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

## $show=home 1. Compare the following numbers $$A=\frac{1}{5}+\frac{2}{5^{2}}+\frac{3}{5^{3}}+\cdots+\frac{2018}{5^{2018}} ; \quad B=\frac{2018}{2019}$$ 2. Suppose that$P$is a point inside a triangle$A B C$so that$\widehat{P B C}=30^{\circ}$,$\widehat{P B A}=8^{\circ}$and$\widehat{P A B}=\widehat{P A C}=22^{\circ}$. Find the value of the angle$\widehat{A P C}$. 3. Find positive solutions of the equation $$\frac{1}{5 x^{2}-x+3}+\frac{1}{5 x^{2}+x+7} +\frac{1}{5 x^{2}+3 x+13}+\frac{1}{5 x^{2}+5 x+21}=\frac{4}{x^{2}+6 x+5}.$$ 4. Given a square$A B C D$with the length of a side$a$. On the sides$A D$and$C D$respectively choose two points$M$,$N$so that$M D+D N=a$. Let$E$be the intersection of two lines$B N$and$A D$. Let$F$be the intersection of two lines$B M$and$C D$. Show that $$M E^{2}-N E^{2}+N F^{2}-M F^{2}=2 a^{2}.$$ 5. Given positive numbers$a, b, c$Find the minimum value of the expression $$P=\frac{a}{\sqrt{a}+\sqrt{b c}}+\frac{b}{\sqrt{b}+\sqrt{c a}}+\frac{c}{\sqrt{c}+\sqrt{a b}} + \frac{9 \sqrt{(a+1)(b+1)(c+1)}}{4(a+b+c)}.$$ 6. Let$a, b, c, d$be positive numbers such that $$\frac{1}{a b}+\frac{1}{b c}+\frac{1}{c d}+\frac{1}{a d}=1 .$$ Show that $$\frac{a b c d}{8}+2 \geq \sqrt{(a+c)\left(\frac{1}{a}+\frac{1}{c}\right)}+\sqrt{(b+d)\left(\frac{1}{b}+\frac{1}{d}\right)}.$$ 7. Find real solutions of the following system of equations $$\begin{cases} x^{3}+2 y^{3} &=2 x^{2}+z^{2} \\ 2 x^{3}+3 x^{2} &=3 y^{3}+2 z^{2}+7 \\ x^{3}+x^{2}+y^{2}+2 x y &=2 x z+2 y z+2\end{cases}$$ 8. Given a triangle$A B C$inscribed in a circle$(O)$. Assume that$B O$and$C O$intersect the altitude$A D$of the triangle respectively at$E$and$F$. Let$I$and$J$respectively be the centers of the circles$(A C F)$and$(A B E)$. Two points$K$,$H$are on$A B$,$A C$respectively so that$J K \parallel A O \parallel  I H$. Suppose that$I J$intersects$A B$and$A C$at$M$and$N$. Show that the intersection between$M H$and$N K$is on the midsegment, which is opposite to the vertex$A$, of the triangle$A B C$. 9. Solve the equation $$8^{x}+27^{\frac{1}{x}}+2^{x+1} \cdot 3^{\frac{x+1}{x}}+2^{x} \cdot 3^{\frac{2 x+1}{x}}=125.$$ 10. Let$[x]$be the maximal integer which does not exceed$x$and let$\{x\}=x-[x]$. Consider the sequence$\left(u_{n}\right)$with $$u_{n}=\left\{\frac{2^{2 n+1}+n^{2}+n+2}{2^{n+1}+2}\right\}.$$ Find the number of terms of the sequence$\left(u_{n}\right)$satisfying $$\frac{2526.2^{n-99}}{2^{n}+1} \leq u_{n} \leq \frac{23}{65}.$$ 11. Find all functions$f: \mathbb{N}^{*} \rightarrow \mathbb{R} \backslash\{0\}$so that $$f(1)+f(2)+\cdots+f(n)=\frac{f(n) f(n+1)}{2}, \forall n \in \mathbb{N}^{*}.$$ 12. Given a triangle$A B C$with$A B+A C=2 B C$. Let$I_{a}$be the center of the excircle corresponding to the angle$A$. The circle$\left(A, A I_{a}\right)$intersects$B C$at$E$and$F$with$E$is on the ray$C B$and$F$is on the ray$B C$. The circle$\left(E B I_{\alpha}\right)$meets$A B$at$M$and the circle$\left(F C I_{\alpha}\right)$meets$A C$at$N$. Show that$B C N M$is both a cyclic and tangential quadrilateral. ##$type=three$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: 2019 Issue 509
2019 Issue 509
Mathematics & Youth