## $hide=mobile$type=ticker$c=12$cols=4$l=0$sr=random$b=0 ### Issue 379 1. Find all pairs of integers$a$,$b$such that $$a^{2}+a b+b^{2}=a^{2} b^{2}$$ 2. Let$ABC$be an isosceles triangle (at vertex$A$) such that$\widehat{B A C} \geq 90^\circ$. Choose a point$M$on$A C$, and let$A H$and$C K$be the altitudes from$A$and$C$onto$B M$respectively$(H, K$are the feet of these altitudes) such that$B H=H K+K C$. Find the angle$B A C$. 3. Solve for$x$$\sqrt{\frac{5 \sqrt{2}+7}{x+1}}+4 x=3 \sqrt{2}-1$$
4. Find $a$, $b$ such that the maximum value of the expression $$|||x+1|-2|-(a x+b)|$$ where $-3 \leq x \leq 4$ is smallest possible.
5. Let $A B C D$ be a rectangular, let $I$ be the midpoint of $C D$, and let $E$ be a point on $A B$. The altitude onto $D E$ through $I$ meets $D E$ and $A D$ at $M$ and $H$ respectively. The altitude onto $C E$ through $I$ meets $C E$ and $B C$ at $N$ and $K$, respectively. $E I$ intersects with $H K$ at $G$. Prove that
a) The points $E$, $G$, $N$, $K$, $B$ lie on the same circle.
b) The points $E$, $G$, $M$, $H$, $A$ lie on the same circle.
6. Let $A B C$ be a triangle. Prove the inequality $$\cos A \cos B \cos C \leq \frac{1}{8} \cos (B-C) \cos (C-A) \cos (A-B)$$
7. Solve for $x$ $$3^{x}\left(4^{x}+6^{x}+9^{x}\right)=25^{x}+2.16^{x}$$
8. Prove that the union of six hemispheres whose diameters are the sides of a given tetrahedron must contain the tetrahedron itself.
9. Let $A B C$ be a triangle. Choose the pair of points $A_{1}$, $A_{2}$; $B_{1}$, $B_{2}$; $C_{1}$, $C_{2}$ on the sides $B C$, $C A$ and $A B$ respectively such that $A_{1} A_{2} B_{1} B_{2} C_{1} C_{2}$ is a convex hexagon with equal opposite sides and the triangles $A A_{1} A_{2}$, $B B_{1} B_{2}$, $CC_{1}C_{2}$ have the same areas. Prove that the lines $A_{1} B_{2}$, $B_{1} C_{2}$, $C_{1} A_{2}$ are colinear.
10. Let $a$, $b$, $c$ be non - negative real numbers such that $$\sqrt[3]{a^{3}+b^{3}}+\sqrt[3]{b^{3}+c^{3}}+\sqrt[3]{c^{3}+a^{3}}+a b c=3.$$ Prove that the smallest value of the expression $$P=\frac{a^{3}}{b^{2}+c^{2}}+\frac{b^{3}}{c^{2}+a^{2}}+\frac{c^{3}}{a^{2}+b^{2}}$$ is $\sqrt[3]{32}m$ where $m$ is the real root of the equation $t^{3}+54 t-162=0$.
11. Let $f: \mathbb{N} \rightarrow \mathbb{R}$ be the function with initial values $f(0)=0$, $f(1)=1$ such that $$f(n+2)-2011 f(n+1)+f(n)=0.$$ What is the probability that $f(n)$ is a prime number, where $n$ is a randomly chosen number from the set of integers $\{0,1,2, \ldots, 2008\}$?
12. Find all functions $f(x)$ such that it is continuous on $[0 ; 1]$, differentiable on the open interval $(0 ; 1)$ and the following two conditions are satisfied $$f(0)=f(1)=\frac{2009}{11},\quad 20 f(x)+11 f(x)+2009 \leq 0,\, \forall x \in(0 ; 1).$$

### Issue 380

1. Compare $\dfrac{1}{6}$ with $$A=\frac{1}{5}-\frac{1}{7}+\frac{1}{17}-\frac{1}{31}+\frac{1}{65}-\frac{1}{127}$$
2. Let $ABC$ be an isosceles triangle (at vertex $C$) with $\widehat{A C B}=100^{\circ}$. $M$ is a point chosen on the ray $C A$ such that $C M=A B$. Find the measure of the angle $\widehat{C M B}$.
3. Find all triple $(x, y, z)$ of integers such that $$\begin{cases}y^{3} &=x^{3}+2 x^{2}+1 \\ x y &=z^{2}+2\end{cases}$$
4. Find all pair of numbers $x$ and $y$ such that the following conditions hold $x>1$, $0<y<1$, $x+y \leq \sqrt{5}$, $\dfrac{1}{x}+\dfrac{1}{y} \leq \sqrt{5}$ and $\dfrac{x}{x+1}+\dfrac{y}{1-y} \leq \sqrt{5}$.
5. Let $A B C D$ be an isosceles trapezoid inscribed inside a circle $\left(O_{1}: R\right)$ and circumscribes the circle $\left(O_{2} ; r\right)$. Let $d=O_{1} O_{2}$. Prove that $$\frac{1}{r^{2}} \geq \frac{2}{R^{2}+d^{2}} .$$ When does equality occur?
6. Prove that in any triangle $A B C$, the following inequality holds $$\frac{\cot A \cdot \cot B \cdot \cot C}{\sin A \cdot \sin B \cdot \sin C} \leq\left(\frac{2}{3}\right)^{3}$$
7. There are $17$ ornament betel-nut trees around a circular pond. How many ways are there to chopped off $4$ trees with the condition that no two consecutive trees be removed?
8. Solve the following system of equations with parameter $a$ $$\begin{cases} 2 x\left(y^{2}+a^{2}\right) &=y\left(y^{2}+9 a^{2}\right) \\ 2 y\left(z^{2}+a^{2}\right) &=z\left(z^{2}+9 a^{2}\right) \\ 2 z\left(x^{2}+a^{2}\right) &=x\left(x^{2}+9 a^{2}\right)\end{cases}$$
9. Let $\left(x_{n}\right)$, $n=0,1,2, \ldots$ be a sequence given by the following recursive formula $$x_{0}=a,\quad x_{n+1}=x_{n}+\sin x+2 \pi,\, n=0,1,2, \ldots$$ where $a, x \in \mathbb{R}$. Prove that the limit $\displaystyle\lim_{n \rightarrow+\infty} \frac{x_{1}+\ldots+x_{n}}{n^{2}}$ exists and find its exact value.
10. Prove that
a) $\displaystyle\sum_{k=0}^{n}\left(\frac{k}{n}-x\right)^{2} C_{n}^{k} x^{k}(1-x)^{n-k}=\frac{x(1-x)}{n},\, \forall x \in \mathbb{R}$
b) $\displaystyle\sum_{k=0}^{n}\left|\frac{k}{n}-x\right|C_{n}^{k} x^{k}(1-x)^{n-k} \leq \frac{1}{2 \sqrt{n}},\, \forall x \in[0 ; 1]$.
11. Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that $$f\left(n^{2}\right)=f(m+n) f(n-m)+m^{2},\, \forall m, n \in \mathbb{R}$$
12. On a given plane, choose a point $A$ outside a circle whose center is at a point $O$ and whose radius equals $R$. A chord $M N$ with constant length moves on the circle such that the two line segments $M N$ and $O A$ always intersect. Determine the positions of $M$ and $N$ such that the sum $A M+A N$ is
a) greatest possible.
b) smallest possible.

### Issue 381

1. Rearrange the following rational numbers in increasing order $$\frac{1005}{2002}, \frac{1007}{2006}, \frac{1009}{2010}, \frac{1011}{2014}$$
2. In an isosceles triangle $A B C$ (at vertex $A$ ), choose a point $M$ such that $$\widehat{M A C}=\widehat{M B A}=\widehat{M C B}.$$ Compare the areas of the triangles $A B M$ and $C B M$.
3. Determine the sum of all rational numbers of the form $\dfrac{a}{b}$ where $a$, $b$ are natural divisors of $27000$ and $\gcd(a, b)=1$.
4. Given $a$, $b$, $c$ such that $a>0$, $b>c$, $a^{2}=b c$, $a+b+c=a b c$. Prove the inequalities $$a \geq \sqrt{3},\quad b \geq \sqrt{3},\quad 0<c \leq \sqrt{3}.$$
5. Let $ABCD$ be a quadrilateral where $\widehat{A B C}=\widehat{A D C}=90^{\circ}$ and $\widehat{B C D}<90^{\circ}$. Choose a point $E$ on the opposite ray of $A C$ such that $D A$ is the angle-bisector of $B D E$. Let $M$ be chosen arbitrarily between $D$ and $E$, choose another point $N$ on the opposite ray of $B E$ such that $\widehat{N C B}=\widehat{M C D}$. Prove that $M C$ is the angle bisector of $D M N$.
6. Prove that the equation $x^{3}+3 y^{3}=5$ has infinitely many rational solutions.
7. If $a$, $b$, $c$ are the length of the sides of a triangle, prove that $$\frac{1}{\sqrt{a b+a c}}+\frac{1}{\sqrt{b c+b a}}+\frac{1}{\sqrt{c a+c b}} \geq \frac{1}{\sqrt{a^{2}+b c}}+\frac{1}{\sqrt{b^{2}+a c}}+\frac{1}{\sqrt{c^{2}+a b}}.$$
8. Let $A B C D . A^{\prime} B^{\prime} C^{\prime} D^{\prime}$ be a parallelepiped and let $S_{1}$, $S_{2}$, $S_{3}$ denote the areas of the sides $A B C D$, $A B B^{\prime} A^{\prime}$ and $A D D^{\prime} A^{\prime}$ respectively. Given that the sum of squares of the areas of all sides of the tetrahedron $A B^{\prime} C D^{\prime}$ equals $3$, find the smallest possible value of the following expression $$T=2\left(\frac{1}{S_{1}}+\frac{1}{S_{2}}+\frac{1}{S_{3}}\right)+3\left(S_{1}+S_{2}+S_{3}\right)$$

### Issue 382

1. Compare $\dfrac{2009}{2008^{2}}$ with the sum (consisting of $2010$ terms) $$\frac{1}{2009}+\frac{2}{2009^{2}}+\frac{3}{2009^{3}}+\ldots+\frac{2009}{2009^{2009}}+\frac{2010}{2009^{2010}}$$
2. Find a root of the polynomial $P(x)=x^{3}+a x^{2}+b x+c$ given that it has at least one root and $a+2 b+4 c=-\dfrac{1}{2}$.
3. Let $a_{1}$, $a_{2}$, $a_{3}$, $a_{4}$, $a_{5}$, $a_{6}$, $a_{7}$, $a_{8}$, $a_{9}$ be non negative real numbers whose sum equals $1$. Put $S_{k}=a_{k}+a_{k+1}+a_{k+2}+a_{k+3}$ $(k=1,2, \ldots, 6)$. Determine the smallest possible value of $$M=\max \left\{S_{1}, S_{2}, S_{3}, S_{4}, S_{5}, S_{6}\right\}.$$
4. Let $m$, $n$, $a$, $b$ and $c$ be real numbers such that the following conditions hold $$\begin{cases}m^{1000}+n^{1000} &=a \\ m^{2000}+n^{2000} &=\dfrac{2 b}{3}\\ m^{5000}+n^{5000} &=\dfrac{c}{36}\end{cases}.$$ Find a formula relating $a$, $b$ and $c$ which does not involve $m$, $n$.
5. Let $A H$ be the altitude from $A$ of a triangle $A B C$. Choose a point $D$ on the half-plane created by $B C$ which contains $A$ such that $D B=D C=\dfrac{A B}{\sqrt{2}}$. Prove that the lengths of the line segments $B D$, $D H$ and $H A$ are the side lengths of a right triangle.
6. Determine the maximum possible value of $x^{2}+y^{2}$ where $x$ and $y$ are two integers chosen arbitrarily within the interval $[-2009 ; 2009]$ such that $$\left(x^{2}-2 x y-y^{2}\right)^{2}=4.$$
7. Consider two polynomials with real coefficients $$P(x)=x^{n}+a_{n-1} x^{n-1}+\ldots+a_{1} x+a_{0}$$ and $Q(x)=x^{2}+x+2009$. Given that $P(x)$ has $n$ distinct real roots but $P(Q(x))$ does not have any real solution. Prove that $P(2009)>\dfrac{1}{4^{n}}$.
8. Let $ABCDEF$ be a regular hexagon and let $G$ be the midpoint of $B F$. Choose a point $I$ on $B C$ such that $B I=B G$. Let $H$ be a point on $I G$ such that $\widehat{C D H}=45^{\circ}$ and $K$ is a point on $E F$ such that $\widehat{D K E}=45^{\circ}$. Prove that $D H K$ is an equilateral triangle.

### Issue 383

1. Let $a$, $b$, $c$ be integers such that $$A=\frac{a^{2}+b^{2}+c^{2}-a b-b c-c a}{2}$$ is a perfect square. Prove that $a=b=c$.
2. Let $A B C$ be a triangle in which $\widehat{B A C}=75^{\circ}$. Given that the altitude $A H$ has length $A H=\dfrac{B C}{2}$. Prove that $A B C$ is an isosceles triangle.
3. Let $x$, $y$ be two nonnegative integers where $x>1$ and $2 x^{2}-1=y^{15}$. Prove that $x$ is divisible by $15$.
4. Let $A B C D$ be a parallelogram. The ray $D x$ from $D$ is perpendicular to $D C$ and lies on the half-plane divided by $C D$ which does not contains $B$. Choose a point $E$ on $D x$ such that $D E=D C$. Draw an isosceles right triangle $B E F$ (right angle at $F$) such that $F$ and $D$ are on the same half-plane divided by $B C \cdot E H$ is the altitude onto $B C$. Prove that $F$ $D$ and $H$ are collinear.
5. Let $a$, $b$, $c$ be three sides of a right triangle, $a$ is the hypotenuse. Prove that the equation $-a x^{2}+b x+c=0$ has two distinct roots $x_{1}$, $x_{2}$ such that $-\sqrt{2}<x_{1}<x_{2}<\sqrt{2}$.
6. Consider a convex polygon with $2009$ vertices. A scissor is used to cut along all of its diagonals, thereby dividing the original polygon into smaller convex polygons. How many vertices are there of a resulting polygon with the greatest number of edges?
7. Determine the smallest value of the expression $$T=\frac{a b+b c}{a^{2}-b^{2}+c^{2}}+\frac{b c+c a}{b^{2}-c^{2}+a^{2}}+\frac{c a+a b}{c^{2}-a^{2}+b^{2}}$$ where $a$, $b$, $c$ are three sides of a triangle $A B C$ and $a b c=1$.
8. Let $A B C D . A_{1} B_{1} C_{1} D_{1}$ be a cube. On its three skew edges, choose three points $M$, $N$, $P$. Determine the positions of the points $M$, $N$, $P$ such that the triangle $M N P$ has a) The smallest perimeter possible. b) The smallest area possible.

### Issue 384

1. Replace the distinct letters by distinct numbers such that the following expression becomes a true equality $$\mathrm{VE}+\mathrm{TRUONG}+\mathrm{SA}=22 \times 12 \times 2009.$$
2. It is well-known that the two right triangles whose side lengths are positive integers $(5,12,13)$ and $(6,8,10)$ possess additional property that the area of each triangle equals its perimeter. Are there other triangles with similar properties?
3. Suppose given $1003$ nonzero rational numbers in which any quadruple form a proportion. Prove that at least $1000$ numbers are equal.
4. Let $x$, $y$, $z$ be non-negative real numbers such that $x+y+z=1$. Find the maximum valuc of the following expression $$P=(x+2 y+3 z)(6 x+3 y+2 z).$$
5. Let $A B C D$ be a cyclic quadrilateral, inscribed in a circle $(O)$. The angle bisectors of $B A D$ and $B C D$ meet at a point $K$ on the diagonal $B D$. Let $Q$ be the second intersection point (different from $A$) of $A P$ and the circle $(O)$; $M$ and $N$ be respectively the midpoints of $B D$ and $C P$. The line through $C$ and parallel to $A D$ meets $A M$ at $P$. Prove that a) $S_{A BQ}=S_{A D Q}$. b) $DN$ is perpendicular to $C P$.
6. For each natural number $n,$ let $p(n)$ be its largest odd divisor. Determine the sum $$\sum_{n=2006}^{4012} p(n)$$
7. Solve for $x$ $$\sqrt{3 x-2}=-4 x^{2}+21 x-22$$
8. Let $A B C$ be a triangle whose circumcircle is $(O)$ and such that $A C<A B$. The tangent lines to $(O)$ at $B$, $C$ intersect at $T$. The line through $A$ and perpendicular to $A T$ meet $B C$ at $S$. Choose the points $B_{1}$ and $C_{1}$ on $S T$ such that $T B_{1}=T C_{1}=T B$ and that $C_{1}$ lies between $S$ and $T$. Prove that $A B C$ and $A B_{1} C_{1}$ are similar.

### Issue 385

1. Let $S(n)$ be denote the sum of the digits of $n$. Find a positive integer $n$ such that $S(n)=n^{2}-2009 n+11$.
2. Inside a square $A B C D$ thoose two points $P$, $Q$ such that $B P$ and $D Q$ are parallel and $B P^{2}+D O^{2}=P Q^{2}$. Find the measure of the angle $P A Q$.
3. Compare $2008$ with the sum $S$ of $2009$ terms $$S=\frac{2008+2007}{2009+2008}+\frac{2008^{2}+2007^{2}}{2009^{2}+2008^{2}}+\ldots +\frac{2008^{2009}+2007^{2009}}{2009^{2009}+2008^{2009}}.$$
4. Let $a$, $b$, $c$ be three positive numbers such that $a+b+c=1$. Find the least value of the expression $$P=\frac{9}{1-2(a b+b c+c a)}+\frac{2}{a b c}.$$
5. Let $B$, $C$ be two fixed points on the circle $(\omega)$ such that $B C$ does not pass through the center of $(\omega) .$ On the major arc $B C$, choose a point $A$ differs from $B$ and $C$. Another point $M$ moves on the line segment $B C$. The lines passing through $M$ and parallel to $A B$, $A C$ intersect $A C$ and $A B$ at $F$ and $E$, respectively. When $M$ moves on the line segment $B C$ and for each point $A$, let $x$ be the least possible length of $EF$. Find the positions of $A$ and $M$ such that $x$ is greatest possible.
6. Find the greatest and the least value of the expression $$A=\frac{y^{2}}{25}+\frac{t^{2}}{144}$$ where $x$, $y$, $z$, $t$ satisfy the system of equations $$\begin{cases}x^{2}+y^{2}+2 x+4 y-20 &=0 \\ t^{2}+z^{2}-2 t-143 &= 0\\ x t+y z-x+t+2 z-61 &\geq 0\end{cases}$$
7. Consider the sequence $\left(u_{n}\right)$ defined as follow $$u_{0}=9,\, u_{1}=161,\quad u_{n}=18 u_{n-1}-u_{n-2},\,\forall n=2,3, \ldots$$ Prove that for any $n$, $\dfrac{u_{n}^{2}-1}{5}$ is always a perfect square.
8. Let $P$ be an arbitrary point inside a given triangle $A B C$. Let $A'$, $B'$, $C'$ be the orthogonal projection of $P$ on $B C$, $C A$, $A B$ respectively. Let $I$ be the incenter and $r$ be the inradius of the triangle $A B C$. Find the least value of the expression $$P A^{\prime}+P B^{\prime}+P C^{\prime}+\frac{P I^{2}}{2 r}$$

### Issue 386

1. Find the last two decimal digits of the sum $2008^{2009}+2009^{2008}$
2. Let $ABC$ be an isosceles right triangle with the right angle at $A$. Let $G$ be a point on $A B$ such that $A G=\dfrac{1}{3} A B$, let $M$ be the midpoint of $B C$ and $E$ be the foot of the altitude from $M$ to $CG$. The two lines $M G$ and $A C$ meet at $D$. Prove that $D E=B C$.
3. Find all integer solutions of the equation $$4 x^{4}+2\left(x^{2}+y^{2}\right)^{2}+x y(x+y)^{2}=132$$
4. Let $a$, $b$ and $c$ satisfy the conditions $a \leq b \leq c$ and $$a+b+c=\frac{1}{a}+\frac{1}{b}+\frac{1}{c}.$$ Find the least value of the expression $P=a b^{2} c^{3}$.
5. A triangle $A B C$ inscribed in a circle centered at $O$, radius $R$, $A D$ is its anglebisector. Let $E$, $F$ be the circumeenters pf the triangles $A B D$ and $A C D$. respectively. Given that $B C=a$. Determine the area of the quadrilateral $A E O F$.
6. Let $A B C$ be a triangle with incenter $I$ such that its centroid $G$ lies inside $(I)$. Let $a$, $b$, $c$ be the lengths of the sides $B C$, $A C$, $A B$, respectively. Find the greatest and least value of the following expression $$P=\frac{a^{2}+b^{2}+c^{2}}{a b+b c+c a}$$
7. Let $x$, $y$, $z$ be positive numbers satisfying $$x^{2}+y^{2}+z^{2}=\frac{1-16 x y z}{4}.$$ Find the least value of the expression $$S=\frac{x+y+z+4 x y z}{1+4 x y+4 y z+4 z x}.$$
8. Solve for $x$ $$2^{x}+5^{x}=2-\frac{x}{3}+44 \log _{2}\left(2+\frac{131 x}{3}-5^{x}\right)$$
9. Let $A B C$ be a right triangle, right angle at $A$, $M$ is the midpoint of $B C$. Construct a right angle $P M Q$ with $P \in A B$, $Q \in A C$. Prove that $$P Q^{2} \geq A P \cdot C Q+A Q \cdot B P$$
10. Let $a_{n}$ be the last non-zero digit (counting from left to right) when expressing $n !$ in the decimal number system. Is the sequence $\left(a_{n}\right)$ for $n=1.2 .3 \ldots$ periodic? (That is, there exist the positive integers $T$ and $N$ such that $a_{i+T}=a_{i} \forall i \geq N$).
11. Given $n$ non-negative numbers $a_{1}, a_{2}, \ldots, a_{n}$ $(n \geq 3)$ satisfying $$a_{1}^{2}+a_{2}^{2}+\ldots+a_{n}^{2}=1.$$ Prove that $$\frac{1}{\sqrt{3}}\left(a_{1}+a_{2}+\ldots+a_{n}\right) \geq a_{1} a_{2}+a_{2} a_{3}+\ldots+a_{n} a_{1}.$$ When does equality occur?
12. Let $\left(x_{n}\right)(n=1,2, \ldots)$ be a sequence given by $$x_{1}=a \ (a>1),\, x_{2}=1,\quad x_{n+2}=x_{n}-\ln x_{n},\,\forall n \in \mathbb{N}^{*}.$$ Put $\displaystyle S_{n}=\sum_{k=5}^{n-1}(n-k) \ln \sqrt{x_{2 k-1}}$ $(n \geq 2)$. Find $\displaystyle\lim_{n \rightarrow \infty}\left(\frac{S_{n}}{n}\right)$.

### Issue 387

1. Prove that there exists a $2009$-digits multiple of $2007$ so that it is formed by four digits $0,2,7$ and $9$ and the sum of its digits is $7209$.
2. Let $a_{1}, a_{2}, \ldots, a_{n}$ be $n$ odd integers $(n>2007)$ satisfying the following condition $$a_{1}^{2}+a_{2}^{2}+\ldots+a_{2005}^{2}=a_{2006}^{2}+a_{2007}^{2}+\ldots+a_{n}^{2}.$$ Determine the least possible value of $n$ and construct an example of such a collection $\left(a_{1}, a_{2}, \ldots, a_{n}\right)$ for the smallest value found above.
3. Determine the value of $S$, given that $$S=\frac{2^{3}-1}{2^{3}+1} \times \frac{3^{3}-1}{3^{3}+1} \times \ldots \times \frac{2009^{3}-1}{2009^{3}+1}$$
4. Let $A B C$ be a right triangle with right angle at $A$ and $A C>A B$. Choose a poind $D$ on $A C$ such that $A B=A D .$ Let $E$ be the foot of the altitude from $D$ onto $B C .$ The line passing through $A$ and parallel to $B C$ meets $D E$ at $H .$ Prove that $$A H<\frac{\sqrt{2}}{2} A C.$$
5. Let $A B$ be a fixed chord of a given circle $(O)$ and $E$ is a point moving on $A B$ (but distinct from $A$ and $B$). From $E$, draw another chord $C D$. $P$ and $Q$ are two points on the rays $D A$ and $D B$ respectively such that $P$ is the reflection of $Q$ through $E$. Prove that the circle $(I)$ passing through $C$ and touches $P Q$ at $E$ always passes through a fixed point.
6. Of all pentagons whose sum of the squares of its diagonals equals $1,$ which one has the smallest possible sum of cube of its sides.
7. Solve the system of equations $$\begin{cases}\sqrt{2 x}+2 \sqrt[4]{6-x}-y^{2} &=2 \sqrt{2} \\ \sqrt[4]{2 x}+2 \sqrt{6-x}+2 \sqrt{2} y &=8+\sqrt{2}\end{cases}$$
8. Prove that $$\frac{\sin A}{\tan \frac{B}{2}}+\frac{\sin B}{\tan \frac{C}{2}}+\frac{\sin C}{\tan \frac{A}{2}} \geq \frac{9}{2}$$ where $A$, $B$, $C$ are the measures of the angles of a triangle. When does equality occur?.
9. In a triangle $A B C$, let $P$ be a point such that $P A=P B+P C$. $R$ is the midpoint of the chord $A B$ (the one that contains $P$) of the circumcircle of the triangle $A B P$ and $S$ is the midpoint of the chord $A C$ (containing $P$) of the circumcircle of the triangle $A C P$. Prove that circumcircles of the triangles $B P S$ and $CPR$ touch each other.
10. Does there exist a function $\mathbb{R} \rightarrow \mathbb{R}$ such that
• $f$ is continuous on $\mathbb{R}$; and
• $f(x+2008)(f(x)+\sqrt{2009})=-2010, \forall x \in \mathbb{R} . ?$
11. Let $\left(u_{n}\right)(n=1,2, \ldots)$ be a sequence given by $$u_{1}=2,1;\quad u_{n+1}=\frac{-2 u_{n}^{2}+5 u_{n}}{-u_{n}^{2}+2 u_{n}+1},\,\forall n=1,2, \ldots$$ Find $\displaystyle\lim_{n\to\infty} \left(u_{1}+u_{2}+\ldots+u_{n}\right)$.
12. $A B C$ and $A^{\prime} B^{\prime} C^{\prime}$ are two triangles in a given plane whose inradii are $r, r^{\prime}$ respectively. Let $R'$ be the radius of the circumcircle of the triangle $A^{\prime} B^{\prime} C^{\prime} .$ Prove that $$\left(\sin \frac{B^{\prime}}{2}+\sin \frac{C^{\prime}}{2}\right) P A +\left(\sin \frac{C^{\prime}}{2}+\sin \frac{A^{\prime}}{2}\right) P B +\left(\sin \frac{A^{\prime}}{2}+\sin \frac{B^{\prime}}{2}\right) P C \geq \frac{6 r r^{\prime}}{R^{\prime}}$$ for any point $P$ in the same plane. When does equalitiy occur?

### Issue 388

1. Prove that a number of the form $(\overline{33 \ldots 3})^{2}$ with $k$ $(k>0)$ digits $3$ can always be written as the difference between of the number whose digits are 1 and the one whose digits are $2$.
2. Find all real numbers $a$ such that $|3 a-2| \leq 1$ and $A=\dfrac{3 a-1}{4 a^{4}+a^{2}}$ is an integer.
3. Find the greatest value of the expression $$P=3 \sqrt{x}+8 \sqrt{y}$$ where $x, y$ are two non-negative numbers satisfying $17 x^{2}-72 x y+90 y^{2}-9=0$
4. Solve the equation $$\sqrt{x-2}+\sqrt{4-x}+\sqrt{2 x-5}=2 x^{2}-5 x$$
5. From a point $A$ outside a given circle witlı center $O$, construct two tangent lines $A B$ and $A C$ with $B, C$ are the tangency points. On $O B$, choose $N$ such that $B N=2 O N$. The perpendicular bisector of the line segment $C N$ meets $O A$ at $M .$ Determine the ratio $\dfrac{A M}{A O}$
6. Find the least value of the expression $$A=\frac{1}{a^{4}(b+1)(c+1)}+\frac{1}{b^{4}(c+1)(a+1)}+\frac{1}{c^{4}(a+1)(b+1)}$$ where $a, b, c$ are positive real numbers satisfying $a b c=1$.
7. Let $a, b$ be two real numbers in the interval $(0 ; 1) .$ A sequence $\left(u_{n}\right)$ $(n=0,1,2, \ldots)$ is given by $$u_{0}=a, u_{1}=b,\quad u_{n+2}=\frac{1}{2010} \cdot u_{n+1}^{4}+\frac{2009}{2010} \sqrt[4]{u_{n}},\,\forall n \in \mathbb{N}.$$ Prove that $\left(u_{n}\right)$ has a finite limit, and find that limit.
8. Let $l_a$, $l_b$, $l_c$ be respectively the lengths of the interior angle-bisectors from the three vertices $A$, $B$, $C$ of a triangle $A B C$. Let $R$ be its circumradius. Prove that $$\frac{l_{a}+l_{b}+l_{c}}{R} \leq 2\left(\cos \frac{A}{2} \cos \frac{B}{2}+\cos \frac{B}{2} \cos \frac{C}{2}+\cos \frac{C}{2} \cos \frac{A}{2}\right).$$ When does the equality occur?
9. Let $A B$ be a fixed chord on a given circle $(O) .$ A point $C$ moves on the circle and let $M$ be a point on the zigzag $A C B$ (consisting of two line segments $A C$ and $C B)$ such that it divides this zigzag into two parts of equal length. Find the locus of $M$ when $C$. moves around the circle $(O)$.
10. Let $m, n, d$ be positive integers such that $m<n$ and $\gcd(d, m)=1$, $\gcd(d, n)=1$. (Here, $\gcd(a, b)$ denotes the greatest common divisor of $a$ and $b$.) Prove that in an arithmetic progression of $n$ terms with common difference $d$, there always exist two distinct terms whose product is a multiple of $m n$.
11. A function $f: \mathbb{R} \rightarrow \mathbb{R}$ is given with the following properties
• $f(0)=1$
• $f(x) \leq 1,\, \forall x \in \mathbb{R}$
• $\displaystyle f\left(x+\frac{11}{24}\right)+f(x)=f\left(x+\frac{1}{8}\right)+f\left(x+\frac{1}{3}\right)$
Put $\displaystyle F(x)=\sum_{n=0}^{2009} f(x+n)$. Find $F(2009)$
1. Given two positive degrees polynomials with real coefficients $$P(x)=x^{n}+a_{1} x^{n-1}+\ldots+a_{n-1} x+a_{n}$$ and $$Q(x)=x^{m}+b_{1} x^{m-1}+\ldots+b_{m-1} x+b_{m}$$ where $Q(x)$ has exactly $m$ real roots and $P(x)$ is divisible by $Q(x)$. Prove that if there exists a $k \in\{1,2, \ldots, m\}$ such that $\left|b_{k}\right|>C_{m}^{k} .2009^{k},$ then there also exists an $i \in\{1,2, \ldots, n\}$ such that $|a,|>2008$

### Issue 389

1. Prove that the number of decimal digits in $2008^{2009}+2^{2009}$ and $2008^{2009}$ are equal.
2. Find the least value of the expression $A=1-x y,$ where $x$ and $y$ are real numbers satisfying the following condition $$x^{2009}+y^{2009}=2 x^{1004} y^{1004}$$
3. Does there exist a positive integer number $n$ such that $n^{6}+26^{n}=21^{2009} ?$
4. Let $A B C$ be a right triangle with right angle at $A .$ On the sides $A B, B C$ and $C^{\prime} A$, choose $D, E$ and $F$ respectively such that $D E \perp B C$ and $D E=D F$. $M$ is the midpoint of EF. Prove that $\widehat{B C M}=\widehat{B F E}$.
5. Let $x, y, z$ be real numbers in the interval $(0 ; 1)$. Prove that $$\frac{1}{x(1-y)}+\frac{1}{y(1-z)}+\frac{1}{z(1-x)} \geq \frac{3}{x y z+(1-x)(1-y)(1-z)}.$$ When does the equality occur?
6. Solve the equation $$\sqrt[3]{x^{2}+4 x+3}+\sqrt[3]{4 x^{2}-9 x-3} -\sqrt[3]{3 x^{2}-2 x+2}+\sqrt[3]{2 x^{2}-3 x-2}$$
7. Let $A_{1} A_{2} \ldots A_{n}$ be a convex polygon $(n \geq 3)$ circumscribed around the circle centered at $J$. Prove that for any point $M$, $$\sum_{i=1}^{n} \cos \frac{A_{i}}{2}\left(M A_{i}-J A_{i}\right) \geq 0$$
8. Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ satifying the condition $$f(x y+f(z))=\frac{x f(y)+y f(x)}{2}+z$$ for all $x, y, z$ in $\mathbb{R}$.
9. Let $a, b, c$ be positive numbers. Prove that $$\frac{a^{2}}{b}+\frac{b^{2}}{c}+\frac{c^{2}}{a} \geq \sqrt{a^{2}-a b+b^{2}}+\sqrt{b^{2}-b c+c^{2}}+\sqrt{c^{2}-c a+a^{2}}.$$
10. Let $A B C$ be an acute triangle and the altitudes $A D, B E, C F$ meet at $H .$ On $D E$ choose a point $K$ such that $D K=D H$. On $D H,$ choose a point $I$ such that $\widehat{I K D}=90^{\circ} .$ Prove that the circle centered at $I$ with radius $I K$ touches the circle whose diameter is $B C$.
11. Let $\left(u_{n}\right)$ be a sequence of positive numbers. Put $S_{n}=u_{1}^{3}+u_{2}^{3}+u_{3}^{3}+\ldots+u_{n}^{3}$ for $n=1,2, \ldots$. Assume that $$u_{n+1} \leq\left(\left(S_{n}-1\right) u_{n}+u_{n-1}\right) \frac{1}{S_{n+1}},$$ for any $n=2,3, \ldots$ Find $\displaystyle\lim_{n\to\infty} u_{n}$
12. Let $p$ be a prime number and $m, n, q$ be natural numbers satisfying $2 \leq n \leq m$ and $(p, q)=1 .$ Prove that $C_{q p^{n}}^{n}$ is divisible by $p^{n+n+1}$ (The binomial coefficient $C_{n}^{k}$ is the number of ways of picking $k$ unordered elements from a set of $n$ elements).

### Issue 390

1. Find all triples $(a, b, c)$ of positive integers such that $(a+b+c)^{2}-2 a+2 b$ is a perfect square.
2. Given a triangle $A B C$ with $A B=A C$ and $\widehat{B A C}=80^{\circ} .$ Choose a point $I$ inside the triangle so that $\widehat{I A C}=10^{\circ}$, $\widehat{I C A}=20^{\circ} .$ Find the measure of the angle $\widehat{C B I}$.
3. Let $G$ and $I$ be respectively the centroid and the incenter of a given triangle $A B C .$ Prove that if $A B^{2}-A C^{2}=2\left(I B^{2}-I C^{2}\right)$ then $G I$ is parallel to $B C$.
4. Solve the equation $$\left(x^{2}+1\right)\left|x^{2}+2 x-1\right|+6 x\left(1-x^{2}\right)=\left(x^{2}+1\right)^{2}$$
5. Let $\left(a_{n}\right)$ be a sequence given by $$a_{1}=1,\quad a_{n+1}=\sqrt{a_{n}\left(a_{n}+1\right)\left(a_{n}+2\right)\left(a_{n}+3\right)+2},\,\forall n \in \mathbb{N}^{*}.$$ Compare $\dfrac{1}{2}$ with the sum $$S=\frac{1}{a_{1}+2}+\frac{1}{a_{2}+2}+\frac{1}{a_{3}+2}+\ldots+\frac{1}{a_{2009}+2}.$$
6. Let $a, b, c$ be positive real numbers such that $a+b+c=6 .$ Prove that $$\frac{a}{\sqrt{b^{3}+1}}+\frac{b}{\sqrt{c^{3}+1}}+\frac{c}{\sqrt{a^{3}+1}} \geq 2$$
7. Find all triangles whose inradius equal 3 and the side lengths form the first three terms of an arithmetic progression with common difference $d$ distinct from $0 .$
8. Let $A B C$ be a triangle with $A B=3 R$, $B C=R \sqrt{7}, C A=2 R .$ Let $M$ be an arbitrary point on the spherical surface $(C ; R)$. Find the least value of $M A+2 M B$.
9. There are $294$ people in a meeting. Those who are acquainted shake hands with each other. Knowing that if $A$ shakes hands with $B$ then one of them shakes hands at most 6 times. What is the greatest number of possible handshakes?
10. Given a positive integer $m$, find all functions $f: \mathbb{N} \rightarrow \mathbb{N}$ such that for every $x, y \in \mathbb{N}$ we have
• If $f(x)=f(y)$ then $x=y$
• $f(f(f(\ldots)))=x+y$. Here, $f$ appears $m$ times on the left hand side.
11. Let $f(x)$ be a continuous function on the closed interval $[0 ; 1],$ and differentiable on the open interval $(0 ; 1)$ such that $f(0)=0$ $f(1)=1 .$ Prove that for two arbitrary real numbers $k_{1}, k_{2},$ there exist two distinct numbers $a, b$ in the open interval $(0 ; 1)$ such that $$\frac{k_{1}}{f(a)}+\frac{k_{2}}{f(b)}=k_{1}+k_{2}.$$
12. Let $A B C$ be a triangle with orthocenter $H$. Prove that the common tangent, distinct from $A H,$ of the incircles of the triangles $A B H$ and $A C H$ passes through the midpoint of $B C$.

## $hide=mobile$type=ticker$c=36$cols=2$l=0$sr=random$b=0 ##$hide=mobile

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,74,Bắc Bộ,2,Bắc Giang,60,Bắc Kạn,4,Bạc Liêu,15,Bắc Ninh,59,Bắc Trung Bộ,3,Bài Toán Hay,5,Balkan,40,Baltic Way,32,BAMO,1,Bất Đẳng Thức,69,Bến Tre,70,Benelux,15,Bình Định,64,Bình Dương,37,Bình Phước,49,Bình Thuận,42,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,12,Cao Quang Minh,1,Câu Chuyện Toán Học,43,Caucasus,3,CGMO,11,China - Trung Quốc,25,Chọn Đội Tuyển,492,Chu Tuấn Anh,1,Chuyên Đề,125,Chuyên SPHCM,7,Chuyên SPHN,26,Chuyên Trần Hưng Đạo,3,Collection,8,College Mathematic,1,Concours,1,Cono Sur,1,Contest,671,Correspondence,1,Cosmin Poahata,1,Crux,2,Czech-Polish-Slovak,28,Đà Nẵng,49,Đa Thức,2,Đại Số,20,Đắk Lắk,73,Đắk Nông,14,Đan Phượng,1,Danube,7,Đào Thái Hiệp,1,ĐBSCL,2,Đề Thi,1,Đề Thi HSG,2197,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,63,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,11,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,39,Gia Viễn,2,Giải Tích Hàm,1,Giảng Võ,1,Giới hạn,2,Goldbach,1,Hà Giang,5,Hà Lan,1,Hà Nam,40,Hà Nội,259,Hà Tĩnh,93,Hà Trung Kiên,1,Hải Dương,65,Hải Phòng,55,Hậu Giang,12,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 10,123,HSG 10 2010-2011,4,HSG 10 2011-2012,6,HSG 10 2012-2013,5,HSG 10 20122-2023,1,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 2021-2022,3,HSG 10 2022-2023,7,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,4,HSG 10 Chuyên SPHN,5,HSG 10 Đắk Lắk,2,HSG 10 Đồng Nai,4,HSG 10 Gia Lai,2,HSG 10 Hà Nam,4,HSG 10 Hà Tĩnh,15,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 PTNK,5,HSG 10 Quảng Nam,1,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 11,134,HSG 11 2010-2011,5,HSG 11 2011-2012,6,HSG 11 2012-2013,8,HSG 11 2013-2014,5,HSG 11 2014-2015,9,HSG 11 2015-2016,3,HSG 11 2016-2017,7,HSG 11 2017-2018,5,HSG 11 2018-2019,7,HSG 11 2019-2020,5,HSG 11 2020-2021,6,HSG 11 2021-2022,2,HSG 11 2022-2023,3,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,12,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,2,HSG 11 Hà Tĩnh,12,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 Nam,1,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,3,HSG 11 Vĩnh Phúc,10,HSG 12,639,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 2020-2021,52,HSG 12 2021-2022,34,HSG 12 2022-2023,38,HSG 12 An Giang,7,HSG 12 Bà Rịa Vũng Tàu,12,HSG 12 Bắc Giang,17,HSG 12 Bạc Liêu,3,HSG 12 Bắc Ninh,13,HSG 12 Bến Tre,19,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,21,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,16,HSG 12 Hải Dương,14,HSG 12 Hải Phòng,20,HSG 12 Hậu Giang,3,HSG 12 Hòa Bình,10,HSG 12 Hưng Yên,9,HSG 12 Khánh Hòa,3,HSG 12 KHTN,26,HSG 12 Kiên Giang,12,HSG 12 Kon Tum,2,HSG 12 Lai Châu,4,HSG 12 Lâm Đồng,11,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,7,HSG 12 Phú Thọ,17,HSG 12 Phú Yên,12,HSG 12 Quảng Bình,13,HSG 12 Quảng Nam,9,HSG 12 Quảng Ngãi,6,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,13,HSG 12 Tuyên Quang,2,HSG 12 Vĩnh Long,7,HSG 12 Vĩnh Phúc,22,HSG 12 Yên Bái,6,HSG 9,574,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 2014-2015,41,HSG 9 2015-2016,39,HSG 9 2016-2017,44,HSG 9 2017-2018,47,HSG 9 2018-2019,50,HSG 9 2019-2020,20,HSG 9 2020-2021,53,HSG 9 2021-2022,58,HSG 9 2022-2023,38,HSG 9 An Giang,8,HSG 9 Bà Rịa Vũng Tàu,7,HSG 9 Bắc Giang,13,HSG 9 Bạc Liêu,1,HSG 9 Bắc Ninh,13,HSG 9 Bến Tre,10,HSG 9 Bình Định,11,HSG 9 Bình Dương,7,HSG 9 Bình Phước,14,HSG 9 Bình Thuận,5,HSG 9 Cà Mau,1,HSG 9 Cần Thơ,4,HSG 9 Cao Bằng,2,HSG 9 Chuyên SPHN,2,HSG 9 Đà Nẵng,11,HSG 9 Đắk Lắk,11,HSG 9 Đắk Nông,3,HSG 9 Điện Biên,3,HSG 9 Đồng Nai,8,HSG 9 Đồng Tháp,10,HSG 9 Gia Lai,9,HSG 9 Hà Giang,4,HSG 9 Hà Nam,9,HSG 9 Hà Nội,26,HSG 9 Hà Tĩnh,17,HSG 9 Hải Dương,15,HSG 9 Hải Phòng,7,HSG 9 Hậu Giang,5,HSG 9 Hòa Bình,3,HSG 9 Hưng Yên,11,HSG 9 Khánh Hòa,6,HSG 9 Kiên Giang,15,HSG 9 Kon Tum,9,HSG 9 Lai Châu,1,HSG 9 Lâm Đồng,14,HSG 9 Lạng Sơn,9,HSG 9 Lào Cai,4,HSG 9 Long An,9,HSG 9 Nam Định,9,HSG 9 Nghệ An,21,HSG 9 Ninh Bình,14,HSG 9 Ninh Thuận,3,HSG 9 Phú Thọ,13,HSG 9 Phú Yên,10,HSG 9 Quảng Bình,14,HSG 9 Quảng Nam,11,HSG 9 Quảng Ngãi,13,HSG 9 Quảng Ninh,16,HSG 9 Quảng Trị,10,HSG 9 Sóc Trăng,9,HSG 9 Sơn La,5,HSG 9 Tây Ninh,16,HSG 9 Thái Bình,10,HSG 9 Thái Nguyên,5,HSG 9 Thanh Hóa,18,HSG 9 Thừa Thiên Huế,8,HSG 9 Tiền Giang,6,HSG 9 TPHCM,11,HSG 9 Trà Vinh,2,HSG 9 Tuyên Quang,6,HSG 9 Vĩnh Long,11,HSG 9 Vĩnh Phúc,12,HSG 9 Yên Bái,5,HSG Cấp Trường,90,HSG Quốc Gia,110,HSG Quốc Tế,16,HSG11 2021-2022,3,HSG11 2022-2023,1,Hứa Lâm Phong,1,Hứa Thuần Phỏng,1,Hùng Vương,2,Hưng Yên,41,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,345,Iran,13,Jakob,1,JBMO,41,Jewish,1,Journal,30,Junior,38,K2pi,1,Kazakhstan,1,Khánh Hòa,29,KHTN,61,Kiên Giang,72,Kim Liên,1,Kon Tum,24,Korea - Hàn Quốc,5,Kvant,2,Kỷ Yếu,46,Lai Châu,10,Lâm Đồng,46,Lăng Hồng Nguyệt Anh,1,Lạng Sơn,35,Langlands,1,Lào Cai,34,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,13,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,45,Nam Phi,1,National,276,Nesbitt,1,Newton,4,Nghệ An,71,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,59,Ninh Thuận,25,Nội Suy Lagrange,2,Nội Suy Newton,1,Nordic,21,Olympiad Corner,1,Olympiad Preliminary,2,Olympic 10,129,Olympic 10/3,6,Olympic 10/3 Đắk Lắk,6,Olympic 11,118,Olympic 12,50,Olympic 23/3,2,Olympic 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MOlympiad.NET: Mathematics and Youth Magazine Problems 2009
Mathematics and Youth Magazine Problems 2009