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2009 Issue 384

  1. Replace the distinct letters by distinct numbers such that the following expression becomes a true equality $$\mathrm{VE}+\mathrm{TRUONG}+\mathrm{SA}=22 \times 12 \times 2009.$$
  2. It is well-known that the two right triangles whose side lengths are positive integers $(5,12,13)$ and $(6,8,10)$ possess additional property that the area of each triangle equals its perimeter. Are there other triangles with similar properties?
  3. Suppose given $1003$ nonzero rational numbers in which any quadruple form a proportion. Prove that at least $1000$ numbers are equal.
  4. Let $x$, $y$, $z$ be non-negative real numbers such that $x+y+z=1$. Find the maximum valuc of the following expression $$P=(x+2 y+3 z)(6 x+3 y+2 z).$$
  5. Let $A B C D$ be a cyclic quadrilateral, inscribed in a circle $(O)$. The angle bisectors of $B A D$ and $B C D$ meet at a point $K$ on the diagonal $B D$. Let $Q$ be the second intersection point (different from $A$) of $A P$ and the circle $(O)$; $M$ and $N$ be respectively the midpoints of $B D$ and $C P$. The line through $C$ and parallel to $A D$ meets $A M$ at $P$. Prove that a) $S_{A BQ}=S_{A D Q}$. b) $DN$ is perpendicular to $C P$.
  6. For each natural number $n,$ let $p(n)$ be its largest odd divisor. Determine the sum $$\sum_{n=2006}^{4012} p(n)$$
  7. Solve for $x$ $$\sqrt{3 x-2}=-4 x^{2}+21 x-22$$
  8. Let $A B C$ be a triangle whose circumcircle is $(O)$ and such that $A C<A B$. The tangent lines to $(O)$ at $B$, $C$ intersect at $T$. The line through $A$ and perpendicular to $A T$ meet $B C$ at $S$. Choose the points $B_{1}$ and $C_{1}$ on $S T$ such that $T B_{1}=T C_{1}=T B$ and that $C_{1}$ lies between $S$ and $T$. Prove that $A B C$ and $A B_{1} C_{1}$ are similar.

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2006,1,2007,12,2008,12,2009,12,2010,12,2011,12,2012,12,2013,12,2014,12,2015,12,2016,12,2017,12,2018,11,2019,12,2020,12,2021,4,Anniversary,4,
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Mathematics & Youth: 2009 Issue 384
2009 Issue 384
Mathematics & Youth
https://www.molympiad.org/2020/09/2009-issue-384.html
https://www.molympiad.org/
https://www.molympiad.org/
https://www.molympiad.org/2020/09/2009-issue-384.html
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