2010 Issue 395

  1. Compare the following two numbers $$A=\frac{2^{2009}+1}{2^{2010}+1} \text { and } B=\frac{2^{2010}+1}{2^{2011}+1}.$$
  2. Let $A B C$ be a triangle with $\widehat{B A C}=45^{\circ}$, $A M$ is its median, $A D$ is the angle bisector of the triangle $M A C$, draw $D K$ perpendicular to $A B$ ($K$ lies on $A B$). $A M$ cuts $D K$ at $I$. Prove that if $A M$ is the angle bisector of $\widehat{B A D}$ then $B I$ is the angle bisector of $\widehat{A B D}$.
  3. Find all positive numbers $a$ and $b$ such that $\dfrac{a^{2}+b}{b^{2}-a}$ and $\dfrac{b^{2}+a}{a^{2}-b}$ are both integers.
  4. Find the minimum value of the expression $P=a+b+c$ given that $3 \leq a, b,c \leq 5$ and $a^{2}+b^{2}+c^{2}=50$.
  5. Let $A B C$ be a triangle. The angle bisector of $\widehat{B A C}$ cuts the angle bisector of $\widehat{A B C}$ at $I$ and meets $B C$ at $E$. The line perpendicular to $A E$ at $E$ meets the arc $\widehat{B I C}$ of the circumcircle of the triangle $B I C$ at $H$. Prove that $A H$ touches the arc $\widehat{B I C}$.
  6. Let $I$ be the incenter of the triangle $A B C$ with $B C=a$, $C A=b$ and $A B=c$. The lines $A I$, $B I$, $C I$ cut the circumcircle of $A B C$ at $A^{\prime}$, $B^{\prime}$, $C^{\prime}$ respectively. Prove that $$(p-a) I A^{\prime 2}+(p-b) I B^{\prime 2}+(p-c) I C^{\prime 2}=\frac{1}{2} a b c$$ where $p=\dfrac{1}{2}(a+b+c)$.
  7. Let $A B C$ ($B C=a$, $A C=b$, $A B=c$) be a triangle where $A$, $B$, $C$ satisfying the condition $$\cos A+\cos B=2 \cos C.$$ Prove the inequality $$c \geq \frac{8}{9} \max \{a, b\}.$$ When does the equality occur?
  8. Solve the equation $$x^{\log _{7} 11}+3^{\log _{7} x}=2 x.$$
  9. Solve the equation $$\sqrt[3]{3 x+4}=x^{3}+3 x^{2}+x-2$$
  10. Let $X$ be the subset of the set $\{1,2,\ldots, 2010\}$ satisfying conditions $|X|=62$ and for every $x \in X$, there exist $a, b \in X \cup\{0 ; 2011\}$ ($a$ and $b$ differ from $x$) such that $x=\dfrac{a+b}{2}$. Prove that there exist two elements $x$, $y$ in $X$ such that $|x-y| \geq 11$ and $\dfrac{x+y}{2}$ is not in $X$.
  11. Make a torus-shaped chessboard by first identifying a pair of opposite edges of an $n \times n$ chessboard to get a cylinder and then identifying the opposite bases of the resulting cylinder. Prove that it is possible to place $n$ queens on this torus chessboard so that none of them are able to capture any other using the standard chess queen's moves if and only if $(n, 6)=1$. (A queen can capture another if they share the same row, column or diagonal.)
  12. Let ABCDEF be an inscribed hexagon, $A C$ is parallel to $D F$ and $B E$ is the circumdiameter. $A B$ cuts $E F$ at $M$ and $B C$ cuts $D E$ at $N$; $I$ is the intersection point of $A N$ and $CM$. Prove that $E I$ is perpendicular to $A C$.




Mathematics & Youth: 2010 Issue 395
2010 Issue 395
Mathematics & Youth
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