2008 Issue 373

  1. Write the numbers $1,2,3, \ldots, 2007$ in an arbitrary order and let $A$ be the resulting number. Can $A+2008^{2007}+2009$ be a perfect square?.
  2. Consider the following two polynomials $$f(x)=(x-2)^{2008}+(2 x-3)^{2007}+2006 x$$ and $$g(y)=y^{2009}-2007 y^{2008}+2005 y^{2007}.$$ Let $s$ be denote the sum of all the coefficients of $f(x)$ (after expansion). Find $s$, and the value of $g(s)$.
  3. Find all positive integer solutions of the following system of two equations $$\begin{cases} x+y+z &= 15 \\ x^{3}+y^{3}+z^{3} &= 495\end{cases}.$$
  4. Let $a, b, c$ be non-negative real numbers such that $a^{2}+b^{2}+c^{2}=1 .$ Find the maximum value of the expression $$(a+b+c)^{3}+a(2 b c-1)+b(2 a c-1)+c(2 a b-1).$$
  5. Given a triangle $A B C$ where $\widehat{A B C}$ is not a right angle. Let $A H$ and $A M$ denote, the altitude and the median throught vertex $A$. Choose a point $E$ on the ray $A B$ and $F$ on the ray $A C$ such that $M E$ $=M F=M A .$ Let $K$ be reflection point of $H$ over $M .$ Prove that the four points $E$, $M$, $K$ and $F$ lie on a single circle.
  6. Solve for $x$ $$\sqrt{x+\sqrt{x^{2}-1}}=\frac{9 \sqrt{2}}{4}(x-1) \sqrt{x-1}.$$
  7. Prove that in any acute triangle $A B C$, the following inequality holds $$\frac{\tan A}{\tan B}+\frac{\tan B}{\tan C}+\frac{\tan C}{\tan A} \geq \frac{\sin 2 A}{\sin 2 B}+\frac{\sin 2 B}{\sin 2 C}+\frac{\sin 2 C}{\sin 2 A}$$
  8. The incircle of a triangle $A B C$ meets $B C$, $C A$ and $A B$ respectively at $A_{1}$, $B_{1}$, $C_{1}$. Let $p$, $S$, $R$ be respectively, half of the perimeter, the area and the circumradius of $A B C$. Let $p_{1}$ be half of the perimeter of $A_{1} B_{1} C_{1}$. Prove the inequality $$p_{1}^{2} \leq \frac{p S}{2 R}.$$ When does equality occur?
  9. Let $A(A \subset \mathbb{N})$ be a non-empty set satisfying the condition: If $a \in A$ then $4 a$ and $[\sqrt{a}]$ are also in $A([x]$ is the integer part of $x$). Prove that $A=\mathbb{N}$.
  10. Let $a$ be a natural number which is greater than $3$ and consider the sequence $\left(u_{n}\right)$ $(n=1,2, \ldots)$ defined inductively by $u_{1}=a$ and $$u_{n+1}=u_{n}-\left[\frac{u_{n}}{2}\right]+1,\,\forall n=1,2, \ldots.$$ Prove that there exists $k \in \mathbb{N}^{*}$ such that $u_{n}=u_{k}$ for all $n \geq k$.
  11. Find all polynomials with real coefficients $P(x)$, $Q(x)$ and $R(x)$ such that $$\sqrt{P(x)}-\sqrt{Q(x)}=R(x),\,\forall x.$$
  12. Let $A B C D$ be a tetrahedron with the centroid $G$ and the circumradius $R$. Prove that $$G A+G B+G C+G D+4 R \geq \frac{2}{\sqrt{6}}(A B+A C+A D+B C+C D+D B).$$




Mathematics & Youth: 2008 Issue 373
2008 Issue 373
Mathematics & Youth
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