2009 Issue 386

  1. Find the last two decimal digits of the sum $2008^{2009}+2009^{2008}$
  2. Let $ABC$ be an isosceles right triangle with the right angle at $A$. Let $G$ be a point on $A B$ such that $A G=\dfrac{1}{3} A B$, let $M$ be the midpoint of $B C$ and $E$ be the foot of the altitude from $M$ to $CG$. The two lines $M G$ and $A C$ meet at $D$. Prove that $D E=B C$.
  3. Find all integer solutions of the equation $$4 x^{4}+2\left(x^{2}+y^{2}\right)^{2}+x y(x+y)^{2}=132$$
  4. Let $a$, $b$ and $c$ satisfy the conditions $a \leq b \leq c$ and $$a+b+c=\frac{1}{a}+\frac{1}{b}+\frac{1}{c}.$$ Find the least value of the expression $P=a b^{2} c^{3}$.
  5. A triangle $A B C$ inscribed in a circle centered at $O$, radius $R$, $A D$ is its anglebisector. Let $E$, $F$ be the circumeenters pf the triangles $A B D$ and $A C D$. respectively. Given that $B C=a$. Determine the area of the quadrilateral $A E O F$.
  6. Let $A B C$ be a triangle with incenter $I$ such that its centroid $G$ lies inside $(I)$. Let $a$, $b$, $c$ be the lengths of the sides $B C$, $A C$, $A B$, respectively. Find the greatest and least value of the following expression $$P=\frac{a^{2}+b^{2}+c^{2}}{a b+b c+c a}$$
  7. Let $x$, $y$, $z$ be positive numbers satisfying $$x^{2}+y^{2}+z^{2}=\frac{1-16 x y z}{4}.$$ Find the least value of the expression $$S=\frac{x+y+z+4 x y z}{1+4 x y+4 y z+4 z x}.$$
  8. Solve for $x$ $$2^{x}+5^{x}=2-\frac{x}{3}+44 \log _{2}\left(2+\frac{131 x}{3}-5^{x}\right)$$
  9. Let $A B C$ be a right triangle, right angle at $A$, $M$ is the midpoint of $B C$. Construct a right angle $P M Q$ with $P \in A B$, $Q \in A C$. Prove that $$P Q^{2} \geq A P \cdot C Q+A Q \cdot B P$$
  10. Let $a_{n}$ be the last non-zero digit (counting from left to right) when expressing $n !$ in the decimal number system. Is the sequence $\left(a_{n}\right)$ for $n=1.2 .3 \ldots$ periodic? (That is, there exist the positive integers $T$ and $N$ such that $a_{i+T}=a_{i} \forall i \geq N$).
  11. Given $n$ non-negative numbers $a_{1}, a_{2}, \ldots, a_{n}$ $(n \geq 3)$ satisfying $$a_{1}^{2}+a_{2}^{2}+\ldots+a_{n}^{2}=1.$$ Prove that $$\frac{1}{\sqrt{3}}\left(a_{1}+a_{2}+\ldots+a_{n}\right) \geq a_{1} a_{2}+a_{2} a_{3}+\ldots+a_{n} a_{1}.$$ When does equality occur?
  12. Let $\left(x_{n}\right)(n=1,2, \ldots)$ be a sequence given by $$x_{1}=a \ (a>1),\, x_{2}=1,\quad x_{n+2}=x_{n}-\ln x_{n},\,\forall n \in \mathbb{N}^{*}.$$ Put $\displaystyle S_{n}=\sum_{k=5}^{n-1}(n-k) \ln \sqrt{x_{2 k-1}}$ $(n \geq 2)$. Find $\displaystyle\lim_{n \rightarrow \infty}\left(\frac{S_{n}}{n}\right)$.




Mathematics & Youth: 2009 Issue 386
2009 Issue 386
Mathematics & Youth
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