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2019 Issue 504

  1. Find the natural number $n$ given that $n^{5}+n+1$ has only one prime factor.
  2. Given a right isosceles triangle $A B C$ with the vertex angle $A$. Let $M$, $N$, $I$ respectively be the midpoints of $A B$, $A C$ and $N C$. Assume that $K$ is the perpendicular projection of $N$ on $B C$. Show that $A K$, $B I$ and $C M$ are concurrent.
  3. Given positive numbers $a$, $b$, $c$, $d$ such that $a+b=c+d=2019$ and $a b \geq c d$. Find the minimum value of the expression $$P=\frac{a+3 \sqrt[3]{b}+2}{\sqrt[3]{c}+\sqrt[3]{d}}.$$
  4. Given a triangle $A B C$ with the angle $\hat{A} > 90^{\circ}$. Choose a point $I$ inside the line segment $B C$ so that $B A$ intersects the circumcircle $\Delta A C I$ at $D$ $(D \neq A)$ and $C A$ intersects the circumcircle $\Delta A B I$ at $E$ $(E \neq A)$, $B E$ intersects $CD$ at $N$. Let $M$ be the midpoint of $B C$, $M A$ intersects the circumcircle $\triangle B N C$ at $F$. Show that $A$, $D$, $E$, $F$, $N$ both belong to a circle.
  5. Solve the system of equations $$\begin{cases} x^{4}+3 x &=y^{4}+y \\ x^{2}-y^{2} &=2\end{cases}.$$
  6. Find the sum of the squares of all real roots of the equation $$x^{5}+2018 x^{2}+2019=x^{4}+2019 x^{3}+2020 x.$$
  7. Given $3$ positive numbers $x, y, z$ such that $x y z=1$. Prove that $$\frac{1}{x^{k+1}(y+z)}+\frac{1}{y^{k+1}(z+x)}+\frac{1}{z^{k+1}(x+y)} \geq \frac{3}{2} \quad \left(k \in \mathbb{N}^{*}\right)$$
  8. Given two equilateral triangles $A B C$ and $A B^{\prime} C^{\prime}$ with the same orientation. Let $K$ be the second intersection between the circumcircles of $A B C$ and $A B^{\prime} C^{\prime}$. Let $M$ be the intersection between $B C^{\prime}$ and $C B^{\prime}$. Show that $M A=M K$.
  9. Determine the coefficient of $x$ in the polynomial expansion of $$(1+x)(1+2 x)^{2} \ldots(1+n x)^{n}.$$
  10. Given the real sequence $\left\{x_{n}\right\}$ determined as follows $$x_{1}=1,\quad x_{n+1}=x_{n}+\frac{1}{2 x_{n}},\, \forall n \geq 1.$$ Show that $\left[9 x_{81}\right]=81$ (where $[x]$ denotes the integral part of $x$ ).
  11. Given an integer $k \geq 2$ and aninteger $n \geq \dfrac{k(k+1)}{2}$. Find the maximal positive integer $m$ so that among $n$ arbitrary distinct positive integers which do not exceed $m$ there always exist $k+1$ numbers of which some number is equal to the sum of the remaining ones.
  12. Given a non-right triangle $A B C$. The altitudes $B B^{\prime}$ and $C C^{\prime}$ intersect at $H$. Let $M$ be the midpoint of $A H$. Let $K$ be an arbitrary point on $B^{\prime} C^{\prime}$ ($K$ is different from $B^{\prime}$, $C^{\prime}$). The line $A K$ intersects $M B^{\prime}$, $M C^{\prime}$ respectively at $E$, $F$. Let $N$ be the intersection between $B E$ and $C F$. Show that $K$ is the orthocenter of the triangle $N B C$.

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Mathematics & Youth: 2019 Issue 504
2019 Issue 504
Mathematics & Youth
https://www.molympiad.org/2020/09/june-2019-issue-504.html
https://www.molympiad.org/
https://www.molympiad.org/
https://www.molympiad.org/2020/09/june-2019-issue-504.html
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