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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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    Name

    Abel,5,Albania,2,AMM,2,Amsterdam,5,An Giang,40,Andrew Wiles,1,Anh,2,APMO,21,Austria (Áo),1,Ba Đình,2,Ba Lan,1,Bà Rịa Vũng Tàu,72,Bắc Bộ,2,Bắc Giang,59,Bắc Kạn,3,Bạc Liêu,15,Bắc Ninh,58,Bắc Trung Bộ,3,Bài Toán Hay,5,Balkan,40,Baltic Way,32,BAMO,1,Bất Đẳng Thức,69,Bến Tre,68,Benelux,15,Bình Định,62,Bình Dương,36,Bình Phước,48,Bình Thuận,41,Birch,1,BMO,40,Booklet,12,Bosnia Herzegovina,3,BoxMath,3,Brazil,2,British,16,Bùi Đắc Hiên,1,Bùi Thị Thiện Mỹ,1,Bùi Văn Tuyên,1,Bùi Xuân Diệu,1,Bulgaria,6,Buôn Ma Thuột,2,BxMO,14,Cà Mau,21,Cần Thơ,25,Canada,40,Cao Bằng,11,Cao Quang Minh,1,Câu Chuyện Toán Học,43,Caucasus,3,CGMO,11,China - Trung Quốc,25,Chọn Đội Tuyển,491,Chu Tuấn Anh,1,Chuyên Đề,125,Chuyên SPHCM,7,Chuyên SPHN,26,Chuyên Trần Hưng Đạo,2,Collection,8,College Mathematic,1,Concours,1,Cono Sur,1,Contest,666,Correspondence,1,Cosmin Poahata,1,Crux,2,Czech-Polish-Slovak,26,Đà Nẵng,48,Đa Thức,2,Đại Số,20,Đắk Lắk,72,Đắk Nông,13,Đan Phượng,1,Danube,7,Đào Thái Hiệp,1,ĐBSCL,2,Đề Thi,1,Đề Thi HSG,2126,Đề Thi JMO,1,DHBB,28,Điện Biên,12,Định Lý,1,Định Lý Beaty,1,Đỗ Hữu Đức Thịnh,1,Do Thái,3,Doãn Quang Tiến,5,Đoàn Quỳnh,1,Đoàn Văn Trung,1,Đống Đa,4,Đồng Nai,62,Đồng Tháp,62,Du Hiền Vinh,1,Đức,1,Dương Quỳnh Châu,1,Dương Tú,1,Duyên Hải Bắc Bộ,28,E-Book,31,EGMO,29,ELMO,19,EMC,10,Epsilon,1,Estonian,5,Euler,1,Evan Chen,1,Fermat,3,Finland,4,Forum Of Geometry,2,Furstenberg,1,G. Polya,3,Gặp Gỡ Toán Học,30,Gauss,1,GDTX,3,Geometry,14,GGTH,30,Gia Lai,38,Gia Viễn,2,Giải Tích Hàm,1,Giảng Võ,1,Giới hạn,2,Goldbach,1,Hà Giang,4,Hà Lan,1,Hà Nam,38,Hà Nội,258,Hà Tĩnh,87,Hà Trung Kiên,1,Hải Dương,64,Hải Phòng,54,Hậu Giang,11,Hậu Lộc,1,Hélènne Esnault,1,Hilbert,2,Hình Học,33,HKUST,7,Hòa Bình,31,Hoài Nhơn,1,Hoàng Bá Minh,1,Hoàng Minh Quân,1,Hodge,1,Hojoo Lee,2,HOMC,5,HongKong,8,HSG 10,116,HSG 10 2010-2011,4,HSG 10 2011-2012,6,HSG 10 2012-2013,5,HSG 10 2013-2014,4,HSG 10 2014-2015,5,HSG 10 2015-2016,2,HSG 10 2016-2017,5,HSG 10 2017-2018,3,HSG 10 2018-2019,3,HSG 10 2019-2020,8,HSG 10 2020-2021,2,HSG 10 2021-2022,2,HSG 10 2022-2023,3,HSG 10 Bà Rịa Vũng Tàu,2,HSG 10 Bắc Giang,1,HSG 10 Bạc Liêu,2,HSG 10 Bắc Ninh,3,HSG 10 Bình Định,1,HSG 10 Bình Dương,1,HSG 10 Bình Thuận,3,HSG 10 Chuyên SPHN,5,HSG 10 Đắk Lắk,2,HSG 10 Đồng Nai,4,HSG 10 Gia Lai,2,HSG 10 Hà Nam,3,HSG 10 Hà Tĩnh,13,HSG 10 Hải Dương,9,HSG 10 KHTN,9,HSG 10 Kon Tum,1,HSG 10 Nghệ An,1,HSG 10 Ninh Thuận,1,HSG 10 Phú Yên,2,HSG 10 Quảng Trị,2,HSG 10 Thái Nguyên,8,HSG 10 Thanh Hóa,1,HSG 10 Trà Vinh,5,HSG 10 Vĩnh Phúc,14,HSG 1015-2016,3,HSG 11,117,HSG 11 2010-2011,4,HSG 11 2011-2012,5,HSG 11 2012-2013,7,HSG 11 2013-2014,4,HSG 11 2014-2015,8,HSG 11 2015-2016,2,HSG 11 2016-2017,5,HSG 11 2017-2018,4,HSG 11 2018-2019,5,HSG 11 2019-2020,5,HSG 11 2020-2021,5,HSG 11 2021-2022,1,HSG 11 An Giang,1,HSG 11 Bà Rịa Vũng Tàu,1,HSG 11 Bắc Giang,4,HSG 11 Bạc Liêu,2,HSG 11 Bắc Ninh,4,HSG 11 Bình Định,11,HSG 11 Bình Dương,3,HSG 11 Bình Thuận,1,HSG 11 Cà Mau,1,HSG 11 Đà Nẵng,9,HSG 11 Đồng Nai,1,HSG 11 Hà Nam,1,HSG 11 Hà Tĩnh,10,HSG 11 Hải Phòng,1,HSG 11 Kiên Giang,4,HSG 11 Lạng Sơn,11,HSG 11 Nghệ An,6,HSG 11 Ninh Bình,2,HSG 11 Quảng Bình,9,HSG 11 Quảng Ngãi,8,HSG 11 Quảng Trị,3,HSG 11 Sóc Trăng,1,HSG 11 Thái Nguyên,8,HSG 11 Thanh Hóa,4,HSG 11 Trà Vinh,1,HSG 11 Tuyên Quang,1,HSG 11 Vĩnh Long,2,HSG 11 Vĩnh Phúc,10,HSG 12,623,HSG 12 2009-2010,2,HSG 12 2010-2011,39,HSG 12 2011-2012,44,HSG 12 2012-2013,58,HSG 12 2013-2014,53,HSG 12 2014-2015,44,HSG 12 2015-2016,36,HSG 12 2016-2017,47,HSG 12 2017-2018,58,HSG 12 2018-2019,44,HSG 12 2019-2020,43,HSG 12 2020-2021,51,HSG 12 2021-2022,34,HSG 12 2022-2023,25,HSG 12 An Giang,7,HSG 12 Bà Rịa Vũng Tàu,11,HSG 12 Bắc Giang,17,HSG 12 Bạc Liêu,3,HSG 12 Bắc Ninh,13,HSG 12 Bến Tre,18,HSG 12 Bình Định,16,HSG 12 Bình Dương,8,HSG 12 Bình Phước,8,HSG 12 Bình Thuận,8,HSG 12 Cà Mau,8,HSG 12 Cần Thơ,7,HSG 12 Cao Bằng,5,HSG 12 Chuyên SPHN,9,HSG 12 Đà Nẵng,3,HSG 12 Đắk Lắk,20,HSG 12 Đắk Nông,1,HSG 12 Điện Biên,3,HSG 12 Đồng Nai,20,HSG 12 Đồng Tháp,18,HSG 12 Gia Lai,13,HSG 12 Hà Nam,4,HSG 12 Hà Nội,15,HSG 12 Hà Tĩnh,15,HSG 12 Hải Dương,14,HSG 12 Hải Phòng,19,HSG 12 Hậu Giang,3,HSG 12 Hòa Bình,10,HSG 12 Hưng Yên,9,HSG 12 Khánh Hòa,2,HSG 12 KHTN,26,HSG 12 Kiên Giang,11,HSG 12 Kon Tum,2,HSG 12 Lai Châu,4,HSG 12 Lâm Đồng,10,HSG 12 Lạng Sơn,8,HSG 12 Lào Cai,16,HSG 12 Long An,17,HSG 12 Nam Định,7,HSG 12 Nghệ An,12,HSG 12 Ninh Bình,11,HSG 12 Ninh Thuận,6,HSG 12 Phú Thọ,16,HSG 12 Phú Yên,12,HSG 12 Quảng Bình,12,HSG 12 Quảng Nam,9,HSG 12 Quảng Ngãi,5,HSG 12 Quảng Ninh,20,HSG 12 Quảng Trị,9,HSG 12 Sóc Trăng,4,HSG 12 Sơn La,5,HSG 12 Tây Ninh,6,HSG 12 Thái Bình,11,HSG 12 Thái Nguyên,13,HSG 12 Thanh Hóa,18,HSG 12 Thừa Thiên Huế,18,HSG 12 Tiền Giang,3,HSG 12 TPHCM,12,HSG 12 Tuyên Quang,2,HSG 12 Vĩnh Long,6,HSG 12 Vĩnh Phúc,22,HSG 12 Yên Bái,6,HSG 9,533,HSG 9 2009-2010,1,HSG 9 2010-2011,21,HSG 9 2011-2012,44,HSG 9 2012-2013,44,HSG 9 2013-2014,36,HSG 9 2014-2015,40,HSG 9 2015-2016,39,HSG 9 2016-2017,42,HSG 9 2017-2018,47,HSG 9 2018-2019,50,HSG 9 2019-2020,20,HSG 9 2020-2021,53,HSG 9 2021-2022,57,HSG 9 2022-2023,1,HSG 9 An Giang,8,HSG 9 Bà Rịa Vũng Tàu,7,HSG 9 Bắc Giang,12,HSG 9 Bạc Liêu,1,HSG 9 Bắc Ninh,12,HSG 9 Bến Tre,9,HSG 9 Bình Định,10,HSG 9 Bình Dương,6,HSG 9 Bình Phước,13,HSG 9 Bình Thuận,5,HSG 9 Cà Mau,1,HSG 9 Cần Thơ,4,HSG 9 Cao Bằng,1,HSG 9 Chuyên SPHN,2,HSG 9 Đà Nẵng,10,HSG 9 Đắk Lắk,11,HSG 9 Đắk Nông,2,HSG 9 Điện Biên,3,HSG 9 Đồng Nai,7,HSG 9 Đồng Tháp,10,HSG 9 Gia Lai,8,HSG 9 Hà Giang,3,HSG 9 Hà Nam,9,HSG 9 Hà Nội,25,HSG 9 Hà Tĩnh,16,HSG 9 Hải Dương,14,HSG 9 Hải Phòng,7,HSG 9 Hậu Giang,4,HSG 9 Hòa Bình,3,HSG 9 Hưng Yên,9,HSG 9 Khánh Hòa,4,HSG 9 Kiên Giang,15,HSG 9 Kon Tum,8,HSG 9 Lai Châu,1,HSG 9 Lâm Đồng,13,HSG 9 Lạng Sơn,9,HSG 9 Lào Cai,3,HSG 9 Long An,9,HSG 9 Nam Định,8,HSG 9 Nghệ An,19,HSG 9 Ninh Bình,13,HSG 9 Ninh Thuận,3,HSG 9 Phú Thọ,12,HSG 9 Phú Yên,8,HSG 9 Quảng Bình,13,HSG 9 Quảng Nam,11,HSG 9 Quảng Ngãi,12,HSG 9 Quảng Ninh,15,HSG 9 Quảng Trị,9,HSG 9 Sóc Trăng,8,HSG 9 Sơn La,4,HSG 9 Tây Ninh,16,HSG 9 Thái Bình,9,HSG 9 Thái Nguyên,5,HSG 9 Thanh Hóa,17,HSG 9 Thừa Thiên Huế,8,HSG 9 Tiền Giang,6,HSG 9 TPHCM,10,HSG 9 Trà Vinh,2,HSG 9 Tuyên Quang,5,HSG 9 Vĩnh Long,11,HSG 9 Vĩnh Phúc,12,HSG 9 Yên Bái,4,HSG Cấp Trường,89,HSG Quốc Gia,109,HSG Quốc Tế,16,HSG11 2021-2022,3,Hứa Lâm Phong,1,Hứa Thuần Phỏng,1,Hùng Vương,2,Hưng Yên,39,Hương Sơn,2,Huỳnh Kim Linh,1,Hy Lạp,1,IMC,26,IMO,57,IMT,2,IMU,2,India - Ấn Độ,47,Inequality,13,InMC,1,International,340,Iran,13,Jakob,1,JBMO,41,Jewish,1,Journal,30,Junior,38,K2pi,1,Kazakhstan,1,Khánh Hòa,26,KHTN,61,Kiên Giang,71,Kim Liên,1,Kon Tum,23,Korea - Hàn Quốc,5,Kvant,2,Kỷ Yếu,46,Lai Châu,10,Lâm Đồng,44,Lăng Hồng Nguyệt Anh,1,Lạng Sơn,35,Langlands,1,Lào Cai,33,Lê Hải Châu,1,Lê Hải Khôi,1,Lê Hoành Phò,4,Lê Hồng Phong,5,Lê Khánh Sỹ,3,Lê Minh Cường,1,Lê Phúc Lữ,1,Lê Phương,1,Lê Quý Đôn,1,Lê Viết Hải,1,Lê Việt Hưng,2,Leibniz,1,Long An,49,Lớp 10 Chuyên,666,Lớp 10 Không Chuyên,347,Lớp 11,1,Lục Ngạn,1,Lượng giác,1,Lương Tài,1,Lưu Giang Nam,2,Lưu Lý Tưởng,1,Lý Thánh Tông,1,Macedonian,1,Malaysia,1,Margulis,2,Mark Levi,1,Mathematical Excalibur,1,Mathematical Reflections,1,Mathematics Magazine,1,Mathematics Today,1,Mathley,1,MathLinks,1,MathProblems Journal,1,Mathscope,8,MathsVN,5,MathVN,1,MEMO,12,Menelaus,1,Metropolises,4,Mexico,1,MIC,1,Michael Atiyah,1,Michael Guillen,1,Mochizuki,1,Moldova,1,Moscow,1,MYM,25,MYTS,4,Nam Định,44,Nam Phi,1,National,276,Nesbitt,1,Newton,4,Nghệ An,69,Ngô Bảo Châu,2,Ngô Việt Hải,1,Ngọc Huyền,2,Nguyễn Anh Tuyến,1,Nguyễn Bá Đang,1,Nguyễn Đình Thi,1,Nguyễn Đức Tấn,1,Nguyễn Đức Thắng,1,Nguyễn Duy Khương,1,Nguyễn Duy Tùng,1,Nguyễn Hữu Điển,3,Nguyễn Minh Hà,1,Nguyễn Minh Tuấn,9,Nguyễn Nhất Huy,1,Nguyễn Phan Tài Vương,1,Nguyễn Phú Khánh,1,Nguyễn Phúc Tăng,2,Nguyễn Quản Bá Hồng,1,Nguyễn Quang Sơn,1,Nguyễn Song Thiên Long,1,Nguyễn Tài Chung,5,Nguyễn Tăng Vũ,1,Nguyễn Tất Thu,1,Nguyễn Thúc Vũ Hoàng,1,Nguyễn Trung Tuấn,8,Nguyễn Tuấn Anh,2,Nguyễn Văn Huyện,3,Nguyễn Văn Mậu,25,Nguyễn Văn Nho,1,Nguyễn Văn Quý,2,Nguyễn Văn Thông,1,Nguyễn Việt Anh,1,Nguyễn Vũ Lương,2,Nhật Bản,4,Nhóm $\LaTeX$,4,Nhóm Toán,1,Ninh Bình,58,Ninh Thuận,24,Nội Suy Lagrange,2,Nội Suy Newton,1,Nordic,21,Olympiad Corner,1,Olympiad Preliminary,2,Olympic 10,127,Olympic 10/3,6,Olympic 10/3 Đắk Lắk,6,Olympic 11,118,Olympic 12,50,Olympic 23/3,2,Olympic 24/3,10,Olympic 24/3 Quảng Nam,10,Olympic 27/4,23,Olympic 30/4,57,Olympic KHTN,7,Olympic Sinh Viên,76,Olympic Tháng 4,12,Olympic Toán,332,Olympic Toán Sơ Cấp,3,Ôn Thi 10,2,PAMO,1,Phạm Đình Đồng,1,Phạm Đức Tài,1,Phạm Huy Hoàng,1,Pham Kim Hung,3,Phạm Quốc Sang,2,Phan Huy Khải,1,Phan Quang Đạt,1,Phan Thành Nam,1,Pháp,2,Philippines,8,Phú Thọ,31,Phú Yên,39,Phùng Hồ Hải,1,Phương Trình Hàm,11,Phương Trình Pythagoras,1,Pi,1,Polish,32,Problems,1,PT-HPT,14,PTNK,55,Putnam,27,Quảng Bình,57,Quảng Nam,51,Quảng Ngãi,44,Quảng Ninh,56,Quảng Trị,38,Quỹ Tích,1,Riemann,1,RMM,13,RMO,24,Romania,37,Romanian Mathematical,1,Russia,1,Sách Thường Thức Toán,7,Sách Toán,70,Sách Toán Cao Học,1,Sách Toán THCS,7,Saudi Arabia - Ả Rập Xê Út,9,Scholze,1,Serbia,17,Sharygin,28,Shortlists,56,Simon Singh,1,Singapore,1,Số Học - Tổ Hợp,28,Sóc Trăng,32,Sơn La,21,Spain,8,Star Education,1,Stars of Mathematics,11,Swinnerton-Dyer,1,Talent Search,1,Tăng Hải Tuân,2,Tạp Chí,17,Tập San,3,Tây Ban Nha,1,Tây Ninh,36,Thạch Hà,1,Thái Bình,42,Thái Nguyên,58,Thái Vân,2,Thanh Hóa,74,THCS,2,Thổ Nhĩ Kỳ,5,Thomas J. 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    MOlympiad.NET: Mathematics and Youth Magazine Problems 2006
    Mathematics and Youth Magazine Problems 2006
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