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2009 Issue 383

  1. Let $a$, $b$, $c$ be integers such that $$A=\frac{a^{2}+b^{2}+c^{2}-a b-b c-c a}{2}$$ is a perfect square. Prove that $a=b=c$.
  2. Let $A B C$ be a triangle in which $\widehat{B A C}=75^{\circ}$. Given that the altitude $A H$ has length $A H=\dfrac{B C}{2}$. Prove that $A B C$ is an isosceles triangle.
  3. Let $x$, $y$ be two nonnegative integers where $x>1$ and $2 x^{2}-1=y^{15}$. Prove that $x$ is divisible by $15$.
  4. Let $A B C D$ be a parallelogram. The ray $D x$ from $D$ is perpendicular to $D C$ and lies on the half-plane divided by $C D$ which does not contains $B$. Choose a point $E$ on $D x$ such that $D E=D C$. Draw an isosceles right triangle $B E F$ (right angle at $F$) such that $F$ and $D$ are on the same half-plane divided by $B C \cdot E H$ is the altitude onto $B C$. Prove that $F$ $D$ and $H$ are collinear.
  5. Let $a$, $b$, $c$ be three sides of a right triangle, $a$ is the hypotenuse. Prove that the equation $-a x^{2}+b x+c=0$ has two distinct roots $x_{1}$, $x_{2}$ such that $-\sqrt{2}<x_{1}<x_{2}<\sqrt{2}$.
  6. Consider a convex polygon with $2009$ vertices. A scissor is used to cut along all of its diagonals, thereby dividing the original polygon into smaller convex polygons. How many vertices are there of a resulting polygon with the greatest number of edges?
  7. Determine the smallest value of the expression $$T=\frac{a b+b c}{a^{2}-b^{2}+c^{2}}+\frac{b c+c a}{b^{2}-c^{2}+a^{2}}+\frac{c a+a b}{c^{2}-a^{2}+b^{2}}$$ where $a$, $b$, $c$ are three sides of a triangle $A B C$ and $a b c=1$.
  8. Let $A B C D . A_{1} B_{1} C_{1} D_{1}$ be a cube. On its three skew edges, choose three points $M$, $N$, $P$. Determine the positions of the points $M$, $N$, $P$ such that the triangle $M N P$ has a) The smallest perimeter possible. b) The smallest area possible.

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Mathematics & Youth: 2009 Issue 383
2009 Issue 383
Mathematics & Youth
https://www.molympiad.org/2020/09/2009-issue-383.html
https://www.molympiad.org/
https://www.molympiad.org/
https://www.molympiad.org/2020/09/2009-issue-383.html
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