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

## $show=home 1. Let $$M=\frac{1}{10}+\frac{1}{20}+\frac{1}{35}+\frac{1}{56}+\frac{1}{84}+\frac{1}{120}+\ldots$$ a) Is the fraction$\dfrac{1}{15400}$a term of$M$? Why?. b) Compute the sum of the$8$first terms of$M$. 2. Given a triangle$A B C$with$A B < A C$. The angle bisector of$\widehat{B A C}$intersects the perpendicular bisector of$B C$at$M$. Let$H$,$K$, and$I$respectively be the perpendicular projections of$M$on$A B$,$A C$and$B C$. Prove that$H$,$I$,$K$is collinear. 3. Find integer solutions of the equation $$x^{3}=4 y^{3}+x^{2} y+y+13$$ 4. Given a quadrilateral$A B C D$inscribed in the circle with the diameter$A C$. Let$M$be the point on$A B$such that$A M=A D$. The lines$D M$and$B C$intersects at$N,$and the$A N$intersects the circle at$K$. Let$H$be the perpendicular projection of$D$on$A C$. Assume that$A B$intersects$N H$and$C K$at$P$and$Q .$Show that $$\frac{1}{M P}=\frac{1}{M A}+\frac{1}{M Q}.$$ 5. Solve the system of equations $$\begin{cases} 3 x^{3}+6 x+2 &=2 y^{3} \\ 3 y^{3}+6 y+2 &=2 z^{3} \\ 3 z^{3}+6 z+2 &=2 x^{3}\end{cases}$$ 6. Solve the inequality $$x^{3}+6 x^{2}+9 x \leq \sqrt{x+4}-2$$ 7. Let$a, b, c$be positive numbers with the product is equal to$1$. Find the minimum value of the expression $$P=\frac{1}{a^{2017}+a^{2015}+1}+\frac{1}{b^{2017}+b^{2015}+1}+\frac{1}{c^{2017}+c^{2015}+1}$$ 8. Given a triangle$A B C$inscribed in the circle$(O)$. The tangent lines of$(O)$at$B$and$C$intersect at$P .$The line which goes through$A$and is parallel to$B P$intersects$B C$at$M$. The line which goes through$A$and is parallel to$B C$intersects$B P$at$N$. Suppose that$I$is the intersections between$A P$and$M N$. Prove that four points$B$,$I$,$O$,$C$lie on a circle. 9. Given the equation$x^{3}+m x^{2}+n=0$. Find$m$,$n$so the the equation has three distinct non-zero real roots$u$,$v$,$t$satisfying $$\frac{u^{4}}{u^{3}-2 n}+\frac{v^{4}}{v^{3}-2 n}+\frac{t^{4}}{t^{3}-2 n}=3$$ 10. Let$p$be an odd prime and$a_{1}, a_{2}, \ldots, a_{p}$is an arithmetic progression with the common difference$d$which is not divisible by$p$. Prove that$\prod_{i=1}^{p}\left(a_{i}+a_{1} a_{2} \ldots a_{p}\right): p^{2}$11. Find all functions$f: \mathbb{R} \rightarrow \mathbb{R}$such that $$f(x+f(y))=f\left(x+y^{2018}\right)+f\left(y^{2018}-f(y)\right),\, \forall x, y \in \mathbb{R}.$$ 12. Given an acute triangle$A B C$with the altitudes$B E$,$C F$. Let$S T$be a chord of the circumcircle of$A E F$. Two circles which go through$S$,$T$is tangent to$B C$respectively at$P$and$Q$. Prove that the intersection between$P E$,$Q F$lies on the circumcircle of$A E F$. ##$type=three$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: 2018 Issue 492
2018 Issue 492
Mathematics & Youth