2010 Issue 399

  1. Find a four digits perfect square, given that all four digits are distinct, and if these digits are written in reverse order, the result is also a perfect square, and is divisible by the original number. 
  2. Determine all possible choices of three integers $x$, $y$ and $z$ such that $$x^{2}+y^{2}+z^{2}+3<x y+3 y+2 z.$$
  3. Let $A B C$ be a triangle where the length of the altitudes $A H$ is $6 \mathrm{cm}, B H$ is $3 \mathrm{cm}$ and the measure of angle $C A H$ is three times the measure of angle $B A H$. Find the area of this triangle. 
  4. Find the greatest value of the expression $$M=\frac{232 y^{3}-x^{3}}{2 x y+24 y^{2}}+\frac{783 z^{3}-8 y^{3}}{6 y z+54 z^{2}}+\frac{29 x^{3}-27 z^{3}}{3 x z+6 x^{2}}$$ where $x, y$ and $z$ are positive numbers satisfying the condition $x+2 y+3 z=\dfrac{1}{4}$ 
  5. Let $A B C$ be a right triangle with right angle at $A$ and $A B<A C$. Let $H$ be the projection of $A$ onto $B C$, let $M$ be the point reflection of $H$ across $A B$. $M C$ meets the circumcircle of triangle $A B H$ at $P$ $(P \neq M)$, $H P$ meets the circumcircle of triangle $A P C$ at $N$ $(N \neq P)$. Let $E$ and $K$ be respectively the intersections of $A B$ and $B C$ with the circumcircle of triangle $A P C$ $(E \neq A, K \neq C)$. Prove that
    a) $E N$ is parallel to $B C$.
    b) $H$ is the midpoint of $B K$. 
  6. Find the interger part of $A$, where $$A=\sqrt{2}+\sqrt[3]{\frac{3}{2}}+\sqrt[4]{\frac{4}{3}}+\ldots+2010 \sqrt{\frac{2010}{2009}}$$
  7. Find the least value of the following expression $$A=\frac{x}{1+y^{2}}+\frac{y}{1+x^{2}}+\frac{z}{1+t^{2}}+\frac{t}{1+z^{2}}$$ where $x, y, z, t$ are nonnegative real numbers satisfying $x+y+z+t=k$ ($k$ is a given positive number).
  8. Let $A B C$ be a given triangle and $M$ is a point which is not on its sides. Prove that $$P_{A /(M C B)}=P_{B /(M C A)}=P_{A /(M A B)}$$ if and only if $M$ is the centroid of triangle $A B C$. ($P_{T /(X Y Z))}$ is the power of the point $T$ with respect to the circle through $X$, $Y$, $Z$.)
  9. Let $A B C$ be a triangle. A circle intersects with the sides $B C$, $C A$ and $A B$ at pairs of two points $(M, N)$; $(P, Q)$ and $(S, T)$ respectively, where $M$ lies between $B$ and $N$; $P$ lies between $C$ and $Q,$ and $S$ lies between $A$ and $T$. Let $K$, $H$, $L$ be respectively the intersections of $S N$ and $Q M$; $Q M$ and $T P$; $T P$ and $S N$. Prove that the lines $A K$, $B H$, $C L$ are concurrent.
  10. Let $\left(a_{n}\right)$ be a sequence of numbers such that $$a_{0}=10,\quad \left(6-a_{n}\right)\left(16+a_{n-1}\right)=96,\, n=0,1,2, \ldots$$ Find the sum $$S=\frac{1}{a_{0}}+\frac{1}{a_{1}}+\ldots+\frac{1}{a_{2010}}$$
  11. Find all functions $f: \mathbb{R}^{+} \rightarrow \mathbb{R}^{+}$ such that the following equality holds $$f(x+y)+f(x y)=x+y+x y,\,\forall x, y \in \mathbb{R}^{+}.$$
  12. Let $A B C$ be a triangle. Prove that $$\cos A \cos B \cos C+8 \sqrt{3} \cos \frac{A}{2} \cos \frac{B}{2} \cos \frac{C}{2} \geq 24 \cos ^{2} \frac{A}{2} \cos ^{2} \frac{B}{2} \cos ^{2} \frac{C}{2}-1$$




Mathematics & Youth: 2010 Issue 399
2010 Issue 399
Mathematics & Youth
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