Mathematics and Youth Magazine Problems 2004

Issue 319

1. Find solutions in positive integers of the equation $$3^{x}+1=(y+1)^{2}.$$
2. Let be given a triangle $A B C$ with $B C=2 A B$. Let $M$ be the midpoint of $B C$ and $D$ be the midpoint of $B M$. Prove that $A C=2 A D$.
3. Find the prime divisor $p<300$ of the number $2^{37}-1=137438953471$.
4. Solve the equation $$\left(16 x^{4 n}+1\right)\left(y^{4 n}+1\right)\left(z^{4 n}+1\right)=32 x^{2 n} y^{2 n} z^{2 n}$$ where $n$ is a given positive integer.
5. Let $a, b, c$ be real numbers satisfying the conditions $$a b c>0,\quad |a b+b c+c a|=2 \sqrt{2004 a b c}.$$ Prove that $$(a+b-2004)(b+c-2004)(c+a-2004) \leq 0 .$$
6. $A B C$ is an arbitrary not acute triangle. Put $A B=c$, $B C=a$, $C A=b$. Find the least value of the expression $\dfrac{(a+b)(b+c)(c+a)}{a b c}$.
7. The circle $(O, R)$ with center $O$, radius $R$ cuts the circle $\left(O^{\prime}, R\right)$ with center $O^{\prime}$, radius $R^{\prime}$ at $A$ and $B$. From a point $C$ on the opposite ray of the ray $A B$, draw the tangents $C D$ and $C E$ to the circle $(O, R)$ ($D$ and $E$ are tangent points, $E$ lies inside $\left(O^{\prime}, R^{\prime}\right)$. $A D$ and $A E$ cut again the circle $\left(O^{\prime}, R\right)$ respectively at $M$ and $N$. Prove that the line $D E$ passes through the midpoint of $M N$.
8. Consider the colourings of a rectangular board of size $m \times n(m+n \geq 3)$ such that $k$ little squares of the board are coloured and each not coloured little squares has at least a common point with a coloured little square. Find the least value of $k$.
9. Prove that $$x^{2} y+y^{2} z+z^{2} x \leq x^{3}+y^{3}+z^{3} \leq 1+\frac{1}{2}\left(x^{4}+y^{4}+z^{4}\right)$$ for non negative real numbers $x, y, z$ satisfying the condition $x+y+z=2$.
10. Find all functions $f: \mathbb N^{*} \rightarrow \mathbb N^{*}$ satisfying the condition $$2\left(f\left(m^{2}+n^{2}\right)\right)^{3}=f^{2}(m) \cdot f(n)+f^{2}(n) \cdot f(m)$$ for all distinct $m, n \in \mathbb N^{*}$.
11. Let $A D$, $B E$, $C F$ be the altitudes of an acute triangle $A B C$. Let $M$, $N$, $P$ be respectively the points of intersection of the segments $A D$ and $E F$, $B E$ and $F D$, $C F$ and $D E$. Let $S$ denote the area of triangle. Prove that $$\frac{1}{S_{A B C}} \leq \frac{S_{M N P}}{S_{D E F}^{2}} \leq \frac{1}{8 \cos A \cdot \cos B \cdot \cos C \cdot S_{A B C}}$$
12. Consider arbitrary regular quadrilateral pyramids $S A B C D$, the measure $\alpha$ of the flat angle $\widehat{A S B}$ of which satisfies $0<\alpha \leq$ $60^{\circ}$. Let $\varphi$ be the measure of the bihedral angle of side $S A$. Determine $\alpha$ so that the expression $P=\cos 3 \varphi-9 \cos \varphi$ attains its greatest value.

Issue 320

1. Write the sum of $18$ fractions $$1, \frac{1}{2}, \frac{1}{3}, \ldots, \frac{1}{17}, \frac{1}{18}$$ in the form of an irreducible fraction $\dfrac{a}{b}$. Prove that $b$ is divisible by $2431$.
2. Given a triangle $A B C$ with $A B > A C$, the foot of its altitude $A H$ lies inside $B C$. The angled-bisectors of $\widehat{A B C}$ and of $\widehat{A C B}$ cut $A H$ respectively at $E$ and $F$. Prove that $B E>E F+F C$.
3. Find positive integers $a \geq b \geq c$ and $x \geq y \geq z$ so that $$\begin{cases} a+b+c &=x y z \\ x+y+z &=a b c \end{cases}$$
4. Solve the equation $$(x-2) \sqrt{x-1}-\sqrt{2} x+2=0.$$
5. Find the greatest value of the expression $$\sqrt{4 x-x^{3}}+\sqrt{x+x^{3}}$$ where $0 \leq x \leq 2$.
6. The circle $(O)$ with center $O$ cuts the circle $\left(O^{\prime}\right)$ with center $O^{\prime}$ at $P$ and $Q$. Their common tangent (nearer to $P$) touches $(O)$ at $A$, $\left(O^{\prime}\right)$ at $B$. Let $C$ be the point of intersection of the tangents to $(O)$ at $P$ with the circle $\left(O^{\prime}\right)$. Let $D$ be the point of intersection of the tangents to $\left(O^{\prime}\right)$ at $P$ with the circle $(O)$. Let $M$ be the point such that $A B$ and $P M$ have common midpoint. The line $A P$ cuts $B C$ at $E$ and the line $B P$ cuts $A D$ at $F$. Prove that $A M B E Q F$ is a hexagone inscribed in a circle.
7. Construct a triangle $A B C$ with given $P$, $Q$, $R$ so that $B$ is the midpoint of $A P$, $C$ is the midpoint of $B Q$, $A$ is the midpoint of $C R$.
8. Prove the following equalities for positive integer $n$.
   a) $\displaystyle \sum_{k=1}^{n} \frac{(-1)^{k-1}}{2 k-1} C_{n}^{k} \cdot C_{n+k-1}^{k-1}=1$.
   b) $\displaystyle \sum_{k=1}^{n} \frac{(-1)^{k-1} \cdot k^{n}}{2 k-1} C_{n}^{k}=\frac{(n !)^{2} \cdot 2^{n}}{(2 n) !}$
9. Solve the following system of equations of $n$ unknowns $$\begin{cases}\sqrt{x_{1}}+\sqrt{x_{2}}+\ldots+\sqrt{x_{n}} &=n \\ \sqrt{x_{1}+8}+\sqrt{x_{2}+8}+\ldots+\sqrt{x_{11}+8} &=3 n\end{cases}$$ ($n$ is a given positive integer). Generalize the problem.
10. Find the greatest value of the function $$f(x)=\sqrt{2} \sin x+\sqrt{15-10 \sqrt{2} \cos x}$$
11. Let $r$ and $R$ be respectively the inradius and the circumradius of triangle $A B C$. Let $p$ and $p^{\prime}$ be respectively the perimeter of $\triangle A B C$ and $\triangle A^{\prime} B^{\prime} C^{\prime}$ where $A^{\prime}$, $B^{\prime}$, $C^{\prime}$ are the touching points of $B C$, $C A$, $A B$ with the incircle. Prove that $$\frac{r}{R} \leq \frac{p^{\prime}}{p} \leq \frac{1}{2}$$
12. Let be given a cube $A B C D A_{1} B_{1} C_{1} D_{1}$ with side $A B=a$. From a point $E$ on the side $C D$ ($E$ distinct from $C$, $D$) draw a line cutting the lines $A A_{1}$ and $B_{1} C_{1}$ respectively at $M$ and $N$. From $M$ draw a line cutting the lines $B C$ and $C_{1} D_{1}$ respectively at $F$ and $P$. Determine the position of $E$ so that the perimeter of triangle $M N P$ attains its least value and calculate this least value.

Issue 321

1. Write the number $2003^{2004}$ as a sum of positive integers. What is the remainder of the division by $3$ of the sum of the cubes of these integers?
2. Simplify the expression $$\frac{(a-2)(a-1002)}{a(a-b)(a-c)}+\frac{(b-2)(b-1002)}{b(b-a)(b-c)}+\frac{(c-2)(c-1002)}{c(c-a)(c-b)}$$ where $a, b, c$ are distinct numbers such that $a b c \neq 0$.
3. Let $a, b, c, d$ be positive numbers. Prove that
   a) $\displaystyle \frac{a}{b}+\frac{b}{c}+\frac{c}{a} \geq \frac{a+b+c}{\sqrt[3]{a b c}}$.
   b) $\displaystyle \frac{a^{2}}{b^{2}}+\frac{b^{2}}{c^{2}}+\frac{c^{2}}{d^{2}}+\frac{d^{2}}{a^{2}} \geq \frac{a+b+c+d}{\sqrt[4]{a b c d}}$.
4. Find a necessary and sufficient condition on the number $m$ so that the following system of equations has a unique solution $$\begin{cases} x^{2} &=(2+m) y^{3}-3 y^{2}+m y \\ y^{2} &=(2+m) z^{3}-3 z^{2}+m z \\ z^{2} &=(2+m) x^{3}-3 x^{2}+m x\end{cases}$$
5. Let $A B C D$ be a trapezoid inscribed in a circle with radius $R=3cm$ such that $B C=2 cm$, $A D=4cm$. Let $M$ be the point on side $A B$ such that $M B=3 M A$. Let $N$ be the midpoint of $C D$. The line $M N$ cuts $A C$ at $P$. Calculate the area of the quadrilateral $A P N D$.
6. Let be given three positive integers $m$, $n, p$ such that $n+1$ is divisible by $m$. Find a formula to calculate the number of $p$-uples of positive integers $\left(x_{1}, x_{2}, \ldots, x_{p}\right)$ satisfying the conditions: the sum $x_{1}+x_{2}+\ldots+x_{p}$ is divisible by $m$ and $x_{1}, x_{2}, \ldots, x_{p}$ are not greater than $n$.
7. $a$, $b$ are arbitrary positive numbers such that the equation $x^{3}-a x^{2}+b x-a=0$ has three roots greater than $1$. Determine $a$, $b$ so that the expression $\dfrac{b^{n}-3^{n}}{a^{n}}$ attains its least value and find this value.
8. The incircle of triangle $A B C$ touches the sides $B C$, $C A$, $A B$ respectively at $D$, $E$, $F$. Prove that $$\frac{D E}{\sqrt{B C . C A}}+\frac{E F}{\sqrt{C A \cdot A B}}+\frac{F D}{\sqrt{A B \cdot B C}} \leq \frac{3}{2}.$$

Issue 322

1. Find all integers $x$ satisfying $$|x-3|+|x-10|+|x+101|+|x+990|+|x+1000|=2004.$$
2. Let $A B C$ be a triangle with its median $A M$. Let $O_{1}$, $O_{2}$. be the incenters of triangles $A B M$, $A C M$ respectively. Prove that $M O_{1}=M O_{2}$ when and only when $A B=A C$.
3. Consider the six pairs of marbles selected from a set of four given marbles, and consider the sum of masses of two marbles of each pair. Let $a, b, c, d, e, f$ be these sums. Determine the mass of each marble, known that $$a+b+c+d+e+f=a^{3}+b^{3}+c^{3}+d^{3}+e^{3}+f^{3}=6.$$
4. Find the least value of the expressions $$\frac{a^{6}}{b^{3}+c^{3}}+\frac{b^{6}}{c^{3}+a^{3}}+\frac{c^{6}}{a^{3}+b^{3}}$$ where $a, b, c, d$ are positive real numbers satisfying the condition $a+b+c=1$.
5. The circumcircle of triangle $A B C$ has center $O$ and diameter $A D$. Let $I$ be the incenter of triangle $A B C$. The lines $A I$, $D I$ cut again the circumcircle at $H, K$ respectively. Draw the line IJ perpendicular to $B C$ at $J$. Prove that $H$, $K$, $J$ are collinear.
6. Prove the inequality $$\frac{1}{2}\left(\sum_{i=1}^{n} x_{i}+\sum_{i=1}^{n} \frac{1}{x_{i}}\right) \geq n-1+\frac{n}{\sum_{i=1}^{n} x_{i}}$$ where $x_{i}(i=1,2, \ldots, n)$ are positive real numbers satisfying $\sum_{i=1}^{n} x_{i}^{2}=n$ and $n$ is an integer greater than $1$.
7. The sequence of numbers $\left(u_{n}\right)(n=$ $1,2,3, \ldots)$ is defined by $$u_{n}=\sum_{k=1}^{n} \frac{1}{(k !)^{2}},\,\forall n=1,2,3, \ldots.$$ Prove that this sequence has a limit and this limit is an irrational.
8. Let $S A B C$ be a tetrahedron. The points $M$, $N$, $P$ lie respectively on the sides $S A$, $S B$, $S C$ so that $A M=B N=C P$ ($M$, $N$, $P$ are distinct from the vertices $S$, $A$, $B$, $C)$. Let $G$ be the centroid of triangle $M N P$. Prove that $G$ lies on a fixed line when $M$, $N$, $P$ move on $S A$, $S B$, $SC$ respectively.

Issue 323

1. Compare $\dfrac{1}{16}$ with the following sum $A$ of 11 numbers $$A=\frac{1}{5^{2}}+\frac{2}{5^{3}}+\ldots+\frac{n}{5^{n+1}}+\ldots+\frac{11}{5^{12}}$$
2. Prove that $$\frac{a}{2 a+b+c}+\frac{b}{2 b+c+a}+\frac{c}{2 c+a+b} \leq \frac{3}{4}$$ where $a, b, c$ are positive integers.
3. Consider the following sum $A$ of $50$ numbers $$A=\frac{1}{\sqrt{1}+\sqrt{2}}+\frac{1}{\sqrt{3}+\sqrt{4}}+\ldots+\frac{1}{\sqrt{2 n-1}+\sqrt{2 n}}+\ldots+\frac{1}{\sqrt{99}+\sqrt{100}}.$$ Find the greatest integer not exceeding $A$.
4. Prove that $$\frac{a}{\sqrt[3]{b^{3}+c^{3}}}+\frac{b}{\sqrt[3]{c^{3}+a^{3}}}+\frac{c}{\sqrt[3]{a^{3}+b^{3}}}<2 \sqrt[3]{4}$$ where $a, b, c$ are the lengths of the sides of a triangle.
5. Let $A B C D$ be a given convex quadrilateral. On the lines $B C$, $A D$, take respectively the points $E$, $F$ so that $A E || C D$, $C F || A B$. Prove that the quadrilateral $A B C D$ circumscribes about a circle when and only when the quadrilateral $A E C F$ circumscribes about a circle.
6. Prove that $$a^{\log _{b} c}+b^{\log _{c} a}+c^{\log _{a} b} \geq 3 \sqrt[3]{a b c}$$ where $a, b, c$ are numbers greater than $1$.
7. Prove that for every acute triangle $A B C$, we have $$\frac{1}{3}(\cos 3 A+\cos 3 B)+\cos A+\cos B+\cos C \geq \frac{5}{6}$$
8. In space, let be given a fixed line $d$ and a fixed point $A$ not lying on $d$. A right angle $x M y$ moves so that its side $M x$ passes through $A$ and its side $M y$ cuts orthogonally $d$. Find the locus of $M$.

Issue 324

1. How many digits does contain the decimal representation of the number $2^{100}$ ? What is the first digit on the left in this representation?
2. Let $A B C$ be a triangle with $\widehat{A B C}=70^{\circ}$, $\widehat{A C B}=50^{\circ}$. On the side $A C$, take $M$ so that $\widehat{A B M}=20^{\circ}$, on the side $A B$, take $N$ so that $\widehat{A C N}=10^{\circ}$. Let $P$ be the point of intersection of $B M$ and $C N$. Prove that $M N=2 P M$.
3. Solve the system of equations $$\begin{cases}x^{3}+y &=2 \\ y^{3}+x &=2\end{cases}$$
4. Prove the inequality $$\sqrt{a^{4}+b^{4}+c^{4}}+\sqrt{a^{2} b^{2}+b^{2} c^{2}+c^{2} a^{2}} \geq \sqrt{a^{3} b+b^{3} c+c^{3} a}+\sqrt{a b^{3}+b c^{3}+c a^{3}}$$ where $a, b, c$ are non negative real numbers.)
5. Let $A B C$ be a triangle right at $A$. For every point $K$ on the side $A C$, construct the circle $(K)$ with center $K$, touching the line $B C$ at $E$. Draw the line $B D$ touching the circle $(K)$ at $D$ (distinct from $E)$. Let $M$, $N$, $P$ and $Q$ be the midpoints of $A B$, $A D$, $B D$ and $M P$ respectively. Let $S$ be the point of intersection of $Q N$ and $B D$. Find the line on which moves the point $S$ when $K$ moves on the side $A C$?.
6. Let $f(x)$ be a polynomial of degree 2003 with $$f(k)=\frac{k^{2}}{k+1},\,\forall k=1,2,3, \ldots 2004.$$ Calculate $f(2005)$.
7. Prove that $$4 x^{2}+4 y^{2} \leq x y+y z+z x+5 z^{2}$$ where $x, y, z$ are positive real numbers satisfying the conditions $x \leq y \leq z$. When does equality occur?
8. Let $r_{a}$, $r_{b}$, $r_{c}$ be the radii of the escribed circles in angles $A, B, C$ of the triangle $A B C$ respectively. Prove that  $$r_{a} \sin (A / 2)+r_{b} \sin (B / 2)+r_{c} \sin (C / 2) \leq \frac{r_{a}^{3}+r_{b}^{3}+r_{c}^{3}}{6}\left(\frac{1}{r_{a}^{2}}+\frac{1}{r_{b}^{2}}+\frac{1}{r_{c}^{2}}\right) $$

Issue 325

1. Prove that the sum $$A=\frac{2004}{2003^{2}+1}+\frac{2004}{2003^{2}+2}+\ldots+\frac{2004}{2003^{2}+n}+\ldots+\frac{2004}{2003^{2}+2003}$$ ($2003$ terms) is not an integer.
2. Let $A B C D$ be a rectangle with $A B=2 A D$ and let $M$ be the midpoint of the segment $A B$. Let $H$ be the point on side $A B$ such that $\widehat{A D H}=15^{\circ}$. The lines $C H$ and $D M$ intersect at $K$. Compare the lengths of the segments $D H$ and $D K$.
3. Solve the system of equations $$\begin{cases}x^{3}(2+3 y) &=1 \\ x\left(y^{3}-2\right) &=3\end{cases}$$
4. Find the least value of the expression $$A=\frac{1}{x^{3}+y^{3}}+\frac{1}{x y}$$ where $x, y$ are positive real numbers satisfying $x+y=1$.
5. Let be given a convex quadrilateral $A B C D$. $O$ is the midpoint of side $B C$, $E$ is symmetric to $D$ with respect to $O$. A point $M$ moves on the side $A D$. The line $E M$ cuts $O A$ at $I$. The line passing through $I$, parallel to $B C$, cuts $A B$ and $A C$ respectively at $K$ and $H$. Prove that the expression $$\frac{A B}{A K}+\frac{A C}{A H}-\frac{A D}{A M}$$ takes constant value.
6. Let be given $a>1$. Find all triples $(x, y, z)$ such that $|y| \geq 1$ and $$\log _{a}^{2}(x y)+\log _{a}\left(x^{3} y^{3}+x y z\right)^{2}+\frac{8+\sqrt{4 z-y^{2}}}{2}=0$$
7. Find the greatest value of the expression $a c+b d+c d$ where $a, b, c, d$ are real numbers satisfying the conditions $a^{2}+b^{2}=4$ and $c+d=4$.
8. The circles $C_{1}$, $C_{2}$, $C_{3}$ internally touch the circle $C$ respectively at $A_{1}$, $A_{2}$, $A_{3}$ and they externally touch each other. Let $B_{1}$, $B_{2}$, $B_{3}$ be respectively the touching point of $C_{2}$ and $C_{3}$, of $K_{3}$ and $C_{1}$, of $C_{1}$ and $C_{2}$. Prove that the lines $A_{1} B_{1}$, $A_{2} B_{2}$, $A_{3} B_{3}$ are concurrent.

Issue 326

1. Factorize $2003^{2004}$ in the product of two natural numbers $a$ and $b$. Is the sum $a+b$ divisible by $2004$?
2. The positive integers $a, b, c, d$ satisfy the conditions $a^{2}+c^{2}=1$ and $\dfrac{a^{4}}{b}+\dfrac{c^{4}}{d}=\dfrac{1}{b+d}$. Prove that $$\frac{a^{2004}}{b^{1002}}+\frac{c^{2004}}{d^{1002}}=\frac{2}{(b+d)^{1002}}.$$
3. Find the least prime number $p$ such that $p$ can be written in ten sums of the forms $$p=x_{1}^{2}+y_{1}^{2}=x_{2}^{2}+2 y_{2}^{2}=x_{3}^{2}+3 y_{3}^{2}=\ldots=x_{10}^{2}+10 y_{10}^{2},$$ where $x_{i}, y_{i}$ $(i=1,2, \ldots, 10)$ are positive integers.
4. Solve the equation $$\sqrt[3]{3 x+1}+\sqrt[3]{5-x}+\sqrt[3]{2 x-9}-\sqrt[3]{4 x-3}=0$$
5. Prove that $$\frac{1}{a(b+1)}+\frac{1}{b(c+1)}+\frac{1}{c(a+1)} \geq \frac{3}{a b c+1}$$ for arbitrary positive numbers $a, b, c$.
6. Prove that for every polygon, there exist at least two sides such that the measures $a$, $b$ of which satisfy the conditions $a \leq b \leq 2 a$.
7. From a point $P$ at the outside of a circle with center $O$, draw two tangents $P A$, $P B$ to the circle. Let $M$ and $N$ be respectively the midpoints of $A P$ and $O P$. The line $B M$ cuts again the circle at $K$. Prove that $K N \perp A K$.
8. Find integer-solutions of the equation of two unknowns $$x^{y^{x}}=y^{x^{y}}.$$
9. Prove that $$x y+\max \{x, y\} \leq \frac{3 \sqrt{3}}{4}$$ for arbitrary real nonnegative numbers $x, y$ satisfying the condition $x^{2}+y^{2}=1$.
10. Find all functions $f: \mathbb R^{+} \rightarrow \mathbb R^{+}$ satisfying the condition $$x f(x f(y))=f(f(y)),\,\forall x, y \in \mathbb R^{+}.$$
11. Let $R$ and $r$ bc respectively the circumradius and the inradius of a triangle $A B C$. and let $I$ be its incenter. Prove that $$\frac{1}{I A . I B}+\frac{1}{I B \cdot I C}+\frac{1}{I C . I A} \leq \frac{5 R+2 r}{8 R r^{2}}$$
12. Let $A B C D$ be a regular tetrahedron with side $a$. Let $H$ and $K$ be the midpoints of $A B$ and $C D$ respectively. An arbitrary plane containing the line $H K$ cuts the sides $B C$ and $A D$ at $E$ and $F$ respectively. Prove that $E F \perp$ $H K$. Find the least value of the area of the quadrilateral $H E K F$.

Issue 327

1. Can my friend write $7$ distinct $7$-digit numbers so that a) for writing each number, he uses $7$ distinct digits $1,2,3,4,5,6,7$. b) the sum of the $7^{\text {th }}$ powers of some (distinct) numbers among them is equal to the sum of the $7^{\text {h }}$ powers of the others?
2. Prove that $$\frac{1}{65}<\frac{1}{5^{3}}+\frac{1}{6^{3}}+\ldots+\frac{1}{n^{3}}+\ldots+\frac{1}{2004^{3}}<\frac{1}{40}$$ (the sum consists of $2000$ terms).
3. Find all integers $x$ such that $x^{3}-2 x^{2}+7 x-7$ is divisible by $x^{2}+3$.
4. Solve the equation $$4 x^{2}-4 x-10=\sqrt{8 x^{2}-6 x-10}.$$
5. Prove that $$\left(1+\frac{1}{a^{3}}\right)\left(1+\frac{1}{b^{3}}\right)\left(1+\frac{1}{c^{3}}\right) \geq \frac{729}{512}$$ where $a, b, c$ are positive real numbers satisfying $a+b+c=6$.
6. The circle $\left(O_{1}\right)$ with center $O_{1}$, radius $R_{1}$ cuts the circle $\left(O_{2}\right)$ with center $O_{2}$, radius $R_{2}$ at the points $A$ and $B$. The tangent to $\left(O_{1}\right)$ at $A$ cuts $\left(O_{2}\right)$ at $C$. The tangent to $\left(O_{2}\right)$ at $A$ cuts $\left(O_{1}\right)$ at $D .$ Let $M$ be the point of intersection of $A B$ and $C D$, let $N$ be the midpoint of $C D$. Prove that $\widehat{C A M}=\widehat{D A N}$ and $\dfrac{M C}{M D}=\dfrac{R_{2}^{2}}{R_{1}^{2}}$.
7. The quadrilateral $A B C D$ is inscribed in a circle with radius $R$ and circumscribes about a circle with radius $r$. Prove that $R \geq r \sqrt{2}$.
8. The sequences $\left(x_{n}\right)$ and $\left(y_{n}\right)$ $(n=1,$, $2,3, \ldots)$ are defined by $x_{1}=-1$, $y_{1}=1$ and $$x_{n+1}=-3 x_{n}^{2}-2 x_{n} y_{n}+8 y_{n}^{2},\, y_{n+1}=2 x_{n}^{2}+3 x_{n} y_{n}-2 y_{n}^{2},\,\forall n=1,2,3 \ldots$$ Find all prime numbers $p$ such that $x_{p}+y_{p}$ is not divisible by $p$.
9. The positive real numbers $a, b, c, d$ satisfy the conditions $a \leq b \leq c \leq d$ and $b c \leq a d$. Prove that $$a^{b} b^{c} c^{d} d^{a} \geq a^{d} b^{a} c^{b} d^{c}.$$
10. For each positive integer $n$, consider the function $$f_{n}(x)=e^{-x}\sum_{m=0}^{n} \frac{x^{m}}{m !},$$ defined on the set of positive real numbers. a) Prove that for every positive real numbers $k$ with $0<k<1$ and for every positive integer $n$, the equation $f_{n}(x)=k$ has a unique root. b) Let $\alpha_{n}$ be the above mentioned root. Find $\displaystyle\lim_{n \rightarrow+\infty} \frac{1}{\alpha_{n}}$.
11. Let be given a triangle $A B C$ with $B C=a$, $C A=b$, $A B=c$ and with circumradius $R$. Let $l_{a}$, $l_{b}$, $l_{c}$ be respectively the measure of the angled bisector of the angle $A$, $B$, $C$ and let $r_{a}$, $r_{b}$, $r_{c}$ be respectively the radius of the escribed circle in the angle $A$, $B$, $C$. Prove that $$\frac{l_{a}^{2} \cdot l_{b}^{2} \cdot l_{c}^{2}}{a^{2} b^{2} c^{2}} \leq\left(\frac{r_{a}+r_{b}+r_{c}}{6 R}\right)^{3}$$
12. Let $A_{1} A_{2} A_{3} A_{4}$ be a tetrahedron, circumscribing about a sphere with center $O$. Let $B_{i}$ be the touching point of the sphere with the face opposite to the vertex $A_{i}$ $(i=1,2,3,4)$. Prove that among the angles formed by a pair of distinct rays $O B_{1}$, $O B_{2}$, $O B_{3}$, $O B_{4}$ there exists an angle $\alpha$ with $$\sin \alpha \leq \frac{2 \sqrt{2}}{3}.$$

Issue 328

1. Compare the numbers $2^{3^{2^{3}}}$ and $3^{2^{3^{2}}}$.
2. Calculate the following sum of 2004 numbers $$f\left(\frac{1}{2005}\right)+f\left(\frac{2}{2005}\right)+\ldots+f\left(\frac{2004}{2005}\right)$$ where $f(x)=\dfrac{100^{x}}{100^{x}+10}$.
3. Find positive integer solutions of the equation $$(n+1)(2 n+1)=10 m^{2}$$
4. Find all positive integers $n$ such that the polynomial with $n+1$ terms $$P(x)=x^{4 n}+x^{4(n-1)}+\ldots+x^{8}+x^{4}+1$$ is divisible by the polynomial with $n+1$ terms $$Q(x)=x^{2 n}+x^{2(n-1)}+\ldots+x^{4}+x^{2}+1.$$
5. Find the greatest value of the expression $$T=\frac{a^{2}+1}{b^{2}+1}+\frac{b^{2}+1}{c^{2}+1}+\frac{c^{2}+1}{a^{2}+1}$$ where $a, b, c$ are non negative real numbers satisfying $a+b+c=1$.
6. Let $A B C$ be a triangle with acute angle $A$ and $A C=2 A B$. The angle bisector $A D$ cuts the altitude $B H$ at $K$ ($D$ lies on $B C$, $H$ on $A C)$. The line $C K$ cuts $A B$ at $E$. Prove that $\triangle A B C$ is right at $B$ when and only when the areas of the triangles $B D E$ and $H D E$ are equal.
7. On the side $A B$ of an equilateral triangle $A B C$ take a point $N$, on the side $A C$ take a point $M$ so that $A N>N B$ and $A M>M C$. The line $B M$ cuts $C N$ at $H$. Let $P$ and $Q$ be respectively the orthocenters of $\triangle A B M$ and $\triangle A C N$. Prove that $B N=C M$ when and only when $H P=H Q$.
8. Find the least prime number $p$ such that $\left[(3+\sqrt{p})^{2 n}\right]+1$ is divisible by $2^{n+1}$ for every natural number $n$, where $[x]$ denotes the greatest integer not exceeding $x$.
9. Prove that $$\left(\frac{a}{b+c}\right)^{k}+\left(\frac{b}{c+a}\right)^{k}+\left(\frac{c}{a+b}\right)^{k} \geq \frac{3}{2^{k}}$$ where $a, b, c, k$ are positive real numbers and $k \geq \dfrac{2}{3}$.
10. Find all positive real numbers $a$ such that there exist a positive real number $k$ and a function $f: \mathbb R \rightarrow \mathbb R$ satisfying the condition $$\frac{f(x)+f(y)}{2} \geq f\left(\frac{x+y}{2}\right)+k \cdot|x-y|^{a}$$ for all real numbers $x, y$.
11. The altitudes $A D$, $B E$, $C F$ of an acute triangle $A B C$ intersect at $H$ so that $A H>H D$, $B H>H E$, $C H>H F$. Prove that $$\tan^{2} A+\tan^{2} B+\tan^{2} C>6$$
12. Let be given $n$ dinstinct points $A_{1}$, $A_{2}, \ldots, A_{n}$. Prove that $$\sum_{i=1}^{n} \widehat{A_{i}A_{i+1}A_{i+2}} \geq \pi \quad \text{and} \quad \sum_{i=1}^{n} \widehat{A_{i} Q A_{i+1}} \leq(n-1) \pi$$ where $A_{n+1}$ is considered as $A_{1}, A_{n+2}$ is considered as $A_{2}$ and $Q$ is an arbitrary point distinct from $A_{1}, A_{2}, \ldots, A_{n}$.

Issue 329

1. Let $p$ and $q$ be two primes satisfying $p>q>3$ and $p-q=2$. Prove that $p+q$ is divisible by $12$.
2. Find the greatest value of the expression $$P=(a-b)^{4}+(b-c)^{4}+(c-a)^{4}$$ where $a, b, c$ are real numbers not less than $1$ and not greater than $2$.
3. Prove that the following sum (of $1999$ terms) $$s=1^{100}-2^{100}+3^{100}-4^{100}+\ldots+n^{100}-(n+1)^{100}+\ldots-1998^{100}+1999^{100}$$ is divisible by $201899$.
4. Solve the equation $$x=(2004+\sqrt{x})(1-\sqrt{1-\sqrt{x}})^{2}.$$
5. Prove that $$\frac{a}{a+\sqrt{(a+b)(a+c)}}+\frac{b}{b+\sqrt{(b+c)(b+a)}}+\frac{c}{c+\sqrt{(c+a)(c+b)}} \leq 1$$ where $a, b, c$ are positive real numbers.
6. Let $M N P Q$ be a quadrilateral inscribed in a circle and let $E$ be the point of intersection of $M P$ and $N Q$. Let $K$ be a point on the segment $M E$ ($K$ distinct from $M$, $E$). The tangent at $E$ to the circumcircle of triangle $N E K$ cuts the lines $Q M$ and $Q P$ respectively at $F$ and $G$. Prove that $$\dfrac{E G}{E F}=\dfrac{K P}{K M}$$
7. Consider the triangles $A B C$ with given perimeter $a+b+c=k$ (const), $a=B C$, $b=C A$, $c=A B$. Find the greatest value of the expression $$T=\frac{a b}{a+b+2 c}+\frac{b c}{2 a+b+c}+\frac{a c}{a+2 b+c}$$
8. Let $a$, $b$ be two real numbers distinct from $0$. Consider the sequence of numbers $\left(u_{n}\right)(n=0,1,2, \ldots)$ defined by $$u_{0}=0,\, u_{1}=1,\quad u_{n+2}=a u_{n+1}-b u_{n},\,\forall n=2,3, \ldots$$ Prove that if there exist four consecutive terms of the sequence that are integers then all terms of the sequence are intergers.
9. Find all values of the parameter $p$ so that the roots $x_{1}, x_{2}, x_{3}$ of the equation $$x^{3}-3 x^{2}-p x-1=0$$ satisfy the conditions $$\frac{1}{2005}<\frac{1}{\left(x_{1}-1\right)^{3}}+\frac{1}{\left(x_{2}-1\right)^{3}}+\frac{1}{\left(x_{3}-1\right)^{3}}<\frac{1}{2004}$$
10. Given positive numbers $a_{i}$, $b_{i}$ $(i=1,2, \ldots, n)$. Prove that $$\frac{a_{1}^{r}}{b_{1}^{s}}+\frac{a_{2}^{r}}{b_{2}^{s}}+\ldots+\frac{a_{n}^{r}}{b_{n}^{s}} \geq \frac{\left(a_{1}+a_{2}+\ldots+a_{n}\right)^{r}}{n^{r-s-1}\left(b_{1}+b_{2}+\ldots+b_{n i}\right)^{s}}$$ where $r, s$ are positive rational numbers and $r \geq s+1$.
11. Suppose that the quadrilateral $A B C D$ is inscribed in a circle with center $O$ with radius $R$ and the opposite rays of the rays $B A$, $D A$, $C B$, $C D$ touch a circle with center $I$ and radius $r$. Prove that by putting $d=O I$, we have $$\frac{1}{(d+R)^{2}}+\frac{1}{(d-R)^{2}}=\frac{1}{r^{2}}$$
12. For a tetrahedron $A B C D$ with $A B=C D$, $A C=B D$, $A D=B C$, let $\varphi_{1}$, $\varphi_{2}$, $\varphi_{3}$ be respectively the measures of the dihedral angles with sides $B C$, $C A$, $A B$. Prove that $$\cos \frac{\varphi_{1}}{2} \cdot \cos \frac{\varphi_{2}}{2} \cdot \cos \frac{\varphi_{3}}{2}=\frac{\sqrt{\cos A \cdot \cos B \cdot \cos C}}{\sin A \cdot \sin B \cdot \sin C}$$ where $A$, $B$, $C$ denote the angles of triangle $A B C$.

Issue 330

1. Find the integers $x$, $y$, $z$ satisfying the equalities $$\frac{x}{y}+\frac{y}{z}+\frac{z}{x}=\frac{y}{x}+\frac{z}{y}+\frac{x}{z}=x+y+z=3$$
2. Let $A B C$ be a triangle with $\widehat{A C B}=50^{\circ}, \widehat{B A C}=100^{\circ}$, let $M$ be the point on the side $A B$ such that $A M=A C$. Compare $C M$ with $A B$.
3. Find all integer roots of the equation $$x^{y}+y^{z}+z^{x}=2(x+y+z).$$
4. Solve the equation $$\sqrt{\sqrt{3}-x}=x \sqrt{\sqrt{3}+x}$$
5. Prove the inequality $$\frac{a^{3}+b^{3}+c^{3}}{2 a b c}+\frac{a^{2}+b^{2}}{c^{2}+a b}+\frac{b^{2}+c^{2}}{a^{2}+b c}+\frac{c^{2}+a^{2}}{b^{2}+a c} \geq \frac{9}{2}$$ where $a, b, c$ are positive real numbers.
6. Let be given a triangle $A B C$ with $A B=A C$. From every point $M$ on the side $B C$, draw $M P \perp A B$ and $M Q \perp A C$ ($P$, $Q$ lie respectively on the lines $A B$, $A C)$. Prove that the perpendicular bisector of $P Q$ passes through a fixed point when $M$ moves on the side $B C$.
7. Let $A B C$ be a triangle with the altitude $A H$ ($H$ distinct from $B$, $C$). Draw $H E \| A C$, $H M \perp A B$ ($E$ and $M$ lie on the line $A B$), draw $H F \| A B$, $H N \perp A C$ ($F$ and $N$ lie on the line $A C$). Prove that the lines $E F$, $M N$ and $B C$ are concurrent.
8. Find the greatest and the least values of the expression $P=x^{y^{z}}$ where $x, y, z$ are integers greater than $2$ and satisfy $x+y+z=20$.
9. Let $M$ and $m$ be respectively the greatest value and the least value of the function $$f(x)=\cos (2002 x)+k \cos (x+\alpha)$$ where $k$, $\alpha$ are real parameters. Prove that $$M^{2}+m^{2} \geq 2.$$
10. Let be given a postive integer $n$. Consider a continuous function $f(x):[0 ; n] \rightarrow \mathbb R$ satisfying $f(0)=f(n)$. Prove that there exist $n$ couples of numbers $a_{i}, b_{i}$ $(i=1,2, \ldots, n)$ belonging to $[0 ; n]$ such that $b_{i}-a_{i}$ are positive integers and $f\left(a_{i}\right)=f\left(b_{i}\right)$ for all $i=1,2, \ldots, n$
11. In plane let be given a line $x y$, a segment $A B$ perpendicular to $x y$ at $A$, a point $C$ on the ray $A x$, a point $D$ on the ray $A y$ ($C$, $D$ distinct from $A$). Draw $A E \perp B C$ ($E$ lies on $B C$), $A F \perp B D$ ($F$ lies on $B D$). A line passing through the midpoint $Q$ of $A B$ cuts the lines $x y$, $B C$, $B D$ respectively at $P$, $M$, $N$. Prove that $P$, $E$, $F$ are collinear when and only when $Q$ is the midpoint of $M N$.
12. Given a regular tetrahedron $A_{1} A_{2} A_{3} A_{4}$. Let $d_{i}(i=1,2,3,4)$ be the distance from a point $M$ in space to the face opposite to vertex $A_{i}$ of the tetrahedron $A_{1} A_{2} A_{3} A_{4}$. Prove that $$x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+x_{4}^{2} \leq 9\left(d_{1}^{2}+d_{2}^{2}+d_{3}^{2}+d_{4}^{2}\right)$$ where $x_{i}=M A_{i}$ $(i=1,2,3,4)$.