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

## $show=home 1. Let $n=1234567891011\ldots99100.$ Delete$100$digits so that the remaining digits in the original order, is greatest possible. 2. Find all integers$x,y,z$such that $(x-y)^{3}+3(y-z)^{2}+5|z-x|=35.$ 3. For what values of$a$is the numbers$a+\sqrt{15}$and$\dfrac{1}{a}-\sqrt{15}$are both intergers?. 4. Solve the equation $\sqrt{6-x}+\sqrt{2x+6}+\sqrt{6x-5}=x^{2}-2x-5.$ 5. Let$(O)$be a circle of diameter$AB$. Point$I$is outside the circle,$IH$is the perpendicular line to$AB$through$I$($H$lies between$O$and$A$).$IA$,$IB$meet$(O)$at points$E$and$F$respectively;$EF$meets$AB$at$P$;$EH$meets$(O)$at the second point$M$;$PM$intersects$(O)$at the second point$N$. Let$K$denote the midpoint of$EF$,$O'$is the circumcenter of triangle$HMN$. Prove that$O'H||OK$. 6. Prove that in any triangle$ABC$, the following inequality holds $\left(\frac{h_{a}}{l_{a}}-\sin\frac{A}{2}\right)\left(\frac{h_{b}}{l_{b}}-\sin\frac{B}{2}\right)\left(\frac{h_{c}}{l_{c}}-\sin\frac{C}{2}\right)\leq\frac{r}{4R},$ where$h_{a},h_{b},h_{c}$and$l_{a},l_{b},l_{c}$are the length of the altitudes and the internal angle bisector from$A,B$and$C$respectively; any$R,r$are its circumradius and inradius. 7. Let$S.ABC$be a triangular pyramid where$\widehat{BAC}=90^{0}$,$BC=2a$,$\widehat{ACB}=\alpha$. The plane$(SAB)$is perpendicular to$(ABC)$. Find the volume of this pyramid, given that$SAB$is an isosceles triangle at vertex$S$and$SBC$is a right triangle. 8. Solve the system of equations $\begin{cases}x^{8}y^{8}+y^{4} & =2x\\ 1+x & =x(1+y)\sqrt{xy} \end{cases}.$ 9. Let$A=2013^{30n^{2}+4n+2013}$where$n\in\mathbb{N}$. Let$X$be the set of remainders when dividing$A$by$21$for some natural number$n$. Determine the set$X$. 10. Let$(x_{n})$be a sequence of natural numbers with $x_{1}=2,\quad x_{n+1}=\left[\frac{3}{2}x_{n}\right],\forall n=1,2,3,\ldots,$ where$[x]$denotes the greates integer not exceeding$x$. Prove that in the sequence$(x_{n})$, there are infinitely many odd numbers and infinitely many even numbers. 11. Given a real number$x\geq1$. Find the limit $\lim_{n\to\infty}(2\sqrt[n]{x}-1)^{n}.$ 12. Let$S$denote the area of a given triangle, and let$P$be an arbitrary point.$A',B',C'$are the midpoints of$BC$,$CA$and$AB$respectively;$h_{a},h_{b},h_{c}$denote the corresponding altitudes. Prove that $PA^{2}+PB^{2}+PC^{2}\geq\frac{4}{\sqrt{3}}S\max\left\{ \frac{PA+PA'}{h_{a}},\frac{PB+PB'}{h_{b}},\frac{PC+PC'}{h_{c}}\right\}.$ ##$type=three$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 436
2013 Issue 436
Mathematics & Youth