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2008 Issue 378

  1. Find all natural numbers $a$ such that both $a+593$ and $a-159$ are perfect squares.
  2. Let $A B C$ be a right triangle, with right angle at $A$ and $\widehat{A C B}=15^{\circ}$. Let $B C=a$, $A C=b$, $A B=c .$ Prove that $a^{2}=4 b c$.
  3. Find all intergers $x, y, z, t$ such that $$x^{2008}+y^{2008}+z^{2008}=2007 . t^{2008}.$$
  4. Prove the inequality $$\left(\frac{4}{a^{2}+b^{2}}+1\right)\left(\frac{4}{b^{2}+c^{2}}+1\right)\left(\frac{4}{c^{2}+a^{2}}+1\right) \geq 3(a+b+c)^{2}$$ where $a, b, c$ are positive numbers such that $a^{2}+b^{2}+c^{2}=3$.
  5. Let $A B C$ be an acute triangle. Choose a point $D,$ different from $B$ and $C,$ on the side $B C .$ Prove that the vertex $A$ and the centers of the circumcircles of the triangles $A B D$, $A C D$ and $A B C$ lie on the same circle.
  6. Find the coefficient of $x^{2}$ in the expansion of $$\left(\left(\ldots\left((x-2)^{2}-2\right)^{2}-\ldots-2\right)^{2}-2\right)^{2}$$ given that the number 2 occurs 1004 times in the expression above and there are 2008 round brackets.
  7. Find the smallest value of the following expression $$\frac{\sqrt{a_{1}+2008}+\sqrt{a_{2}+2008}+\ldots+\sqrt{a_{n}+2008}}{\sqrt{a_{1}}+\sqrt{a_{2}}+\ldots+\sqrt{a_{n}}}$$ where $n$ is a given positive natural number and $a_{1}, a_{2}, \ldots, a_{n}$ are non-negative real numbers such that $a_{1}+a_{2}+\ldots+a_{n}=n$
  8. Let $(O)$ be a circle centered at $O$ and fixed diameter $A B .$ Let $\Delta$ be a straight line which touches $(O)$ at $A .$ Choose a point $M$ on the circle $(O),$ different from $A$ and $B .$ The tangent line with $(O)$ through $M$ meets $\Delta$ at $C$. Let $(I)$ be the circle through $M$ and touches $\Delta$ at $C$. Let $C D$ be the diameter of $(I)$. Prove that
    a) $D O C$ is an isosceles triangle.
    b) The line through $D$ and perpendicular to $B C$ always passes through a fixed point when $M$ moves on the circle $(O)$.
  9. Does there exist a sequence of positive integers $a_{2003}>a_{2002}>\ldots>a_{2}>a_{1}$ with $a_{1}=2003$ such that the following two conditions are satisfied
    • All integers in the interval $\left(2003 ; a_{2 \times 13}\right)$ are either a member of this sequence or a non-prime.
    • $A=\dfrac{2004}{a_{1}}+\dfrac{2004}{a_{2}}+\ldots+\dfrac{2004}{a_{2003}}$ is an integer?.
    1. Find all continuous functions $f, g, h$ on $\mathbb{R}$ such that $$f(x+y)=g(x)+h(y)$$ for all real numbers $x$, $y$.
    2. Let $H$ and $O$ denote the orthocenter and circumcenter respectively of a triangle $A B C$. Prove the inequality $$3 R-2 O H \leq H A+H B+H C \leq 3 R+O H$$ where $R$ is its circumradius.
    3. Let $A B C D . A_{1} B_{1} C_{1} D_{1}$ be a cubic whose side is a. Let $N$, $M$ be two points on $A B_{1}$ and $B C_{1}$ respectively such that the angle between $M N$ and the plane $(A B C D)$ is $60^{\circ}$. Prove that $$M N \geq 2 a(\sqrt{3}-\sqrt{2}).$$ When does equality occur?.

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    Mathematics & Youth: 2008 Issue 378
    2008 Issue 378
    Mathematics & Youth
    https://www.molympiad.org/2020/09/2008-issue-378.html
    https://www.molympiad.org/
    https://www.molympiad.org/
    https://www.molympiad.org/2020/09/2008-issue-378.html
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