Mathematics and Youth Magazine Problems 2003

Issue 307

1. Find the measures of the sides of all triangles satisfying the conditions : these measures are whole numbers, the perimeter and the area are expressed by equal numbers.
2. Find the least value of the expression $$(\sqrt{1+a}+\sqrt{1+b})(\sqrt{1+c}+\sqrt{1+d})$$ where $a, b, c, d$ are positive numbers satisfying the condition $a b c d=1$.
3. Solve the system of equations $$\begin{cases}\sqrt{x}+\sqrt{y}+\sqrt{z} &=3 \\ (1+x)(1+y)(1+z) &=(1+\sqrt[3]{x y z})^{3}\end{cases}$$
4. Let $A B C D$ be a parallelogram. Take a point $M$ on the side $A B$, a point $N$ on the side $C D$. Let $P$ be the point of intersection of $A N$ and $D M$, $Q$ be the point of intersection of $B N$ and $C M$. Prove that $P Q$ passes through a fixed point when $M$ and $N$ move respectively on $A B$ and $C D$.
5. Let $A B C D$ be a convex quadrilateral. Prove that $$ \min \{A B, B C, C D, D A\} \leq \frac{\sqrt{A C^{2}+B D^{2}}}{2} \leq  \max \{A B, B C, C D, D A\}$$
6. Find all natural numbers $n \geq 3$ such that in the coordinate plane, there exists a regular $n$-polygon all vertices of which have integral coordinates.
7. The sequence of numbers $\left(u_{n}\right)$ $(n=1,2,3, \ldots)$ is defined by $$u_{n}=\sum_{k=1}^{n} \frac{(-1)^{k-1}}{k},\,\forall n=1,2,3 \ldots.$$ Prove that the sequence has limit and find the limit.
8. Find the greatest value of the expression $$x_{1}^{3}+x_{2}^{3}+\ldots+x_{n}^{3}-x_{1}^{4}-x_{2}^{4}-\ldots-x_{1}^{4}$$ ($n$ is a given number), where the numbers $x_{i}$ $(i=1,2, \ldots, n)$ satisfy the conditions $0 \leq x_{i} \leq 1$ $(i=1,2, \ldots, n)$ and $x_{1}+x_{2}+\ldots+x_{n}=1$.
9. In a triangle $A B C$, let $m_{u}$, $m_{b}$, $m_{c}$ be respectively the measures of the medians issued from $A$, $B$, $C$, let $r_{a}$, $r_{b}$, $r_{c}$ be respectively the radii of the escribed circles in the angles $A$, $B$, $C$. Prove that $$r_{a}^{2}+r_{b}^{2}+r_{c}^{2} \geq m_{a}^{2}+m_{b}^{2}+m_{c}^{2}.$$ When does equality occur?
10. Let $H$ and $O$ be respectively the orthocenter and the circumcenter of the triangle $A B C$ of a tetrahedron $S A B C$ such that $S A$, $S B$, $S C$ are orthogonal each to the others. Prove that $$\frac{O H^{2}}{S H^{2}}+2=\frac{1}{4 \cos A \cdot \cos B \cdot \cos C}$$ where $\cos A$, $\cos B$, $\cos C$ are cosinus of the angles of triangle $A B C$.

Issue 308

1. There are three bells in the laboratory. The first bell rings every $4$ minutes, the second every $12$ minutes, the third every $16$ minutes. The three bells ring simultaneously at $7^{\text {h }} 30$ in the morning.
   a) At what next time do the three bells ring simultaneously?
   b) From $7^{\mathrm{h}} 30$ to $11^{\mathrm{h}} 30$ p.m., how many times do only two bells ring simultaneously?
2. a) In the figure let $\widehat{B E C}=30^{\circ}$, $\widehat{A C D}=70^{\circ}$, $\widehat{C D E}=110^{\circ}$ and $\widehat{B A C}=\widehat{C E D}=\widehat{A B E}$ and justify the answer.
   b) How many triangles are there in the figure? Write down these triangles.
3. Find the integer-solutions of the equation $$4(a-x)(x-b)+b-a=y^{2}$$ where $a, b$ are given integers, $a>b$.
4. Prove that the equation $$(n+1) x^{n+2}-3(n+2) x^{n+1}+a^{n+2}=0$$ (where $n$ is a given even positive integer and $a>3$) has no solution.
5. Prove that $$\sqrt{\frac{a}{b}}+\sqrt{\frac{b}{a}}+\frac{3(\sqrt{a}+\sqrt{b})}{\sqrt{a+b}}>6$$
6. Let $A B C$ be a triangle ; on the opposite rays of the rays $B A, C A$ take respectively the points $E, F$ (distinct from $B, C)$ $B F$ cuts $C E$ at $M$. Prove that $$\frac{M B}{M F}+\frac{M C}{M E} \geq 2 \sqrt{\frac{A B \cdot A C}{A F \cdot A E}}$$ When does equality occur?
7. Let be given a convex quadrilateral $A B C D$. $A B$ cuts $C D$ at $E$, $A D$ cuts $B C$ at $F$. The diagonals $A C$, $B D$ intersect at $O$. Let $M$, $N$, $P$, $Q$ be respectively the midpoints of $A B$, $B C$, $C D$, $DA$. $CF$ cuts $M P$ at $H$, $O E$ cuts $N Q$ at $K$. Prove that $H K$ is parallel to $E F$.
8. Let $x, y, p$ be integers such that $p>1$ and $x^{2002}$, $y^{2003}$ are divisible by $p$. Prove that $1+x+y$ is not divisible by $p$.
9. Let $a_{1}, a_{2}, \ldots a_{n}$ and $b_{1}, b_{2}, \ldots, b_{n}$ be positive numbers. Prove that
   a) $\displaystyle \frac{d_{1}^{r}}{b_{1}^{r-1}}+\frac{c_{2}}{b_{2}^{r-1}}+\ldots+\frac{a_{n}^{r}}{b_{n}^{r-1}} \geq \frac{\left(a_{1}+a_{2}+\ldots+a_{n}\right)^{r}}{\left(b_{1}+b_{2}+\ldots+b_{n}\right)^{r-1}}$ where $r$ is a rational number, $r>1$;
   b) $\displaystyle \frac{a_{1}^{s}}{b_{1}}+\frac{a_{2}^{s}}{b_{2}}+\ldots+\frac{a_{n}^{s}}{b_{n}} \geq \frac{\left(a_{1}+a_{2}+\ldots+a_{n}\right)^{s}}{n^{s-2}\left(b_{1}+b_{2}+\ldots+b_{n}\right)}$ where $s$ is a rational number, $s \geq 2$.
10. The sequence of numbers $\left(v_{n}\right)$ is defined by $$v_{0}=1,\quad v_{n}=\frac{-1}{3+v_{n-1}},\,\forall n=1,2,3, \ldots$$ Prove that the sequence has a limit and find this limit.
11. $M$ is a point inside an acute triangle $A B C$ with $B C=a$, $C A=b$, $A B=c$. Let $D$, $E$, $F$ be respectively the orthogonal projections of $M$ on the sides $B C$, $C A$, $A B$. Find the greatest value of the expression $$P=a \cdot M E \cdot M F + b \cdot M F \cdot M D+c \cdot M D \cdot M E$$ and determine the position of $M$ where this expression assumes its greatest value. 
12. Let $S_{A}$, $S_{B}$, $S_{C}$, $S_{D}$ be respectively the areas of the faces $B C D$, $C D A$, $D A B$, $A B C$ of a tetrahedron $A B C D$ and $R$ be the radius of the circumscribed sphere of $A B C D$. Prove that $$R \geq \frac{T}{S_{A}+S_{B}+S_{C}+S_{D}}$$ where $$T^{2}=A B^{2} S_{A} S_{B}+A C^{2} S_{A} S_{C}+A D^{2} S_{A} S_{D}+B C^{2} S_{B} S_{C}+B D^{2} S_{B} S_{D}+C D^{2} S_{C} S_{D}.$$ When does equality occur?

Issue 309

1. Compare the value of the expression $$A=\frac{3}{4}+\frac{8}{9}+\frac{15}{16}+\ldots+\frac{9999}{10000}$$ with the numbers $98$ and $99$.
2. Let $A B C$ be a triangle with $\widehat{A B C}=30^{\circ}$, $\widehat{A C B}=20^{\circ}$. The perpendicular bisector of $A C$ cuts $B C$ at $E$ and cuts the ray $B A$ at $F$. Prove that $A F=E F$ and $A C=B E$.
3. Find the intergers $x, y$ satisfying $$\frac{x+2 y}{x^{2}+y^{2}}=\frac{7}{20}.$$
4. Prove that $$\frac{a+b}{2} \geq \sqrt{a b}+\frac{(a-b)^{2}(3 a+b)(a+3 b)}{8(a+b)\left(a^{2}+6 a b+b^{2}\right)}$$ for positive numbers $a, b$.
5. Solve the inequation $$(x-1) \sqrt{x^{2}-2 x+5}-4 x \sqrt{x^{2}+1} \geq 2(x+1)$$
6. On a line, take three distinct points $A$, $B$, $C$ in this order. Draw the tangents $A D$, $A E$ to the circle with diameter $B C$ ($D$ and $E$ are the touching points). Draw $D H \perp C E$ at $H$. Let $P$ be the midpoint of $D H$. The line $C P$ cuts again the circle at $Q$. Prove that the circle passing through the three points $A$, $D$, $Q$ is tangent to the line $A C$.
7. $M$ is a point on the incircle of triangle $A B C$. Let $K$, $H$, $J$ be respectively the orthogonal projections of $M$ on the lines $A B$, $B C$, $C A$. Determine the position of $M$ so that the sum $M K+M H+M J$.
   a) attains its greatest value
   b) attains its least value
8. Determine all sequences of positive integers $\left(x_{n}\right)$ $(n=1,2,3, \ldots)$ satisfying $$x_{1}=1,\, x_{2}>1,\quad x_{n+2}=\frac{1+x_{n+1}^{4}}{x_{n}},\,\forall n=1,2,3, \ldots$$
9. Solve the system of equations $$\begin{cases}\log _{2}(1+3 \cos x) &=\log _{3}(\sin y)+2 \\ \log _{2}(1+3 \sin y) &=\log _{3}(\cos x)+2\end{cases}$$
10. Let $n$ be a given positive integer. Find the least number $t=t(n)$ such that for all real numbers $x_{1}, x_{2}, \ldots, x_{n}$, holds the following inequality $$\sum_{k=1}^{n}\left(x_{1}+x_{2}+\ldots+x_{k}\right)^{2} \leq t\left(x_{1}^{2}+x_{2}^{2}+\ldots+x_{n}^{2}\right)$$
11. Prove that for arbitrary triangle $A B C$, $$\sin A \cos B+\sin B \cos C+\sin C \cos A \leq \frac{3 \sqrt{3}}{4}.$$ When does equality occur?
12. In space, let be given $n$ distinct points $A_{1}, A_{2}, \ldots, A_{n}$ $(n \geq 2)$ and $n$ points $K_{1}, K_{2}, \ldots, K_{n}$ ($K_{i}$ does not coincide with $A_{i}$ for every $i=1,2, \ldots, n)$. Prove that there exist $n$ spheres (with positive radii) $S_{i}$ $(i=1,2, \ldots, n)$ satisfying simultaneously the following two conditions
    a) the spheres do not intersect each others,
    b) the products $P_{A_{i} / S_{i}} \cdot P_{K_{i} / S_{i}}$ are negative numbers for all $i=1,2, \ldots, n$, where $P$ denotes the power of a point with respect to a sphere.

Issue 310

1. Do there exist three positive integers $a, b, c$ such that the sum $$\frac{a}{a+b}+\frac{b}{b+c}+\frac{c}{c+a}$$ is an integer?
2. Let $A B C$ be a triangle with $\widehat{B A C}=90^{\circ}$, $\widehat{A B C}=50^{\circ}$. Construct the triangle $B C D$ such that $\widehat{B C D}=\widehat{C B D}=30^{\circ}$ and $D$ is outside of triangle $A B C$. The lines $A B$ and $C D$ intersect at $E$. The lines $A C$ and $B D$ intersect $F$. Calculate the angles $\widehat{A E F}$ and $\widehat{A F E}$.
3. Find integral solutions of the equation $$x^{3}-y^{3}=x y+2003.$$
4. Solve the system of equations $$\begin{cases} x+y+z &=-1 \\ x y z &=1 \\ \dfrac{x}{y^{2}}+\dfrac{y}{z^{2}}+\dfrac{z}{x^{2}} &=\dfrac{y^{2}}{x}+\dfrac{z^{2}}{y}+\dfrac{x^{2}}{z}\end{cases}$$
5. Find the least value of the expression $$\frac{a^{3}}{1+b}+\frac{b^{3}}{1+a},$$ where $a$ and $b$ are positive numbers satisfying the condition $a b=1$.
6. Let $A B C D$ a convex quadrilateral such that the diagonals $C A$ and $D B$ are the angled-bisectors of $\widehat{B C D}$ and $\widehat{A D C}$. Let $E$ be the point of intersection of $C A$ and $D B$. Prove that $$E C \cdot E D=E A \cdot E B+E A \cdot E D+E B \cdot E C$$ when and only when $\widehat{A E D}=45^{\circ}$.
7. Let be given a semi-circle with diameter $A B$ and a fixed point $C$ on the segment $A B$ ($C$ distinct from $A$, $B$). $M$ is a point on the semi-circle. The line passing through $M$, perpendicular to $M C$, cuts the tangents of the semi-circle at $A$ and $B$ respectively at $E$ and $F$. Find the least value of the area of triangle $C E F$ when $M$ moves on the semi-circle.
8. Prove that $$(a+b)(b+c)(c+a) \geq 2(1+a+b+c)$$ where $a, b, c$ are positive numbers satisfying the condition $a b c=1$.
9. Solve the following equation with parameter $m$ $$x^{3}+5 x^{2}+(5 m+1) x+m^{2}=\left(x^{2}-x+1\right)^{2}$$
10. Prove the inequalities $$\frac{2 x^{n}}{1+x^{n+1}} \leq\left(\frac{1+x}{2}\right)^{n-1} \leq \frac{x^{n}-1}{n(x-1)}$$ for positive number $x \neq 1$ and positive integer $n$.
11. Prove that for arbitrary acute triangle $A B C$ $$\tan A+\tan B+\tan C+6(\sin A+\sin B+\sin C) \geq 12 \sqrt{3}.$$
12. Let $S A B C$ be a tetrahedron with $S A=B C$, $S B=C A$, $S C=A B$. A plane passing through the incenter of triangle $A B C$ cuts the rays $S A$, $S B$, $S C$ respectively at $M$, $N$, $P$. Prove that $$S M+S N+S P \geq S A+S B+S C.$$

Issue 311

1. Compare the fractions $$A=\frac{2003^{2003}+1}{2003^{2004}+1}, \quad B=\frac{2003^{2002}+1}{2003^{2003}+1}$$ Can you do it by distinct methods?
2. Let $A B C$ be a triangle. Construct the segment $B D$ so that $\widehat{A B D}=60^{\circ}$, $B D=B A$ and the ray $B A$ lies between the rays $B C$, $B D$. Construct the segment $B E$ so that $\widehat{C B E}=60^{\circ}$, $B E=B C$ and the ray $B C$ lies between the rays $B A$, $B E$. Let $M$ be the midpoint of $D E$, $P$ be the point of intersection of the perpendicular bisectors of the segments $B A$ and $B D$. Calculate the angles of triangle $C M P$.
3. Let $x_{1}$, $x_{2}$ be the solutions of the equation $x^{2}-4 x+1=0$. Prove that for every positive integer $n$, $x_{1}^{n}+x_{2}^{n}$ can be written as the sum of the squares of three consecutive integers.
4. Find all triples of numbers $a, b, c$ satisfying $$\left(a^{2}+1\right)\left(b^{2}+2\right)\left(c^{2}+8\right)=32 a b c.$$
5. Find the least value of the expression $$\frac{a^{8}}{\left(a^{2}+b^{2}\right)^{2}}+\frac{b^{8}}{\left(b^{2}+c^{2}\right)^{2}}+\frac{c^{8}}{\left(c^{2}+a^{2}\right)^{2}}$$ where $a, b, c$ are positive numbers satisfying the condition $a b+b c+c a=1$.
6. Let $A B C$ be a triangle, right at $A$. Construct a square $E F G D$ so that $E$, $F$ lie on the side $B C$ and $G$, $D$ lie respectively on the sides $A C$, $A B$. Let $R_{1}$, $R_{2}$, $R_{3}$ be respectively the inradii of the triangles $B D E$, $C G F$, $A D G$. Prove that the area of $E F G D$ assumes its greatest value when and only when $$R_{1}^{2}+R_{2}^{2}=R_{3}^{2}.$$
7. A chord $D E$ of the circumcircle of triangle $A B C$ cuts the incircle of $A B C$ at $M$ and $N$. Prove that $D E \geq 2 M N$.
8. Each domino consists of two squares having a common side, on its first and second squares are marked $x$ and respectively $y$ dots. Thid momino is denoted by $(x, y)$ or $(y, x) .$ For every non ordered pair of number $x, y$, there's only one domino $(x, y)$. Divide the set of dominoes with $1 \leq x \leq 5,1 \leq y \leq 5$ into three groups, each group is presented as a closed circuit where the numbers $a, b, c, d$, $a \mid b] b \mid c$ are not necessarily distinct. $a|e| e \mid d\rfloor d$ How many such dividings are there?
9. Prove that for every $x \in\left[0 ; \frac{\pi}{2}\right]$, holds $$\sin x \leq \frac{4}{\pi} x-\frac{4}{\pi^{2}} x^{2}.$$
10. Prove that for every triangle $A B C$, hold $$\sqrt{3} \leq \frac{\cos (A / 2)}{1+\sin (A / 2)}+\frac{\cos (B / 2)}{1+\sin (B / 2)}+\frac{\cos (C / 2)}{1+\sin (C / 2)}<2$$
11. Le $A B C D$ be a convex a quadrilateral, the diagonals $A C$ and $B D$ of which are orthogonal. The lines $B C$ and $A D$ intersect at $I$, the lines $A B$ and $C D$ intersect at $J$. Prove that the quadrilateral $B D I J$ is inscribable when and only when $A B \cdot C D=A D \cdot B C$. 
12. The tetrahedron $A B C D$ has its four altitudes concurrent at a point $H$ inside the tetrahedron. Let $M$, $N$, $P$ be respectively the midpoints of $B C$, $C D$, $D B$. Let $R$ and $R_{1}$ be respectively the circumradii of the tetrahedra $A B C D$ and $H M N P$. Prove that $R=2 R_{1}$.

Issue 312

1. Find the the first three digits from the left side of the number $$1^{1}+2^{2}+3^{3}+\ldots+999^{999}+1000^{1000} .$$
2. Prove that if the numbers $a, b, c, x, y, z$ satisfy the condition $$\frac{b z+c y}{x(-a x+b y+c z)}=\frac{c x+a z}{y(a x-b y+c z)}=\frac{a y+b x}{z(a x+b y-c z)}$$ then $$\frac{x}{a\left(b^{2}+c^{2}-a^{2}\right)}=\frac{y}{b\left(a^{2}+c^{2}-b^{2}\right)}=\frac{z}{c\left(a^{2}+b^{2}-c^{2}\right)}$$
3. Given an integer $n>1$ such that $3^{n}-1$ is divisible by $n$, show that $n$ is even.
4. Find all positive solutions of the system of equations $$\begin{cases}\dfrac{3 x}{x+1}+\dfrac{4 y}{y+1}+\dfrac{2 z}{z+1} &=1 \\ 8^{9} \cdot x^{3} \cdot y^{4} \cdot z^{2} &=1\end{cases}$$
5. Prove that $$\left(a^{3}+b^{3}+c^{3}\right)\left(\frac{1}{a^{3}}+\frac{1}{b^{3}}+\frac{1}{c^{3}}\right) \geq \frac{3}{2}\left(\frac{b+c}{a}+\frac{c+a}{b}+\frac{a+b}{c}\right)$$ where $a, b, c$ are positive real numbers.
6. Prove that $$(a+b+c)\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)+\frac{3(a-b)(b-c)(c-a)}{a b c} \geq 9$$ where $a, b, c$ are the lengths of the sides of a triangle.
7. The circle inscribed a triangle $A B C$ meets the sides $B C$, $C A$, $A B$ at $D$, $E$, $F$, respectively. Let $H$ be the projection of $D$ on $E F$. Show that $\widehat{B H D}=\widehat{C H D}$.
8. Let $p_{1}, p_{2}, \ldots, p_{n}$ and $q_{1}, q_{2}, \ldots, q_{m}$ be $m+n$ different prime numbers such that $p_{i} \geq n+1$ $(i=1,2, \ldots, n)$, $q_{j} \geq m+1$ $(j=1,2, \ldots, m)$. Put $$P=p_{1} p_{2} \ldots p_{n} \quad \text{and} \quad Q=q_{1} q_{2} \ldots q_{m}.$$ Prove that the equation $P^{s} x+Q^{t} y=1$, where $s$, $t$ are fixed positive integers, have infinitely many integral solutions $\left(x_{0}, y_{0}\right)$ such that $\left(P, x_{0}\right)=1$ and $\left(Q, y_{0}\right)=1$.
9. Solve the equation $$\left(16 \cos ^{4} x+3\right)^{4}=2048 \cos x-768.$$
10. Prove that $$\frac{x_{1}^{m}}{x_{2}^{n}+x_{3}^{n}+\ldots+x_{n}^{n}}+\ldots+\frac{x_{n}^{m}}{x_{1}^{n}+x_{2}^{n}+\ldots+x_{n-1}^{n}} \geq \frac{n}{n-1} \sqrt[n]{\left(\frac{2003}{n}\right)^{m-n}}$$ where $m \geq n>1$ are integers and $x_{i}$ $(i=1,2, \ldots, n)$ are positive numbers such that $$x_{1}^{n}+x_{2}^{n}+\ldots+x_{n}^{n}\geq 2003 .$$
11. Show that for a triangle $A B C$ with three acute angles, $$\frac{\cos A \cos B}{\sin 2 C}+\frac{\cos B \cos C}{\sin 2 A}+\frac{\cos C \cos A}{\sin 2 B} \geq \frac{\sqrt{3}}{2}$$
12. Construct in the base plane $(A B C)$ of a tetrahedron $S A B C$ the triangles $A_{1} B C$, $B_{1} C A$, $C_{1} A B$ on the same side as $\triangle A B C$ and $A_{2} B C$, $B_{2} C A$, $C_{2} A B$ on the opposite side as $A B C$ such that $$\Delta A_{1} B C=\Delta A_{2} B C=\Delta S B C,\,\Delta B_{1} C A=\Delta B_{2} C A=\triangle S C A,\,\Delta C_{1} A B=\Delta C_{2} A B=\Delta S A B.$$ Prove that $\dfrac{R_{1}}{R_{2}}=\dfrac{r}{r_{S}}$, where $R_{1}$, $R_{2}$ are the radii of the circumcircles of $A_{1} B_{1} C_{1}$, $A_{2} B_{2} C_{2}$ and $r$, $r_{S}$ are the radii of the spheres inscribed and escribed the tetrahedron $S A B C$ opposite to $S$, respectively.

Issue 313

1. Let $$A=\frac{1}{14}+\frac{1}{29}+\ldots+\frac{1}{n^{2}+(n+1)^{2}+(n+2)^{2}}+\ldots+\frac{1}{1877}.$$ Prove that $0,15<A<0,25$.
2. The median $B M$ and the angled-bisector $C D$ of a triangle $A B C$ intersect at $K$ such that $K B=K C$. Calculate the measures of the angles $\widehat{A B C}$, $\widehat{A C B}$, knowing that the measure of $\widehat{B A C}$ is $105^{\circ}$.
3. Prove that the number $A=2^{n}+6^{n}+8^{n}+9^{n}$ ($n$ is a positive integer) is divisible by $5$ when and only when $n$ is not divisible by $4$.
4. Solve the equation $$2 \sqrt[3]{x+2}+\sqrt[3]{3 x+1} \geq \sqrt[3]{2 x-1}$$
5. Prove that if the polynomial $$x^{3}+a x^{2}+b x+c$$ has $3$ distinct roots then the polynomial $$x^{3}-b x^{2}+a c x-c^{2}$$ has also $3$ distinct roots.
6. The measures of the sides $B C$, $C A$, $A B$ of a triangle $A B C$ are respectively $a$, $b$, $c$ and the measures of its altitudes $A A^{\prime}$, $B B^{c}$, $C C^{\prime}$ are respectively $h_{a}$, $h_{b}$, $h_{c}$. Prove that the triangle $A B C$ is equilateral when and only when $$\sqrt{a+h_{a}}+\sqrt{b+h_{b}}+\sqrt{c+h_{c}}=\sqrt{a+h_{b}}+\sqrt{b+h_{c}}+\sqrt{c+h_{a}}$$
7. A triangle $A B C$ has $\dfrac{1}{4} A C<A B<4 A C$. A line passing through its centroid $G$ cuts the sides $A B$, $A C$ respectively at $E$, $F$. Determine the position of $E$ so that $A E+A F$ attains its least value.
8. Find the least value of the expression $$P =\frac{x_{1}^{3}}{x_{2}+x_{3}+\ldots+x_{n}}+\frac{x_{2}^{3}}{x_{1}+x_{3}+\ldots+x_{n}}+\ldots +\frac{x_{n}^{3}}{x_{1}+x_{2}+\ldots+x_{n-1}}$$ where $x_{1}, x_{2}, \ldots, x_{n}$ are real positive numbers satisfying the condition $$x_{1} x_{2}+x_{2} x_{3}+\ldots+x_{n-1} x_{n}+x_{n} x_{1}=s$$ ($s$ is a given constant).
9. The polynomial $P(x)$ satisfies the conditions $$P(2003)=2003 !,\quad x \cdot P(x-1) \equiv (x-2003) \cdot P(x).$$ Prove that the polynomial $f(x)=$ $P^{2}(x)+1$ is irreducible.
10. The sequence of real numbers $\left(x_{n}\right)$ $(n=0,1,2, \ldots)$ is defined by $$x_{0}=a,\quad x_{n+1}=2 x_{n}^{2}-1,\,\forall n=0,1,2, \ldots$$ Find all values of $a$ so that $x_{n}<0$ for all $n=0,1,2, \ldots$.
11. Let $M$ be a point inside the triangle $A B C$. Let $d_{1}$, $d_{2}$, $d_{3}$ be respectively the distances from $M$ to the lines $B C$, $C A$, $A B$, let $R$ and $r$ be respectively the circumradius and the inradius of $\triangle A B C$. Prove that $$\frac{M A \cdot M B \cdot M C}{d_{1} \cdot d_{2} \cdot d_{3}} \geq \frac{4 R}{r}$$
12. The altitudes of a tetrahedron $A B C D$ are concurrent at a point $H$. Prove that $$(H A+H B+H C+H D)^{2} \leq A B^{2}+A C^{2}+A D^{2}+B C^{2}+B D^{2}+C D^{2}$$

Issue 314

1. Calculate $\frac{A}{B}$, where $$\begin{align}A&=\frac{1}{2.32}+\frac{1}{3.33}+\ldots+\frac{1}{n(n+30)}+\ldots+\frac{1}{1973.2003},\\ B&=\frac{1}{2.1974}+\frac{1}{3.1975}+\ldots+\frac{1}{n(n+1972)}+\ldots+\frac{1}{31.2003}.\end{align}$$
2. Find the value of the expression $S=x^{4}+y^{4}+z^{4}$, knowing that $x+y+z=0$ and $x^{2}+y^{2}+z^{2}=1$.
3. Prove that for every integer $n>1$ $$\frac{1}{3 \sqrt{2}}+\frac{1}{4 \sqrt{3}}+\ldots+\frac{1}{(n+1) \sqrt{n}}<\sqrt{2}$$
4. Solve the equation $$x^{2}-2 x+3=\sqrt{2 x^{2}-x}+\sqrt{1+3 x-3 x^{2}}$$
5. Find the least value of the expression $$\frac{a}{1+b-a}+\frac{b}{1+c-b}+\frac{c}{1+a-c}$$ where $a, b, c$ are positive numbers satisfying the condition $a+b+c=1$.
6. A tangent to the incircle with center $I$ and with radius $r$ of a triangle $A B C$ cuts the circumcircle of $\triangle A B C$ at $M$ and $N$. Prove that $M N>2 r$.
7. Let be given a rhombus $A B C D$ with $\widehat{B A D}=60^{\circ}$. $P$ is a point on the line $A B$, dinstinct from $A$ and $B$. The lines $C P$ and $D A$ intersect at $E$. The lines $D P$ and $B E$ intersect at $M$. Find the locus of $M$ when $P$ moves on the line $A B$.
8. Let be given an integer $k \geq 2$. Prove that $\left(k^{2003}\right) !$ is divisible by $$(k !)\left(1+k+k^{2}+\ldots+k^{2002}\right)$$ where $n !$ denotes $1.2.3...n$.
9. Find the greatest value of the expression $x^{3}+y^{3}+(9-x-y)^{3}$, where $x, y$ are positive integers satisfying the conditions $y \leq x \leq 5$, $x+y \leq 8$ and $y+\dfrac{x+1}{2} \geq 5$.
10. The periodic functions $f(x): \mathbb R \rightarrow \mathbb R$ and $g(x): \mathbb R \rightarrow \mathbb R$ satisfy the condition $$\lim_{x \rightarrow+\infty}(f(x)-g(x))=0.$$ Prove that $f(x)=g(x)$ for every real number $x$.
11. Let $A D$, $B E$, $C F$ be the altitudes of an acute triangle $A B C$. Let $R$ and $r$ be respectively the circumradii of the triangles $A B C$ and $D E F$. Prove that $$\sin ^{2} A+\sin ^{2} B+\sin ^{2} C=2+\frac{r}{R}$$
12. Let $S_{A}$, $S_{B}$, $S_{C}$, $S_{D}$ be respectively the areas of the faces opposite to the vertices $A$, $B$, $C$, $D$ of a tetrahedron $A B C D$. Denote the measures of the dihedral angles with sides $B C$, $D A$; $A C$, $B D$; $A B$ $C D$ respectively by $\alpha$, $\alpha^{\prime}$; $\beta$, $\beta^{\prime}$; $\gamma$, $\gamma^{\prime}$. Prove the relation $$S_{A}^{2}+S_{B}^{2}+S_{C}^{2}+S_{D}^{2}=2 T$$ where $$T=S_{A}S_{D} \cos \alpha+S_{B} S_{D} \cos \beta+S_{C} S_{D} \cos \gamma+S_{B} S_{C} \cos \alpha^{\prime}+S_{A} S_{C} \cos \beta^{\prime}+S_{A} S_{B} \cos \gamma^{\prime}.$$

Issue 315

1. Let $A$ be the product of the consecutive integers from $1$ to $1001$ and $B$ be the product of the consecutive integers from $1002$ to $2002$. Is $A+B$ divisible by $2003$? Can you do it by distinct methods?
2. Prove that the sum $$A=\frac{1}{3^{2}}-\frac{1}{3^{4}}+\ldots+\frac{1}{3^{4 n-2}}-\frac{1}{3^{4 n}}+\ldots-\frac{1}{3^{100}}$$ is less than $0$, $1$. Give a generalization of the problem.
3. Find the greatest value of the sum $$T=\frac{1}{a}+\frac{1}{b}+\frac{1}{c},$$ where $a, b, c$ are positive integers satisfying the condition $$\frac{1}{a}+\frac{1}{b}+\frac{1}{c}<1.$$
4. Solve the equation $$\sqrt{\frac{42}{5-x}}+\sqrt{\frac{60}{7-x}}=6.$$
5. Find the greatest value of the expression $x y+y z+z x$, where $x, y, z$ are positive real numbers satisfying the conditions $x \geq y \geq z$ and $32-3 x^{2}=z^{2}=16-4 y^{2}$.
6. Let be given an angle $\widehat{x A y}=90^{\circ}$ and a point $M$ inside the angle. Let $H$ and $K$ be respectively the projections of $M$ on $A x$ and $A y$. On the line passing through $M$ perpendicular to $H K$ take a point $P$ such that $P M=H K$. Find the locus of $P$ when $M$ moves inside the angle $x A y$.
7. Let be given an acute triangle $A B C$ inscribed in a circle with center $O$ and radius $R$. Let $D$, $E$, $F$ be respectively the points of intersection of the lines $A O$ and $B C$, $B O$ and $A C$, $C O$ and $A B$. Prove that $$A D+B E+C F \geq \frac{9 R}{2}.$$
8. Find all natural number a so that there exists a natural number $n>1$ such that $a^{n}+1$ is divisible by $n^{2}$.
9. Prove the inequality $$\frac{\sin ^{n+2} x}{\cos ^{n} x}+\frac{\cos ^{n+2} x}{\sin ^{n} x} \geq 1$$ where $0<x<\dfrac{\pi}{2}$ and $n$ is a positive integer.
10. The sequences of numbers $\left(u_{n}\right)$ and $\left(v_{n}\right)$ $(n=0,1,2, \ldots)$  are defined by $$u_{0}=2001,\, u_{1}=2002,\, v_{0}=v_{1}=1,\quad u_{n+2}=2002 \sqrt{\frac{u_{n}}{v_{n+1}^{2001}}},\, v_{n+2}=2002 \sqrt{\frac{v_{n}}{u_{n+1}^{2001}}}$$ for every $n=0,1,2, \ldots$. Prove that these sequences have finite limits and find these limits.
11. Let be given a triangle $A B C$ with $B C=a$, $C A=b$, $A B=c$ and $\widehat{B}>\widehat{C}$. Prove that a necessary and sufficient condition for $\widehat{A}=2(\hat{B}-\widehat{C})$ is $$(b-c)(b+c)^{2}=a^{2} b.$$
12. Let $R$ be the radius of the circumscribed sphere of the tetrahedron $A_{1} A_{2} A_{3} A_{4}$ and let $S_{i}$ $(i=1,2,3,4)$ be the area of the face opposite to the vertex $A_{i}$ $(i=1,2,3,4)$. $M$ is an arbitrary point in space. Prove that $$\frac{M A_{1}}{S_{1}}+\frac{M A_{2}}{S_{2}}+\frac{M A_{3}}{S_{3}}+\frac{M A_{4}}{S_{4}} \geq \frac{2 \sqrt{3}}{R}.$$

Issue 316

1. Consider all $7$-digit numbers, each of which is formed by seven distinct digits belonging to $\{1,2,3,4,5,6,7\}$.
   a) Are there three numbers $a, b, c$ among them such that $a+b=c$?
   b) Are there two distinct numbers $a$, $b$ among them so that $a$ is divisible by $b$?
2. Let $A B C$ be a triangle with $\widehat{A B C}=30^{\circ}$, $\widehat{B A C}=130^{\circ}$ and let $A x$ be the opposite ray of the ray $A B$. The angled-bisector of $\widehat{A B C}$ cuts the angled-bisector of $\widehat{C A x}$ at $D$. The line $B A$ cuts the line $C D$ at $E$. Compare the measures of $A C$ and $C E$.
3. Find all pairs of prime numbers $p$, $q$ so that $$p^{3}-q^{5}=(p+q)^{2}.$$
4. Solve the system of equations $$\begin{cases} 20 \cdot \dfrac{y}{x^{2}}+11 y &=2003 \\ 20\cdot \dfrac{z}{y^{2}} + 11 z&=2003 \\ 20 \cdot \dfrac{x}{z^{2}}+11 x &=2003\end{cases}$$
5. Find the least value of the expression $$\frac{a}{\sqrt{b}}+\frac{b}{\sqrt{c}}+\frac{c}{\sqrt{a}},$$ where $a, b, c$ are positive numbers satisfying the condition $a+b+c \geq 3$.
6. Let $A B C$ be a triangle with $B C=a$, $A B=A C=b$ and suppose that the angledbisector of $\widehat{A C B}$ cuts the side $A B$ at $D$ so that $C D+D A=a$. Prove that $$a^{3}+b^{3}=3 a b^{2}.$$
7. Let $D$ be a point on the the side $B C$ of triangle $A B C$ ($D$ distinct from $B$, $C)$ and let $E$ and $F$ be respectively the incenters of triangle $A B D$ and triangle $A C D$. Prove that if the four points $B$, $C$, $E$, $F$ lie on a circle then $$\frac{A D+D B}{A D+D C}=\frac{A B}{A C}.$$
8. Prove that for every natural number $k>0$, the number $(\sqrt{2}+\sqrt{3})^{2 k}$ can be written in the form $a_{k}+b_{k} \sqrt{6}$ where $a_{k}$, $b_{k}$ are positive integers. Find the relations defining the sequences $\left(a_{k}\right)$, $\left(b_{k}\right)$, $k=1,2,3, \ldots$. Prove that for every $k \geq 2$, $a_{k-1} \cdot a_{k+1}-6 b_{k}^{2}$ is a constant not depending on $k$.
9. Find the greatest value of the expression $$\left(x^{a_{1}}+x^{a_{2}}+\ldots+x^{a_{n}}\right)\left(\frac{1}{x^{a_{1}}}+\frac{1}{x^{a_{2}}}+\ldots+\frac{1}{x^{a_{n}}}\right)$$ where $x$ is an arbitrary positive number, $x \neq 1$, $n$ is a given even natural number, $a_{1}, a_{2}, \ldots, a_{n}$ are given numbers belonging to the segment $[m, m+1]$, where $m$ is a given natural number.
10. Consider the equation $$x^{2}-2 x+\cos \alpha=0$$ where the parameter $\alpha$ belongs to the interval $(0 ; \pi / 2)$.
    a) Prove that the equation has two positive roots $x_{1}$, $x_{2}$.
    b) Let $f(x)$ be the trinomial with leading coefficient $1$, the two roots of which are $t_{1}=x_{1}^{x_{1}}+x_{2}^{x_{1}}$ and $t_{2}=x_{1}^{x_{2}}+x_{2}^{x_{2}}$. Find all value $c$ such that $f(c)<0$ for all $\alpha$ in the interval $(0 ; \pi / 2)$
11. Let $A B C$ be a triangle with $B C=a$, $C A=b$, $A B=c$, let $O$ and $R$ be respectively its circumcenter and circumradius, let $I_{a}$, $I_{b}$, $I_{c}$ be respectively the centers of its escribed circles in the angles $A$, $B$, $C$ and let $r$ be the inradius of $\triangle A B C$. Prove that $$\frac{1}{2 R} \leq \frac{O I_{a}}{(a+b)(a+c)}+\frac{O I_{b}}{(b+c)(b+a)}+\frac{O I_{c}}{(c+a)(c+b)} \leq \frac{1}{4 r}.$$
12. Consider the tetrahedra $A B C D$ with opposite congruent sides ($A B=C D$, $A C=B D$, $A D=B C$) and the dihedral angles $\alpha$, $\beta$, $\gamma$ respectively of sides $B C$, $C A$, $A B$ are acute. Find the least value of the expression $$T=\sqrt{\cos ^{2} \alpha+\frac{1}{\cos ^{2} \alpha}}+\sqrt{\cos ^{2} \beta+\frac{1}{\cos ^{2} \beta}}+\sqrt{\cos ^{2} \gamma+\frac{1}{\cos ^{2} \gamma}}.$$

Issue 317

1. Calculate the following sums
   a) $A=1.2+2.3+\ldots+n(n+1)+\ldots+98.99$.
   b) $B=1.99+2.98+\ldots+n(100-n)+\ldots+98.2+99.1$.
2. Find all pairs of rational numbers $x, y$ such that both numbers $x+y$ and $\dfrac{1}{x}+\dfrac{1}{y}$ are integers.
3. Find the remainder of the division of $13376^{2003 !}$ by $2000$, where $n!$ is the product of the $n$ integers from $1$ to $n$.
4. Prove that $$\frac{1}{a^{2}+2 b^{2}+3}+\frac{1}{b^{2}+2 c^{2}+3}+\frac{1}{c^{2}+2 a^{2}+3} \leq \frac{1}{2}$$ where $a, b, c$ are real numbers satisfying the condition $a b c=1$. When does equality occur?
5. Let be given the real numbers $a, b, c, x, y$, $z$ satisfying the conditions $a x^{2003}=b y^{2003}=c z^{2003}$ and $x y+y z+z x=x y z \neq 0$. Prove that $$\sqrt[2003]{a x^{2012}+b y^{2002}+c z^{2012}}=\sqrt[2003]{a}+\sqrt[2003]{b}+\sqrt[2003]{c}.$$
6. Let be given a triangle $A B C$. Construct outside of $A B C$ the parallelograms $A B E F$ and $A C P Q$ so that $A F=A C$, $A Q=A B$. Let $D$ be the point of intersection of $B P$ and $C E$. The lines $Q D$ and $F D$ cut $B C$ respectively at $M$ and $N$. Calculate the ratio $\dfrac{MN}{BC}$.
7. A quadrilateral $A B C D$ with $A B>A C$ circumscribes about a circle with center $O$. Let $E$ and $F$ be the points of intersection of $B D$ with the circle. The line $O H$ passing through $O$ cuts orthogonally $A C$ at $H$. Prove that $\widehat{B H E}=\widehat{D H F}$.
8. Let be given positive integers $m, n, k$ with $n>m$. Prove that the number of positive integral solutions of the system of equations $$\begin{cases}x_{1}+x_{2}+\ldots+x_{n} &=y_{1}+y_{2}+\ldots+y_{m}+1 \\ x_{1}+x_{2}+\ldots+x_{n} & \leq n k\end{cases}$$ is equal to $\displaystyle \sum_{i=0}^{n(k-1)} C_{n-1+i}^{n-1} \cdot C_{n-2+i}^{m-1}$.
9. For every positive integer $k$, consider the sequence of numbers $\left(x_{n}^{k}\right)(n=1,2, \ldots)$ defined by $$x_{1}^{k}=1,\quad x_{n}^{k}=\sum_{i=1}^{n} \frac{i^{k}}{i !},\,\forall n=2,3, \ldots$$ a) Prove that the sequence $\left(x_{n}^{k}\right)(n=1,2, \ldots)$ has a finite limit for every positive integer $k$. b) Put $\displaystyle E_{k}=\lim _{n \rightarrow \infty} x_{n}^{k}$. Prove that $y_{k}=\dfrac{E_{k}}{E_{1}}$ is a positive integer for every positive integer $k$.
10. Let $a_{1}, a_{2}, \ldots, a_{n}$ be real numbers distinct from 0 and $u_{1}, u_{2}, \ldots, u_{n}$ be positive real numbers such that $u_{1}<u_{2}<\ldots<u_{n}$, and let $f(x)=\sum_{i=1}^{n} a_{i} \cos \left(u_{i} x\right)$ be a periodic function defined on $R$. Prove that $\dfrac{u_{i}}{u_{1}}$ is a rational number for every $i=2,3, \ldots, n$.
11. Let be given a convex polygon $A_{1} A_{2} \ldots A_{n}$ $(n \geq 3)$ and $M$ be a point inside the polygon. Let $\alpha_{i}=\widehat{M A_{i} A_{i+1}}$ ($i=1,2, \ldots, n$ and $A_{n+1}$ is considered as $A_{1}$). Prove that $$\min _{1 \leq i \leq n}\left\{\alpha_{i}\right\} \leq \frac{(n-2) \pi}{2 n}.$$
12. The sides $O A$, $O B$, $O C$ of a tetrahedron. $O A B C$ are orthogonal each to the others. The incircle of triangle $A B C$ touches the sides $B C$, $C A$, $A B$ respectively at $D$, $E$, $F$. Let $\alpha$, $\beta$, $\gamma$ be respectively the measures of the dihedral angles with sides $B C$, $C A$, $A B$ of the tetrahedron. Prove that
    a) $V_{O D E F}=\dfrac{1}{3} O A \cdot O B \cdot O C \cdot \sin \dfrac{A}{2} \cdot \sin \dfrac{B}{2} \cdot \sin \dfrac{C}{2}$ where $A$, $B$, $C$ denote the angles of triangle $A B C$.
    b) $\tan\alpha \cdot \tan\beta \cdot \tan\gamma \geq 2 \sqrt{2}$. When does equality occur ?

Issue 318

1. Can the sum of the digits of a perfect square be equal to one of the following numbers
   a) $2003$;
   b) $2004$;
   c) $2007$?
2. Let $A B C$ be a triangle with $A B=A C$, $\widehat{B A C}=$ $90^{\circ}$ and $M$ be the midpoint of $B C$. $D$ is a point on the ray $B C$ ($D$ distinct from $B$, $M)$. Draw the line $B K$ perpendicular to $A D$ at $K$. Prove that $K M$ is the interior or exterior angled bisector of $\Delta B K D$ issued from the vertex $K$.
3. Prove that if $2 n$ is the sum of two perfect squares (greater than $1$) then $n^{2}+2 n$ can be written as the sum of four distinct perfect squares (greater than $1$).
4. Solve the system of equations $$\begin{cases} x^{2}(y+z)^{2} &=\left(3 x^{2}+x+1\right) y^{2} z^{2} \\ y^{2}(z+x)^{2} &=\left(4 y^{2}+y+1\right) z^{2} x^{2} \\ z^{2}(x+y)^{2} &=\left(5 z^{2}+z+1\right) x^{2} y^{2} \end{cases}$$
5. Prove the inequality $$\left(1+a^{n+1}\right)\left(1+b^{n+1}\right)\left(1+c^{n+1}\right) \geq\left(1+a b^{n}\right)\left(1+b c^{n}\right)\left(1+c a^{n}\right)$$ where $a, b, c$ are positive numbers and $n$ is a positive integer, $n \leq 4$.
6. Let be given a trapezoid $A B C D$ with $A B || C D$ and $A B \perp B D$. The diagonals $A C$ and $B D$ intersect at $G$. On the line $C E$ perpendicular to $A C$, take the point $E$ so that $C E=A G$ and the line $C D$ does not cut the segment $G E$. On the ray $D C$, take the point $F$ such that $D F=G B$. Prove that $G F$ is perpendicular to $E F$.
7. The circle with center $O$, radius $R$ is tangent to the circle with center $O^{\prime}$, radius $R^{\prime}$ $(R>R^{\prime})$ at the point $A$. The ray $A x$ of a right angle $x A y$ cuts the circle with center $O$ again at $B$ and the ray $A y$ cuts the circle with center $O'$ again at $C$. Let $H$ be the projection of $A$ on $B C$. Prove that when the right angle $x A y$ rotates around the point $A$, the point $H$ moves on a circle.
8. Find all positive integers $k$ so that the equation $$x^{3}+y^{3}+z^{3}=k x^{2} y^{2} z^{2}$$ has positive integral solution and solve this equation.
9. Prove that the sequence of numbers $$s_{n}=\sum_{k=1}^{n} k \sin \frac{k}{n^{3}}(n=1,2,3, \ldots)$$ has finite limit and find this limit.
10. Find all functions $f: \mathbb R \rightarrow \mathbb R$ satisfying the condition $$f(x)=\max _{y \in \mathbb R}\left\{x y^{2003}+y x^{2003}-f(y)\right\},\,\forall x \in \mathbb R.$$
11. Let $A B C D$ be an inscribable quadrilateral such that the circle with diameter $C D$ cuts the segments $A C$, $A D$, $B C$, $B D$ respectively at $A_{1}$, $A_{2}$, $B_{1}$, $B_{2}$ and the circle with diameter $A B$ cuts the segments $C A$, $C B$, $D A$, $D B$ respectively at $C_{1}$, $C_{2}$, $D_{1}$, $D_{2}$. Prove that there exists a circle touching the four lines $A_{1} A_{2}$, $B_{1} B_{2}$, $C_{1} C_{2}$, $D_{1} D_{2}$.
12. In space, consider four rays $P a$, $P b$, $P c$, $P d$ so that no three of them are coplanar and $\widehat{a P b}=\widehat{c P d}$, $\widehat{b P c}=\widehat{d P a}$, $\widehat{c P a}=\widehat{b P d}$.
    a) Prove that there exist such four rays.
    b) Let $\alpha$, $\beta$, $\gamma$, $\delta$ be respectively the angles formed by the rays $P a$, $P b$, $P c$, $P d$ with a fifth given ray $P t$. Find the set of all rays $P x$ such that $$\cos \widehat{a P x}+\cos \widehat{b P x}+\cos \widehat{c P x}+\cos \widehat{d P x}=\cos \alpha+\cos \beta+\cos \gamma+\cos \delta.$$