# Mathematics and Youth Magazine Problems 2006

### Issue 343

1. Find the numbers $x$ and $y$ satisfying the condition $$|x-2005|+|x-2006|+|y-2007|+|x-2008|=3.$$
2. Let $A B C$ be a triangle with $\widehat{B A C}=55^{\circ}$ $\widehat{A B C}=115^{\circ} .$ On the bisector of angle $A C B$ take the point $M$ so that $\widehat{M A C}=25^{\circ} .$ Calculate the measure of angle $\angle B M C$.
3. Find the natural numbers $x, y, z$ satisfying the following conditions
• $x^{3}+y^{3}=2 z^{3}$.
• $x+y+z$ is a prime number.
4. Solve the equation $$\sqrt[3]{x+86}-\sqrt[3]{x-5}=1.$$
5. Find the least value of the expression $$A=\frac{a^{4}}{(b-1)^{3}}+\frac{b^{4}}{(a-1)^{3}}$$ where $a$, $b$ are numbers greater than $1$, satisfying the condition $a+b \leq 4$.
6. Let $A B C$ be an triangle with $B C=a$, $A B=A C=b$ $(a>b)$. Suppose that the measure of the angled bisector $B D$ is equal to $b$. Prove that $$\left(1+\frac{a}{b}\right)\left(\frac{a}{b}-\frac{b}{a}\right)=1.$$
7. Let $A B C$ be a triangle with angled bisectors $A A_{1}$, $B B_{1}$, $C C_{1}$. Suppose that $\widehat{A_{1} B_{1} C_{1}}=90^{\circ}$. Calculate the measure of angle $A B C$.
8. For every positive number $x$, let $a(x)$ denote the number of prime numbers not exceeding $x$ and for every positive integer $m,$ let $b(m)$ denote the number of prime divisors of $m$ Prove that for every positive integer $n,$ we have $$a(n)+a\left(\frac{n}{2}\right)+\ldots+a\left(\frac{n}{n}\right)=b(1)+b(2)+\ldots+b(n).$$
9. Solve the equation $$\sqrt[3]{x^{2}}-2 \sqrt[3]{x}-(x-4) \sqrt{x-7}-3 x+28=0.$$
10. Not using calculators, find the exact measure of acute angle $x$ satisfying $$\cos x=\frac{1}{\sqrt{1+(\sqrt{6}+\sqrt{2}-\sqrt{3}-2)^{2}}}.$$
11. Let $A B C$ be a triangle satisfying the condition $a^{2}=4 S c o \operatorname{tg} A,$ where $B C=a$ and $S$ is the area of $\triangle A B C .$ Let $O$ and $G$ be respectively the circumcenter and the centroid of triangle $A B C .$ Calculate the measure of the angle formed by the lines $A G$ and $O G .$.
12. Let $A B C D$ be a tetrahedron such that its altitudes are concurrent. Let $R$ and $r$ be respectively the circumradius and the inradius of the tetrahedron $ABCD$. Let $R_A$, $R_B$, $R_C$, $R_D$ be respectively the circumradii of the tetrahedra $OBCD$, $OACD$, $OABD$, $OABC$ where $O$ is the circumcenter of the tetrahedron $A B C D$. Prove that
a) $\displaystyle \frac{1}{R_{A}^{2}}+\frac{1}{R_{B}^{2}}+\frac{1}{R_{C}^{2}}+\frac{1}{R_{D}^{2}} \geq \frac{16}{9 R^{2}}$.
b) $\displaystyle \frac{R_{A}}{\sqrt{3 R^{2}+4 R_{A}^{2}}}+\frac{R_{B}}{\sqrt{3 R^{2}+4 R_{B}^{2}}}+\frac{R_{C}}{\sqrt{3 R^{2}+4 R_{C}^{2}}}+\frac{R_{D}}{\sqrt{3 R^{2}+4 R_{D}^{2}}} \leq \frac{\sqrt 3}{3}\frac{R}{r}$.

### Issue 344

1. Find natural number $n$ such that the sum of $2 n$ terms $$\frac{1}{1.3}+\frac{1}{2.4}+\frac{1}{3.5}+\ldots+\frac{1}{(2 n-1)(2 n+1)}+\frac{1}{2 n(2 n+2)}$$ is equal to $\dfrac{14651}{19800}$.
2. Let $A B C$ be an isosceles right angled triangle. Let $M$ be the midpoint of the hypotenuse $B C$, $E$ be the orthogonal projection of $M$ on the line $C G,$ where $G$ is the point on the side $A B$ such that $A G=\dfrac{1}{3} A B$. The lines $M G$ and $A C$ intersect at $D$. Compare the lengths of the segments $D E$ and $B C$.
3. Solve the equation $$\frac{2+\sqrt{x}}{\sqrt{2}+\sqrt{2+\sqrt{x}}}+\frac{2-\sqrt{x}}{\sqrt{2}-\sqrt{2-\sqrt{x}}}=\sqrt{2}.$$
4. Solve the system of equations $$\begin{cases}3 x^{3}-y^{3} &= \dfrac{1}{x+y} \\ x^{2}+y^{2} &=1\end{cases}.$$
5. Pind the least value of the expression $$M=\frac{a b^{2}+b c^{2}+c a^{2}}{(a b+b c+c a)^{2}}$$ where $a$, $b$, $c$ are positive numbers satisfying the condition $a^{2}+b^{2}+c^{2}=3$.
6. Let $X$ be a point on the side $A B$ of a parallelogram $A B C D$. The line passing through $X,$ parallel to $A D$ cuts $A C$ at $M$ and cuts $B D$ at $N .$ The line $X D$ cuts $A C$ at $P$ and the line $X C$ cuts $B D$ at $Q .$ Prove that $$\frac{M P}{A C}+\frac{N Q}{B D} \geq \frac{1}{3}.$$ When does equality occur?
7. Let $A B C$ be a triangle with altitudes $A M$, $B N$ and with circumcircle $(O) .$ Let $D$ be a point on $(O),$ such that $D$ is distinct from $A$, $B$ and $D A$ is not parallel to $B N .$ The line $D A$ intersects the line $B N$ at $Q$. The line $D B$ intersects the line $A M$ at $P$. Prove that when $D$ moves on the circle $(O)$. the midpoint of the segment PQ lies on a fixed line.
8. Let $p$ be a given odd prime number Prove that the difference $$\sum_{j=0}^{p}\left(\begin{array}{c} p \\ j \end{array}\right)\left(\begin{array}{c} p+j \\ j \end{array}\right)-\left(2^{p}+1\right)$$ is divisible by $p^{2}$, where $\left(\begin{array}{l}p \\ j\end{array}\right)$ is binomial coefficient.
9. Consider the sequence $\left(f_{n}(x)\right)$ $(n=0,1,2, \ldots)$ of functions defined on $[0: 1]$ such that $$f_{0}(x)=0,\quad f_{n+1}(x)=f_{n}(x)+\frac{1}{2}\left(x-\left(f_{n}(x)\right)^{2}\right),\,\forall n=0,1,2, \ldots$$ Prove that $\dfrac{n x}{2+n \sqrt{x}} \leq f_{n}(x) \leq \sqrt{x}$ for all $n \in \mathrm{N}$, $x \in[0 ; 1]$
10. Consider the polynomial $P(x)=x^{2}-1$. Find the number of distinct real roots of the equation $$P(P(\ldots, P(x)) \ldots)=0$$ where there are $2006$ notations $P$ on the left hand side of the equation.
11. Suppose that $A_{1} B_{1} C_{1}$, $A_{2} B_{2} C_{2}$, $A_{3} B_{3} C_{3}$ are three triangles satisfying the conditions $$\widehat{C_{1}}=\widehat{C_{2}}=\widehat{C_{3}},\quad A_{1} B_{1}=A_{2} B_{2}=A_{3} B_{3},\\ B_{1} C_{1}+C_{2} A_{2}=B_{2} C_{2}+C_{3} A_{3}=B_{3} C_{3}+C_{1} A_{1}.$$ Prove that these three triangles are congruent.
12. Consider a convex hexagon $A B C D E F$ inscribed in a circle. The diagonal $B F$ cuts $A E$, $A C$ respectively at $M$, $N$. The diagonal $B D$ cuts $C A$, $C E$ respectively at $P$, $Q$. The diagonal $D F$ cuts $E C$, $EA$ respectively at $R$, $S$. Prove that $M Q$, $N R$ and $P S$ are concurrent.

### Issue 345

1. Let $$A=\left(1-\frac{1}{1+2}\right)\left(1-\frac{1}{1+2+3}\right) \cdot\left(1-\frac{1}{1+2+3+\ldots+n}\right)$$ (consisting of $n-1$ factors) and $B=\dfrac{n+2}{n}$. Calculate $\dfrac{A}{B}$.
2. Let $A B C$ be an isosceles triangle $(A B=A C)$ and $O$ be a point inside $A B C$ such that $\widehat{A O B}<\widehat{A O C}$. Compare the measures of $OB$ and $O C$.
3. Find the numbers $x$ such that $$\frac{\sqrt{x}}{x\sqrt{x}-3 \sqrt{x}+ 3}$$ is an integer.
4. Find the greatest value of the expression $$P=\frac{x}{1+y^{2}}+\frac{y}{1+x^{2}}$$ where $x$, $y$ are non negative real numbers not exceeding $\dfrac{\sqrt{2}}{2}$.
5. Prove that $$\frac{3 \sqrt{3}}{4} \leq \frac{b c}{a(1+b c)}+\frac{c a}{b(1+c a)}+\frac{a b}{c(1+a b)} \leq \frac{a+b+c}{4}$$ where $a, b, c$ are positive real numbe satisfying the condition $a+b+c=a b c$. When do equalities occur?
6. Two arbitrary points $E$, lie respectively on the sides $A B$, $A C$ of a triangle $A B C$ so that $\dfrac{A E}{E B}=\dfrac{C D}{D A}$. The lines $B D$, $C E$ intersect at $M$. Determine the positions of $E$ and $D$ so that the area of triangle $B M C$ attains its greatest value and calculate this value in terms of the area of triangle $A B C$.
7. Let $A B C$ be a triangle inscribed in a circle $(O)$. The bisector of angle $B A C$ cuts the circle $(O)$ at $A$ and $D .$ The circle with center $D$ and radius $D B$ cuts the line $A B$ at $B$ and $Q$, cuts the line $A C$ at $C$ and $P$. Prove that the line $A O$ is perpendicular to the line $P Q$.
8. Determine non empty subsets $A$, $B$, $C$ of the set $N=\{0,1,2, \ldots\}$ satisfying the following conditions
• $A \cap B=B \cap C=C \cap A=\varnothing$;
• $A \cup B \cup C=N$;
• if $a \in A, b \in B, c \in C$ then $a+c \in A$ $b+c \in B, a+b \in C$
1. Prove that $$\left|x_{1}+x_{2}+\ldots+x_{2007}\right| \leq \frac{2007}{3}$$ where $x_{1}, x_{2}, \ldots, x_{2007}$ are real numbers belonging to the segment $[-1 ; 1],$ so that the sum of their cubes is equal to $0$. When does equality occur?
2. Find all functions $f: \mathbb{Z} \rightarrow \mathbb{Z}$ satisfying the following conditions $f(1)>0$ and $$f(f(m)-n)=f\left(m^{2}\right)+f(n)-2 n f(m),\,\forall m, n \in \mathbb{Z} .$$
3. Let $A A_{1}$, $B B_{1}$, $C C_{1}$ be the medians of a triangle $A B C$. Prove that if the radii of the incircles of the triangles $B C B_{1}$, $C A C_{1}$, $A B A_{1}$ are all equal then $A B C$ is an equilateral triangle.
4. Let be given a sphere with center $O$ and radius $R$. A pyramid $S . A B C$ moves so that the sides $S A$, $S B$, $S C$ touch the sphere $(O)$ respectively at $A$, $B$, $C$ and so that $\widehat{A S B}=90^{\circ}$, $\widehat{B S C}=60^{\circ}$, $\widehat{C S A}=120^{\circ}$. Find the locus of the apex $S$.

### Issue 346

1. Compare the number $\dfrac{1}{1002}$ with the following sum (consisting of $2006$ terms) $$A=\frac{2}{2005+1}+\frac{2^{2}}{2005^{2}+1}+\ldots+\frac{2^{n+1}}{2005^{2^{n}}+1}+\ldots+\frac{2^{2006}}{2005^{2^{2005}}+1}.$$
2. Let $a, b, c$ be three distinct integers different from $0$ such that $a+b+c=0$. Calculate the value of the expression $$P=\left(\frac{a-b}{c}+\frac{b-c}{a}+\frac{c-a}{b}\right)\left(\frac{c}{a-b}+\frac{a}{b-c}+\frac{b}{c-a}\right).$$
3. Let $a, b, c$ be positive real numbers satisfying $a b c \leq 1$. Prove that $$\frac{a}{c}+\frac{b}{a}+\frac{c}{b} \geq a+b+c.$$ When does equality occur?
4. Solve the equation $$2 \sqrt{2 x+4}+4 \sqrt{2-x}=\sqrt{9 x^{2}+16}.$$
5. Find the least value of the expression $$\left(x^{2}+1\right) \sqrt{x^{2}+1}-x \sqrt{x^{4}+2 x^{2}+5}+(x-1)^{2}.$$
6. Let $A B C$ be a triangle with obtuse angle $\widehat{A B C}$. Prove that $$\sin (x+y)=\sin x \cdot \cos y+\sin y \cdot \cos x$$ where $x=\widehat{B A C}$ and $y=\widehat{B C A}$.
7. Let $A B C D$ be a cyclic quadrilateral such that the sides $A B$, $C D$ are not parallel and let $I$ be the point of intersection of its diagonals. Let $M$, $N$ be respectively the midpoints of $B C$, $C D$. Prove that if $N I$ is perpendicular to $A B$ then $M I$ is perpendicular to $A D$.
8. Let $a, b, c, d, e, f$ be six positive integers satisfying $a b c=d e f$. Prove that $$a\left(b^{2}+c^{2}\right)+d\left(e^{2}+f^{2}\right)$$ is a composite number.
9. Consider all quadratic trinomials $f(x)=a x^{2}+b x+c$ ($a, b, c$ are integers, $a>0)$ having two distinct roots belonging to the interval $(0 ; 1)$. Find the trinomial such that the coefficient $a$ attains its least value.
10. Prove that $$a b+b c+c a \geq 8\left(a^{2}+b^{2}+c^{2}\right)\left(a^{2} b^{2}+b^{2} c^{2}+c^{2} a^{2}\right)$$ where $a, b, c$ are nonnegative real numbers satisfying $a+b+c=1$.
11. The incircle $(I)$ of a triangle $A_{1} A_{2} A_{3}$ has radius $r$ and touches the sides $A_{2} A_{3}$, $A_{3} A_{1}$, $A_{1} A_{2}$ respectively at $M_{1}$, $M_{2}$, $M_{3}$. Let $\left(I_{i}\right)$ be the circle touching the sides $A_{i} A_{j}$, $A_{j} A_{k}$ and externally touching $(I)$ ($i, j, k \in\{1,2,3\}$, $i \neq j \neq k \neq i$). Let $K_{1}$, $K_{2}$, $K_{3}$ be the touching points respectively of $\left(I_{1}\right)$ with $A_{1} A_{2}$, of $\left(I_{2}\right)$ with $A_{2} A_{3}$, of $\left(I_{3}\right)$ with $A_{3} A_{1}$. Put $A_{i} I_{1}=a_{i}$, $A_{i} K_{i}=b_{i}$ $(i=1,2,3)$. Prove that $$\frac{1}{r} \sum_{i=1}^{3}\left(a_{i}+b_{i}\right) \geq 2+\sqrt{3}.$$When does equality occur?
12. Let be given a sphere with center $O$ and a chord $A B$, not passing through $O$. Let $M M^{\prime}$, $N N^{\prime}$, $P P^{\prime}$ be three chords (not coinciding with $A B$) passing through the midpoint $I$ of $A B$. Let $E$, $E^{\prime}$ be the points of intersection of the line $A B$ respectively with the planes $(MNP)$, $\left(M^{\prime} N^{\prime} P^{\prime}\right)$. Prove that $I E=I E^{\prime}$.

### Issue 347

1. Compare $\dfrac{5}{8}$ with $\left(\dfrac{389}{401}\right)^{10}$.
2. Let $E$, $F$ be points respectively on the sides $A C$, $A B$ of a triangle $A B C$ such that $\widehat{A B E}=\dfrac{1}{3} \widehat{A B C}$, $\widehat{A C F}=\dfrac{1}{3} \widehat{A C B}$. The lines $B E$ and $C F$ intersect at $O$. Suppose that $O E=O F$. Prove that $A B=A C$ or $\widehat{B A C}=90^{\circ}$.
3. Find integral solutions of the system of equations $$\begin{cases}4 x^{3}+y^{2} &=16 \\ z^{2}+y z &=3\end{cases}$$
4. Consider all quadratic equations $a x^{2}+b x+c=0$ having two roots belonging to the segment $[0 ; 2]$. Find the greatest value of the expression $$P=\frac{8 a^{2}-6 a b+b^{2}}{4 a^{2}-2 a b+a c}.$$
5. Consider all triangles $A B C$ such that the measures $a, b, c$ of their sides satisfy the relation $$1964 a b+15 b c+10 c a=2006 a b c.$$ Find the least value of the expression $$M=\frac{1974}{p-a}+\frac{1979}{p-b}+\frac{25}{p-c}$$ where $p$ is the semiperimeter of triangle $A B C$.
6. Let be given a quadrilateral $A B C D$. Take two points $M, P$ respectively on the sides $A B$, $A C$ such that $\dfrac{A M}{A B}=\dfrac{C P}{C D}$. Find the locus of the midpoints $I$ of the segments $M P$ when $M$, $P$ moves respectively on $A B$, $A C$.
7. Let $A B C$ be a triangle with $\widehat{B A C}=135^{\circ}$ and $A M$, $B N$ be two of its altitudes ($M$ on $B C$, $N$ on $C A$). The line $M N$ cuts the perpendicular bisector of $A C$ at $P$. Let $D$ and $E$ be the midpoints respectively of $N P$ and $B C$. Prove that $A B C$ is a right isosceles triangle.
8. Let be given $167$ sets $A_{1}, A_{2}, \ldots, A_{167}$ satisfying the following conditions
• $\sum_{i=1}^{167}\left|A_{i}\right|=2004$;
• $\left|A_{j}\right|=\left|A_{i} \| A_{i} \cap A_{j}\right|$ for all $i, j \in\{1,2, \ldots,167\}$ and $i \neq j$.
Calculate $\left|\bigcup_{i=1}^{67} A_{i}\right|$, where $|A|$ denotes the number of elements of the set $A$.
9. Find all continuous functions $f$, defined on $\mathbb{R}$, satisfying the condition $$f_{3}(x)+f(x)=2 x,\,\forall x \in \mathbb{R}$$ where $f_{3}(x)=f(f(f(x)))$.
10. Find the least value of the expression $$H=\frac{x^{2}}{y+z}+\frac{y^{2}}{z+x}+\frac{z^{2}}{x+y},$$ where $x, y, z$ are positive numbers satisfying $$\sqrt{x^{2}+y^{2}}+\sqrt{y^{2}+z^{2}}+\sqrt{z^{2}+x^{2}}=2006.$$
11. Let $A B C$ be an acute triangle with altitudes $A D$, $B E$, $C F$ and let $O$ be its circumcenter. Let $M$, $N$, $P$ be the midpoints respectively of the segments $B C$, $C A$, $A B$. Let $D_{1}$, $E_{1}$, $F_{1}$ be the reflexions respectively of $D$ in $M$, of $E$ in $N$, of $F$ in $P_{1}$. Prove that $O$ lies inside the triangle $D_{1} E_{1} F_{1}$.
12. Let $G_{1}$, $G_{2}$, $G_{3}$, $G_{4}$ be the centroids respectively of the faces $B C D$, $CDA$, $D A B$, $A B C$ of a tetrahedron $A B C D$. Let $A_{1}$, $B_{1}$, $C_{1}$, $D_{1}$ be the points of intersection of the circumsphere of the tetrahedron respectively with $A G_{1}$, $B G_{2}$, $C G_{3}$, $D G_{4}$. Prove that $$\frac{A G_{1}}{A A_{1}}+\frac{B G_{2}}{B B_{1}}+\frac{C G_{3}}{C C_{1}}+\frac{D G_{4}}{D D_{1}} \leq \frac{8}{3} .$$

### Issue 348

1. Find all four-digit numbers $\overline{a b c d}$ satisfying the condition $$\overline{a b c d}=a^{2}+2 b^{2}+3 c^{2}+4 d^{2}+2006.$$
2. Let $A B C$ be a right-angled triangle with right angle at $A .$ On the side $A C$ take the point $E$ so that $\widehat{E B C}=2 \widehat{A B E}$. On the segment $B E$ take the point $M$ such that $E M=B C$. Compare the measures of the angles $\widehat{M B C}$ and $\widehat{B M C}$.
3. Solve the equation $$\frac{1}{4 x-2006}+\frac{1}{5 x+2004}=\frac{1}{15 x-2007}-\frac{1}{6 x-2005}.$$
4. Prove that $$a(b+c)+b(c+a)+c(a+b)+2\left(\frac{1}{1+a^{2}}+\frac{1}{1+b^{2}}+\frac{1}{1+c^{2}}\right) \geq 6$$ for arbitrary numbers $a, b, c$ not less than $1$.
5. Find the greatest value of the expression $$P=3 x y+3 y z+3 z x-x y z$$ where $x, y, z$ are positive numbers satisfying the condition $x^{3}+y^{3}+z^{3}=3$.
6. Let be given a triangle $A B C$. $P$ is a point on the line $B C$. On the opposite ray of the ray $A P$, take the point $D$ such that $A D=\dfrac{B C}{2}$. Let $E$ and $F$ be the midpoints respectively of the segments $D B$ and $D C$. Prove that the circle with diameter $E F$ passes through a fixed point when $P$ moves on the line $B C$.
7. Let $A B C$ be a triangle with $A B=A C=a$. Construct a circle with center $A$, with radius $R$ $(R<a)$. From $B$ and $C$, draw the tangents $B M$, $C N$ to this circle ($M$ and $N$ are touching points) so that they are not symmetric with respect to the altitude $A H$ of triangle $A B C$. Let $I$ be the point of intersection of $B M$ and $C N$.
a) Find the locus of $I$ when $R$ varies.
b) Prove that $I B \cdot I C=\left|a^{2}-d^{2}\right|$ where $A I=d$.
8. Let be given $n$ real positive numbers $a_{1}, a_{2}, \ldots, a_{n}$ satisfying the condition $$\sum_{i=1}^{k} a_{i} \leq \sum_{i=1}^{k} i(i+1),\,\forall k=1,2, \ldots, n.$$ Prove that $$\frac{1}{a_{1}}+\frac{1}{a_{2}}+\ldots+\frac{1}{a_{n}} \geq \frac{n}{n+1}$$
9. Determine the number of distinct real roots belonging to the interval $(0 ; 2 \pi)$ of the equation $$e^{2 \cos ^{2} x}\left(8 \sin ^{6} x-12 \sin ^{4} x+10 \sin ^{2} x\right)=e+\frac{5}{2}.$$
10. Find all polynomials $P(x)$ with real coefficients satisfying the condition $$P(x) \cdot P\left(2 x^{2}\right)=P\left(x^{3}+x\right),\,\forall x \in \mathbb{R}.$$
11. Let $O$ be the point of intersection of the diagonals $A C, B D$ of a convex quadrilateral $A B C D$. Let $G_{1}$ and $G_{2}$ be the centroids respectively of the triangles $O A B$ and $O C D$. Let $H_{1}$ and $H_{2}$ be the orthocenters respectively of the triangles $O B C$ and $O D A$. Prove that $G_{1} G_{2}$ is perpendicular to $H_{1} H_{2}$
12. Let $I$ and $r$ be respectively the center and the radius of the sphere inscribed in al tetrahedron $A B C D$. Let $r_{A}$, $r_{B}$, $r_{C}$, $r_{D}$ be the radii of the spheres inscribed respectivelly in the tetrahedra $I B C D$, $I A C D$, $I A B D$, $I A B C$. Prove the inequality $$\frac{1}{r_{A}}+\frac{1}{r_{B}}+\frac{1}{r_{C}}+\frac{1}{r_{D}} \leq \frac{4+\sqrt{6}}{r}.$$

### Issue 349

1. Let $S$ be the following sum of $2006$ terms $$S=\frac{2}{2^{1}}+\frac{3}{2^{2}}+\ldots+\frac{n+1}{2^{n}}+\ldots+\frac{2007}{2^{2006}} .$$ Compare $S$ with $3$.
2. Let $A B C$ be a triangle with its two medians $A D$, $B E$ meeting at $M$. Prove that if $$\widehat{A M B} \leq 90^{\circ} then A C+B C>3 A B.$$
3. Prove that for every given positive integer $r$ less than $59$, there exists a unique positive integer $n$ less than $59$ such that $\left(2^{n}-r\right)$ is divisible by $59$.
4. Solve the equation $$2 x^{2}-5 x+2=4 \sqrt{2\left(x^{3}-21 x-20\right)}.$$
5. Prove that $$4 a b c \left[\frac{1}{(a+b)^{2} c}+\frac{1}{(b+c)^{2} a}+\frac{1}{(c+a)^{2} b}\right]+\frac{a+c}{b}+\frac{b+c}{a}+\frac{a+b}{c} \geq 9$$ for arbitrary positive real numbers $a, b, c$.
6. Let $A B C$ be a right-angled triangle, right at $B$ and $A B=2 B C$. Let $D$ be the point on side $A C$ such that $B C=C D$, let $E$ be the point on side $A B$ such that $A D=A E$. Prove that $A D^{2}=A B \cdot B E$.
7. In plane, let be given two lines $\Delta_{1}$, $\Delta_{2}$ intersecting at $O$. A point $M$ moves in plane so that $O M$ is equal to a constant $R$ and $M$ does not lie on $\Delta_{1}$, $\Delta_{2}$. Let $H$, $K$ be the orthogonal projections of $M$ on $\Delta_{1}$, $\Delta_{2}$ respectively. Find the locus of the incenter of triangle $M H K$.
8. Let be given three prime numbers $p_{1}$, $p_{2}, p_{3}$ $\left(p_{1}<p_{2}<p_{3}\right)$. Put $$A=\left\{n \mid n \in \mathbb{N}^{*}, 1 \leq n \leq p_{1} p_{2} p_{3}, p_{1} \nmid n, p_{2} \nmid n, p_{3} \nmid n\right\}.$$ Prove that $|A| \geq 8$ ($|A|$ denotes the number of elements of the set $A$). When does equality occur?
9. Let be given six real numbers $a, b, c$, $a_{1}$, $b_{1}$, $c_{1}$ $\left(a a_{1} \neq 0\right)$ satisfying the condition $$\left(\frac{c}{a}-\frac{c_{1}}{a_{1}}\right)^{2}+\left(\frac{b}{a}-\frac{b_{1}}{a_{1}}\right) \cdot \frac{b c_{1}-c b_{1}}{a a_{1}}<0.$$ Prove that each of the following equations $a x^{2}+b x+c=0$ and $a_{1} x^{2}+b_{1} x+c_{1}=0$ has two distinct roots and by representing these roots on the number line, the roots of one equation alternate with the roots of the other equation.
10. Find all polynomials with real coefficients $P(x)$ satisfying the condition $$P(x) \cdot P(x+1)=P\left(x^{2}+2\right),\,\forall x \in \mathbb{R}$$
11. Let $A A_{1}$, $B B_{1}$, $C C_{1}$ be the inner angled bisectors of triangle $A B C$ and $A_{2}$, $B_{2}$, $C_{2}$ be the touching points of the incircle of triangle $A B C$ with the sides $B C$, $C A$, $A B$ respectively. Let $S$, $S_{1}$, $S_{2}$ be the areas of triangles $A B C$, $A_{1} B_{1} C_{1}$, $A_{2} B_{2} C_{2}$ respectively. Prove that $$\frac{3}{S_{1}}-\frac{2}{S_{2}} \leq \frac{4}{S}.$$
12. Let $Sxyz$ be a trihedral angle with $\widehat{x S y}=121^{\circ}$, $\widehat{x S z}=59^{\circ}$. $A$ is a point on $S x$, $O A=a$. On the ray bisecting the angle $\widehat{z S y}$, take the point $B$ such that $S B=a \sqrt{3}$. Calculate the measures of the angles of triangle $S A B$.

### Issue 350

1. Prove that $2005^{2007^{2006}}+2006^{2005^{2007}}+2007^{2006^{2005}}$ is divisible by $102$.
2. Consider the sum of $n$ terms $$S_{n}=1+\frac{1}{1+2}+\frac{1}{1+2+3}+\ldots+\frac{1}{1+2+\ldots+n}$$ for $n \in \mathbb{N}^{*}$. Find the least rational number $a$ such that $S_{n}<a$ for all $\in \mathbb{N}^{*}$.
3. Find all solutions $(x, y)$ of the equation $$\left(x^{2}+4 y^{2}+28\right)^{2}=17\left(x^{4}+y^{4}+14 y^{2}+49\right)$$ such that $x, y$ are natural numbers.
4. Solve the following system of equations $$\begin{cases}\dfrac{1}{x}+\dfrac{1}{y+z} &=\dfrac{1}{2} \\ \dfrac{1}{y}+\dfrac{1}{x+z} &=\dfrac{1}{3} \\ \dfrac{1}{z}+\dfrac{1}{x+y} &=\dfrac{1}{4}\end{cases}$$
5. Find the greatest value and the least value of the expression $$P=\sqrt{2 x+1}+\sqrt{3 y+1}+\sqrt{4 z+1}$$ where $x, y, z$ are arbitrary non negative real numbers satisfying the condition $x+y+z=4$.
6. Let $M$ be a point inside an acute triangle $A B C$ satisfying the condition $\widehat{M B A}=\widehat{M C A}$. Let $K$ and $L$ be the feet of the perpendiculars respectively to $A B$ and $A C$ passing through $M$. Prove that $K$ and $L$ are in equal distances from the midpoint of $B C$ and the median issued from $M$ of triangle $M K L$ passes through a fixed point when $M$ moves inside triangle $A B C .$
7. Let be given a right-angled triangle $A B C$, right at $A$ and $A H$ be its altitude issued from $A$. A circle passing through $B$ and $C$ cuts $A B$ and $A C$ at $M$ and $N$ respectively. Consider the rectangle $A M D C$. Prove that $H N$ is perpendicular to $H D$.
8. Let $a$ be a natural number greater than 1. Consider a non empty subset $A$ of $N$ satysfying the condition: If $k \in A$ then $k+2 a \in A$ and $\left[\frac{k}{a}\right] \in A([x]$ denotes the integral part of $x$). Prove that $A=N$.
9. Find all continuous functions $f: \mathbb{R} \rightarrow \mathbb{R}$ satisfying the condition $$9 f(8 x)-9 f(4 x)+2 f(2 x)=100 x,\,\forall x \in \mathbb{R}.$$
10. Find the greatest value and the least value of the expression $$P=a(b-c)^{3}+b(c-a)^{3}+c(a-b)^{3}$$ where $a, b, c$ are arbitrary non negative real numbers satisfying the condition $a+b+c=1$.
11. Let $I$ and $G$ be respectively the incenter and the centroid of a triangle $A B C$. Let $R_{1}$, $R_{2}$, $R_{3}$ be the circumradii respectively of the triangles $I B C$, $I C A$, $I A B$ and let $R_{1}^{\prime}$, $R_{2}^{\prime}$, $R_{3}^{\prime}$ be the circumradii respectively of the triangles $G B C$, $G C A$, $G A B$. Prove that $$R_{1}^{\prime}+R_{2}^{\prime}+R_{3}^{\prime} \geq R_{1}+R_{2}+R_{3} .$$
12. Let $A B C D$ be a tetrahedron, the measures of its sides are: $B C=a$, $D A=a_{1}$, $C A=b$, $D B=b_{1}$, $A B=c$, $D C=c_{1}$ and let $G$ be its centroid. The sphere circumscribing $A B C D$ cuts $A G$, $B G$, $C G$, $D G$ respectively at $A_{1}$, $B_{1}$, $C_{1}$, $D_{1}$; let $R$ be its radius. Prove that $$\frac{4}{R} \leq \frac{1}{G A_{1}}+\frac{1}{G B_{1}}+\frac{1}{G C_{1}}+\frac{1}{G D_{1}} \leq \frac{2 \sqrt{3}}{3}\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{a_{1}}+\frac{1}{b_{1}}+\frac{1}{c_{1}}\right).$$

### Issue 351

1. Consider the product of $11$ factors $$T=(5 a+2006 b)(6 a+2005 b)(7 a+2004 b) \ldots(15 a+1996 b)$$ where $a$, $b$ are given integers. Prove that if $T$ is divisible by $2011$ then $T$ is divisible by $2011^{11}$.
2. Calculate the sum of 2006 terms $$S=\frac{3^{3}+1^{3}}{2^{3}-1^{3}}+\frac{5^{3}+2^{3}}{3^{3}-2^{3}}+\frac{7^{3}+3^{3}}{4^{3}-3^{3}}+\ldots+\frac{4013^{3}+2006^{3}}{2007^{3}-2006^{3}}$$
3. Find the prime number $p$ such that $2005^{2005}-p^{2006}$ is divisible by $2005+p$
4. Solve the system of equations $$\begin{cases}x+y+z+t & =12 \\ x^{2}+y^{2}+z^{2}+t^{2} & =50 \\ x^{3}+y^{3}+z^{3}+t^{3} & =252 \\ x^{2} t^{2}+y^{2} z^{2} & =2 x y z t\end{cases}$$
5. Find the least value of the expression $$P=\frac{a b+b c+c a}{a^{2}+b^{2}+c^{2}}+\frac{(a+b+c)^{3}}{a b c},$$ where $a, b, c$ are positive real numbers.
6. Let be given a not obtuse triangle $A B C$ with its three altitudes $A A_{1}$, $B B_{1}$, $C C_{1}$ and its orthocenter $H$. Prove that $$H A^{2}+H A_{1}^{2}+H B^{2}+H B_{1}^{2}+H C^{2}+H C_{1}^{2} \geq \frac{5}{2}\left(H A \cdot H A_{1}+H B \cdot H B_{1}+H C \cdot H C_{1}\right)$$
7. Let be given five concyclic points $A$, $B$, $C$, $D$, $E$ and let $M$, $N$, $P$, $Q$ be the orthogonal projections of $E$ respectively on the lines $A B$, $B C$, $C D$, $D A$. Prove that the orthogonal projections of $E$ on the lines $M N$, $N P$, $P Q$, $Q M$ are collinear.
8. Prove that $(2 n+1)^{n+1} \leq(2 n+1) ! ! \pi^{n}$ for every natural number $n$, where $(2 n+1) ! !$ denotes the product of the first $n+1$ positive odd integers.
9. Solve the equation $$x^{3}-3 x=\sqrt{x+2}.$$
10. Let $f(x)$ be a continuous function defined on $[0 ; 1]$ satisfying the conditions $$f(0)=0,\, f(1)=1,\quad 6 f\left(\frac{2 x+y}{3}\right)=5 f(x)+f(y),\,\forall x \geq y ; x, y \in[0 ; 1].$$ Calculate $f\left(\dfrac{8}{23}\right)$.
11. Calculate the measures of the angles of a triangle $A B C$ satisfying the condition $$\frac{h_{a}}{m_{b}}+\frac{h_{b}}{m_{a}}=\frac{4}{\sqrt{3}}$$ where $m_{a}, m_{b}$ are the measures of its two medians and $h_{a}$, $h_{a}$ are the measures of its two altitudes issued respectively from the vertices $A$, $B$.
12. Let be given an equifaced tetrahedron $A B C D$ ($A B=C D$, $A C=B D$, $B C=A D$) and let $V$, $R$, $r$ be respectively its volume, its circumradius, its inradius. Prove that $$\frac{243 V^{2}}{512 R^{6}} \leq \cos A \cdot \cos B \cdot \cos C \leq \frac{9}{8}\left(\frac{r}{R}\right)^{2}$$ where $A$, $B$, $C$ are the angles of triangle $A B C$. When do equalities occur?

### Issue 352

1. Find a $5$-digit number such that by multiplying it by $2$ we obtain a $6$-digit number with six distinct nonzero digits and by multiplying it respectively by $5,6,7,8,11$ we obtain five $6$-digit numbers such that the digits of each number are the six above mentioned nonzero digits but written in another order.
2. Let $a, b, c, d, m, n$ be positive integers such that $a b=c d$. Prove that the number $$A=a^{2 n+1}+b^{2 m+1}+c^{2 n+1}+d^{2 m+1}$$ is a composite number.
3. Find integral solutions of the equation $$x^{5}-y^{5}-x y=32 .$$
4. Let be given positive numbers $a, b, c$ satisfying the condition $a b c \geq 1$. Prove that $$\frac{a}{\sqrt{b+\sqrt{a c}}}+\frac{b}{\sqrt{c+\sqrt{a b}}}+\frac{c}{\sqrt{a+\sqrt{b c}}} \geq \frac{3}{\sqrt{2}}.$$
5. Find real numbers $x, y$ satisfying the conditions $$x+y \geq 4,\quad \left(x^{3}+y^{3}\right)\left(x^{7}+y^{7}\right)=x^{11}+y^{11}.$$
6. Let $A B C D$ be a convex quadrilateral and let $E$, $F$ be the midpoints respectively of $A D$, $B C$. The lines $A F$, $B E$ intersect at $M$, the lines $C E$, $D F$ intersect at $N$. Find the least value of $$P=\frac{M A}{M F}+\frac{M B}{M E}+\frac{N C}{N E}+\frac{N D}{N F} .$$
7. Let $A$, $B$, $C$ be three points lying on a circle with center $O$ and radius $R$ so that $$C B-C A=R,\quad C A \cdot C B=R^{2} .$$ Calculate the measures of the angles of triangle $A B C$.
8. The sequence of numbers $\left(a_{i}\right)$ $(i=1,2,3, \ldots)$ is defined by $$a_{1}=1,\, a_{2}=-1,\quad a_{n}=-a_{n-1}-2 a_{n-2},\,\forall n=3,4, \ldots$$ Calculate the value of the expression $$A=2 a_{2006}^{2}+a_{2006} \cdot a_{2007}+a_{2007}^{2}.$$
9. Let $N_{m}$ be the set of all integers not less then a given integer $m$. Find all functions $f: N_{m} \rightarrow N_{m}$ satisfying the condition $$f\left(x^{2}+f(y)\right)=y+(f(x))^{2},\,\forall x, y \in N_{m}.$$
10. Suppose that the system of equations $$\begin{cases}x^{2}+x y+x &=1 \\ y^{2}+x y+x+y &=1\end{cases}$$ has a unique solution $\left(x_{0}, y_{0}\right)$ with $x_{0}>0$, $y_{0}>0$. Prove that $$\frac{1}{x_{0}}+\frac{1}{y_{0}}=8 \cos ^{3} \frac{\pi}{7}.$$
11. The measures of the sides of a triangle $A B C$ are $B C=a$, $C A=b$, $A B=c$ and the measures of its altitudes issued respectively from $A$, $B$, $C$ are $h_{a}$, $h_{b}$, $h_{c}$. Take $A_{1}$ on the side $B C$ so that the incircles of triangles $A B A_{1}$, $A C A_{1}$ have equal radii $r_{A}$. One defines $r_{B}$, $r_{C}$ analogously. Prove that $$2\left(r_{A}+r_{B}+r_{C}\right)+p \leq h_{a}+h_{b}+h_{c}$$ where $p$ is the semiperimeter of triangle $A B C$.
12. Let be given a triangular pyramid $S.MNP$ such that $$\widehat{M S N}+\widehat{N S P}+\widehat{P S M}=180^{\circ}.$$ Prove that $\cos \alpha+\cos \beta+\cos \gamma=1$ where $\alpha$, $\beta$, $\gamma$ are the measures of the dihedral angles with sides $S M$, $S N$, $S P$ respectively.

### Issue 353

1. Find $2 n$-digit number of the form $\overline{a_{1} a_{2} \ldots a_{2 n-1} a_{2 n}}$ satisfying the condition $$\overline{a_{1} a_{2} \ldots a_{2 n-1} a_{2 n}}=a_{1} \cdot a_{2}+\ldots+a_{2 n-1} \cdot a_{2 n}+2006.$$
2. Do there exist three numbers $a, b, c$ satisfying $$\frac{a}{b^{2}-c a}=\frac{b}{c^{2}-a b}=\frac{c}{a^{2}-b c}$$
3. Find all positive integers $x, y, z$ satisfying simultaneously the two conditions
• $\dfrac{x-y \sqrt{2006}}{y-z \sqrt{2006}}$ is a rational number,
• $x^{2}+y^{2}+z^{2}$ is a prime number.
4. Find the greatest value and the least value of the expression $P=x y z$ where $x, y, z$ are real numbers satisfying $$\frac{8-x^{4}}{16+x^{4}}+\frac{8-y^{4}}{16+y^{4}}+\frac{8-z^{4}}{16+z^{4}} \geq 0.$$
5. Prove that $$\frac{2}{9} \leq a^{3}+b^{3}+c^{3}+3 a b c < \frac{1}{4}$$ where $a, b, c$ are the measures of three sides of a triangle with perimeter $a+b+c=1$.
6. Consider convex quadrilateral $A A^{\prime} C^{\prime} C$ such that the lines $A C$, $A^{\prime} C^{\prime}$ intersect at a point $I$. Take a point $B$ on the side $A C$ and a point $B^{\prime}$ on the side $A^{\prime} C^{\prime}$. Let $O$ be the point of intersection of the lines $A C^{\prime}$, $A^{\prime} C$; let $P$ be that of $A B^{\prime}$, $A^{\prime} B$; let $Q$ be that of $B C^{\prime} \cdot B^{\prime} C$. Prove that the points $P$, $O$, $Q$ are collinear.
7. Let be given an isosceles triangle $A B C$ with $A B=A C$. Take a point $D$ on the side $A B$ and a point $E$ on the side $A C$ so that $D E=B D+C E$. The bisector of angle $B D E$ cuts the side $B C$ at $I$. a) Find the measure of angle $\angle D I E$. b) Prove that the line $D I$ passes through a fixed point when $D$ moves on $A B$ and $E$ moves on $A C$.
8. Find all positive integers $n$ greater than 1 such that every integer $k$, $1<k<n$ satisfying $\gcd(k, n)=1$, is a prime.
9. Find all polynomials $P(x)$ satisfying the condition $$P\left(x^{2006}+y^{2006}\right)=(P(x))^{2006}+(P(y))^{2006}$$ for all real numbers $x, y$.
10. Solve the equation $$2 \sqrt{x^{2}-\frac{1}{4}+\sqrt{x^{2}-\frac{1}{4}+\sqrt{\ldots+\sqrt{x^{2}-\frac{1}{4}+\sqrt{x^{2}+x+\frac{1}{4}}}}}}=2 x^{3}+3 x^{2}+3 x+1$$ where on the left side there are $2006$ signs of radical.
11. Let be given a quadrilateral $A B C D$ inscribed in a circle with center $O$, radius $R$. The lines $A B$, $C D$ intersect at $P$, the lines $A D$, $B C$ intesect at $Q$. Prove that $$\overrightarrow{O P} \cdot \overrightarrow{O Q}=R^{2}.$$
12. Let $M$ be a point lying inside the tetrahedron $A B C D$. The lines $M A$, $M B$, $M C$, $M D$ cut the faces $B C D$, $C D A$, $D A B$, $A B C$ respectively at $A^{\prime}$, $B^{\prime}$, $C^{\prime}$, $D^{\prime}$. Prove that the volume of the tetrahedron $A^{\prime} B^{\prime} C^{\prime} D^{\prime}$ does not exceed $\dfrac{1}{27}$ that of the tetrahedron $A B C D$.

### Issue 354

1. a) Find all natural number, each of which can be written as the sum of two relatively prime integers greater than $1$.
b) Find all natural numbers, each of which can be written as the sum of three pairwise relatively prime integers greater than $1$.
2. Let $A B C$ be a triangle with acute angle $\widehat{A B C}$. Let $K$ be a point on the side $A B$, and $H$ be its orthogonal projection on the line $B C$. A ray $B x$ cuts the segment $KH$ at $E$ and cuts the line passing through $K$ parallel to $B C$ at $F$. Prove that $\widehat{A B C}=3 \widehat{C B F}$ when and only when $E F=$ $2 B K$.
3. Find all natural numbers $n$ such that the product of the digits of $n$ is equal to $$(n-86)^{2}\left(n^{2}-85 n+40\right).$$
4. Prove that $a b+b c+c a<\sqrt{3} d^{2}$, where $a, b, c, d$ are real numbers satisfying the following conditions $$0<a, b, c<d,\quad \frac{1}{a}+\frac{1}{b}+\frac{1}{c}-\sqrt{\frac{1}{a^{2}}+\frac{1}{b^{2}}+\frac{1}{c^{2}}}=\frac{2}{d}.$$
5. Solve the equation $$x^{4}+2 x^{3}+2 x^{2}-2 x+1=\left(x^{3}+x\right) \sqrt{\frac{1-x^{2}}{x}}.$$
6. Let $A B C D$ be a square with sides equal to $a$. On the side $A D$, take the point $M$ such that $A M=3 M D$. Draw the ray $B x$ cutting the side $C D$ at $I$ such that $\widehat{A B M}=\widehat{M B I}$. The angle bisector of $\widehat{C B I}$ cuts the side $C D$ at $N$. Calculate the area of triangle $B M N$.
7. Let $B C$ be a fixed chord (which is not a diameter) of a circle. On the major arc $B C$ of the circle, take a point $A$ not coinciding with $B$, $C$. Let $H$ be the orthocenter of triangle $A B C$. The second points of intersection of the line $B C$ with the circumcircles of triangles $A B H$ and $A C H$ are $E$ and $F$ respectively. The line $E H$ cuts the side $A C$ at $M$ and the line $F H$ cuts the side $A B$ at $N$. Determine the position of $A$ so that the measure of the segment $M N$ attains its least value.
8. How many are there natural $9$-digit numbers with $3$ distinct odd digits, $3$ distinct even digits and every even digit in each number appears exactly two times (in this number).
9. For every positive integer $n$, consider the function $f_{n}$ defined on $\mathbb{R}$ by $$f_{n}(x)=x^{2 n}+x^{2 n-1}+\ldots+x^{2}+x+1$$ a) Prove that the function $f_{n}$ attains its least value at a unique value $x_{n}$ of $x$.
b) Let $S_{n}$ be the least value of $f_{n}$. Prove that
• $S_{n}>\dfrac{1}{2}$ for all $n$ and there does not exist a real number $a>\dfrac{1}{2}$ such that $S_{n}>a$ for all $n$.
• $\left(S_{n}\right)$ $(n=1,2, \ldots)$ is a decreasing sequence and $\lim S_{n}=\dfrac{1}{2}$.
• $\displaystyle\lim_{n\to\infty} x_{n}=-1$.
10. Let $$A=\sqrt{x^{2}+\sqrt{4 x^{2}+\sqrt{16 x^{2}+\sqrt{100 x^{2}+39 x+\sqrt{3}}}}}.$$ Find the greatest integer not exceeding $A$ when $x=20062007$.
11. Let $A B C$ be a triangle with $B C=d$ $C A=b$, $A B=c$, with inradius $r$ and with incenter $I$. Let $A_{1}$, $B_{1}$, $C_{1}$ be respectively the touching points of the sides $B C$, $C A$, $A B$ with the incircle. The rays $I A$, $I B$, $I C$ cut the incircle respectively at $A_{2}$, $B_{2}$, $C_{2}$. Let $B_{i} C_{i}=a_{1}$, $C_{1} A_{i}=b_{i}$, $A_{i} B_{i}=c_{i}$ $(\mathrm{i}=1,2)$. Prove that $$\frac{a_{2}^{3} b_{2}^{3} c_{2}^{3}}{a_{1}^{2} b_{1}^{2} c_{1}^{2}} \geq \frac{216 r^{6}}{a b c}.$$ When does equality occur?
12. Let $O A B C$ be a tetrahedron with $$\widehat{A O B}+\widehat{B O C}+\widehat{C O A}=180^{\circ}.$$ $O A_{1}$, $O B_{1}$, $O C_{1}$ are internal angle bisectors respectively of the triangles $O B C$, $O C A$, $O A B$; $O A_{2}$, $O B_{2}$, $O C_{2}$ are internal angle bisectors respectively of the triangles $O A A_{1}$, $O B B_{1}$, $O C C_{1}$. Prove that $$\left(\frac{A A_{1}}{A_{2} A_{1}}\right)^{2}+\left(\frac{B B_{1}}{B_{2} B_{1}}\right)^{2}+\left(\frac{C C_{1}}{C_{2} C_{1}}\right)^{2} \geq(2+\sqrt{3})^{2}.$$ When does equality occur?
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