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

## $show=home 1. Find all natural numbers$N$so that the sum of its factors is equal to$2 N$and the product of its factors is equal to$N^{2}$. 2. Given natural numbers$a, b, c$such that$\dfrac{a}{3}=\dfrac{b}{5}=\dfrac{c}{7}$. Prove that $$\frac{2019 b-2020 a}{2019 c-2020 b}>1.$$ 3. Solve the system of equations $$\begin{cases} x^{2} &=2 z-1 \\ y^{2} &=x z \\ z^{2} &=2 y-1\end{cases}.$$ 4. Given an acute triangle$A B C$. Draw the altitudes$C H$,$B K$($H$,$K$is respectively on$A B$and$A C$). Choose two points$P$and$Q$on the ray$C H$and the ray$B K$respectively such that$\widehat{P A Q}=90^{\circ}$. Draw$A M$perpendicular to$P Q$($M$is on$P Q$). Show that$M B$is perpendicular to$M C$. 5. Let$a, b, c$be positive numbers satisfying$a b+b c+c a=8$. Find the minimum value of the expression $$P=3\left(a^{2}+b^{2}+c^{2}\right)+\frac{27(a+b)(b+c)(c+a)}{(a+b+c)^{3}}.$$ 6. Let$x, y, z$be positive numbers so that$x \geq z$. Find the minimum value of the expression $$P=\frac{x z}{y^{2}+y z}+\frac{y^{2}}{x z+y z}+\frac{x+2 z}{x+z}.$$ 7. Find the integral solutions of the equation $$\tan \frac{3 \pi}{x}+4 \sin \frac{2 \pi}{x}=\sqrt{x}.$$ 8. Given a triangle$A B C$inscribed in a circle$(O)$and$(I)$is the incircle of the triangle. Let$M$be the midpoint of$B C$and$X$the midpoint of the arc$\widehat{B A C}$of$(O)$. Let$P$,$Q$respectively be the perpendicular projections of$M$on$C I$,$B I$. Show that$X I \perp P Q$. 9. Given a triangle$A B C$with area$S$and$B C=a$,$C A=b$,$A B=c$. Solve the system of equations (variables$x, y, z$) $$\begin{cases}a^{2} x+b^{2} y+c^{2} z &=4 S \\ x y+y z+z x &=1\end{cases}.$$ 10. For any positive integer$n$show that$n$and$2^{2^{n}}+1$are coprime. 11. The sequences$\left(x_{n}\right)$,$\left(y_{n}\right)$are determined as follows $$x_{1}=3,\, x_{2}=17,\quad x_{n+2}=6 x_{n+1}-x_{n},\,\forall n \in \mathbb{N}^*,$$ $$y_{1}=4,\, y_{2}=24,\quad y_{n+2}=6 y_{n+1}-y_{n},\,\forall n \in \mathbb{N}^*.$$ Show that no term in these sequences$\left(x_{n}\right)$,$\left(y_{n}\right)$is a cube number. 12. Given a right triangle$A B C$, with the right angle$A$, inscribed in a circle$(O)$. The point$A^{\prime}$is the reflection point of$A$over$0$. The point$P$is the perpendicular projection of$A^{\prime}$on the perpendicular bisector of$B C$. Let$H_{a}$,$H_{b}$,$H_{c}$respectively be the orthocenter of$A P A^{\prime}$,$B P A^{\prime}$,$C P A^{\prime}$. Show that the circle$(H_aH_bH_c)$is tangent to the circle$(O)$. ##$type=three$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

Name

2006,1,2007,12,2008,12,2009,12,2010,12,2011,12,2012,12,2013,12,2014,12,2015,12,2016,12,2017,12,2018,11,2019,12,2020,12,2021,6,Anniversary,4,
ltr
item
Mathematics & Youth: 2020 Issue 512
2020 Issue 512
Mathematics & Youth