Find the max and min value of $a\cos^2 x+b\sin x\cos x+c\sin^2 x$.

If the polynomial $P(x)=ax^2+bx+c$ takes value $0$ for three different values of $x$, then prove the polynomial $P(x)$ takes value $0$ for all $x$.

Prove that : $\frac{1}{(\log_{bc} a)^n}+\frac{1}{(\log_{ac} b)^n}+\frac{1}{(\log_{bc} a)^n}\geq 3\cdot2^{n}$ where $a,b,c>1$ and $n$ is natural number.

Solve: \[ \begin{cases}x_{2}x_{3}x_{4}\cdots x_{n}=a_{1}x_{1}\\ x_{1}x_{3}x_{4}\cdots x_{n}=a_{2}x_{2}\\x_{1}x_{2}x_{4}\cdots x_{n}=a_{3}x_{3}\\ \ldots\\x_{1}x_{2}x_{3}\cdots x_{n-1}=a_{n-1}x_{n-1} \end{cases} \]

Let $O$ be the center of $\bigtriangleup ABC$'s circumcircle. $CO$ line intersect $AB$ at $D$ and $BO$ line intersect $AC$ at $E$. If $\angle A=\angle CDE=50$° then find $\angle ADE$

Prove that $1^{2011}+2^{2011}+3^{2011}+...+2012^{2011} $ is divisible by $2025078$.

If $a,b,c$ are positive real numbers and satisfy: $\frac{a_1}{b_1}=\frac{a_2}{b_2}=...=\frac{a_n}{b_n}$ then prove that :$ \sum_{i=1}^{n} a^{2}_i \cdot \sum_{i=1}^{n} b^{2}_i =(\sum_{i=1}^{n} a_{i}b_{i})^2$

Let $ABC$ be a triangle inscribed in a circle of radius $1$. If the triangle's sides are integer numbers, then find that triangle's sides.