2007 Issue 361

  1. The number $(\overline{9 x})^{8}$, where $x \in\{0,1,2, \ldots, 9\}$ contains how many digits when written in decimal form?
  2. Let $H$ be the orthocenter of an acute triangle $A B C$ and denote by $M$ the midpoint of $B C$. The perpendicular line to $M H$ through $H$ meets $A B$ and $A C$ at $P$ and $Q$ respectively. Prove that $H P=H Q$.
  3. Prove that if $a, b, c$ and $d$ are distinct positive integers such that the sum $$\frac{a}{a+b}+\frac{b}{b+c}+\frac{c}{c+d}+\frac{d}{d+a}$$ is also an integer then the product $a b c d$ is a perfect square.
  4. Compare the values of the following two numbers $$A=\min _{|y| \leq 1} \max _{|x| \leq 1}\left(x^{2}+y x\right) ,\quad B=\max _{|x| \leq 1} \min _{|y| \leq 1}\left(x^{2}+y x\right).$$
  5. Solve the system of equations $$\begin{cases}\sqrt{x}+\sqrt{y}+\sqrt{z}-\dfrac{1}{\sqrt{x}}-\dfrac{1}{\sqrt{y}}-\dfrac{1}{\sqrt{z}} &=\dfrac{8}{3} \\ x+y+z+\dfrac{1}{x}+\dfrac{1}{y}+\dfrac{1}{z} &=\dfrac{118}{9} \\ x \sqrt{x}+y \sqrt{y}+z \sqrt{z}-\dfrac{1}{x \sqrt{x}}-\dfrac{1}{y \sqrt{y}}-\dfrac{1}{z \sqrt{z}} &=\dfrac{728}{27}\end{cases}.$$
  6. Let $A B C D$ be a parallelogram. Let $M$ be a point in the plane spanned by the parallelogram $A B C D$ such that $\widehat{M D A}=\widehat{M B A}$. Prove that the two triangles $M A B$ and $M C D$ share a common orthocenter.
  7. Let $AD$, $BE$, $CF$ be the three bisectors of a triangle $A B C$ ($D \in B C$,$ E \in C A$, $F \in A B)$. Denote by $O$ the circumcenter of this triangle and by $R$ its. Let $O_{1}$, $O_{2}$ and $O_{3}$ be the circumcenters of the triangles $A B D$. $B C E$ and $A C F$ respectively. Prove that $$\frac{3}{2} R \leq O O_{1}+O O_{2}+O O_{3}<2 R.$$
  8. Let $a, b, c$ be positive real numbers such that $\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c} \leq 1 .$ Find the smallest possible value of $$P=[a+b]+[b+c]+[c+a]$$ where $[x]$ is the largest integers which is smaller than $x$.
  9. Find all funtions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that the following equality holds for all $x, y \in \mathbb{R}$ $$f\left(x^{3}-y\right)+2 y\left(3 f^{2}(x)+y^{2}\right)=f(y+f(x)).$$
  10. Prove that in any triangle $A B C$, the following inequality holds $$-1<6 \cos A+3 \cos B+2 \cos C<7$$
  11. Let ABC be a triangle with circumcircle $(O ; R)$. Let $E$ be the midpoint of $A B$ and denote by $F$ the point on $A C$ such that $\dfrac{A F}{A C}=\dfrac{1}{3} .$ Construct a parallelogram $A E M F$ Prove that $$M A+M B+M C \leq \sqrt{11\left(R^{2}-O M^{2}\right)}.$$ Now suppose that $(O ; R)$ is fixed. Construct a triangle $A B C,$ inscribed in $(O, R)$ such that $$M A+M B+M C=\sqrt{11\left(R^{2}-O M^{2}\right)}.$$
  12. Let $A B C D$ be a tetrahedron whose sides $D A$, $D B$ and $D C$ are pairwise perpendicular. Denote by $x, y, z$ the angles formed from sides $A B$, $B C$ and $C A$ respectively. Prove that $$\left(2+\tan ^{2} x\right)\left(2+\tan ^{2} y\right)\left(2+\tan ^{2} z\right) \geq 64.$$




Mathematics & Youth: 2007 Issue 361
2007 Issue 361
Mathematics & Youth
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