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

## $show=home 1. The first$2013$natural numbers from 1 to 2013 are writeen in a line in some order. Substract one from the first number, two from the second ... and 2013 from the$2013^{\text{th}}$number. Is the product of the resulting$2013$numbers odd or even?. 2. Let$ABC$be an acute triangle with orthocenter$O$.$AO$meets$BC$at$D$. Points$E$and$F$are on sides$AB$and$AC$respectively such that$DE=DB$,$DF=DC$. Prove that$DA$is the angle bisector of angle$EDF$. 3. Find all positive integers$a,b$($a\geq2$,$b\geq2$) so that$a+b$is amultiple of$4$and $\frac{a(a-1)+b(b-1)}{(a+b)(a+b-1)}=\frac{1}{2}.$ 4. Find$x,y$such that $\begin{cases} x\sqrt{x}+y\sqrt{y} & =2\\ x^{3}+2y^{2} & \leq y^{3}+2x\end{cases}.$ 5. Given a circle centered at$O$, and diameter$AB$. Point$C$, different from$A$and$B$, is chosen on circle$(O)$. Point$P$on$AB$such that$BP=AC$. The perpendicular from$P$to$AC$meet$AC$at$H$. The intenal angle bisector of angle$CAB$intersects circle$(O)$at$E$and intersects$PH$at$F$.$CF$meets circle$(O)$at$N$. Prove that$CN$passes through the midpoint of$AP$. 6. Let$a,b,cthe positive real number. Prove the following inequality \begin{align*} & \left(\frac{1}{a}+\frac{2}{b+c}+\frac{3}{a+b+c}\right)^{2}+\left(\frac{1}{b}+\frac{2}{c+a}+\frac{3}{a+b+c}\right)^{2}\\ & +\left(\frac{1}{c}+\frac{2}{a+b}+\frac{3}{a+b+c}\right)^{2}\geq\frac{81}{a^{2}+b^{2}+c^{2}}.\end{align*} 7. TriangleABC$is inscribed in circle$(O)$, another circle$(O')$touches$AB$,$AC$at$P,Q$respectively and touches circle$(O)$at other points$M$,$N$. Points$E$,$D$,$F$are the perpendicular feet of point$S$on$AM$,$MN$,$NA$respectively. Prove that$DE=DF$. 8. The real numbers$a,b,c$satisfying the condition that the polynomial $P(x)=x^{4}+ax^{3}+bx^{2}+cx+1$ has at least one real root. Determine all triple$(a,b,c)$such that$s^{2}+b^{2}+c^{2}$is smallest possible. 9. Let$a$and$B$be two real numbers such that$a^{p}-b^{p}$is a positive integers for all prime number$p$. Prove that$a$and$b$are integers. 10. The sequence$\{u_{n}\}$is given recursively as follows $u_{1}=\frac{1}{1+a},\quad\frac{1}{u_{n+1}}=\frac{1}{u_{n}^{2}}-\frac{1}{u_{n}}+1,\,\forall n\geq1$ where$a\in\mathbb{R},a\ne-1$. Let $$S_{n}=u_{1}+u_{2}+\ldots+u_{n},\quad P_{n}=u_{1}u_{2}\ldots u_{n}.$$ Determine the value of the following expression$aS_{n}+P_{n}$. 11. Determine all funtions$f:\mathbb{R}^{+}\to\mathbb{R}^{+}$such that a)$f$is a decreasing function on$\mathbb{R}^{+}$. b)$f(2x)=2012^{-x}f(x)$,$\forall x\in\mathbb{R}^{+}$where$\mathbb{R}^{+}=(0,+\infty)$. 12. Let$ABC$be a triangle inscribed in circle$(O)$.$AD$is a diameter of$(O)$. Point$E$belongs to the opposite ray of ray$DA$. The perpendicular through$E$o$AD$meets$BC$at$T$.$TP$is a tangent line to$(O)$such that$P$and$A$are on opposite sides of$BC$;$AP$meets$TE$at$Q$.$M$is the midpoint of$AQ$;$TM$meets$AB$,$AC$at$X$,$Y$respectively. Prove that$M$is the midpoint of$XY$. ##$type=three$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: 2013 Issue 432
2013 Issue 432
Mathematics & Youth