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

## $show=home 1. Find the largest possible perfect square of the form$4^{27}+4^{1020}+4^{x}$, where$x$is a natural number. 2. Let$ABC$be a right triangle with right angle at vertex$A$and$\widehat{ABC}=54^{0}$. The median$AM$meets the internal angle-bisector$CD$at$E$. Prove that$CE=AB$. 3. Find all prime numbers$p$such that$p-1$and$p+1$each has exactly$6$divisors. 4. Solve the system of equations $\begin{cases} 3\sqrt{x}+2\sqrt{y}+\sqrt{z} & =\dfrac{1}{6}\sqrt{xyz}\\ 6\sqrt{xy}+2\sqrt{yz}+3\sqrt{zx} & =108+18\sqrt{x+4}+12\sqrt{y+9}+6\sqrt{z+36} \end{cases}.$ 5. Let$AB$and$AC$be the tangent lines to a circle$(O)$through an external point$A$($B$and$C$are the poins of tangency). The median$BM$of triangle$ABC$intersects$(O)$at$D$, the ray$AD$meets$(O)$at$E$. Prove that$BE||AC$. 6. Given that$a,b,care the sides of a triangle, prove the inequality \begin{align*} & \sqrt{a^{2}-(b-c)^{2}}+\sqrt{b^{2}-(c-a)^{2}}+\sqrt{c^{2}-(a-b)^{2}}\\ \leq & \sqrt{ab}+\sqrt{bc}+\sqrt{ca}\leq a+b+c \end{align*} 7. The circle(O)$is inscribed in a triangle$ABC$. The tangents to$(O)$which are parallel to the sides of the triangle are drawn, they intersect these sides at points$M,N,P,Q,R$and$S$($M,S\in AB$,$N,P\in AC$,$Q,R\in BC$). Let$l_{1},l_{2},l_{3}$be the lengths of the internal angle-bisectors from$A,B,C$of triangles$AMN$,$BSR$and$CPQ$respectively. Prove that $\frac{1}{l_{1}^{2}}+\frac{1}{l_{2}^{2}}+\frac{1}{l_{3}^{2}}\geq\frac{81}{p^{2}},$ where$p$denotes the semiperimeter of triangle$ABC$. 8. It is given that in a triangle$ABC$,$\tan\dfrac{B}{2}\tan\dfrac{C}{2}=\dfrac{1}{3}$. Solve for$x$$x^{2}+x-\cos A-\frac{1}{4}\cos(B-C)=0.$ 9. Find all real numbers$x$such that $\left\{ \frac{x^{2}+1}{x^{2}+x+1}\right\} =\frac{1}{2}$ where$\{a\}$denote te fractional part of$a$, that is$\{a\}=a-[a]$. 10. Consider the sequence$(x_{n})$, $x_{n+1}=x_{n}+\frac{1}{x_{n}}+\frac{2}{x_{n}^{2}}+\ldots+\frac{2012}{x_{n}^{2012}}\quad(n\in\mathbb{N}^{*})$ where$x_{1}>0$is given. Determine all values of$\alpha$such that the sequence$(nx_{n}^{\alpha})$has a non-zero limit. 11. Find all functions$f:\mathbb{R}\to\mathbb{R}$such that $f(xf(y)+3y^{2})+f(3xy+y)=f(3y^{2}+x)+4xy-x+y$ for all$x,y\in\mathbb{R}$. 12. Let$G$be the centroid of a tetrahedron$ABCD$. Points$X,Y<Z,T$are chosen on the faces$(BCD)$,$(CDA)$,$(DAB)$, and$(ABC)$respectively such that$XY$,$YZ$,$ZT$,$TX$are parallel to$GA$,$GB$,$GC$and$GD$. Determine the volume ratio of the two tetrahedrons$ABCD$and$XYZT$. ##$type=three$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: 2013 Issue 434
2013 Issue 434
Mathematics & Youth