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

## $show=home 1. Find all natural numbers$x,y,z$such that $2010^{x}+2011^{y}=2012^{z}.$ 2. The natural numbers$a_{1},a_{2},\ldots,a_{100}$satisfy the equation $\frac{1}{a_{1}}+\frac{1}{a_{2}}+\ldots+\frac{1}{a_{100}}=\frac{101}{2}.$Prove that there are at least two equal numbers. 3. Let$a,b,c$be positive real numbers. Prove the inequality $\frac{(a+b)^{2}}{ab}+\frac{(b+c)^{2}}{bc}+\frac{(c+a)^{2}}{ca}\geq9+2\left(\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}\right).$ 4. Solve the equation $4x^{2}+14x+11=4\sqrt{6x+10}.$ 5. In a triange$ABC$, te incircle$(I)$meets$BC$,$CA$at$D$,$E$respectively. Let$K$be the point of reglection of$D$through the midpoint of$BC$, the line through$K$and perpendicular to$BC$meets$DE$at$L$,$N$is the midpoint of$KL$. Prove that$BN$and$AK$are orthogonal. 6. Determine the maximum value of the expression $A=\frac{mn}{(m+1)(n+1)(m+n+1)}$ where$m,n$are natural numbers. 7. Triangle$ABC$($AB>AC$) is inscribed in circle$(O)$. The exterior angle bisector of$BAC$meets$(O)$at another point$E$;$M,N$are the midpoints of$BC$,$CA$respectively;$F$os the perpendicular foot of$E$on$AB$,$K$is the intersection of$MN$and$AE$. Prove that$KF$and$BC$are parallel. 8. Solve the equation $\sin^{2n+1}x+\sin^{n}2x+(\sin^{n}x-\cos^{n}x)^{2}-2=0$ where$n$is a given positive integer. 9. Find all polynomials$P(x)$such that $P(2)=12,\quad P(x^{2})=x^{2}(x^{2}+1)P(x),\:\forall x\in\mathbb{R}.$ 10. Let$r_{1},r_{2},\ldots,r_{n}$be$n$rational numbers such that$0<r_{i}\leq\dfrac{1}{2}$,${\displaystyle \sum_{i=1}^{n}r_{i}=1}$($n>1$), and let$f(x)=[x]+\left[x+\dfrac{1}{2}\right]$. Find the greatest value of the expression${\displaystyle P(k)=2k-\sum_{i=1}^{n}f(kr_{i})}$where$k$runs over the integers$\mathbb{Z}$(the notation$[x]$means the greatest integer not exceeding$x$). 11. Suppose that$f:\mathbb{R}\to\mathbb{R}$is a continuous funtion such that$f(x)+f(x+1006)$is a rational number if and only if$x\in\mathbb{R}$, $f(x+20)+f(x+12)+f(x+2012)$ is itrational. Prove that$f(x)=f(x+2012)$for all$x\in\mathbb{R}$. 12. Prove the following inequality $\frac{m_{a}}{h_{a}}+\frac{m_{b}}{h_{b}}=\frac{m_{c}}{h_{c}}\leq1+\frac{R}{r},$ where$m_{a},b_{b},m_{c}$are medians;$h_{a},h_{b},h_{c}$are the altitudes from$A,B,C$and$R,r$are the circumradius and inradius, respectively. ##$type=three$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

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Mathematics & Youth: 2012 Issue 416
2012 Issue 416
Mathematics & Youth