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

## $show=home 1. Find positive integers$x, y$such that$x^{y}+y^{x}=100$2. Given an acute triangle$A B C$. Outside the triangle, we construct two isosceles right triangles with the right angles$A A B D$and$A C E .$Let$J$and$K$respectively be the perpendicular projections of$D$and$E$on$B C$. Prove that$A J K$is an isosceles right triangle. 3. Solve the equation $$\frac{1}{\sqrt{3 x^{2}+x^{3}}}+2 \sqrt{\frac{x}{3 x+1}}=\frac{3}{2}.$$ 4. Given a right triangle$A B C$with the right angle$A$and$A C=2 A B$. Let$A H$be an altitude of$A B C$. On the opposite ray of$A H$choose the point$K$such that$A H=2 A K$. Find the sum$\widehat{A K B}+\widehat{H A B}$. 5. Given real numbers$x, y$satisfying$y-2 x+4<0$. Find the minimum value of the expression $$P=x^{2}-4 y+\frac{4\left(x^{2}-4 y\right)}{(y-2 x+4)^{2}}.$$ 6. Find all positive numbers$x, y, z$satisfying $$\begin{cases}x^{2}+y^{2}+z^{2}+x y z &=4 \\ \left(\dfrac{1}{x^{9}}+\dfrac{1}{y^{9}}+\dfrac{1}{z^{9}}\right)(1+2 x y z) &=9\end{cases}.$$ 7. Given positive numbers$a, b, c$such that$a^{3}+b^{3}+c^{3}=3$. Find the minimum value of the expression $$M=\frac{a^{2}}{b}+\frac{b^{2}}{c}+\frac{c^{2}}{a}.$$ 8. Let$A B C$be inscribed in a circle$(O)$and assume that$I$is the incenter of$A B C$. Assume that$M$is the midpoint of$A I$. Choose$K$on$B C$such that$I K \perp I O$. Choose$Q$on$(O)$such that$A Q \parallel I K$. Suppose that$Q M$intersects$B C$at$H$. Choose$G$on$A K$such that$H G \parallel Q I$.$Q K$intersects$(O)$at$L$other than$Q$. Show that$\widehat{G I K}=\widehat{I A L}$. 9. For each natural number$n$, let $$A_{n}=\left[\frac{n+3}{4}\right]+\left[\frac{n+5}{4}\right]+\left[\frac{n}{2}\right]$$ (the notation$[x]$denotes the maximal integer which does not exceed$x$). Find all natural numbers$n$such that$\dfrac{n^{6}-1}{A_{n}}$is a perfect square. 10. We call the trace of a triangle the ratio between the length of its shortest side and the length of its longest side. Given a square with the side equals 2019 length units. Does there exist a positive integer$n$and a decomposition of this square into$n$triangles satisfying both following conditions • Any side of any of these triangles is less than and equal the side of the square, • The sum of all the traces of these$n$triangles does not exceed$\dfrac{n^{2}+6 n-9}{n^{2}}$. 11. The sequence$\left\{u_{n}\right\}_{n \in \mathbb{N}^{*}}$is determined as follows $$u_{1}=1,\quad u_{n+1}=\sqrt{n(n+1)}\frac{u_{n}}{u_{n}^{2}+n},\, \forall n=1,2 \ldots.$$ a) Find$\displaystyle \lim _{n \rightarrow+\infty} u_{n}$. b) Find$\left[\dfrac{\sqrt{2018}}{u_{2018}}\right]$where$[x]$denotes the maximal integer which does not exceed$x$. 12. Given an acute triangle$A B C$. The points$M$,$N$are on the line segment$B C$and the points$P$,$Q$respectively are on the line segments$CA$,$A B$such that$MNPQ$is a square. The incircles of$A P Q$,$B Q M$,$C P N$are respectively tangent to$P Q$,$Q M$,$P N$at$X$,$Y$,$Z$. Prove that$A X \perp Y Z$. ##$type=three$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 499
2019 Issue 499
Mathematics & Youth