2009 Issue 379

  1. Find all pairs of integers $a$, $b$ such that $$a^{2}+a b+b^{2}=a^{2} b^{2}$$
  2. Let $ABC$ be an isosceles triangle (at vertex $A$) such that $\widehat{B A C} \geq 90^\circ$. Choose a point $M$ on $A C$, and let $A H$ and $C K$ be the altitudes from $A$ and $C$ onto $B M$ respectively $(H, K$ are the feet of these altitudes) such that $B H=H K+K C$. Find the angle $B A C$.
  3. Solve for $x$$$\sqrt{\frac{5 \sqrt{2}+7}{x+1}}+4 x=3 \sqrt{2}-1 $$
  4. Find $a$, $b$ such that the maximum value of the expression $$|||x+1|-2|-(a x+b)|$$ where $-3 \leq x \leq 4$ is smallest possible.
  5. Let $A B C D$ be a rectangular, let $I$ be the midpoint of $C D$, and let $E$ be a point on $A B$. The altitude onto $D E$ through $I$ meets $D E$ and $A D$ at $M$ and $H$ respectively. The altitude onto $C E$ through $I$ meets $C E$ and $B C$ at $N$ and $K$, respectively. $E I$ intersects with $H K$ at $G$. Prove that
    a) The points $E$, $G$, $N$, $K$, $B$ lie on the same circle.
    b) The points $E$, $G$, $M$, $H$, $A$ lie on the same circle.
  6. Let $A B C$ be a triangle. Prove the inequality $$\cos A \cos B \cos C \leq \frac{1}{8} \cos (B-C) \cos (C-A) \cos (A-B)$$
  7. Solve for $x$ $$3^{x}\left(4^{x}+6^{x}+9^{x}\right)=25^{x}+2.16^{x}$$
  8. Prove that the union of six hemispheres whose diameters are the sides of a given tetrahedron must contain the tetrahedron itself.
  9. Let $A B C$ be a triangle. Choose the pair of points $A_{1}$, $A_{2}$; $B_{1}$, $B_{2}$; $C_{1}$, $C_{2}$ on the sides $B C$, $C A$ and $A B$ respectively such that $A_{1} A_{2} B_{1} B_{2} C_{1} C_{2}$ is a convex hexagon with equal opposite sides and the triangles $A A_{1} A_{2}$, $B B_{1} B_{2}$, $CC_{1}C_{2}$ have the same areas. Prove that the lines $A_{1} B_{2}$, $B_{1} C_{2}$, $C_{1} A_{2}$ are colinear.
  10. Let $a$, $b$, $c$ be non - negative real numbers such that $$\sqrt[3]{a^{3}+b^{3}}+\sqrt[3]{b^{3}+c^{3}}+\sqrt[3]{c^{3}+a^{3}}+a b c=3.$$ Prove that the smallest value of the expression $$P=\frac{a^{3}}{b^{2}+c^{2}}+\frac{b^{3}}{c^{2}+a^{2}}+\frac{c^{3}}{a^{2}+b^{2}}$$ is $\sqrt[3]{32}m$ where $m$ is the real root of the equation $t^{3}+54 t-162=0$.
  11. Let $f: \mathbb{N} \rightarrow \mathbb{R}$ be the function with initial values $f(0)=0$, $f(1)=1$ such that $$f(n+2)-2011 f(n+1)+f(n)=0.$$ What is the probability that $f(n)$ is a prime number, where $n$ is a randomly chosen number from the set of integers $\{0,1,2, \ldots, 2008\}$?
  12. Find all functions $f(x)$ such that it is continuous on $[0 ; 1]$, differentiable on the open interval $(0 ; 1)$ and the following two conditions are satisfied $$f(0)=f(1)=\frac{2009}{11},\quad 20 f(x)+11 f(x)+2009 \leq 0,\, \forall x \in(0 ; 1).$$




Mathematics & Youth: 2009 Issue 379
2009 Issue 379
Mathematics & Youth
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