Mathematics and Youth Magazine Problems 2010

Issue 391

1. Does there exist a positive integer $k$ such that $2^{k}+3^{k}$ is a perfect square?
2. Let $A B C$ be a triangle and let $M$ be the midpoint of $B C$ such that $A B=5cm$,$ A M=6cm$ and $A C=13cm$. The line through $B$ and perpendicular to $B C$ meets $A M$ at $D,$ the line through $C$ and perpendicular to $B C$ meets $A B$ at $E$. Prove that $C D$ is perpendicular to $M E$
3. Find all pair of real numbers $a$ and $b$ so that $a+b=\dfrac{\sqrt[4]{8}}{2}$ and $A=a^{4}-6 a^{2} b^{2}+b^{4}$ is a positive integer.
4. Let $O$ be the midpoint of a line segment $A B=2 a$. In the half-plane with edge $A B$, draw two rays $A x$, $B y$, both perpendicular to $A B$. Choose $M$ and $N$ on $A x$ and $B y$ respectively such that $M N=A M+B N$. Let $H$ be the foot of the altitude from $O$ onto $M N$. Find the positions of $M$ and $N$ such that the area of the triangle $H A B$ is greatest possible.
5. Without using trigonometry formula, prove the following equalities
   a) $\cos 36^{\circ} \cdot \cos 72^{\circ}=\dfrac{1}{4}$.
   b) $\tan 36^{\circ} \cdot \tan 72^{\circ}=\sqrt{5}$.
6. Solve the equation $$3 x^{4}-4 x^{3}=1-\sqrt{\left(1+x^{2}\right)^{3}}$$
7. Does there exist a polynomial $P(x)$ of degree $2010$ such that $P\left(x^{2}-2010\right)$ is divisible to $P(x)$?.
8. Let $Oxyz$ be a right trihedral with right angle at $O$ and $A$, $B$, $C$ move on the sides $O x$, $O y$ and $O z$ respectively so that the area of the triangle $A B C$ is a constant $S$. Let $S_{1}$, $S_{2}$, $S_{3}$ be the areas of the triangles $O B C$, $OCA$, $OAB$ respectively. Find the greatest value of the expression $$P=\frac{S_{1}}{S+2 S_{1}}+\frac{S_{2}}{S+2 S_{2}}+\frac{S_{3}}{S+2 S_{3}}$$
9. Let $a, b, c$ be positive real numbers. Prove that $$\min \left\{\frac{a b}{c^{2}}+\frac{b c}{a^{2}}+\frac{c a}{b^{2}} ; \frac{a^{2}}{b c}+\frac{b^{2}}{c a}+\frac{c^{2}}{a b}\right\} \geq \max \left\{\frac{a}{b}+\frac{b}{c}+\frac{c}{a} ; \frac{b}{a}+\frac{c}{b}+\frac{a}{c}\right\}$$
10. For a given positive integer $n$, how many $n$ -digit natural numbers can be formed from five possible digits $1,2,3,4$ and $5$ so that an odd numbers of $1$ and even numbers of $2$ are used?
11.  sequence $\left(x_{n}\right)$, $n=0,1, \ldots$ is given by $$x_{0}=\alpha,\quad x_{n}=\sqrt{1+\frac{1}{x_{n}+1}},\, n=0,1, \ldots$$ where $\alpha$ is greater than $1$. Determine $\displaystyle\lim_{n\to\infty} x_{n}$.
12. Let $A B C$ be a triangle with the altitudes $A A^{\prime}$, $B B^{\prime}$, $C C^{\prime}$ meet at $H$. Prove that $$\frac{H A}{H A^{\prime}}+\frac{H B}{H B^{\prime}}+\frac{H C}{H C^{\prime}}+6 \sqrt{3} \geq 6+\frac{a}{H A^{\prime}}+\frac{b}{H B^{\prime}}+\frac{c}{H C^{\prime}}$$

Issue 392

1. Does there exist a pair of integers $x$, $y$ such that $$x^{3}-y^{3}=10 \times 10 \times 2010$$
2. Let $n$ be a natural number, greater than $1$. Prove that $$\frac{1+n}{1+n^{n+1}}>\left(\frac{1+n^{n}}{1+n^{n+1}}\right)^{n}$$
3. Let $a, b, c, x, y, z$ be positive integers satisfying the conditions $$\begin{cases}\dfrac{1}{x}+\dfrac{1}{y}+\dfrac{1}{z} &=2010 \\ a x^{3}=b y^{3}=c z^{3}\end{cases}.$$ Prove that $$x+y+z \geq \frac{3}{670}.$$
4. Let $A B C$ be a triangle whose sides satisfying the relation $$B C^{2}+A B \cdot A C-A B^{2}=0.$$ Determine the sum $\hat{A}+\dfrac{2}{3} \hat{B}$.
5. Two orthogonal diameters $A E$ and $B F$ of a circle center $O$ radius $R$ are given. $A$ point $C$ is chosen on the minor arc $E F$. The chord $A C$ intersects $B F$ at $P$ and the chord $B C$ meets $A E$ at $Q$. Find the area of the quadrilateral $A P Q B$ in term of $R$.
6. Solve the system of equations $$\begin{cases}\sqrt{y^{2}-8 x+9}-\sqrt[3]{x y+12-6 x} &\leq 1 \\ \sqrt{2(x-y)^{2}+10 x-6 y+12}-\sqrt{y} &=\sqrt{x+2}\end{cases}.$$
7. A quadrilateral $A B C D$ is inscribed in a circle centered $I$ and circumscribed another circle with center at $O$. The diagonals $A C$ and $B D$ intersect at $E$. Prove that $E$, $I$ and $O$ are collinear.
8. Let $\left(x_{n}\right)$ be a sequence given by $$x_{1}=5, \quad x_{n+1}=x_{n}^{2}-2,\,\forall n \geq 1 .$$ Find
   a) $\displaystyle\lim _{n \rightarrow+\infty} \frac{x_{n+1}}{x_{1} x_{2} \ldots x_{n}}$.
   b) $\displaystyle\lim _{n \rightarrow+\infty}\left(\frac{1}{x_{1}}+\frac{1}{x_{1} x_{2}}+\ldots+\frac{1}{x_{1} x_{2} \ldots x_{n}}\right)$.
9. Prove that the equation $P(x)=2^{x}$ where $P(x)$ is a polynomial of degree $n,$ has less than $n+1$ roots.
10. Find all continuous functions $f: \mathbb{R} \rightarrow \mathbb{R}$ satisfying the condition $$f(2010 x-f(y))=f(2009 x)-f(y)+x,\, \forall x, y \in \mathbb{R}$$
11. Let $\left(a_{n}\right)$ be a sequence with $$a_{1}=a_{2}=1,\quad a_{n+2}=a_{n+1}+a_{n},\,\forall n \geq 1.$$ Find all pair of positive integers $a$, $b$, $a<b$ so that $a_{n}-2 n a^{n}$ is a multible of $b$ for any $n \geq 1$.
12. $I$ and $R$ are the incenter and the circumradius of a gven triangle $A B C$. $I A$, $I B$, $I C$ intersect the circumcircle at $A_{1}$, $B_{1}$, $C_{1}$ respectively. Prove the inequality $$2 R+\frac{L A+I B+I C}{3} \leq L A_{1}+I B_{1}+I C_{1} \leq \frac{5}{2} R+\frac{I A+I B+I C}{6}$$

Issue 393

1. Let $a_{1}, a_{2}, \ldots, a_{2010}$ be natural numbers such that $$\frac{1}{a_{1}^{11}}+\frac{1}{a_{2}^{11}}+\frac{1}{a_{3}^{11}}+\ldots+\frac{1}{a_{2010}^{11}}=\frac{1005}{1024}.$$ Determine the value of the following expression $$A=\frac{a_{2010}^{6}}{a_{1}^{5}}+\frac{a_{2009}^{6}}{a_{2}^{5}}+\frac{a_{2008}^{6}}{a_{3}^{5}}+\ldots+\frac{a_{1}^{6}}{a_{2010}^{5}}$$
2. In a triangle $A B C$ where $B$ and $C$ are acute angles, let $B D$ and $A H$ be respectively the angle bisector and the altitude. Given that $\widehat{A D B}=\widehat{A H D}=\alpha,$ find the measure of $\alpha$.
3. Find all integer solutions of the equation $$y^{3}=x^{6}+2 x^{4}-1000$$
4. Find the minimum value of the expression $$P=\frac{a}{b+c+d-a}+\frac{b}{c+d+a-b}+\frac{c}{d+a+b-c}+\frac{d}{a+b+c-d}$$ where $a, b, c, d$ are the length of 4 sides of a convex quadrilateral.
5. Let $A B C$ be an isosceles triangle with vertex at $C$. Let $O$, $I$ be its circumcenter and incenter respctively. $D$ is a point chosen on the side $B C$ so that $D O$ is perpendicular to $B I$. Prove that $D I$ is parallel to $A C$.
6. Let $a, b, c$ be positive real numbers satisfying the condition $$\frac{1}{a+b+1}+\frac{1}{b+c+1}+\frac{1}{c+a+1} \geq 1.$$ Prove that $$a+b+c=a b+b c+c a$$
7. Let $A B C D E$ be a cyclic pentagon with $A C \parallel D E$ and $\widehat{A M B}=\widehat{B M C}$ where $M$ is the midpoint of $B D$. Prove that the line $B E$ passes through the midpoint of $A C$.
8. Let $O A B C$ be a tetrahedron with right trihedral angle at vertex $O$. Prove that $$\cot A B \cdot \cot B C+\cot B C \cdot \cot C A+\cot C A \cdot \cot A B \leq \frac{3}{2}$$ where cot $A B$ is the cotangent of the dihedral angle of side $A B$.
9. Let $A B C$ be an acute triangle. The altitudes $B K$ and $C L$ meet at $H$. The line passing through $H$ meets $A B$, $A C$ at $P$, $Q$ respectively. Prove that $H P=H Q$ if and only if $M P=M Q$ where $M$ is the midpoint of $B C$
10. Find all real numbers $k$ and $m$ such that $$k\left(x^{3}+y^{3}+z^{3}\right)+m x y z \geq(x+y+z)^{3}$$ for any non-negative numbers $x, y, z$
11. Let $\left(x_{n}\right)$ be a sequence of real numbers, $n=1,2, \ldots$ satisfying $$\ln \left(1+x_{n}^{2}\right)+n x_{n}=1$$ for any positive integers $n$. Find $$\lim _{n \rightarrow+\infty} \frac{n\left(1+n x_{n}\right)}{x_{n}}$$
12. Find all continuous functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that $$f(x+f(y))=2 y+f(x),\, \forall x, y \in \mathbb{R}$$

Issue 394

1. Find all pair of natural numbers $x, y$ such that $$\left(2^{x}+1\right)\left(2^{x}+2\right)\left(2^{x}+3\right)\left(2^{x}+4\right)-5^{y}=11879$$
2. Let $n$ be a positive integer and let $U(n)=\left\{d_{1} ; d_{2} ; \ldots ; d_{m}\right\}$ be the set of all positive divisors of $n$. Prove that $$d_{1}^{2}+d_{2}^{2}+\ldots+d_{m}^{2} \leq n^{2} \sqrt{n}$$
3. Prove that $$\frac{1}{a^{4}(a+b)}+\frac{1}{b^{4}(b+c)}+\frac{1}{c^{4}(c+a)} \geq \frac{3}{2}$$ where $a$, $b$, $c$ are three positive numbers satisfying $a b c=1$.
4. Solve the equation $$3 \sqrt{x^{3}+8}=2 x^{2}-6 x+4$$
5. Let $A B C D$ be a square, $M$ is a point lying on $C D$ ($M \neq C$, $M \neq D$).  Through the point $C$ draw a line perpendicular to $A M$ at $H$ $B H$ meets $A C$ at $K$. Prove that a) $M K$ is always parallel to a fixed line when $M$ moves on the side $C D$.
   b) The circumcenter of the quadrilateral $ADMK$ lies on a fixed line.
6. Let $a, b, c$ be positive real numbers such that $a b c=1$. Prove that $$\frac{1}{\sqrt{a^{3}+2 b^{3}+6}}+\frac{1}{\sqrt{b^{3}+2 c^{3}+6}}+\frac{1}{\sqrt{c^{3}+2 a^{3}+6}} \leq 1$$
7. Consider all triangles $A B C$ where $A<B<C \leq \frac{\pi}{2}$. Find the least value of the expression $$M=\cot ^{2} A+\cot ^{2} B+\cot ^{2} C +2(\cot A-\cot B)(\cot B-\cot C)(\cot C-\cot A)$$
8. Let $A B C$ be a triangle with $B C=a$, $A C=b$, $A B=c$. A line $d$ passing through its incenter meets $A B$, $A C$, $B C$ respectively at $M$, $N$, $P$. Prove that $$\frac{a}{\overline{B P} \cdot \overline{P C}}+\frac{b}{C N \cdot N A}+\frac{c}{\overline{A M} \cdot \overline{M B}}=\frac{(a+b+c)^{2}}{a b c}$$
9. Let $x, y, z$ be non-zero real numbers such that $x+2 y+3 z=5$ and $2 x y+6 y z+3 x z=8$. Prove that $$1 \leq x \leq \frac{7}{3} ; \frac{1}{2} \leq y \leq \frac{7}{6} ; \frac{1}{3} \leq z \leq \frac{7}{9}$$
10. Solve the system of equations $$\begin{cases} \sqrt[3]{x}+\sqrt[3]{y} &=\sqrt[3]{3(x+y)} \\ 4 x^{3}+6 x^{2}+4 x+1 &=15 y^{4}\end{cases}$$
11. Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ satisfy $$f(f(x)+y)=f(x+y)+x f(y)-x y-x+1$$
12. Suppose that the tetrahedron $A B C D$ satisfies the following conditions: All faces are acute triangles and $B C$ is perpendicular to $AD$. Let $h_{a}$, $h_{d}$ be respectively the lengths of the altitudes from $A$, $D$ onto the opposite faces, and let $2 \alpha\left(0^{\circ}<\alpha<45^{\circ}\right)$ be the measure of the dihedral angle at edge $B C$, $d$ is the distance between $B C$ and $A D$. Prove the inequality $$\frac{1}{h_{a}}+\frac{1}{h_{d}} \leq \frac{1}{d \cdot \sin \alpha}$$

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 $A C$.

Issue 396

1. Let $A=14916 . . .4040100$ be the number obtained by writing the perfect squares $1^{2}, 2^{2}, \ldots, 2010^{2}$ consecutively. Let $B+C$ be the sum obtained by putting the sign "$+$" in between certain two digits of $A$. Is $B+C$ divisible by $9$? Explain your reasoning?
2. Let $A B C$ be a triangle with the altitude $A H$ satisfying $B C=A H \sqrt{2}$. Compute the measure of the angle $\widehat{A C B}$, given that $\widehat{A B C}=67^{\circ} 30^{\prime}$.
3. Let $n_{1}, n_{2}, \ldots, n_{m}$ be a sequence of strictly decreasing natural numbers. For each natural number $n$, put $$P_{n}=2\left(3^{n}+3^{n_{1}}+3^{n_{2}}+\ldots+3^{n_{m}}\right).$$ Does there exist an $n$ with $n>n_{1}$ such that $P_{n}$ is a perfect square?
4. Let $x, y, z$ be real numbers satisfying $x \geq 2$, $y \geq 9$, $z \geq 1945$, $x+y+z=2010$. Find the greatest value of the product $x y z$.
5. Let $A B C$ be a triangle. Let $M$, $N$, $P$ be the points of contact of its incircle $(I)$ with the sides $A B$, $A C$, $B C$ respectively and let $M D$, $N E$, $P F$ be the altitudes of the triangle $M N P$. Prove that $$D A \cdot F B \cdot E C=E A \cdot D B \cdot F C.$$
6. Let $a, b, c$ be positive real numbers satisfying $a^{2}+b^{2}+c^{2}=1$. Prove the inequality $$\frac{1}{1-a b}+\frac{1}{1-b c}+\frac{1}{1-c a} \leq \frac{9}{2(1+9 a b c-a-b-c)}$$
7. Let $\alpha \in\left(0 ; \frac{\pi}{2}\right)$. Find the minimum value of the expression $$P=(\cos \alpha+1)\left(1+\frac{1}{\sin \alpha}\right)+(\sin \alpha+1)\left(1+\frac{1}{\cos \alpha}\right)$$
8. Let $S . A B C$ be a pyramid with $S A=a$, $S B=b$, $S C=c$ and $$\widehat{A S B}=\widehat{B S C}=\widehat{C S A}=\alpha.$$ Compute its volume in term of $a, b, c$ and $\alpha$.
9. Let $n$ be a positive integer. Let $p(n)$ be the product of its nonzero digits. (If $n$ has a single digit then $p(n)=n$). Consider the expression $S=p(1)+p(2)+\ldots+p(999)$. What is the greatest prime divisor of $S ?$
10. Let $(u_n)$ $(n = 0, 1, 2, ...)$ be the sequence given by $$u_{0}=0,\quad u_{n+1}=\frac{u_{n}+2008}{-u_{n}+2010}.$$ a) Prove that the sequence $\left(u_{n}\right)$ $(n=0,1,2, \ldots)$ converges and find limit of $u_{n}$.
    b) Put $\displaystyle T_{n}=\sum_{k=0}^{n} \frac{1}{u_{k}-2008} \cdot$ Find $\displaystyle \lim_{n\to\infty} \frac{T_{n}}{n+2009}$.
11. Solve the equation $$4 \sqrt{x+2}+\sqrt{22-3 x}=x^{2}+8$$
12. Let $A B C$ be an acute triangle with orthocenter $H$. Let $R$, $r$ be its the circumradius and inradius respectively. Prove that $$\max \left\{\frac{H B}{H C}+\frac{H C}{H B^{\prime}} + \frac{H C}{H A}+\frac{H A}{H C^{\prime}} + \frac{H A}{H B}+\frac{H B}{H A}\right\} \geq \frac{2 R}{r}-2$$

Issue 397

1. Compare the following two fractions $$A=\frac{1010^{1010}}{2010^{2010}} \text { and } B=\frac{2010^{2010}}{3010^{3010}}$$
2. Let $A B C$ be a triangle with $\widehat{B A C} \geq 60^{\circ}$. Prove that $A B+A C \leq 2 B C$.
3. Find the remainder when dividing $3^{2^{n}}$ by $2^{n+3}$ where $n$ is a positive integer.
4. Let $A B C$ be a triangle. Construct a parallelogram $AMPN$ so that the points $M$, $N$ are in $A B$, $A C$ respectively; $P$ lies inside the triangle $A B C$. Let $Q$ be the intersection of the line $A P$ and $B C$. Prove that $$\frac{A M \cdot A N \cdot P Q}{A B \cdot A C \cdot A Q} \leq \frac{1}{27}.$$ Find the position of $P$ when the equality occurs.
5. Solve the equation $$\left(x+\frac{5-x}{\sqrt{x}+1}\right)^{2}=\frac{-192(\sqrt{x}+1)}{5 \sqrt{x}-x \sqrt{x}}$$
6. Two circles $\left(O_{1}\right)$ and $\left(O_{2}\right)$ meet at points $K$ and $L$ such that their centers $O_{1}$ and $O_{2}$ lie on the same side of the line $K L .$ The tangent line to $\left(O_{1}\right)$ at $K$ meets $\left(O_{2}\right)$ at $A .$ The tangent line to $\left(O_{2}\right)$ at $K$ meets $\left(O_{1}\right)$ at $B$. Find the area of the triangle $A K B,$ given that $A L=3$, $B L=6$ and $\tan \widehat{A K B}=-\dfrac{1}{2}$.
7. Let $a, b, c$ be positive real numbers satisfying $a+b+c=3$. Prove the following inequality $$\sqrt[3]{a}+\sqrt[3]{b}+\sqrt[3]{c}+5 \geq(a+b)(b+c)(c+a).$$ When does equality occur?
8. Solve the system of equations $$\begin{cases}x&=3 z^{3}+2 z^{2} \\ y&=3 x^{3}+2 x^{2} \\ z&=3 y^{3}+2 y^{2}\end{cases}$$
9. Let $A B C$ be an acute triangle. Let $a$, $b$, $c$ be the side lengths of the triangle and $h_{a}$, $h_{b}$, $h_{c}$ be the length of the corresponding altitudes. Let $r$, $R$ be respectively the inradius and circumradius of this triangle. Prove the inequality $$\frac{9 R}{a^{2}+b^{2}+c^{2}} \leq \frac{1}{h_{a}+\sqrt{h_{b} h_{c}}}+\frac{1}{h_{b}+\sqrt{h_{c} h_{a}}}+\frac{1}{h_{c}+\sqrt{h_{a} h_{b}}} \leq \frac{1}{2 r}$$
10. Let $A$ be the set of $n$ distinct points on the plane $(n \geq 2)$ and $T(A)$ be the set of vectors whose endpoints are in $A$. Find the maximum and minimum value of $|T(A)|$. (where $|T(A)|$ denotes the cardinality of $T(A)$.)
11. Let $f$ be a continuous function on $\mathbb{R}$ satisfying the following two conditions $$f(2012)=2011,\quad f(x)f_{4}(x)=1,\, \forall x \in \mathbb{R}.$$ Denote $f_{n}(x)=\underbrace{f(f \ldots f(x)))}_{n \text { times } f}$. Find $f(2010)$
12. Let $\left(x_{n}\right)$ $(n=1,2, \ldots)$ be a sequence given by $$x_{1}=2,1;\quad x_{n+1}=\frac{x_{n}-2+\sqrt{x_{n}^{2}+8 x_{n}-4}}{2}.$$ For each positive integer $n$, let $\displaystyle y_{n}=\sum_{i=1}^{n} \frac{1}{x_{i+1}^{2}-4}$. Find $\displaystyle\lim_{n \rightarrow+\infty} y_{n}$.

Issue 398

1. Let $n$ be a positive integer so that the first digit of $2^{n}$ and $5^{n}$ are the same. Prove that the number obtained by writing $2^{n}$ and $5^{n}$ consecutively has $n+1$ digits, where the digit 3 appears at least twice.
2. Let $A B C$ be an isosceles triangle with $A B=A C$. Point $E$ on the median $B D$ is chosen so that $\widehat{D A E}=\widehat{A B D}$. Prove that $\widehat{D A E}=\widehat{E C B}$.
3. Find all positive integer solutions of the equation $$x(x+2 y)^{3}-y(y+2 x)^{3}=27$$
4. Find the value of the following expression $$P=\sqrt{12 \sqrt[3]{2}-15}+2 \sqrt{3 \sqrt[3]{4}-3}$$
5. Let $ABC$ be a triangle with $\widehat{A C B}=70^{\circ}$, $\widehat{A B C}=50^{\circ}$. Let $D$, $E$ be respectively the midpoints of $B C$ and $A D$. Draw $EF$ perpendicular to $B C$ ($F$ is on $B C$). Let $M$ be a point on $E F$; let $N$, $P$ be respectively the orthogonal projections of $M$ onto $A C$, $A B$. Given that the three points $N$, $E$, $P$ are colinear, find the measure of angle $M A B$.
6. Find the least value of the expression $$T=3 \sqrt{1+2 x^{2}}+2 \sqrt{40+9 y^{2}}$$ where $x, y$ are non-negative real numbers such that $x+y=1$.
7. Let $A B C$ be an acute triangle. Prove that $$\frac{\cos A}{\cos \frac{B}{2} \cos \frac{C}{2}}+\frac{\cos B}{\cos \frac{C}{2} \cos \frac{A}{2}}+\frac{\cos C}{\cos \frac{A}{2} \cos \frac{B}{2}} \geq 2.$$
8. Let $A B C$ be a triangle. A straight line cut the lines $B C$, $C A$ and $A B$ at $A^{\prime}$, $B^{\prime}$ and $C^{\prime}$ respectively. Let $A^{\prime \prime}$, $B^{\prime \prime}$ and $C^{\prime \prime}$ be the points reflection of $A^{\prime}$, $B^{\prime}$ and $C^{\prime}$ with centers at $A$, $B$ and $C$ respectively. Prove that the area of the triangle $A B C$.
9. The positive integers are colored with either black or white such that the sum of two numbers with different color is painted black, and there are infinitely many numbers with white color. Let $q$ $(q>1)$ be the smallest positive integer with black color. Prove that $q$ is prime.
10. Find all functions $f: \mathbb{N}^{*} \rightarrow \mathbb{N}^{*}$ such that $$f\left(f^{2}(m)+2 f^{2}(n)\right)=m^{2}+2 n^{2},\,\forall m, n \in \mathbb{N}^{*}$$
11. The sequence $\left(x_{n}\right)(n \geq 1)$ of real numbers is defined inductively as follows $$x_{1}=a \in \mathbb{R},\quad x_{n+1}=2 x_{n}^{3}-5 x_{n}^{2}+4 x_{n},\,\forall n \geq 1 .$$ Find all possible values of $a$ such that the sequence $\left(x_{n}\right)$ has finite limit. Determine the limit of $\left(x_{n}\right)$ with respect to each such value of $a$.
12. Let $A B C D$ be a tetrahedron. Find all points $P$ inside the tetrahedron such that $$x d_{A}+y d_{B}+z d_{C}+t d_{D}=c$$ where $x, y, z, t, c$ are given positive constants and $d_{A}$, $d_{B}$, $d_{C}$, $d_{D}$ are respectively the distances from $P$ to the four faces $B C D$, $C D A$, $D A B$, $A B C$ of the tetrahedron.

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 $$\cos A \cos B \cos C+8 \sqrt{3} \cos \frac{A}{2} \cos \frac{B}{2} \cos \frac{C}{2} \geq 24 \cos ^{2} \frac{A}{2} \cos ^{2} \frac{B}{2} \cos ^{2} \frac{C}{2}-1$$

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 $I M$.

Issue 401

1. Let $n$ be a natural number greater than $11$. Does there exist a natural number $x$ so that $n^{2010}<x<n^{2011}$ and the last 2011 digits of $x$ are $0 ?$
2. Let $p$ be a prime number, $a$ and $b$ are natural numbers $(a<b)$ such that the sum of all irreducible fractions with denominator $p$ which lies between $a$ and $b$ is equal to $2011 .$ Find the values of $p$, $a$, $b$.
3. Let $a$ be an $n$ -digits natural number (in decimal system) and $a^{3}$ has $m$ digits. Can $n+m$ be equal to $2011 ?$ Why?
4. Solve the equation $$x+y+z+\sqrt{x y z}=2(\sqrt{x y}+\sqrt{y z}+\sqrt{z x}-2).$$
5. Let $A B C D$ be a parallelogram. The angle-bisector of $B A D$ meets $B C$, $D C$ at $M$, $N$ respectively. Let $E$ be the other intersection point of the circumcircles of the triangles $B C D$ and $C M N .$ Find the measure of angle $A E C$.
6. Find the least value of $$A=\frac{2}{|a-b|}+\frac{2}{|b-c|}+\frac{2}{|c-a|}+\frac{5}{\sqrt{a b+b c+c a}}$$ where $a$, $b$, $c$ are real numbers satisfying $a+b+c=1$ and $a b+b c+c a>0$
7. Solve the system of equations $$\begin{cases}4+9.3^{x^{2}-2 y} &= \left(4+9^{x^{2}-2 y}\right) \cdot 7^{2 y-x^{2}+2} \\ 4^{x}+4 &= 4 x+4 \sqrt{2 y-2 x+4}\end{cases}$$
8. Let $A B C D$ be a square of side $4 a$. $M$, $N$ are points on the spheres $S(D ; a)$, $S(C ; 2 a)$ Determine the minimum value of the sum $M A+2 N B+4 M N$.
9. Let $A B C$ be an acute triangle with $A B \neq A C$. Let $P$ be a point inside the triangle so that $\widehat{P B A}=\widehat{P C A},$ draw lines $P M$ and $P N$ perpendicular to $A B$ and $A C$ respectively. $O$ is the midpoint of $B C$. The angle-bisectors of $B A C$ and $M O N$ intersects at $R .$ Prove that the circumcircles of the triangles $B M R$ and $C N R$ meet at another point on the line segment $B C$.
10. Let $a$ be a positive real number. Let $\left(x_{n}\right)(n=1,2, \ldots)$ be a sequence defined by $$x_{1}=a,\quad x_{n+1}=\frac{x_{n} \sqrt{2+\sqrt{2+\ldots+\sqrt{2}}}}{x_{n}+1},\,\forall n=1,2, \ldots$$ (there are exactly $n$ numbers $2$ in the numerator). Prove that the sequence $\left(x_{n}\right)(n=1,2, \ldots)$ has a finite limit and find this limit.
11. Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ satisfying the condition $$f(x+f(y))=f^{4}(y)+4 x^{3} f(y)+6 x^{2} f^{2}(y)+4 x f^{3}(y)+f(-x)$$ for all $x$, $y$ in $\mathbb{R}$.
12. Let $A B C$ be an equilateral triangle with circumradius $R$ and let $P$ be a point inside the triangle. Prove that $$P A \cdot P B \cdot P C \leq \frac{9}{8} R^{3}.$$

Issue 402

1. Compare $\dfrac{5}{24}$ ưith the sum $$\frac{1}{1.2 .4}+\frac{1}{2.5 .7}+\frac{1}{3.8 .10}+\ldots+\frac{1}{1000.2999 .3001}$$
2. Let $A B C$ be an isosceles right triangle, with right angle at vertex $A .$ Let $D$ be a point inside the triangle such that $\Delta A B D$ is an isosceles triangle and $\widehat{A D B}=150^{\circ} .$ Let $A C E$ be an equilateral triangle so that points $D$ and $E$ are on the different side of the half-plane $A C .$ Prove that $B$, $D$, $E$ are collinear.
3. Find all positive integer numbers $m$ such that the following equation $$x^{2}-m x y+y^{2}+1=0$$ has positive integer roots.
4. Let $A B C$ be an acute triangle and let $A H$, $B K$, $C L$ be its three altitudes. Prove that $$A K \cdot B L \cdot C H=A L \cdot B H \cdot C K=H K \cdot K L \cdot L H.$$
5. Solve the equation. $$\frac{8 x\left(1-x^{2}\right)}{\left(1+x^{2}\right)^{2}}-\frac{2 \sqrt{2} x(x+3)}{1+x^{2}}=5-\sqrt{2}$$
6. Let $a$, $b$, $c$ be three positive real numbers such that $a b c=1$. Prove that $$\dfrac{1}{\sqrt{a^{5}-a^{2}+3 a b+6}}+\dfrac{1}{\sqrt{b^{5}-b^{2}+3 b c+6}} +\frac{1}{\sqrt{c^{5}-c^{2}+3 c a+6}} \leq 1.$$
7. Solve the equation $$2 \sin \left(x+\frac{\pi}{3}\right) +2^{2} \sin \left(x+\frac{2 \pi}{3}\right)+\ldots +2^{2010} \sin \left(x+\frac{2010 \pi}{3}\right)=0.$$
8. Let $O A B C$ be a tetrahedron where $O A$, $O B$, $O C$ are pairwise orthogonal. Let $M$ be a point on the plane containing the base $A B C$. Let $G_{1}$, $G_{2}$, $G_{3}$ be respectively the centroids of triangles $O A B$, $O B C$ and $O C A$. Put $O A=a$, $O B=b$, $O C=c$. Prove the inequality $$M G_{1}^{2}+M G_{2}^{2}+M G_{3}^{2} \geq \frac{a^{2} b^{2} c^{2}}{a^{2} b^{2}+b^{2} c^{2}+c^{2} a^{2}}.$$ When does equality occur?
9. Let $A B C$ be an acute triangle with altitude $A D$. $M$ is a point on $A D$. The lines $B M$, $C M$ meets $A C$, $A B$ at $E$, $F$ respectively. $D E$, $D F$ intersects with the circles whose diameters $A B$, $A C$ at $K$, $L$ respectively. Prove that the line connecting the midpoints of $E F$, $K L$ goes through $A$.
10. Prove that there exist infinitely many triples of positive integers $(a, b, c)$ such that $a b+1$, $b c+1$, $c a+1$ are all square numbers.
11. Find all positive integers $a$, $b$, $c$ so that the equation $$x+3 \sqrt[4]{x}+\sqrt{4-x}+3 \sqrt[4]{4-x}=a+b+c$$ has solution and the expression $P=a b+2 a c+3 b c$ is greatest possible.
12. Find all positive real numbers $a$ such that there exists a positive real number $k$ and a function $f: \mathbb{R} \rightarrow \mathbb{R}$ such that $$\frac{f(x)+f(y)}{2} \geq f\left(\frac{x+y}{2}\right)+k|x-y|^{a}$$ for all real numbers $x$, $y$.