# TÀI LIỆU TOÁN - WWW.MOLYMPIAD.NET - ĐỀ THI TOÁN

## $show=home 1. Let$x,y$and$z$be integers satisfying$x\ne y$and$\frac{x}{y}=\frac{x^{2}+z^{2}}{y^{2}+z^{2}}$. Prove that$x^{2}+y^{2}+z^{2}$is not a prime number. 2. Given a triangle$ABC$with$AB<AC$. Let$AD$be the angle bisector. Let$E$and$F$respectivly be points on the sides$AB$and$AC$such that$BE=CF$. If$G$and$H$respectively are the midpoints of$BF$and$CE$. Prove that$GH\perp AD$. 3. Given$3$different positive integers$a,b$and$c$satisfying$ab+bc+ca\geq674$. Find the minimum values of the expression $$P=\frac{a^{3}+b^{3}+c^{3}}{3}-abc.$$ 4. Given an acute triangle$ABC$with the altitude$AH$. Let$E\in AB$and$F\in AC$such that$HE$is perpendicular to$AB$and$HF$is perpendicular to$AC$. Assume that$HE$and$BF$intersect at$K$and$HF$and$CE$intersect at$G$. Choose$M$and$N$respectively on the line segments$HB$and$HC$such that$EMHK$and$FNHK$are inscribed quadrilaterals. Prove that$KN=GM$. 5. Solve the system of equations $$x^{5}+y^{3}=2z$$ $$y^{5}+z^{3}=2x$$ $$z^{5}+x^{3}=2y$$ 6. Let$a,b$and$c$be nonnegative numbers such that$a+b+c\leq\pi$. Prove that $$0\leq a-\sin a-\sin b-\sin c+\sin\left(a+b\right)+\sin\left(a+c\right)\leq\pi.$$ 7. Let$a,b$and$c$be three sides of a triangle. Prove that $$\frac{3}{2}<\sqrt{\frac{a}{b+c}}+\sqrt{\frac{b}{c+a}}+\sqrt{\frac{c}{a+b}}<\frac{4\pi}{5}.$$ 8. Given a triangle$ABC$with the altitude$AH$($H\in BC$). Let$\left(I\right)$be the circle with the diameter$AH$and the center$I$. The circle$\left(I\right)$intersects$AB$and$AC$respectively at$T$and$S$. The tangent of$\left(I\right)$at$T$and$S$intersect at$M$. Let$N$,$K$,$J$respectively be the intersections of$BC$and$MA$,$BS$and$CT$,$TS$and$IM$. Prove that$NJ$goes through the midpoint of$AK$. 9. Given$a_{1},a_{2},\ldots a_{n}\in\mathbb{R}$,$n\in\mathbb{N}$,$n\geq3$. Find $$\max\left(\min\left\{ \frac{a_{1}+a_{2}}{1+a_{3}^{2}};\frac{a_{2}+a_{3}}{1+a_{3}^{2}};\ldots;\frac{a_{n-2}+a_{n-1}}{1+a_{n}^{2}}; \frac{a_{n-1}+a_{n}}{1+a_{1}^{2}};\frac{a_{n}+a_{1}}{1+a_{2}^{2}}\right\} \right).$$ 10. Let$n\geq3$be a natural number. Prove that$\dfrac{(3n)!}{n!(n+1)!(n+2)!}$is also a natural number. 11. Given two sequences$\left(x_{n}\right)$and$\left(y_{n}\right)$such that $$\begin{cases} \sqrt{y_{n}-x_{n+1}^{2}+1} & =x_{n}\\ \sqrt{3+x_{n+1}^{2}-y_{n}}+\sqrt{10+y_{n-1}-x_{n-1}^{2}} & =\dfrac{5n-1}{n} \end{cases}\quad\forall n\in\mathbb{N}^{*}.$$ Prove that$\left(x_{n}\right)$and$\left(y_{n}\right)$converge. Find the limits. 12. Given an acute triangle$ABC$which is not isosceles and is inscribed in a circle$\left(O\right)$. Let$P$be a point inside the triangle such that$AP\perp BC$. Let$E$and$F$respectivly be the circumcircle of the triangle$AEF$intersects$\left(O\right)$at another point$G$besides$A$. Prove that$GP$,$BE$and$CF$are concurrent. ##$type=three$c=3$source=random$title=oot$p=1$h=1$m=hide$rm=hide ## Anniversary_$type=three$c=12$title=oot$h=1$m=hide\$rm=hide

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Mathematics & Youth: 2016 Issue 466
2016 Issue 466
Mathematics & Youth