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

## $show=home 1. Let $$A=\dfrac{2011^{2011}}{2012^{2012}},\quad B=\dfrac{2011^{2011}+2011}{2012^{2012}+2012}.$$ Which number is greater,$A$or$B$?. 2. Given $A=\sqrt{6+\sqrt{6+\ldots+\sqrt{6}}},\:B=\sqrt{6+\sqrt{6+\ldots+\sqrt{6}}},$ where there are exactly$n$square roots in$A$and$n$cube roots in$B$. Write$[x]$for the greatest integer not exceeding$x$. Determine the value of$\left[\dfrac{A-B}{A+B}\right]$. 3. Find all pairs of natural numbers$x,y$such that $x^{2}-5x+7=3^{y}.$ 4. Prove the inequality $\left(1+\frac{1}{2}\right)\left(1+\frac{1}{2^{2}}\right)\ldots\left(1+\frac{1}{2^{n}}\right)<3.$ 5. Let$ABCD$be aparallelogram. Points$H$and$K$are chosen on lines$AB$and$BC$such that triangles$KAB$and$HCB$are isosceles ($KA=AB$,$HC=CB$). Prove that a) Triangle$KDH$is also isosceles. b) Triangle$KAB$,$BCH$and$KDH$are similar. 6. In a triangle$ABC$with$a=BC$,$b=CA$,$c=AB$,$A_{1}$is the midpoint of$BC$;$O$and$I$are its circumcenter and incenter respectively. Prove that if$AA_{1}$isperpendicular to$OI$then $\min\{b,c\}\leq a\leq\max\{b,c\}.$ 7. The real numbers$x,y$and$z$are such that $\begin{cases}\sqrt{x}\sin\alpha+\sqrt{y}\cos\alpha-\sqrt{z} & =-\sqrt{2(x+y+z)}\\ 2x+2y-13\sqrt{z} & =7 \end{cases},\quad\pi\leq\alpha\leq\frac{3\pi}{2}.$Determine the value of$(x+y)z$. 8. Solve the following system of equations in two variables $\begin{cases}\log_{2}x & =2^{y+2}\\ 2\sqrt{1+x}+xy\sqrt{4+y^{2}} & =0 \end{cases}.$ 9. A collection of prime numbers (each prime can be repeated) is said to be beautiful if their product is exactly ten times their sum. Find all beautiful collections. 10. Points$A,B,C,D,E$in clockwise order, lie on the same circle.$M,N,P,Q$are the feet of perpendicular lines from$E$onto$AB$,$BC$,$CD$,$DA$. Prove that$MN$,$NP$,$PQ$,$QM$are tangent lines to a certain parabole whose focus point if$E$. 11. The sequence$(a_{n})$is defined recursively by the following rules $a_{1}=1,\quad a_{n+1}=\frac{1}{a_{1}+\ldots+a_{n}}-\sqrt{2},\:n=1,2,\ldots.$ Find the limit of the sequence$(b_{n})$where $b_{n}=a_{1}+\ldots+a_{n}.$ 12. Let$\alpha$and$\beta$be two real roots of the equation $4x^{2}-4tx-1=0$ where$t$is a parameter. Let$f(x)=\dfrac{2x-t}{x^{2}+1}$be a funtion defined on the interval$[\alpha;\beta]$, and let $g(t)=\max_{x\in[\alpha;\beta]}f(x)-\min_{x\in[\alpha;\beta]}f(x).$ Prove that if a triple$a,b,c\in\left(0;\frac{\pi}{2}\right)$are such that$\sin a+\sin b+\sin c=1$, then $\frac{1}{g(\tan a)}+\frac{1}{g(\tan b)}+\frac{1}{g(\tan c)}<\frac{3\sqrt{6}}{4}.$ ##$type=three$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: 2012 Issue 415
2012 Issue 415
Mathematics & Youth