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2020 Issue 515

  1. Two stations $A$ and $B$ are $999km$ away. The milestones along the railway from $A$ to $B$ show the distances from that point to $A$ and $B$ as follows $$0 / 999 ; 1 / 998 ; 2 / 997 ; \ldots ; 999 / 0.$$ Among these milestones, how many of them contains only two different digits?
  2. Given a triangle $A B C$ with the side $B C$ is fixed and the vertex $A$ can vary. Draw the perpendicular bisector $A D .$ Through $C$ draw a perpendicular line to $A D$ at $N$. Let $M$ be the midpoint of $A C$. Show that when $A$ is moving, $M N$ always passes through a fixed point.
  3. Suppose that $a$ and $b$ are positive integers so that $(a, 6)=1$ and $3 \mid a+b$. Assume that $p, q$ are prime numbers so that both $p q+a$ and $b p+q$ are also prime numbers. Prove that $a+6$ is a prime number.
  4. Given a circle $(O, R)$. From a point $A$ outside the circle we draw two tangents $A B$, $A C$ ($B$, $C$ are touch points) and a secant $A D E$ $(D$ is in between $A$ and $E)$. The line $B C$ intersects $OA$ at $H$. From $H$ draw the line parallel to $B E$. That line intersects $A B$ at $K$. Show that $B D$ passes through the midpoint of $H K$.
  5. Solve the system of equations $$\begin{cases}x^{3}+x(y+z)^{2} &=26 \\ y^{3}+y(z+x)^{2} &=40 \\ z^{3}+z(x+y)^{2} &=54\end{cases}.$$
  6. Solve the equation $$\sqrt{x^{3}+x+2}=x^{4}-x^{3}-7 x^{2}-x+10.$$
  7. Suppose that $x$, $y$ are positive numbers so that $x+y \leq 6$. Find the minimum value of the expression $$P=x^{2}(6-x)+y^{2}(6-y)+(x+y)\left(\frac{1}{x y}-x y\right).$$
  8. Given a triangle $A B C$. Let $O$ and $I$ be the circumcenter and the incenter of the triangle respectively. Let $D$ be the second intersection between $(O)$ and $AI$. Let $P$ be the intersection between $B C$ and the line which passes though $I$ and perpendicular to $AI$. Let $Q$ be the reflection point of $I$ in $O$. Show that $$\widehat{P A Q}=\widehat{P D Q}=90^{\circ}.$$
  9. Find the limit $$\lim_{n \rightarrow+\infty} \frac{\mid \sqrt[3]{1}]+[\sqrt[3]{2}]+\ldots+\left[\sqrt[3]{n^{3}+3 n^{2}+3 n}\right]}{n^{4}}$$ where $n$ is a positive integer and the notion $[x]$ denote the integer which does not exceed $x$.
  10. Given a positive integer $n$ so that both $6 n+1$ and $20 n+1$ are perfect squares. Show that $58 n+11$ is a composite number.
  11. Suppose that $g:[a, b] \rightarrow R$ is $a$ continuous functions with $g(a) \leq g(b)$ and $f:[a, b] \rightarrow[g(a), g(b)]$ is an increasing function. Show that the equation $f(x)=g(x)$ has at least one solution.
  12. Given a triangle $A B C$. Let $O$ and $H$ be the circumcenter and the orthocenter of the triangle respectively. Let $S$ be the circumcenter of the triangle $O B C$. Denote $K$ and $L$ the reflection points of $S$ in $A B$ and $A C$ respectively. Show that $K L$ passes through the the midpoint of $O H$.

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Mathematics & Youth: 2020 Issue 515
2020 Issue 515
Mathematics & Youth
https://www.molympiad.org/2020/09/2020-issue-515.html
https://www.molympiad.org/
https://www.molympiad.org/
https://www.molympiad.org/2020/09/2020-issue-515.html
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