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2019 Issue 501

  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 $$12 \sqrt[3]{x^{2}+4} \cdot \sqrt{2 x-3}=\left(x^{2}+16 x-12\right) \sqrt[3]{x-1}.$$
  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.

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Mathematics & Youth: 2019 Issue 501
2019 Issue 501
Mathematics & Youth
https://www.molympiad.org/2020/09/2019-issue-501.html
https://www.molympiad.org/
https://www.molympiad.org/
https://www.molympiad.org/2020/09/2019-issue-501.html
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