where $a, b$ and $c$ are positive real numbers, then $\alpha +{\alpha }^2=$

If one root of the equation $t^2-\left(12x\right)t-\left(f\right(x)+64x)=0$ is twice of other, then find the maximum value of the function $f\left(x\right)$, where $x\in R$.

If $p, q, r, s\in R$, then equation $\left(x^2+px+3q\right)\left(-x^2+rx+q\right)\left(-x^2+sx-2q\right)=0$ has

Let $\mathrm{p}$ and $\mathrm{q}$ be real numbers such that $\mathrm{p}\ne 0, {\mathrm{p}}^3\ne \mathrm{q}$ and ${\mathrm{p}}^3\ne -\mathrm{q}$. If $\mathrm{\alpha }$ and $\mathrm{\beta }$ are non zero complex numbers satisfying $\mathrm{\alpha }+\mathrm{\beta }=-\mathrm{p}$ and ${\mathrm{\alpha }}^3+{\mathrm{\beta }}^3=\mathrm{q}$, then a quadratic equation having $\frac{\mathrm{\alpha }}{\mathrm{\beta }}$ and $\frac{\mathrm{\beta }}{\mathrm{\alpha }}$ as its roots is

If $\alpha ,\beta$ are the roots of the equation $a{x}^2+bx+c=0$ and $S_n={\alpha }^n+{\beta }^n,$ then $a{S}_{n+1}+b{S}_n+c{S}_{n-1}=\_\_\_\_\_\_\_\_\_(n\ge 2)$.

Number of real solutions of $\sqrt{x}+\sqrt{x-\sqrt{1-x}}=1$ is: