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

## $show=home 1. Find natural numbers$x$such that $$\frac{x-1}{2018}+\frac{x-7}{503}=\frac{x-3}{1008}+\frac{x-9}{670}.$$ 2. Find all pairs non-negative numbers$(x, y)$such that $$1+3^{x+1}+2.3^{3 x}=y^{3}.$$ 3. Given$0 \leq a \leq 2$,$0 \leq b \leq 2$,$0 \leq c \leq 2$and$a+b+c=3$. Show that $$3 \leq a^{3}+b^{3}+c^{3} \leq 9.$$ 4. From a point$M$which is outside a circle$(O)$we draw a tangent$M A$and a secant$M B C$to$(O)$($B$is in between$M$and$C$). Let$D$,$E$,$K$respectively be the midpoints of$M A$,$M B$,$M E$. Suppose that$H$is the point of reflection of$D$through$K$. The line$H E$intersects$C D$at$N$. Prove that$C N K H$is a cyclic quadrilateral. 5. Solve the system of equations $$\begin{cases} 2 x \sqrt{x}+3 x \sqrt{y}+\sqrt{y} &=138 \\ y \sqrt{y}+6 \sqrt{x}y+8 \sqrt{x} &=213\end{cases}.$$ 6. Solve the equation $$12 \sqrt{x^{2}+4} \cdot \sqrt{2 x-3}=\left(x^{2}+16 x-12\right) \sqrt{x-1}.$$ 7. Find the coefficient of$x^{3}$in the expansion of $$(1+x)(1+2 x)(1+3 x) \ldots(1+n x).$$ 8. Given a triangle$A B C$. Let$B C=a$,$C A=b$,$B A=c$and$r_{a}$,$r_{b}$,$r_{c}$respectively the radii of the excircles relative to$A$,$B$,$C$. Prove that $$\frac{r_{a}}{r_{b}}+\frac{r_{b}}{r_{e}}+\frac{r_{c}}{r_{o}} \geq \frac{a}{b}+\frac{b}{c}+\frac{c}{a}.$$ 9. Given positive numbers$a$,$b$,$c$,$d$such that$a \geq b \geq c \geq d$and$a b c d=1$. Find the smallest constant$k$so that the following inequality holds $$\frac{1}{a+1}+\frac{1}{b+1}+\frac{1}{c+1}+\frac{k}{d+1} \geq \frac{3+k}{2}.$$ 10. Let$n$be an integer which is greater or equal to$6 .$Find the largest positive integer$m$so that among$n$arbitrary distinct positive integers which are less than or equal to$m$there always exist$4$numbers so that one of them is equal to the sum of the remains. 11. Find the smallest integer$k$so that there exist two sequences$\left\{a_{i}\right\},\left\{b_{i}\right\}$satisfying a)$a_{i}, b_{i} \in\left\{1,2018,2018^{2}, 2018^{3}, \ldots\right\}$,$ i=1,2, \ldots, k$b)$a_{i} \neq b_{i}, i=1,2, \ldots, k$c)$a_{i} \leq a_{i+1}, b_{i} \leq b_{i+1}$d)$\displaystyle\sum_{i=1}^{k} a_{i}=\sum_{i=1}^{k} b_{i}$12. Given a triangle$A B C$with$A B+A C=2 B C$. Let$O$be its circumcenter, and$H$its orthocenter. Let$M_{a}$be the midpoint of$B C$. Prove that the circles with diameters$HM_{a}$and$A O$are tangent to each other. ##$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: 2019 Issue 501
2019 Issue 501
Mathematics & Youth