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

## $show=home 1. Given $A=1^{5}+2^{5}+3^{5}+\ldots+2011^{5}.$ Find the last digit of$A$. 2. Let$ABC$be an isosceles right triangle with right angle at$A$. On the half-plane defined by$AB$containing$C$draw an isosceles right triangle$ABD$with right angle at$B$. Let$E$be the midpoint of segment$BD$. Draw$CM$perpendicular to$AE$at$M$. Let$N$be the midpoint of segment$CM$,$K$is the intersection of$BM$and$DN$. Find the measure of the angle$BKD$. 3. Find all positive integer solutions of the equation $3^{x}-32=y^{2}.$ 4. Find all minimal value of the expression $A=\frac{1}{x^{3}+xy+y^{3}}+\frac{4x^{2}y^{2}+2}{xy}$ where$x$and$y$are positive real numbers satisfying$x+y=1$. 5. Let$ABC$be an acute triangle with orthocenter$H$. Prove that$ABC$is an equilateral triangle if and only if $\frac{AH}{BC}=\frac{BH}{CA}=\frac{CH}{AB}.$ 6. Let$ABC$be a triangle with circumcenter$O$, and incenter$I$.$BC$touches the circle$(I)$at$D$. The circle whose diameter is$AI$meets$(O)$at$M$($M\ne A$) and cuts the line passing through$A$parallel to$BC$at$N$. Prove that$MO$passes through the midpoint of$DN$. 7. Solve the system of equations $\begin{cases}\sqrt{xy+(x-y)(\sqrt{xy}+2)}+\sqrt{x} & =y+\sqrt{y}\\(x+1)(y+\sqrt{xy}+x(1-x)) & =4\end{cases}.$ 8. Let$ABC$be an acute triangle. Prove the inequaltiy $\cos^{3}A+\cos^{3}B+\cos^{3}C+\cos A.\cos B.\cos C\geq\frac{1}{2}.$ 9. For each natural number$n$, let$(S_{n})$be the sum of all digits of$n$(in the decimal system). Put$S_{k}(n)=S(S(\ldots(S(n))\ldots))$($k$times). Find all natural numbers$n$such that $S_{1}(n)+S_{2}(n)+\ldots S_{k}(n)+\ldots+S_{223}(n)=n.$ 10. Does there exist a set$X$satisfying the following two conditions •$X$contains$2012$natural numbers. • The sum of any arbitrary elements in$X$is the$k$-th power of a positive integer ($k\geq2$). 11. Find all functions$f:\mathbb{R}\to\mathbb{R}$satisfing $f\left(\frac{xf(y)}{2}\right)+f\left(\frac{yf(x)}{2}\right)=4xy,\:\forall x,y\in\mathbb{R}.$ 12. Fix two circles$(K)$and$(O)$, where$(K)$is inside$(O)$. Two circles$(O_{1})$,$(O_{2})$are moving so that they always externally touch each other at$M$. Both also internally touch$(O)$, and externally touch$(K)$. Prove that$M$belongs to a fixed circle. ##$type=three$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 418
2012 Issue 418
Mathematics & Youth