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2019 Issue 500

  1. Find all integer $x$ such that $$(x-2018)^{3}+(x-2019)^{2}=2020-x.$$
  2. Let $x_{n}=2^{2^{n}}+1$, $n=1,2, \ldots, 2019$. Show that $$\frac{1}{x_{1}}+\frac{2}{x_{2}}+\frac{2^{2}}{x_{3}}+\ldots+\frac{2^{2018}}{x_{2019}}<\frac{1}{3}.$$
  3. Given $0<x<y \leq z \leq 2$ and $3 x+2 y+z=9$. Find the maximum value of the expression $$A=3 x^{2}+2 y^{2}+z^{2}.$$
  4. Given triangle $A B C$ with $A B<B C$. On the side $B C$ choose $D$ so that $C D=A B$. Through $D$ draw the line which is parallel to $A C$. This line intersects the angle bisector of $\widehat{A B C}$ at $I$. From $I$ draw $IH$ perpendicular to $B C$ ($H$ is on $B C$). From $H$ draw $H E$ perpendicular to $A B$ ($E$ is on $A B$) and draw $H F$ perpendicular to $A D$ ($F$ is on $A D$). Prove that $\widehat{A E I}=\widehat{A F I}$.
  5. Solve the system of equations $$\begin{cases} x y(x+y)-x(y-1)-\dfrac{3}{8}\left(y^{2}+1\right) &=0 \\ x y^{2}+x-y &=0 \end{cases}.$$
  6. Find $x \in[-1 ; 0]$ such that $$\sqrt{x+1}+\sqrt[3]{x^{2}+1}=2.$$
  7. Solve the equation $$(2+\sqrt{3})^{|\sin x|}+\left(\frac{\sqrt{2}+\sqrt{6}}{2}\right)^{|\cos x|}=\frac{2+\sqrt{2}+\sqrt{6}}{2}.$$
  8. Given a triangle $A B C$ with $h_{\alpha}$, $h_{b}$, $h_{c}$ and $a$, $b$, $c$ are the lengths of altitudes and the corresponding sides. Prove that $$\frac{a}{h_{c}}+\frac{b}{h_{b}}+\frac{c}{h_{c}} \geq 2\left(\tan \frac{A}{2}+\tan \frac{B}{2}+\tan \frac{C}{2}\right).$$
  9. Find all positive integers $n$ such that $$3 n=2 S^{3}(n)+7 S^{2}(n)+16$$ where $S(n)$ is the sum of all the digits of $n$
  10. Given $0<\lambda<1$ and $2019$ positive numbers $x_{1}, x_{2}, \ldots, x_{2019}$ such that $\displaystyle \sum_{i=1}^{200} x=2019$. Prove that $$673 \leq \sum_{i=1}^{2019} f\left(\frac{2019-\lambda x_{i}}{2019-\lambda}\right) \leq \sum_{i=1}^{2019} f\left(x_{i}\right)$$ where $f(x)=\dfrac{1}{x^{2}+x+1}$.
  11. Two sequences $\left(x_{n}\right)$, $\left(y_{n}\right)$ are determined by $x_{1} \in\left[\dfrac{1}{2},\dfrac{\sqrt{3}}{2}\right]$, $y_{1} \in\left[\dfrac{1}{2}, \dfrac{\sqrt{3}}{2}\right]$ and $$x_{n+1}=\sqrt{1-y_{2}^{2}},\,y_{m+}=\frac{1}{\sqrt{21}} \sqrt{9-5 x_{n}^{2}},\,\forall n \in \mathbb{N}.$$ Show that the sequences $\left(x_{n}\right)$, $\left(y_{n}\right)$ converse and then find $\displaystyle \lim_{n\to\infty} x_{2}$ and $\displaystyle \lim_{n\to\infty} y_{n}$.
  12. Given a triangular pyramid $S . A B C$ with $\widehat{A S B}=60^{\circ}$, $\widehat{B S C}=90^{\circ}$, $\widehat{C S A}=120^{\circ}$; $S A=2$, $S B=9$, $S C=4$. Let $X$ be the set of all values $d$ such that there exists a point $M$ from which the distances to the planes $(S B C)$, $(S C A)$, $(S A B)$, $(A B C)$ are respectively $\sqrt{3} d$, $\dfrac{d}{2}$, $d$, $d$. Find the cardinality of $X$ and also find the largest possible value for $d$.

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2006,1,2007,12,2008,12,2009,12,2010,12,2011,12,2012,12,2013,12,2014,12,2015,12,2016,12,2017,12,2018,11,2019,12,2020,12,2021,6,Anniversary,4,
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Mathematics & Youth: 2019 Issue 500
2019 Issue 500
Mathematics & Youth
https://www.molympiad.org/2020/09/2019-issue-500.html
https://www.molympiad.org/
https://www.molympiad.org/
https://www.molympiad.org/2020/09/2019-issue-500.html
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