2007 Issue 360

  1. Consider the following sequence $$a_{1}=3,\, a_{2}=4,\, a_{3}=6, \ldots, a_{n+1}=a_{n}+n, \ldots$$ a) Is $2006$ a number in the above sequence?
    b) What is the $2007^{\text {th }}$ number in this sequence?
    c) Calculate the sum of the first $100$ numbers in the above sequence?
  2. Let $A B C$ be a right triangle with right angle at $A,$ and $\widehat{A C B}=54^{\circ} .$ Choose a point $E$ on the open ray in opposite direction to $C A$ such that $\widehat{A B E}=54^{\circ} .$ Prove that $B C < A E$
  3. Consider the following sum of $2006$ terms $$S=\sqrt{\frac{2+1}{2}}+\sqrt[3]{\frac{3+1}{3}}+\sqrt[4]{\frac{4+1}{4}}+\ldots+200 \sqrt[2]{\frac{2007+1}{2007}}.$$ Find $[S]$. (Here $[a]$ denote the largest integer which does not exceed $a$.)
  4. Solve the following system of equations $$\begin{cases} x-\dfrac{4}{x} &=2 y-\dfrac{2}{y} \\ 2 x &=y^{3}+3\end{cases}$$
  5. Let $a, b, c$ be numbers, all greater than or equal $-\dfrac{3}{2},$ such that $$a b c+a b+b c+c a+a+b+c \geq 0.$$ Prove that $a+b+c \geq 0$.
  6. Let $M$, $N$ and $P$ be three points outside a given triangle $A B C$ such that $$\angle C A N=\angle C B M=30^{\circ},\,\angle A C N=\angle B C M=20^{\circ},\,\angle P A B=\angle P B A=40^{\circ} .$$ Show that $M N P$ is an equilateral triangle.
  7. Let $A B C$ be a right triangle with right angle at $A$ and let $A H$ be the altitute through $A$. Denote by $I$, $I_{1}$, $I_{2}$ the incenters of the triangles $A B C$, $A H B$ and $A H C$. respectively. Prove that the circumcircle of the triangle $I I_{1} I_{2}$ is exactly the incircle of the triangle $A B C$.
  8. Find all functions $f: \mathbb{R}^{+} \rightarrow \mathbb{R}^{+}$ which satisfies the following identity $$ f(x) \cdot f(y)=f(x+y f(x)), \forall x, y \in \mathbb{R}^{+}$$
  9. Find the largest real number $m$ such that there exists a number $k$ in the interval $[1 ; 2]$ such that the following inequality $$\left(\frac{a}{b}+\frac{b}{c}+\frac{c}{a}\right)^{2 k} \geq(k+1)(a+b+c)\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)+m$$ is satisfied for all positive real numbers $a, b, c$.
  10. Let $\left(u_{k}\right)$ $(k=1,2, \ldots, n,)$ be an increasing sequence and let $A_{n}$ be the set of all positive numbers of the form $u_{i}-u_{j}$ $(1 \leq j<i \leq n)$. Prove that if $A_{n}$ has fewer that $n$ elements, then $\left(u_{k}\right)$ $(k=1,2, \ldots, n)$ forms an arithmetic sequence.
  11. Let $A B C D$ be a quadrilateral with nonparallel opposite sides and let $O$ denote the intersection of the diagonals $A C$ and $B D .$ The circumcircles of the triangles $O A B$ and $O C D$ meet at $X$ and $O$. The circumcircles of the triangles $O A B$ and $O C B$ meet at $Y$ and $O .$ The circles with diameters $A C$ and $B D$ intersect at $Z$ and $T$. Prove that either $X$, $Y$, $Z$, $T$ are colinear or they lie in a circle.
  12. Consider a closed polygon $A_{1} A_{2} \ldots A_{n}$ where each of its $n$ sides is tangent to a sphere $\mathscr{C}$ at center $O$ and radius $R .$ Let $G$ be the centroid of the system of points $A_{i}$, $i=1,2, \ldots, n$ (that is, $\overline{G A_{1}}+\overline{G A_{2}}+\ldots+\overline{G A_{n}}=\overrightarrow{0}$) and denote $\angle A_{i}=\angle A_{i-1} A_{i} A_{i+1}$ (with the convention that $A_{0} \equiv A_{n}$ and $A_{n+1} \equiv A_{1}$.) Prove the inequality $$ \sum_{i=1}^{n} \frac{G A_{i}}{\sin \frac{A_{i}}{2}} \geq \frac{1}{n R} \sum_{1 \leq i<j \leq n} A_{i} A_{j}^{2}$$




Mathematics & Youth: 2007 Issue 360
2007 Issue 360
Mathematics & Youth
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