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Find the non-zero \(x\) satisfying \[\large(2- \sqrt{3})^x+(7-4 \sqrt{3})(2+ \sqrt{3})^x=4(2- \sqrt{3}).\]

Given \(\triangle_{ABC}\). \(I\) is the incenter of \(\triangle_{ABC}\), \(D\);\(E\) are the mipoints of \(AC\);\(AB\),respectively. \(DI\) intersects \(AB\) at \(Q\) and \(EI\) intersects \(AC\) at ...

\(\triangle_{ABC}\) is given that \[3 \times \angle ABC+2 \times \angle BCA=180^{o}\] And three sides of this triangle are 3 consecutive integers. Find the length of these ...

Determine all pairs \((x, y)\) of integers satisfying the equation:\[1+2^{x}+2^{2x+1}=y^{2}\] If your answers are ...

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