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

## $show=home 1. Find prime numbers$x$,$y$,$z$which satisfy the equality $$x^{5}+y^{3}-(x+y)^{2}=3 z^{3}$$ 2. Given a triangle$A B C$. Let$M$be the point on the side$A B$so that$M B=\dfrac{1}{4} A B,$and$I$the point on the side$B C$so that$I C=\dfrac{3}{8} B C$. The point$N$is the reflection of$A$in the point$I$. Show that$M N \parallel A C$. 3. Solve the equation $$\sqrt{x+2020}+\sqrt{x+2021}+\sqrt{x+2022}=0$$ 4. Given an acute triangle$A B C(A B>B C)$inscribed in a circle$(O)$. Let$A D$and$C E$be two altitudes of the triangle$A B C$. Let$I$be the midpoint of$D E$. The ray$A I$intersects$(O)$at$K(K \neq A)$. Show that the circumcenter of the triangle$I D K$lies on$B D$. 5. Find the minimum value of the expression $$\frac{a^{3}}{1-a^{2}}+\frac{b^{3}}{1-b^{2}}+\frac{c^{3}}{1-c^{2}}$$ where$a, b, c$are positive numbers satisfying$a+b+c=1$. 6. Solve the system of equations $$\begin{cases}\dfrac{1}{\sqrt{4 x^{2}+8 x+5}}+\dfrac{1}{\sqrt{4 y^{2}-8 y+5}} &= \dfrac{2}{\sqrt{(x+y)^{2}+1}} \\ \dfrac{1}{\sqrt{x-1}} +\dfrac{1}{\sqrt{y-3}} &= \dfrac{2 \sqrt{5}}{5}\end{cases}.$$ 7. Given real numbers$a$,$b$,$c$satisfying$a+b+c=1$. Show that $$8 a b c-8 \leq(a b+b c+c a+1)^{2}$$ 8. Given a triangle$ABC$inscribed in a circle$(O)$. Let$H$be the orthocenter of the triangle. Let$X$,$Y$,$Z$respectively be the second intersection between the circles$(A O H)$,$(B O H)$,$(C O H)$with$(O)$. Show that$H$is the incenter of the triangle$X Y Z$. (The notation$(U V W)$denotes the circumcircle of the triangle$U V W$). 9. Find all real numbers$x$,$y$,$z$such that $$2^{x^{2}-3 y+z}+2^{y^{2}-3 z+x}+2^{z^{2}-3 x+y}=\frac{3}{2}$$ 10. The integers from$51$to$150$are aranged in a chess board of the size$10 \times 10$. Does there exist an arrangment so that for any pair of numbers$(a ; b)$which are adjacent in a row or in a column at least one of the following equations $$x^{2}-a x+b=0 \quad \text{and} \quad x^{2}-b x+a=0$$ has an integral solution? 11. A sequence$\left(a_{n}\right)\left(n \in \mathbb{N}^{*}\right)$is determined as follows $$a_{1}=1,\, a_{2}=2,\quad a_{n+2}=2 a_{n+1}-p a_{n},\, n \in \mathbb{N}^{*}$$ where$p$is some prime number. Find all possible values for$p$so that there exists a positive integer$m$satisfying$a_{m}=-3$. 12. Given a non-isosceles triangle$A B C$and let$I$,$J$respectively be the center of the incircle of$A B C$and the excircle corresponding to the angle$A$. Let$D$be the second intersection between the circle with the diameter$A I$and the circumcircle of$A B C$. Assume that$E$is the intersection between$A I$and$B C$and$P$is the midpoint of the arc$B A C$. Show that the intersect between$P J$and$D E$lies on the circumcircle of$A B C$. ##$type=three$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 518
2020 Issue 518
Mathematics & Youth