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

## $show=home 1. Compare$A$and$B$where $$A=\frac{1}{2!}+\frac{2}{3!}+\frac{3}{4!}+\ldots+\frac{2015}{2016!}$$ (with$n!$is the usual notation for the product$1,2\ldots,n$) and$B=1,02015$. 2. Given 5 line segments where any 3 of them can form a triangle. Prove that there is at least one acute triangle among the ones which can be formed bt any 3 line segments. 3. Suppose that$a,b$and$c$are positive numbers and$n$is an integer which is greater or equal to 2. Prove that $$\sqrt[n]{\frac{a}{a+nb}}+\sqrt[n]{\frac{b}{b+nc}}+\sqrt[n]{\frac{c}{c+na}}>1.$$ 4. Given a rhombus$ABCD$and let the length of the side$AB$be$2a$. Let$R_{1},R_{2}$respectively be the circumradius of the triangles$ABC$and$ABD$. Prove that$R_{1}R_{2}\geq2a^{2}$. When do we have the equality?. 5. Find integral solutions of the following equation $$\frac{11x}{5}-\sqrt{2x+1}=3y-\sqrt{4y-1}+2.$$ 6. Given a system of equations $$\begin{cases} x\sqrt{y-m}+y\sqrt{z-m}+z\sqrt{x-m} & =6m\sqrt{m},\\ x^{2}+y^{2}+z^{2} & =12m^{2}, \end{cases}$$ where$m$is a positive number. Find$x,y$and$z$in terms$m$. 7. Find the maximum and minimum of the function $$y=f\left(x\right)=\cos x+4\cos\frac{x}{2}+7\cos\frac{x}{4}+6\cos\frac{x}{8}.$$ 8. Given a triangle$ABC$. Let$M$be the midpoint of$BC$. Prove that if$\widehat{BAC}\leq\frac{\pi}{2}$then $$2AM\leq\sqrt{2\left(AB^{2}+AC^{2}\right)}\cos\frac{\widehat{BAC}}{2}.$$ When do we have the equality?. 9. Let$x,y$and$z$be positive numbers such that$xyz=1$. Prove that $$x+y+z+xy+yz+zx\leq 3+\left(\frac{x}{y}\right)^{n}+\left(\frac{y}{z}\right)^{n}+\left(\frac{z}{x}\right)^{n}$$ for every$n\in\mathbb{N}^{*}$. 10. Find the largest$n$for which there exist$n$different positive numbers$x_{1},x_{2},\ldots,x_{n}$such that $$\frac{x_{i}}{x_{j}}+\frac{x_{j}}{x_{i}}+8\left(\sqrt{3}-2\right)\geq\left(7-4\sqrt{3}\right)\left(\frac{1}{x_{i}x_{i}}+x_{i}x_{j}\right)$$ for any two different indices$i,j$. 11. Find all functions$f:\mathbb{R}^{+}\to\mathbb{R}$satisfying $$f\left(\frac{x+y}{2016}\right)=\frac{f\left(x\right)+f\left(y\right)}{2015}$$ for all$x,y\in\mathbb{R}^{+}$. 12. Two circles$\left(O\right)$and$\left(O_{1}\right)$intersect at$B$and$C$, Let$M$be the midpoint of$BC$. Let$A$be a point which varies on$\left(O\right)$but is different from$B$and$C$. The lines$AB$and$CA$respectively intersect$\left(O_{1}\right)$at$F$and$E$. Assume that$P$and$Q$respectively be the perpendicular projections of$M$on$BE$and$CF$. Construct the parallelogram$MPKQ$. Prove that$AK$always goes through a fixed point. ##$type=three$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 469
2016 Issue 469
Mathematics & Youth