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

## $show=home 1. Find all integers$x$,$y$,$z$which satisfy $$3 x^{2}+6 y^{2}+2 z^{2}+3 y^{2} z^{2}-18 x=6.$$ 2. Given an isosceles triangle$A B C$with the vertex angle$A$. Let$H$be the point in the interior domain determined by the angle$A$such that$H B \perp B A$,$H C \perp C A$. On the line segment$B C$we choose$M$such that$B M=\dfrac{1}{4} B C$. Let$N$be the midpoint of$A C$Calculate the angle$\widehat{H M N}$. 3. Find all pairs of integers$(x ; y)$satisfying $$y^{3}-2(x-4) y^{2}+\left(x^{2}-9 x-1\right) y+3 x^{2}+x=0.$$ 4. Given an acute triangle$A B C$and suppose that$B E$and$C F$are the two altitudes. Draw$F H$and$E K$perpendicular to$B C(H, K \in B C)$. Draw$H M$parallel to$A C$and$K N$parallel to$A B(M \in A B, N \in A C)$. Show that$E F \parallel M N$. 5. Solve the equation $$\sqrt{\frac{x-1}{x+1}}+\frac{2 x+6}{(\sqrt{x-1}+\sqrt{x+3})^{2}}=2.$$ 6. Given two positive numbers$a$and$b$such that$a<b$and$a^{b}=b^{a}$. Show that there exists a positve number$c$such that $$a=\left(1+\frac{1}{c}\right)^{c},\quad b=\left(1+\frac{1}{c}\right)^{c+1}.$$ 7. Solve the system of equations $$\begin{cases}\tan x-\tan y &=(1+\sqrt{x+y})^{y}-(1+\sqrt{x+y})^{x} \\ 3^{\sqrt{1-x}}+5^{\sqrt{1-y}} &=2(1+\sqrt{9-10 x+y})\end{cases}.$$ 8. Show that for any triangle$A B C$we always have $$\frac{(b+c) a}{m_{a}^{2}}+\frac{(c+a) b}{m_{b}^{2}}+\frac{(a+b) c}{m_{c}^{2}} \geq 8$$ where$a$,$b$,$c$,$m_{a}$,$m_{b}$,$m_{c}$respectively are the lengths of the sides$B C$,$C A$,$A B$and the corresponding medians. 9. Let$a$,$b$,$c$be positive numbers such that$a+b+c=3$. Find the minimum value of the expression $$M=\sqrt{\frac{a^{5}}{b^{4}}}+\sqrt{\frac{b^{5}}{c^{4}}}+\sqrt{\frac{c^{5}}{a^{4}}}.$$ 10. Find all natural numbers$n$so that$2^{n}+n^{2}+1$is a perfect square. 11. Given a strictly increasing sequence of positive integers$\left(a_{n}\right)$. Let $$S_{n}=\frac{\sqrt{a_{1}}}{\left[a_{1}, a_{2}\right]}+\frac{\sqrt{a_{2}}}{\left[a_{2}, a_{3}\right]}+\ldots+\frac{\sqrt{a_{n}}}{\left[a_{n}, a_{n+1}\right]},\, \forall n=1,2, \ldots$$ (for positive integers$x, y$we denote$[x, y]$the least common multiple (l.c.m.) of$x$and$y$.) Show that the sequence$\left(S_{n}\right)$has the finite limit when$n \rightarrow+\infty$. 12. Given an acute triangle$A B C(A B < A C)$. Two altitudes$B E$and$C F$intersect at$H$. Let$I$be the center of the circle which passes through$A$,$B$and is tangent to$B C$and$J$the center of the circle which passes through$B$,$H$and is tangent to$B C$. Let$M$be the midpoint of$A H$, and$S=E F \cap B C$Show that$S M$bisects$I J$. ##$type=three$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 506
2019 Issue 506
Mathematics & Youth