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2019 Issue 503

  1. Given integers $a, b, c$ satisfying $a=b-c=\dfrac{b}{c}$. Prove that $a+b+c$ is a cube of some integer.
  2. Given an isosceles $A B C$ with the acute vertex angle $A$. Draw the altitude $B H$ ($H$ is on $A C)$. The line through $H$ and parallel to $B C$ and the line through $C$ and parallel to $B H$ intersect at $E .$ Let $M$ be the midpoint of $H E .$ Find the value of the angle $\widehat{A M C}$.
  3. Solve the equation $$x^{2}-3 x+1=-\frac{\sqrt{3}}{3} \sqrt{x^{4}+x^{2}+1}.$$
  4. Given a half circle with the center $O$ and the diameter $A B=2 R$. Let $M$ be a point on the opposite ray of the ray $A B$. Draw a secant $M C D$ of the circle $(C$ is between $M$ and $D$). Assume that $A D$ intersects $B C$ at $I$. Determine the position of $M$ given that $M C I O$ is a cyclic quadrilateral.
  5. Given real numbers $a$, $b$ and $c$ satisfying $\left(1+a^{2}\right)\left(4+b^{2}\right)\left(9+c^{2}\right) \leq 100$. Show that $$-4 \leq 3 a b+2 a c+b c \leq 16.$$
  6. Solve the system of equations $$\begin{cases}\sqrt{x^{2}+x y+y^{2}}+\sqrt{y^{2}+y z+z^{2}}+\sqrt{z^{2}+z x+x^{2}} &=\sqrt{3}(x+y+z) \\ \sqrt{x y z}-(\sqrt{x}+\sqrt{y}+\sqrt{z})+2 &=0 \end{cases}.$$
  7. For any triangle $A B C$, show that $$ \tan \frac{A}{4}+\tan \frac{B}{4}+\tan \frac{C}{4}+\tan \frac{A}{4} \tan \frac{B}{4} + \tan \frac{B}{4} \tan \frac{C}{4}+\tan \frac{C}{4} \tan \frac{A}{4} \leq 3(9-5 \sqrt{3}).$$ When does the equality happen?
  8. Given a triangle $A B C$ inscribed in a circle $(O)$. The medians $A A_{1}$, $B B_{1}$, $C C_{1}$ respectively intersect $(O)$ at $A_{2}$, $B_{2}$, $C_{2}$. Let $A B=c$, $B C=a$, $C A=b$. Show that $$\frac{A_{1} A_{2}}{a}+\frac{B_{1} B_{2}}{b}+\frac{C_{1} C_{2}}{c} \geq \frac{\sqrt{3}}{2}.$$
  9. Given non-negative numbers $a$, $b$, $c$ satisfying $a+b+c=\dfrac{4}{3}$. Prove that $$3\left[a(a-1)^{2}+b(b-1)^{2}+c(c-1)^{2}\right] \geq a b+b c+c a.$$ When does the equality happen?
  10. Find all pairs of positive integers $(n ; k)$ so that $$(n+1)(n+2) \ldots(n+k)-k$$ is a complete square. 
  11. Find all functions $f(x)$ which are continuous on $[a ; b]$, are differentiable on $(a ; b)$, and satisfy $$f^{\prime}(x) \leq \frac{f(b)-f(a)}{b-a},\, \forall x \in(a ; b),$$ where $a$, $b$ are given real numbers with $a < b$.
  12. Given a circle $(O)$ and a point $K$ lying outside the circle. Draw the tangents $K I$, $K J$ to the circle at $I$, $J$. On the opposite ray of $I O$ take an arbitrary point $O$. The circle with center $O'$ radius $O^{\prime} J$ intersects $(O)$ at the other point $A$. $A I$ intersects $\left(O^{\prime}\right)$ at the other point $D$. The line through $K$ perpendicular to $O^{\prime}D$ meets $\left(O^{\prime}\right)$ at $B$ and $C$. Show that $I$ is the center of the incircle of $A B C$.

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Mathematics & Youth: 2019 Issue 503
2019 Issue 503
Mathematics & Youth
https://www.molympiad.org/2020/09/2019-issue-503.html
https://www.molympiad.org/
https://www.molympiad.org/
https://www.molympiad.org/2020/09/2019-issue-503.html
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