2008 Issue 368

  1. Consider $n$ consecutive points $A_{1}, A_{2}, A_{3}, \ldots, A_{n}$ on the same line such that $$A_{1} A_{2}=A_{2} A_{3}=A_{3} A_{4}=\ldots=A_{n-1} A_{n}.$$ Find $n,$ given that there are exactly $2025$ segments on that line whose midpoints are one of these $n$ points.
  2. Let $A B C$ be a right triangle with right angle at $A$. On the halfplane divided by $B C$ which does not contain $A,$ choose the points $D$, $E$ such that $B D$ is orthogonal with $B A$ and $B D=B A$, $B E$ is orthogonal with $B C$ and $B E=B C$. Denote by $M$ the midpoint of $C E .$ Prove that $A$, $D$ and $M$ are colinear.
  3. Find all positive integers $x$, $y$ and $z$ such that $$2 x y-1=z(x-1)(y-1).$$
  4. Solve for $x$ $$4 x-x^{2}=3 \sqrt{4-3 \sqrt{10-3 x}}.$$
  5. Let $A B C$ be a right triangle with right angle at $A$ and let $A D$ be the angle bisector at $A .$ Denote by $M$ and $N$ the bases of the altitudes from $D$ onto $A B$ and $A C$ respectively. $B N$ meets $C M$ at $K$ and $A K$ meets $D M$ at $I$. Find the measure of angle $B I D$.
  6. Let $$f(x)=2009 x^{5}-x^{4}-x^{3}-x^{2}-2006 x+1 .$$ Prove that $f(n)$, $f(f(n))$, $f(f(f(n)))$ are pairwise coprime for any positive integer $n$.
  7. Find $a$, $b$ such that $\max _{0 \leq x \leq 16}|\sqrt{x}+a x+b|$ is smallest possible. Find this minimum value.
  8. The incircle of a triangle $A B C$ touches $B C$, $C A$ and $A B$ at $A^{\prime}$, $B^{\prime},$ and $C^{\prime}$ respectively. Prove that $$A B^{\prime 2}+B C^{\prime 2}+C A^{\prime 2} \geq A B^{\prime} . B^{\prime} C^{\prime}+B C^{\prime} \cdot C^{\prime} A^{\prime}+C A^{\prime} . A^{\prime} B^{\prime} \geq B^{\prime} C^{\prime 2}+C^{\prime} A^{\prime 2}+A^{\prime} B^{\prime 2}.$$ When does equality occur?
  9. Let $k$ be a positive integer. Write $k$ as a product of prime numbers (not necessarily distinct, for instance, $k=18=3.3 .2)$ and let $T(k)$ be the sum of all factors in the factorization above. Find the largest constant $C$ such that $T(k) \geq C \ln k$ for all positive integer $k$.
  10. Let $a, b, c$ be non-negative real numbers whose sum of squares equal $3 .$ Find the maximum value of the following expression $$P=a b^{2}+b c^{2}+c a^{2}-a b c$$
  11. Let $\left(x_{n}\right)(n=1,2, \ldots)$ be a sequence, determined by the following recursive formula $$x_{1}=\frac{1}{2},\quad x_{n+1}=x_{n}-x_{n}^{2}+x_{n}^{3}-x_{n}^{4}+\ldots+x_{n}^{2007}-x_{n}^{2008},\,\forall n \in \mathbb{N}^{*}.$$ Find the limit $\displaystyle \lim_{n \rightarrow+\infty} n x_{n}$.
  12. Let $A B C D$ be a tetrahedron whose altitudes are concurrent. Denote by $R$ the radius of its circumcircle; by $h_{1}$, $h_{2}$, $h_{3}$, $h_{4}$ the lengths of the altitudes corresponding to the vertices $A$, $B$, $C$, $D$ respectively; and by $R_{1}$, $R_{2},$ $R_{3}$, $R_{4}$ the circumcircles's radius of the opposite faces of the vertices $A$, $B$, $C$, $D$ respectively. Prove the following inequality $$\frac{1}{h_1+2 \sqrt{2} R_{1}}+\frac{1}{h_{2}+2 \sqrt{2} R_{2}}+\frac{1}{h_{3}+2 \sqrt{2} R_{3}}+\frac{1}{h_{4}+2 \sqrt{2} R_{4}} \geq \frac{1}{R}.$$ When does equality occur?




Mathematics & Youth: 2008 Issue 368
2008 Issue 368
Mathematics & Youth
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