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

## $show=home 1. Let$m,n$be two positive integers such that$3^{m}+5^{n}$is divisible by$8$. Prove that$3^{n}+5^{m}$is also divisible by$8$. 2. Given a triangle$ABC$with$A$is an obtuse angle. Let$M$be the midpoint of$BC$. Inside$\widehat{BAC}$, draw two rays$Ax$and$Ay$such that$\widehat{BAx}=\widehat{CAy}=22^{0}$. Let$H$be the projection of$B$on$Ax$, and$I$the projection of$C$on$Ay$. Find the angle$HMI$. 3. Solve the following equation $\sqrt{x^{2}+3x+3}+\sqrt{2x^{2}+3x+2}=6x^{2}+12x+8.$ 4. Let$ABC$be a right triangle with the right angle$A$and let$AB=a$,$AC=b$. Two internal angle bisectors$BB_{1}$and$CC_{1}$intersect at$R$,$AR$intersects$B_{1}C_{1}$at$M$. Compute the distance from$M$to$BC$in terms of$a$and$b$. 5. Let$a,b,c$be positive real numbers satisfying$a^{3}+b^{3}+c^{3}=1$. Prove that $\frac{a^{2}+b^{2}}{ab(a+b)^{3}}+\frac{b^{2}+c^{2}}{bc(b+c)^{3}}+\frac{c^{2}+a^{2}}{ca(c+a)^{3}}\geq\frac{9}{4}.$ 6. Express 2015 as a sum of integers$a_{1},a_{2},\ldots,a_{n}$which are greater than$1$such that${\displaystyle \sum_{i=1}^{n}\sqrt[a_{i}]{a_{i}}}$is maximal. 7. Given a quadrilateral$ABCD$and$a,b,c,d$respectively are external angle bisectors of$\widehat{DAB}$,$\widehat{ABC}$,$\widehat{BCD}$,$\widehat{CDA}$. Denote$K=a\cap b$,$L=b\cap c$,$M=c\cap d$,$N=d\cap a$. Prove that the quadrilateral$KLMN$inscribes a circle whose radius is $\frac{KM\cdot LN}{AB+BC+CD+DA}.$ 8. Suppose that the polynomial $f(x)=x^{3}+ax^{2}+bx+c$ has three non-negative solutions. Find the maximal real number$\alpha$such that $f(x)\geq\alpha(x-a)^{2},\quad\forall x\geq0.$ 9. Let$[x]$be the greatest integer not exceeding$x$and let$\{x\}=x-[x]$. Find $$\left\{ \frac{p^{2012}+q^{2016}}{120}\right\}$$ where$p,q$are primes numbers which are greater than 5. 10. Let$x,y,z$be positive real numbers satisfying$x^{3}+y^{2}+z=2\sqrt{3}+1$. Find the minimum value of the expression $P=\frac{1}{x}+\frac{1}{y^{2}}+\frac{1}{z^{3}}.$ 11. Given a sequence$\{a_{n}\}$whose terms are greater than 1 and satisfy $\lim_{n\to\infty}\frac{\ln(\ln a_{n})}{n}=\frac{1}{2014}.$ Let$b_{n}=\sqrt{a_{1}+\sqrt{a_{2}+\ldots+\sqrt{a_{n}}}}$($n\in\mathbb{N}^{*}$). Prove that$\lim_{n\to\infty}b_{n}$is a finite number. 12. Given a triangle$ABC$and$O$is any point inside the triangle. Let$P,Q$and$R$respectively be the projections of$O$on$BC$,$CA$and$AB$respectively. Let$A_{1},B_{1}$and$C_{1}$be arbitrary points other than$A,B,C$on the lines$BC,CA$and$AB$respectively. Let$A_{2},B_{2}$and$C_{2}$are the reflections of$A_{1},B_{1}$and$C_{1}$through the points$P,Q$and$R. Let \begin{align*} Z_{1} & \equiv(AB_{1}C_{1})\cap(BC_{1}A_{1})\cap(CA_{1}B_{1}),\\ Z_{2} & \equiv(AB_{2}C_{2})\cap(BC_{2}A_{2})\cap(CA_{2}B_{2}). \end{align*} Prove thatO$is equidistant from$Z_{1}$and$Z_{2}$. ##$type=three$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: 2014 Issue 448
2014 Issue 448
Mathematics & Youth