2010 Issue 400

  1. Given $$A=\frac{1}{4} \cdot \frac{3}{6} \cdot \frac{5}{8} \cdot . . \frac{995}{998} \cdot \frac{997}{1000}$$ and $$B=\frac{2}{5} \cdot \frac{4}{7} \cdot \frac{6}{9} \ldots \frac{996}{999} \cdot \frac{998}{1001}.$$
    a) Compare $A$ and $B$.
    b) Prove that $A<\dfrac{1}{12900}$.
  2. Let $A B C$ be a triangle, the median $B M$ and the angle bisector $C D$ meets at $J$ and $J B=J C .$ From $A$ draw $A H$ perpendicular to $B C$. Prove that $J M=J H$.
  3. Assume that $n \in \mathbb{N}$, $n \geq 2$. Consider all natural numbers $a_{n}=\overline{11 \ldots 1}$ consisting of exactly $n$ digits $1 .$ Prove that if $a_{n}$ is a prime number then $n$ is a divisor of $a_{n}-1$.
  4. Let $a, b, c, d$ be real numbers in the half-open interval $\left(0 ; \frac{1}{2}\right] .$ Prove that $$\left(\frac{a+b+c+d}{4-a-b-c-d}\right)^{4} \geq \frac{a b c d}{(1-a)(1-b)(1-c)(1-d)}$$
  5. Let $A B C D$ be a square whose side length is $a, M$ is an arbitrary point on $A B$ $(M \neq A, M \neq B)$. $M C$ meets $B D$ at $P$, $M D$ cuts $A C$ at $Q$. Find the maximum value of the area of triangle $M P Q$ and the minimum value of the area of the quadrilateral $C P Q D$.
  6. Solve the equation $$25 x+9 \sqrt{9 x^{2}-4}=\frac{2}{x}+\frac{18 x}{x^{2}+1}$$
  7. Let $A B C$ be a triangle with incenter $I$ and centroid $G$. Let $R_{1}$, $R_{2}$, $R_{3}$ be the circumradii of the triangles $I B C$, $I C A$ and $I A B$ respectively. Let $R_{1}^{\prime}$, $R_{2}^{\prime}$, $R_{3}^{\prime}$ be the of circumradii of the triangles $G B C$, $G C A$ and $G A B$ respectively. Prove that $$R_{1}^{\prime}+R_{2}^{\prime}+R_{3}^{\prime} \geq R_{1}+R_{2}+R_{3}$$
  8. Let $f:|a ; b| \rightarrow \mathbb{R}(0<a<b)$ be a continuous function on $|a ; b|$ and differentiable on $(a ; b)$, $f(x) \neq 0$ for all $x \in(a ; b)$. Prove that there exists $c \in(a ; b)$ so that $$\frac{2}{a-c}<\frac{f^{\prime}(c)}{f(c)}<\frac{2}{b-c}$$
  9. Let $\left(a_{n}\right)(n=1,2, \ldots)$ be a sequence given by $$a_{1}=1,\quad a_{n+1}=1+\frac{1}{a_{n}+1},\,\forall n \in \mathbb{N}^{\circ} .$$ Prove that $$a_{1}^{2}+a_{2}^{2}+\ldots+a_{2010}^{2}<4020$$
  10. Given any set $A \subset \mathbb{R},$ let $A+1$ be the set $A+1=\{a+1 \mid a \in A\} .$ How many subsets $A$ of set $\{1,2, \ldots, n\}(n \geq 1, n \in \mathbb{N})$ are there such that $A \cup(A+1)=\{1,2, \ldots, n\}$?
  11. Find all the functions $f: \mathbb{R}^{+} \rightarrow \mathbb{R}^{+}$ so that $$f(x) f(y)=\beta f(x+y f(x)),\,\forall x, y \in \mathbb{R}^{+}$$ (for a given $\beta \in \mathbb{R}$, $\beta>1$.)
  12. Let $A B C$ be a triangle and $M$ be the midpoint of the arc $B C$ of its circumcircle. Let $I$, $J$, $K$ be the projections of $M$ onto the lines $A B$, $B C$, $C A$ respectively; $X$ is the intersection of $B K$ and $A J$; $L$ is the intersection of $C X$ and $1 J$. The ray $J y$ perpendicular to $M K$ cuts $A L$ at $T$. Prove that $C T$ is perpendicular to $I M$.




Mathematics & Youth: 2010 Issue 400
2010 Issue 400
Mathematics & Youth
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