Mathematics and Youth Magazine Problems 2008

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Issue 367

  1. Let $S$ be the sum of $100$ terms $$S=\frac{1}{1.1 .3}+\frac{1}{2.3 .5}+\frac{1}{3.5 .7}+\frac{1}{4.7 .9}+\ldots+\frac{1}{100.199 .201}.$$ Compare $S$ with $\dfrac{2}{3}$.
  2. Let $A B C$ be an isosceles triangle $(A B=A C)$ such that $\widehat{B A C}<90^{\circ} .$ Let $B D$ and $A H$ be the altitudes. Choose a point $K$ on $B D$ such that $B K=B A$ Find the measure of angle $HAK$.
  3. Given $0<b<a \leq 4$ and $2 a b \leq 3 a+4 b$. Find the maximum value of the expression $a^{2}+b^{2}$.
  4. The equation $$5 x^{6}-16 x^{4}-33 x^{3}-40 x^{2}+8=0$$ has two roots which are reciprocal. Find these roots.
  5. Let $A B C$ be a right triangle with right angle at $A$ and $A C>A B$. Let $O$ be the midpoint of $B C,$ and $I$ be the incenter of the triangle $A B C .$ Suppose that $\widehat{O I B}=90^{\circ},$ find the ratio between three edges of the triangle $A B C$.
  6. Find all pairs of positive integers $(x ; y)$ such that $$y^{x}-1=(y-1) !$$ where $y$ is a prime number.
  7. Let $a, b, c$ be non-negative real numbers. Prove the inequality $$\left(a^{2}+b^{2}+c^{2}\right)^{2} \geq 4(a-b)(b-c)(c-a)(a+b+c).$$ When does equality occur?
  8. Let $A B C$ be an isosceles triangle at vertex $A$ and $B C \leq A C$. Choose a point $M$ on $A B$ (but not the vertices $A$ or $B$) and a point $N$ on $A C$ such that $M N$ touch the incircle of the triangle $A B C$. Find the maximum value of the ratio $\dfrac{A M}{B M \cdot C N}$ when $M$ moves along the edge $A B$.
  9. Let $O$ be the circumcenter of a triangle $A B C .$ Choose a point $P,$ different from $B$ and $C$ on the line connecting $B C$. The circumcircle of $A B C$ meets $A P$ at a point $N$ and the circle whose diameter is $A P$ at a point $E$ ($N$, $E$ are both different from $A$). $B C$ and $A E$ intersect at $M .$ Prove that $M N$ always passes through a fixed point.
  10. A sequence $\left(u_{n}\right)(n=1,2, \ldots)$ is determined by the following recursive formula $$u_{1}=1,\quad u_{n+1}=\frac{16 u_{n}^{3}+27 u_{n}}{48 u_{n}^{2}+9}.$$ Find the largest integer which is smaller than the sum $S$ of $2008$ summands $$S=\frac{1}{4 u_{1}+3}+\frac{1}{4 u_{2}+3}+\ldots+\frac{1}{4 u_{2008}+3}$$
  11. Find the smallest value of the following expression $$P=\frac{1}{\cos ^{6} a}+\frac{1}{\cos ^{6} b}+\frac{1}{\cos ^{6} c}$$ where $a$, $b$ and $c$ form an arithmetic sequence whose common difference is $\dfrac{\pi}{3}$.
  12. Let $f(x)$ be a function defined on $[0 ; 1]$ such that the following properties hold
    • $f(1)=1$
    • $f(x)=\dfrac{1}{3}\left(f\left(\dfrac{x}{3}\right)+f\left(\dfrac{x+1}{3}\right)+f\left(\dfrac{x+2}{3}\right)\right)$ for all $x \in[0 ; 1]$
    • for every $\varepsilon$ positive but can be arbitrarily small, there exists a positive number $\delta_{\varepsilon}\left(\delta_{\varepsilon}\right.$ depends on $\varepsilon$) such that: For all $x, y \in[0 ; 1]$ such that $|x-y|<\delta_{\varepsilon},$ we have $|f(x)-f(y)|<\varepsilon$.
      Prove that $f(x)=1$ for all $x \in[0 ; 1]$.

    Issue 368

    1. Consider $n$ consecutive points $A_{1}, A_{2}, A_{3}, \ldots, A_{n}$ on the same line such that $$A_{1} A_{2}=A_{2} A_{3}=A_{3} A_{4}=\ldots=A_{n-1} A_{n}.$$ Find $n,$ given that there are exactly $2025$ segments on that line whose midpoints are one of these $n$ points.
    2. Let $A B C$ be a right triangle with right angle at $A$. On the halfplane divided by $B C$ which does not contain $A,$ choose the points $D$, $E$ such that $B D$ is orthogonal with $B A$ and $B D=B A$, $B E$ is orthogonal with $B C$ and $B E=B C$. Denote by $M$ the midpoint of $C E .$ Prove that $A$, $D$ and $M$ are colinear.
    3. Find all positive integers $x$, $y$ and $z$ such that $$2 x y-1=z(x-1)(y-1).$$
    4. Solve for $x$ $$4 x-x^{2}=3 \sqrt{4-3 \sqrt{10-3 x}}.$$
    5. Let $A B C$ be a right triangle with right angle at $A$ and let $A D$ be the angle bisector at $A .$ Denote by $M$ and $N$ the bases of the altitudes from $D$ onto $A B$ and $A C$ respectively. $B N$ meets $C M$ at $K$ and $A K$ meets $D M$ at $I$. Find the measure of angle $B I D$.
    6. Let $$f(x)=2009 x^{5}-x^{4}-x^{3}-x^{2}-2006 x+1 .$$ Prove that $f(n)$, $f(f(n))$, $f(f(f(n)))$ are pairwise coprime for any positive integer $n$.
    7. Find $a$, $b$ such that $\max _{0 \leq x \leq 16}|\sqrt{x}+a x+b|$ is smallest possible. Find this minimum value.
    8. The incircle of a triangle $A B C$ touches $B C$, $C A$ and $A B$ at $A^{\prime}$, $B^{\prime},$ and $C^{\prime}$ respectively. Prove that $$A B^{\prime 2}+B C^{\prime 2}+C A^{\prime 2} \geq A B^{\prime} . B^{\prime} C^{\prime}+B C^{\prime} \cdot C^{\prime} A^{\prime}+C A^{\prime} . A^{\prime} B^{\prime} \geq B^{\prime} C^{\prime 2}+C^{\prime} A^{\prime 2}+A^{\prime} B^{\prime 2}.$$ When does equality occur?
    9. Let $k$ be a positive integer. Write $k$ as a product of prime numbers (not necessarily distinct, for instance, $k=18=3.3 .2)$ and let $T(k)$ be the sum of all factors in the factorization above. Find the largest constant $C$ such that $T(k) \geq C \ln k$ for all positive integer $k$.
    10. Let $a, b, c$ be non-negative real numbers whose sum of squares equal $3 .$ Find the maximum value of the following expression $$P=a b^{2}+b c^{2}+c a^{2}-a b c$$
    11. Let $\left(x_{n}\right)(n=1,2, \ldots)$ be a sequence, determined by the following recursive formula $$x_{1}=\frac{1}{2},\quad x_{n+1}=x_{n}-x_{n}^{2}+x_{n}^{3}-x_{n}^{4}+\ldots+x_{n}^{2007}-x_{n}^{2008},\,\forall n \in \mathbb{N}^{*}.$$ Find the limit $\displaystyle \lim_{n \rightarrow+\infty} n x_{n}$.
    12. Let $A B C D$ be a tetrahedron whose altitudes are concurrent. Denote by $R$ the radius of its circumcircle; by $h_{1}$, $h_{2}$, $h_{3}$, $h_{4}$ the lengths of the altitudes corresponding to the vertices $A$, $B$, $C$, $D$ respectively; and by $R_{1}$, $R_{2},$ $R_{3}$, $R_{4}$ the circumcircles's radius of the opposite faces of the vertices $A$, $B$, $C$, $D$ respectively. Prove the following inequality $$\frac{1}{h_1+2 \sqrt{2} R_{1}}+\frac{1}{h_{2}+2 \sqrt{2} R_{2}}+\frac{1}{h_{3}+2 \sqrt{2} R_{3}}+\frac{1}{h_{4}+2 \sqrt{2} R_{4}} \geq \frac{1}{R}.$$ When does equality occur?

    Issue 369

    1. For each integer $n$ greater than $6,$ denote by $A_{n}$ the collection of integers which are less than $n$ and not less than $\dfrac{n}{2} .$ Find $n$ such that there are no perfect square in $A_{n}$.
    2. Let $A B C$ be an acute triangle with $\widehat{B A C}=60^{\circ} . E$ and $F$ are two points on $A C$ and $A B$ respectively such that $\widehat{E B C}=\widehat{F C B}=30^{\circ}$. Prove that $$B F=F E=E C \geq \dfrac{B C}{2}.$$
    3. Find four distinct integers $a, b, c, d$ in the set $\{10 ; 21 ; 37 ; 51\}$ such that $$a b+b c-a d=637.$$
    4. Solve for $x$ $$(x+3) \sqrt{(4-x)(12+x)}=28-x.$$
    5. Let $A B C$ be a triangle with $\widehat{B A C} \neq 45^{\circ}$ and $\widehat{A I O}=90^{\circ}$ where $(O)$ and $(I)$ are its circumcircle and incircle, respectively. Choose a point $D$ on the ray $B C$ such that $B D=A B+A C$. The tangent line through $D$ touches $(O)$ at $E$. The tangent line through $B$ of $(O)$ meets $D E$ at $F$; $C F$ meets $(O)$ at another point denoted by $K$. Let $G$ be the centroid of the triangle $A B C$. Prove that $I G$ is parallel to $E K$.
    6. Let $a_{1}, a_{2}, \ldots, a_{n}$ be positive integers such that the following two properties hold
      • $a_{i}<2008$ for all $i=1,2, \ldots, n$
      • The greatest common divisor of any pair of numbers is greater than $2008$.
      Prove that $\displaystyle\sum_{i=1}^{n} \frac{1}{a_{i}}<2$.
    7. Prove the following inequality $$(3 a+2 b+c)\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right) \leq \frac{45}{2}$$ where $a, b,$ and $c$ are in the interval $[1 ; 2]$ When does equality occur?
    8. Given a triangle $A B C$ with circumcircle $(O)$ and three sides $B C=a$, $C A=b$, $A B^{\prime}=c .$ Denote by $A_{1}$, $B_{1},$ and $C_{1}$ the midpoints of $B C$, $C A,$ and $A B$ respectively; and $A_{2}$, $B_{2}$, $C_{2}$ are the midpoint of the arcs $\widehat{B C}$ (which does not contain $A$), $\widehat{C A}$ (which does not contains $B$), and $\widehat{A B}$ (which does not contain $C$). Draw the circles $\left(Q_{1}\right),\left(O_{2}\right),$ and $\left(O_{3}\right)$ whose diameters are $A_{1} A_{2}$, $B_{1} B_{2},$ and $C_{1} C_{2}$ respectively. Prove the inequality $$\mathcal{P}_{A/\left(O_{1}\right)}+\mathcal{P}_{B /\left(O_{2}\right)}+\mathcal{P}_{C /\left(O_{3}\right)} \geq \frac{(a+b+c)^{2}}{3}.$$ When does equality occur? 
    9. Find all positive integers $x, y, z, n$ such that $$x !+y !+z !=5 . n !$$ where $k !=1 \times 2 \times \ldots \times k$.
    10. Let $a$ and $b$ be two real numbers in the open interval $(0 ; 4) .$ A sequence $\left(a_{n}\right),$ $(n=0,1, \ldots)$ is constructed by the following recursive formula $$a_{0}=a,\, a_{1}=b,\quad a_{n+2}=\frac{2\left(a_{n+1}+a_{n}\right)}{\sqrt{a_{n+1}}+\sqrt{a_{n}}}.$$ Prove that the sequence $\left(a_{n}\right)$ (n=0,1, \ldots)$ converges and find its limit.
    11. Find all functions $f: \mathbb{R}^{+} \rightarrow \mathbb{R}^{+}$ such that $$f(f(x))+f(x)=\left(26^{3^{2008}}+\left(26^{32008}\right)^{2}\right) x.$$
    12. Let $S$ denote the surface area of a given tetrahedron. Prove that the sum of areas of the six angle-bisectors of this tetrahedron does not exceed $\dfrac{\sqrt{6}}{2} S$.

    Issue 370

    1. Prove the inequality $$\frac{3}{2}+\frac{7}{4}+\frac{11}{8}+\frac{15}{16}+\ldots+\frac{4 n-1}{2^{n}}<7$$ where $n$ is an arbitrary positive integer.
    2. Let $A B C$ be a right triangle with right angle at $A$. Suppose $A B=5cm$ and $I C=6cm$ where $I$ is the incenter of $A B C$. Determine the length of $B C$.
    3. Find all positive integers $x, y, z$ such that the following equality holds $$5 x y z=x+5 y+7 z+10.$$
    4. Solve for $x$ $$x^{4}-2 x^{2}-16 x+1=0.$$
    5. Let $A B C$ be an acute triangle whose altitudes $A A^{\prime}$, $B B^{\prime}$, $C C^{\prime}$ meet at $H$ Denote by $A_{1}$, $B_{1},$ and $C_{1}$ the othocenters of the triangles $A B^{\prime} C^{\prime}$, $B C^{\prime} A^{\prime}$ and $C A^{\prime} B^{\prime}$ respectively. Suppose that $H$ is the incenter of the triangle $A_{1} B_{1} C_{1}$. Prove that $A B C$ is an equilateral triangle.
    6. Let $A B C$ be an isosceles triangle with $A B=B C=a$ and $\widehat{A B C}=140^{\circ} .$ Let $A N$ and $A H$ respectively be the anglebisector and the altitude from $A$. Prove that $2 B H \cdot C N=a^{2}$
    7. Find the minimum value of the function $$f(x)=\left(32 x^{5}-40 x^{3}+10 x-1\right)^{2006}+\left(16 x^{3}-12 x+\sqrt{5}-1\right)^{2008}.$$
    8. Prove that the following equation $$A \cdot a^{x}+B \cdot b^{x}=A+B$$ where $a>1$, $0<b<1$, $A, B \in \mathbb{R}$ has at most two solutions.
    9. Let $a_{1}=\dfrac{1}{2}$ and for each $n$ greater than $1,$ let $$a_{n}=\frac{1}{d_{1}+1}+\frac{1}{d_{2}+1}+\ldots+\frac{1}{d_{k}+1}$$ where $d_{1}, d_{2}, \ldots, d_{k}$ is the collection of all distinct positive divisors of $n .$ Prove the inequality $$n-\ln n<a_{1}+a_{2}+\ldots+a_{n}<n.$$
    10. Given $n$ numbers $a_{1}, a_{2}, \ldots, a_{n}$ in $[-1 ; 2]$ such that their total sum is 0. Let $U_{k}=\dfrac{a_{k} \sqrt{4 k-1}}{(4 k-3)(4 k+1)}$ for $k=1,2, \ldots, n .$ Prove that $$\left|U_{1}+U_{2}+\ldots+U_{n}\right|<\frac{\sqrt{n}}{2}$$
    11. Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that $$f(x+y)=x^{2} f\left(\frac{1}{x}\right)+y^{2} f\left(\frac{1}{y}\right),\,\forall x, y \in \mathbb{R}^{*}.$$
    12. Let $P$ be a point on the insphere of a tetrahedron $A B C D$ and let $G_{a}$, $G_{b}$, $G_{c}$, $G_{d}$ be the centroids of the tetrahedra $P B C D$, $PCDA$, $PDAB$, $PABC$, respectively. Prove that $A G_{a}$, $B G_{b}$, $C G_{c}$ and $D G_{d}$ pass through a common point; and find the orbit of this common point when $P$ moves on the insphere of the given tetrahedron $A B C D$.

    Issue 371

    1. Which number is greater? $A=\dfrac{1}{2006}$ or $$B =\frac{1}{2008}+\left(\frac{1}{2008}+\frac{1}{2008^{2}}\right)^{2}+\ldots +\left(\frac{1}{2008}+\frac{1}{2008^{2}}+\ldots+\frac{1}{2008^{2007}}\right)^{2007}.$$
    2. In a triangle $A B C$ with altitude $A D,$ one has $A D=D C=3 B D$. Let $O$ and $H$ be the circumcenter and the orthocenter, respectively. Prove that $\dfrac{O H}{B C}=\dfrac{1}{4}$.
    3. Find all pairs of natural numbers $x$ and $y$ such that $$x^{3}=y^{3}+2\left(x^{2}+y^{2}\right)+3 x y+17.$$
    4. Let $a, b, c$ be positive real numbers. Prove the inequality $$\frac{a^{2}-b^{2}}{\sqrt{b+c}}+\frac{b^{2}-c^{2}}{\sqrt{c+a}}+\frac{c^{2}-a^{2}}{\sqrt{a+b}} \geq 0.$$ When does equality occur?
    5. A circle $\left(S_{1}\right)$ passing through the vertices $A$ and $B$ of a triangle $A B C$ meets $B C$ at another point $D .$ Another circle, $\left(S_{2}\right),$ passing through $B$ and $C$ meets $A B$ at another point $E$ and meets $\left(S_{1}\right)$ at $F$. Prove that if all four points $A$, $C$, $D$ and $E$ lie on the same circle with center at $O$, then $\widehat{B F O}=90^{\circ}$.
    6. Solve the system of equations $$\begin{cases}\dfrac{x}{a-30}+\dfrac{y}{a-4}+\dfrac{z}{a-14}+\dfrac{t}{a-10} &=1 \\ \dfrac{x}{b-30}+\dfrac{y}{b-4}+\dfrac{z}{b-14}+\dfrac{t}{b-10} &=1 \\ \dfrac{x}{c-30}+\dfrac{y}{c-4}+\dfrac{z}{c-14}+\dfrac{t}{c-10} &=1 \\ \dfrac{x}{d-30}+\dfrac{y}{d-4}+\dfrac{z}{d-14}+\dfrac{1}{d-10} &=1\end{cases}$$ where $a, b, c, d$ are distinct numbers, none of which belong to the set $\{4 ; 10 ; 14 ; 30\} .$
    7. Let $a, b, c$ be positive numbers. Prove that $$a^{b+c}+b^{c+a}+c^{a+b} \geq 1$$
    8. Choose six points $D$, $E$, $F$, $G$, $H$, $K$ in that order, on a circle with radius $R$ and center at $O$ such that $D E=F G=H K=R$. $K D$ and $E F$ meet at $A$, $E F$ and $G H$ meet at $B$ and $G H$ meets $K D$ at $C$. Prove that $$OA \cdot B C=O B \cdot C A=O C \cdot A B.$$
    9. Let $A B C D$ be a cyclic quadrilateral, inscribed in a circle centered at $I .$ Prove the following inequality $$(A I+D I)^{2}+(B I+C I)^{2} \leq(A B+C D)^{2}$$ and determine when equality occurs.
    10. Consider the quadratic equation $x^{2}-a x-1=0$ where $a$ is a positive integer. Let $\alpha$ be a positive root of this equation and construct a sequence $\left(x_{n}\right)$ by the following recursive rule $$x_{0}=a, x_{n+1}=\left[\alpha x_{n}\right], n=0,1,2, \ldots$$ Prove that there exists infinitely many integer $n$ such that $x_{n}$ is a multiple of $a$. 
    11. Given a function $f: \mathrm{N} \rightarrow \mathrm{N}$ such that for all $n \in \mathbb{N} $ $$(f(2 n)+f(2 n+1)+1)(f(2 n+1)-f(2 n)-1)=3(1+2 f(n)),\quad f(2 n) \geq f(n).$$ Denote $M=\{m \in f(\mathrm{N}): m \leq 2008\} .$ Find the number of elements of $M$.
    12. For what kind of triangle $A B C$ that the following relation among its sides and its angles holds $$\frac{b c}{b+c}(1+\cos A)+\frac{c a}{c+a}(1+\cos B)+\frac{a b}{a+b}(1+\cos C) \\ = \frac{3}{16}(a+b+c)^{2}+\cos ^{2} A+\cos ^{2} B+\cos ^{2} C.$$

    Issue 372

    1. Let $$A=\frac{1}{1^{2}}+\frac{1}{2^{3}}+\frac{1}{3^{4}}+\ldots+\frac{1}{2007^{2008}}.$$ Prove that $A$ is not an integer.
    2. Let $P(x)=x^{3}-7 x^{2}+14 x-8 .$ Prove that for every natural number $n,$ there exists a triple of distinct intergers $a_{1}$, $a_{2}$, $a_{3}$ such that the following two conditions are satisfied
      • $\left|a_{i}-a_{j}\right|<5, \forall i, j \in\{1,2,3\}$
      •  $P\left(a_{i}\right) \neq 0$ and $5^{n} \mid P\left(a_{i}\right)$ for all $i \in\{1,2,3\}$
    3. Find all pairs of intergers $x$, $y$ such that $$\sqrt[n]{x+\sqrt[n]{x+\ldots+\sqrt[n]{x}}}=y$$ ($m$ times) where $m$, $n$ are positive intergers which are greater than $2$.
    4. Given $x>2$. Prove the inequality $$\frac{x}{2}+\frac{8 x^{3}}{(x-2)(x+2)^{2}}>9.$$
    5. A pentagon $A B C D E$ is inscribed in a circle with center at $O$ and radius $R$ such that $A B=C D=E A=R$. Let $M$, $N$ be respectively the midpoints of $B C$ and $D E$ Prove that $A M N$ is an equilateral triangle.
    6. Do there exist two distinct positive integers $a$, $b$ such that $b^{n}+n$ is $a$ multiple of $a^{n}+n$ for every postive integer $n ?$.
    7. Let $S$ be the set of all pairs of real numbers $(\alpha, \beta)$ such that the equation $$x^{3}-6 x^{2}+\alpha x-\beta=0$$ has three real roots (not necessarily distinct) and they are all greater than $1$. Find the maximum value of $T=8 \alpha-3 \beta$ for $(\alpha, \beta) \in S$.
    8. Let $A B C$ be an acute triangle and denote by $Q$ the center of its Euler's circle. The circumcircle of $A B C,$ which has radius $R,$ meets $A Q, B Q,$ and $C Q$ respectively at $M$, $N$ and $P$ Prove the inequality $$\frac{1}{Q M}+\frac{1}{Q N}+\frac{1}{Q P} \geq \frac{3}{R}.$$
    9. Find the interger part of $$A=\sqrt[n]{1-\frac{x}{2008}}+\sqrt[m]{1+\frac{x}{2008}}$$ where $x$ is a real number in $[-2008 ; 2008]$, and $m$, $n$ are natural numbers, $m \geq n \geq 2$.
    10. Let $S$ be denote the set of all $n$-tuples $(n>1)$ of real numbers $\left(a_{1}, a_{2}, \ldots, a_{n}\right)$ such that $$3\sum_{i=1}^{n} a_i^{2}=502.$$ a) Prove that $\displaystyle \min _{1 \leq \leq j \leq n}\left|a_{j}-a_{i}\right| \leq \sqrt{\frac{2008}{n\left(n^{2}-1\right)}}$.
      b) Give an example of an $n$ -tuple $\left(a_{1}, a_{2}, \ldots, a_{n}\right)$ such that the above condition holds and for which there is an equality in a).
    11. A function $f(x),$ whose domain is the interval $[1 ;+\infty),$ has the following two properties $$f(1)=\frac{1}{2008},\quad f(x)+2007(f(x+1))^{2}=f(x+1),\,\forall x \in[1 ;+\infty).$$ Find the limit $$\lim_{n \rightarrow+\infty}\left(\frac{f(2)}{f(1)}+\frac{f(3)}{f(2)}+\ldots+\frac{f(n+1)}{f(n)}\right).$$
    12. The angle-bisectors $A A_{1}$, $B B_{1}$, $C C_{1}$ of a triangle $A B C$ with perimeter $p$ meet $B_{1} C_{1}$, $C_{1} A_{1},$ and $A_{1} B_{1}$ respectively at $A_{2}$, $B_{2},$ and $C_{2}$. The line through $A_{2}$ and parallel to $B C$ meets $A B$, $A C$ at $A_{3}$, $A_{4}$. Construct the points $B_{3}$, $B_{4}$ and $C_{3}$, $C_{4}$ in a similar way. Prove the inequality $$A B_{4}+B C_{4}+C A_{4}+B A_{3}+C B_{3}+A C_{3} \leq p.$$ When does equality holds?

    Issue 373

    1. Write the numbers $1,2,3, \ldots, 2007$ in an arbitrary order and let $A$ be the resulting number. Can $A+2008^{2007}+2009$ be a perfect square?.
    2. Consider the following two polynomials $$f(x)=(x-2)^{2008}+(2 x-3)^{2007}+2006 x$$ and $$g(y)=y^{2009}-2007 y^{2008}+2005 y^{2007}.$$ Let $s$ be denote the sum of all the coefficients of $f(x)$ (after expansion). Find $s$, and the value of $g(s)$.
    3. Find all positive integer solutions of the following system of two equations $$\begin{cases} x+y+z &= 15 \\ x^{3}+y^{3}+z^{3} &= 495\end{cases}.$$
    4. Let $a, b, c$ be non-negative real numbers such that $a^{2}+b^{2}+c^{2}=1 .$ Find the maximum value of the expression $$(a+b+c)^{3}+a(2 b c-1)+b(2 a c-1)+c(2 a b-1).$$
    5. Given a triangle $A B C$ where $\widehat{A B C}$ is not a right angle. Let $A H$ and $A M$ denote, the altitude and the median throught vertex $A$. Choose a point $E$ on the ray $A B$ and $F$ on the ray $A C$ such that $M E$ $=M F=M A .$ Let $K$ be reflection point of $H$ over $M .$ Prove that the four points $E$, $M$, $K$ and $F$ lie on a single circle.
    6. Solve for $x$ $$\sqrt{x+\sqrt{x^{2}-1}}=\frac{9 \sqrt{2}}{4}(x-1) \sqrt{x-1}.$$
    7. Prove that in any acute triangle $A B C$, the following inequality holds $$\frac{\tan A}{\tan B}+\frac{\tan B}{\tan C}+\frac{\tan C}{\tan A} \geq \frac{\sin 2 A}{\sin 2 B}+\frac{\sin 2 B}{\sin 2 C}+\frac{\sin 2 C}{\sin 2 A}$$
    8. The incircle of a triangle $A B C$ meets $B C$, $C A$ and $A B$ respectively at $A_{1}$, $B_{1}$, $C_{1}$. Let $p$, $S$, $R$ be respectively, half of the perimeter, the area and the circumradius of $A B C$. Let $p_{1}$ be half of the perimeter of $A_{1} B_{1} C_{1}$. Prove the inequality $$p_{1}^{2} \leq \frac{p S}{2 R}.$$ When does equality occur?
    9. Let $A(A \subset \mathbb{N})$ be a non-empty set satisfying the condition: If $a \in A$ then $4 a$ and $[\sqrt{a}]$ are also in $A([x]$ is the integer part of $x$). Prove that $A=\mathbb{N}$.
    10. Let $a$ be a natural number which is greater than $3$ and consider the sequence $\left(u_{n}\right)$ $(n=1,2, \ldots)$ defined inductively by $u_{1}=a$ and $$u_{n+1}=u_{n}-\left[\frac{u_{n}}{2}\right]+1,\,\forall n=1,2, \ldots.$$ Prove that there exists $k \in \mathbb{N}^{*}$ such that $u_{n}=u_{k}$ for all $n \geq k$.
    11. Find all polynomials with real coefficients $P(x)$, $Q(x)$ and $R(x)$ such that $$\sqrt{P(x)}-\sqrt{Q(x)}=R(x),\,\forall x.$$
    12. Let $A B C D$ be a tetrahedron with the centroid $G$ and the circumradius $R$. Prove that $$G A+G B+G C+G D+4 R \geq \frac{2}{\sqrt{6}}(A B+A C+A D+B C+C D+D B).$$

    Issue 374

    1. Find all triple of natural numbers $a, b, c$ less than $20$ such that $$a(a+1)+b(b+1)=c(c+1)$$ where $a$ is a prime number and $b$ is a multiple of $3 .$
    2. Let $f(n)=\left(n^{2}+n+1\right)^{2}+1,$ where $n$ is a positive integer and let $$P_{n}=\frac{f(1) \cdot f(3) \cdot f(5) \ldots f(2 n-1)}{f(2) \cdot f(4) \cdot f(6) \ldots f(2 n)}.$$ Prove the inequality $$P_{1}+P_{2}+\ldots+P_{n}<\frac{1}{2}.$$
    3. Find the maximum value of the expression $$T=\frac{(y+z)^{2}}{y^{2}+z^{2}}-\frac{(x+z)^{2}}{x^{2}+z^{2}},$$ where $x$, $y$, $z$ are real numbers such that $x>y$, $z>0$ and $z^{2} \geq x y$.
    4. Solve for $x$ $$\sqrt[3]{14-x^{3}}+x=2\left(1+\sqrt{x^{2}-2 x-1}\right).$$
    5. An acute triangle $A B C$ is inscribed in a fixed circle with center at $O$. Let $A I$, $B D$ and $C E$ denote the altitudes through $A$, $B$ and $C$ respectively. Prove that the perimeter of the triangle $I D E$ does not change when $A, B,$ and $C$ move on the circle $(O)$ such that the area of the triangle $A B C$ is always equal to $a^{2}$.
    6. Solve the system of equations $$\begin{cases} \sqrt{x^{2}+91} &=\sqrt{y-2}+y^{2} \\ \sqrt{y^{2}+91} &=\sqrt{x-2}+x^{2}\end{cases}.$$
    7. Let $a, b, c$ be real numbers such that $4(a+b+c)-9=0 .$ Find the maximum value of the sum $$S=\left(a+\sqrt{a^{2}+1}\right)^{b}\left(b+\sqrt{b^{2}+1}\right)^{c}\left(c+\sqrt{c^{2}+1}\right)^{a}.$$
    8. Let $I$ and $O$ denote respectively the incenter and the circumcenter of a triangle $A B C .$ Given that $\widehat{A I O}=90^{\circ},$ prove that the area of the triangle $A B C$ is less than $\dfrac{3 \sqrt{3}}{4} A I^{2}$.
    9. Let $a, b, n$ be positive integers, $b>1$ and $a$ is a multiple of $b^{n}-1 .$ Rewritten $a$ to the base $b,$ prove that the resulting contains at least $n$ non-zero digits.
    10. Let $x, y, z$ be real numbers such that $0<z \leq y \leq x \leq 8$ and $$3 x+4 y \geq \max \left\{x y ; \frac{1}{2} x y z-8 z\right\}.$$ Find the maximum value of $$A=x^{5}+y^{5}+z^{5}.$$
    11. Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that $$f\left(f(x)+y^{2}\right)=f^{2}(x)-f(x) f(y)+x y+x.$$
    12. Let $K$ denote the intersection of the two diagonals of a quadrilateral $A B C D$ where $\widehat{A B C}=\widehat{A D C}=90^{\circ} ;$ and $A C=A B+A D .$ Prove that the radii of the inscribed of the triangles $A B K$ and $A D K$ are equal.

    Issue 375

    1. Find a natural number $x$ and two decimal digits $y$, $z$ such that $$\left(5.10^{n}-2\right) x=3 . \overline{y \ldots y z}$$ for any natural number $n>1,$ where $\overline{y \ldots y z}$ (in the decimal system) contains $n-1$ digits $y$.
    2. Prove that for any $x$ and $y$  $$\frac{|x|}{2008+|x|}+\frac{|y|}{2008+|y|} \geq \frac{|x-y|}{2008+|x-y|}$$
    3. Consider an integer $n>2008$ such that both $2 n-4015$ and $3 n-6023$ are perfect squares. Find the remainder of the division of $n$ by 40
    4. Solve the quation $$\sqrt{x-\frac{1}{x}}-\sqrt{1-\frac{1}{x}}=\frac{x-1}{x}.$$
    5. Let $A B C$ be an isosceles triangle with the apex angle at $A$ and $\widehat{B A C}=150^{\circ}$. Construct the triangles $A M B$ and $A N C$ such that the rays $A M$ and $A N$ lie in the angle $B A C$ and $\widehat{A B M}=\widehat{A C N}=90^{\circ}$, $\widehat{M A B}=30^{\circ}$, $\widehat{N A C}=60^{\circ} .$ Let $D$ be a point on $M N$ such that $N D=3 M D$. $B D$ intersects with $A M$ and $A N$ at $K$ and $E,$ respectively. $B C$ and $A N$ meets at $F$. Prove that
      a) $NCE$ is an isosceles triangle;
      b) $K F$ and $C D$ are parallel.
    6. Find all pairs of integers $m$ and $n$, both greater than $1$ such that the following equality $$a^{m+n}+b^{m+n}+c^{m+n}=\frac{a^{m}+b^{m}+c^{m}}{m} \cdot \frac{a^{n}+b^{n}+c^{n}}{n}$$ is true for all real numbers $a$, $b$, $c$ satisfying $a+b+c=0$.
    7. Let $k$ be a positive integer and let $a$, $b$, $c$ be positive real numbers such that $a b c \leq 1$. Prove the equality $$\frac{a}{b^{k}}+\frac{b}{c^{k}}+\frac{c}{a^{k}} \geq a+b+c.$$
    8. Let $C$ be a point on a fixed circle whose diameter is $A B=2 R$ ($C$ is different from $A$ and $B$). The incircle of $A B C$ touches $A B$ and $A C$ at $M$ and $N$, respectively. Find the maximum value of the length of $M N$ when $C$ moves on the given fixed circle.
    9. Let $\left(x_{n}\right)$ $(n=0.1,2, \ldots)$ be a sequence such that $$x_{0}=2,\quad x_{n+1}=\frac{2 x_{n}+1}{x_{n}+2},\,\forall n=0,1,2, \ldots.$$ Determine $\displaystyle \left[\sum_{k=1}^{n} x_{k}\right]$ where $[x]$ denote the largest integer not exceeding $x$.
    10. Prove that if $a$, $b$, $c$ are positive numbers whose product $a b c=1,$ then $$\frac{a}{\sqrt{8 c^{3}+1}}+\frac{b}{\sqrt{8 a^{3}+1}}+\frac{c}{\sqrt{8 b^{3}+1}} \geq 1.$$
    11. Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a function such that $f(0)=0$ and $\dfrac{f(t)}{t}$ is a monotonic function on $\mathbb{R} \backslash\{0\}$. Prove that $$x \cdot f\left(y^{2}-z x\right)+y \cdot f\left(z^{2}-x y\right)+z \cdot f\left(x^{2}-y z\right) \geq 0$$ for all positive numbers $x$, $y$ and $z$.
    12. Let $A_{1} A_{2} A_{3} A_{4}$ be a tetrahedron. Denote by $B_{i}$ $(i=1,2,3,4)$ the feet of the altitude from a given point $M$ onto $A_{i} A_{i+1}$ (where we consider $A_{5}$ as identical to $A_{1}$). Find the smallest value of $\displaystyle\sum_{1 \leq i \leq 4} A_{i} A_{i+1} . A_{i} B_{i}$

    Issue 376

    1. Write the numbers $8^{2008}$ and $125^{2008}$ consecutively. What is the number of decimal digits of the resulting number?
    2. Find a rational number $\dfrac{a}{b}$ such that the following three conditions are satisfied
      • $-\dfrac{1}{2}<\dfrac{a}{b}<-\dfrac{2}{5}$
      • $11 a+5 b=26$
      • $200<|a|+|b|<230$
    3. Find all non-zero natural numbers $n$ such that the number $A=\dfrac{1.3 .5 .7 \ldots(2 n-1)}{n^{n}}$ is an integer, here the numerator of $A$ is the product of the first $n$ odd numbers.
    4. Prove that $$\frac{a}{b}+\frac{b}{c}+\frac{c}{a} \geq \frac{3}{2}(a+b+c-1)$$ where $a, b, c$ are positive real numbers such that $a b c=1 .$ When does equality hold? $?$
    5. In a right triangle $A B C$ with right angle at $A,$ the altitude $A H,$ the median $B M,$ and the angle-bisector $C D$ meet at a common point. Determine the ratio $\dfrac{A B}{A C}$
    6. Solve for $x$ $$\sqrt[3]{x+6}+\sqrt{x-1}=x^{2}-1$$
    7. Let $S$ denote the area of a given triangle $A B C,$ and denote $B C=a$, $C A=b$, $A B=c$. Prove the inequality $$a^{2} b^{2}+b^{2} c^{2}+c^{2} a^{2} \geq 16 S^{2}+\frac{1}{2} a^{2}(b-c)^{2}+\frac{1}{2} b^{2}(c-a)^{2}+\frac{1}{2} c^{2}(a-b)^{2}.$$ When does equality hold?
    8. Given a triangle $A B C$ with three sides $B C=a$, $A C=b$, $A B=c$ such that $a+c=2 b$ let $h_{a}$, $h_{c}$ be the altitudes from $A$ and $C$ respectively; and let $r_{a}$, $r_{c}$ denote the $A$-exradius and $C$-exradius respectively. Prove that $$\frac{1}{r_{a}}+\frac{1}{r_{c}}=\frac{1}{h_{a}}+\frac{1}{h_{c}}.$$
    9. The positive real numbers $a$, $b$, $c$, $x$, $y$ and $z$ are such that $$\begin{cases} c y+b z &=a \\ a z+c x &=b \\ b x+a y &=c\end{cases}.$$ Find the smallest possible value of the expression $$P=\frac{x^{2}}{1+x}+\frac{y^{2}}{1+y}+\frac{z^{2}}{1+z}.$$
    10. Let $f$ be a continuous function on $\mathbb{R}$ such that $f(2010)=2009$ and $f(x) \cdot f_{4}(x)=1$ for all $x \in \mathbb{R}$ (where $\left.f_{4}(x)=f(f f(f(x)))\right)$. Determine the value of $f(2008)$.
    11. Let $u_{1}, u_{2}, \ldots, u_{n}$ $(n>2)$ be a sequence of positive real numbers such that
      • $\dfrac{1004}{k}=u_{1} \geq u_{2} \geq \ldots \geq u_{n} \quad$ for some positive integer $k$;
      • $u_{1}+u_{2}+\ldots+u_{n}=2008$. Show that it is possible to select $k$ elements from the set $\left(u_{n}\right)$ such that in this collection of $k$ numbers, the smallest one is at least half of the largest.
    12. Consider the circle $(O)$ and three colinear points $X$, $Y$, $H$ that are not on this circle such that $\overline{H X} \cdot \overline{H Y} \neq \mathscr{P}_{H/(O)} .$ A straight line $d$ through $H$ meets $(O)$ at two points $M$ and $N$. $M X$ and $N Y$ intersect with $(O)$ again at $P$ and $Q$ respectively. Show that as the line $d$ through $H$ varies, the line connecting $P$ and $Q$ always passes through a fixed point.

    Issue 377

    1. Let $$A=\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\ldots+\frac{1}{2007}+\frac{1}{2008},$$ $$B=\frac{2007}{1}+\frac{2006}{2}+\frac{2005}{3}+\ldots+\frac{2}{2006}+\frac{1}{2007}.$$ Determine $\dfrac{B}{A}$.
    2. Let $A B C$ be a right isosceles triangle with right angle at $A$. $M$ is an arbitrary point on the side $B C$ (M differs from $B$, $C$ as well as the midpoint of $B C$). The altitudes from $B$ and $C$ onto $A M$ meet $A M$ at $H$ and $K$, respectively. The line through $C$ and parallel to $A M$ meets $B H$ at $N$, $A N$ meets $C K$ at $P$, $B P$ intersects with $A M$ at $I$ Prove that $I B=I P$.
    3. Let $a, b, c, d$ and $e$ be natural numbers such that $$a^{4}+b^{4}+c^{4}+d^{4}+e^{4}=2009^{2008}.$$ Prove that abcde is a multiple of $10^{4}$.
    4. Let $a, b, c$ be positive real numbers such that $a \geq b \geq c .$ Prove the inequality $$a^{2} b(a-b)+b^{2} c(b-c)+c^{2} a(c-a) \geq 0.$$
    5. Let $A M$ and $B N$ be two tangent lines from two points $A$, $B$ ($A$ differs from $B$) outside the circle $(O)$ ($M$, $N$ are on the circle, $M$ and $N$ are different). Prove that if $A M=B N$, then the line $M N$ is either parallel to $A B$ or passes through its midpoint.
    6. A number is said to be a beautiful number if it is a composite number and but it is not a multiple of either $2$, $3$ or $5$ (for example, the three smallest beautiful numbers are $49,77$ and $91$). How many beautiful numbers which are less than $1000 ?$
    7. Let $D$ and $E$ be two points on the side $B C$ of a triangle $A B C$ such that $\dfrac{B D}{C D}=2\dfrac{C E}{B E}$. The circumcircle of $A D E$ meets $A B$ and $A C$ at $M$ and $N,$ respectively. Prove that regardless of the positions of the points $D$ and $E$ on $B C,$ the centroid of the triangle $A M N$ lies on a fixed line.
    8. Find the smallest real number $k$ such that the following inequality holds for all nonnegative real numbers $a, b, c$ $$\frac{a+b+c}{3} \leq \sqrt[3]{a b c}+k \cdot \max \{|a-b|, b-c|,| c-a \mid\}.$$
    9. Determine all triple of real numbers $x, y, z$ such that $$x^{6}+y^{6}+z^{6}-6\left(x^{4}+y^{4}+z^{4}\right)+10\left(x^{2}+y^{2}+z^{2}\right) - 2\left(x^{3} y+y^{3} z+z^{3} x\right)+6(x y+y z+z x)=0.$$
    10. Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that $$f\left(x^{3}-y\right)+2 y\left(3 f^{2}(x)+y^{2}\right)=f(y+f(x)),\, \forall x, y \in \mathbb{R}.$$
    11. Consider the sequence $\left(u_{n}\right)$ $(n=1,2,\ldots)$ given by the following recursive formula $$u_{1}=u_{2}=1,\quad u_{n+1}=4 u_{n}-5 u_{n-1},\,\forall n \geq 2.$$ Prove that $\displaystyle \lim_{n \rightarrow+\infty}\left(\frac{u_{n}}{a^{n}}\right)=0$ for all real number $a>\sqrt{5}$.
    12. Choose three points $A_{1}$, $B_{1}$, $C_{1}$ on the sides of a triangle $A B C$, $A_{1} \in B C$, $B_{1} \in A C$, $C_{1} \in A B$ such that $A A_{1}$, $B B_{1}$, $C C_{1}$ meet at a common point. Again, choose three points $A_{2}$, $B_{2}$, $C_{2}$ on the sides of the triangle $A_{1} B_{1} C_{1}$, $A_{2} \in B_{1} C_{1}$, $B_{2} \in A_{1} C_{1}$, $C_{2} \in A_{1} B_{1}$. Prove that the three lines $A A_{2}$, $B B_{2}$, $C C_{2}$ meet at a common point if and only if so do $A_{1} A_{2}$, $B_{1} B_{2}$, $C_{1} C_{2}$.

    Issue 378

    1. Find all natural numbers $a$ such that both $a+593$ and $a-159$ are perfect squares.
    2. Let $A B C$ be a right triangle, with right angle at $A$ and $\widehat{A C B}=15^{\circ}$. Let $B C=a$, $A C=b$, $A B=c .$ Prove that $a^{2}=4 b c$.
    3. Find all intergers $x, y, z, t$ such that $$x^{2008}+y^{2008}+z^{2008}=2007 . t^{2008}.$$
    4. Prove the inequality $$\left(\frac{4}{a^{2}+b^{2}}+1\right)\left(\frac{4}{b^{2}+c^{2}}+1\right)\left(\frac{4}{c^{2}+a^{2}}+1\right) \geq 3(a+b+c)^{2}$$ where $a, b, c$ are positive numbers such that $a^{2}+b^{2}+c^{2}=3$.
    5. Let $A B C$ be an acute triangle. Choose a point $D,$ different from $B$ and $C,$ on the side $B C .$ Prove that the vertex $A$ and the centers of the circumcircles of the triangles $A B D$, $A C D$ and $A B C$ lie on the same circle.
    6. Find the coefficient of $x^{2}$ in the expansion of $$\left(\left(\ldots\left((x-2)^{2}-2\right)^{2}-\ldots-2\right)^{2}-2\right)^{2}$$ given that the number 2 occurs 1004 times in the expression above and there are 2008 round brackets.
    7. Find the smallest value of the following expression $$\frac{\sqrt{a_{1}+2008}+\sqrt{a_{2}+2008}+\ldots+\sqrt{a_{n}+2008}}{\sqrt{a_{1}}+\sqrt{a_{2}}+\ldots+\sqrt{a_{n}}}$$ where $n$ is a given positive natural number and $a_{1}, a_{2}, \ldots, a_{n}$ are non-negative real numbers such that $a_{1}+a_{2}+\ldots+a_{n}=n$
    8. Let $(O)$ be a circle centered at $O$ and fixed diameter $A B .$ Let $\Delta$ be a straight line which touches $(O)$ at $A .$ Choose a point $M$ on the circle $(O),$ different from $A$ and $B .$ The tangent line with $(O)$ through $M$ meets $\Delta$ at $C$. Let $(I)$ be the circle through $M$ and touches $\Delta$ at $C$. Let $C D$ be the diameter of $(I)$. Prove that
      a) $D O C$ is an isosceles triangle.
      b) The line through $D$ and perpendicular to $B C$ always passes through a fixed point when $M$ moves on the circle $(O)$.
    9. Does there exist a sequence of positive integers $a_{2003}>a_{2002}>\ldots>a_{2}>a_{1}$ with $a_{1}=2003$ such that the following two conditions are satisfied
      • All integers in the interval $\left(2003 ; a_{2 \times 13}\right)$ are either a member of this sequence or a non-prime.
      • $A=\dfrac{2004}{a_{1}}+\dfrac{2004}{a_{2}}+\ldots+\dfrac{2004}{a_{2003}}$ is an integer?.
      1. Find all continuous functions $f, g, h$ on $\mathbb{R}$ such that $$f(x+y)=g(x)+h(y)$$ for all real numbers $x$, $y$.
      2. Let $H$ and $O$ denote the orthocenter and circumcenter respectively of a triangle $A B C$. Prove the inequality $$3 R-2 O H \leq H A+H B+H C \leq 3 R+O H$$ where $R$ is its circumradius.
      3. Let $A B C D . A_{1} B_{1} C_{1} D_{1}$ be a cubic whose side is a. Let $N$, $M$ be two points on $A B_{1}$ and $B C_{1}$ respectively such that the angle between $M N$ and the plane $(A B C D)$ is $60^{\circ}$. Prove that $$M N \geq 2 a(\sqrt{3}-\sqrt{2}).$$ When does equality occur?.



      Abel,5,Albania,2,AMM,2,Amsterdam,4,An Giang,45,Andrew Wiles,1,Anh,2,APMO,21,Austria (Áo),1,Ba Lan,1,Bà Rịa Vũng Tàu,77,Bắc Bộ,2,Bắc Giang,62,Bắc Kạn,4,Bạc Liêu,18,Bắc Ninh,53,Bắc Trung Bộ,3,Bài Toán Hay,5,Balkan,41,Baltic Way,32,BAMO,1,Bất Đẳng Thức,69,Bến Tre,72,Benelux,16,Bình Định,65,Bình Dương,38,Bình Phước,52,Bình Thuận,42,Birch,1,BMO,41,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,15,Cà Mau,22,Cần Thơ,27,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,515,Chu Tuấn Anh,1,Chuyên Đề,125,Chuyên SPHCM,7,Chuyên SPHN,30,Chuyên Trần Hưng Đạo,3,Collection,8,College Mathematic,1,Concours,1,Cono Sur,1,Contest,675,Correspondence,1,Cosmin Poahata,1,Crux,2,Czech-Polish-Slovak,28,Đà Nẵng,50,Đa Thức,2,Đại Số,20,Đắk Lắk,76,Đắk Nông,15,Danube,7,Đào Thái Hiệp,1,ĐBSCL,2,Đề Thi,1,Đề Thi HSG,2249,Đề Thi JMO,1,DHBB,30,Điện Biên,15,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,Đồng Nai,64,Đồng Tháp,63,Du Hiền Vinh,1,Đức,1,Dương Quỳnh Châu,1,Dương Tú,1,Duyên Hải Bắc Bộ,30,E-Book,31,EGMO,30,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,40,Gia Viễn,2,Giải Tích Hàm,1,Giới hạn,2,Goldbach,1,Hà Giang,5,Hà Lan,1,Hà Nam,45,Hà Nội,255,Hà Tĩnh,91,Hà Trung Kiên,1,Hải Dương,70,Hải Phòng,57,Hậu Giang,14,Hélènne Esnault,1,Hilbert,2,Hình Học,33,HKUST,7,Hòa Bình,33,Hoài Nhơn,1,Hoàng Bá Minh,1,Hoàng Minh Quân,1,Hodge,1,Hojoo Lee,2,HOMC,5,HongKong,8,HSG 10,126,HSG 10 2010-2011,4,HSG 10 2011-2012,7,HSG 10 2012-2013,8,HSG 10 2013-2014,7,HSG 10 2014-2015,6,HSG 10 2015-2016,2,HSG 10 2016-2017,8,HSG 10 2017-2018,4,HSG 10 2018-2019,4,HSG 10 2019-2020,7,HSG 10 2020-2021,3,HSG 10 2021-2022,4,HSG 10 2022-2023,11,HSG 10 2023-2024,1,HSG 10 Bà Rịa Vũng Tàu,2,HSG 10 Bắc Giang,1,HSG 10 Bạc Liêu,2,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,10,HSG 10 KHTN,9,HSG 10 Nghệ An,1,HSG 10 Ninh Thuận,1,HSG 10 Phú Yên,2,HSG 10 PTNK,10,HSG 10 Quảng Nam,1,HSG 10 Quảng Trị,2,HSG 10 Thái Nguyên,9,HSG 10 Vĩnh Phúc,14,HSG 1015-2016,3,HSG 11,135,HSG 11 2009-2010,1,HSG 11 2010-2011,6,HSG 11 2011-2012,10,HSG 11 2012-2013,9,HSG 11 2013-2014,7,HSG 11 2014-2015,10,HSG 11 2015-2016,6,HSG 11 2016-2017,8,HSG 11 2017-2018,7,HSG 11 2018-2019,8,HSG 11 2019-2020,5,HSG 11 2020-2021,8,HSG 11 2021-2022,4,HSG 11 2022-2023,7,HSG 11 2023-2024,1,HSG 11 An Giang,2,HSG 11 Bà Rịa Vũng Tàu,1,HSG 11 Bắc Giang,4,HSG 11 Bạc Liêu,3,HSG 11 Bắc Ninh,2,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,12,HSG 11 Quảng Nam,1,HSG 11 Quảng Ngãi,9,HSG 11 Quảng Trị,3,HSG 11 Sóc Trăng,1,HSG 11 Thái Nguyên,8,HSG 11 Thanh Hóa,3,HSG 11 Trà Vinh,1,HSG 11 Tuyên Quang,1,HSG 11 Vĩnh Long,3,HSG 11 Vĩnh Phúc,11,HSG 12,668,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 2015-2016,37,HSG 12 2016-2017,46,HSG 12 2017-2018,55,HSG 12 2018-2019,43,HSG 12 2019-2020,43,HSG 12 2020-2021,52,HSG 12 2021-2022,35,HSG 12 2022-2023,42,HSG 12 2023-2024,23,HSG 12 2023-2041,1,HSG 12 An Giang,8,HSG 12 Bà Rịa Vũng Tàu,13,HSG 12 Bắc Giang,18,HSG 12 Bạc Liêu,3,HSG 12 Bắc Ninh,13,HSG 12 Bến Tre,19,HSG 12 Bình Định,17,HSG 12 Bình Dương,8,HSG 12 Bình Phước,9,HSG 12 Bình Thuận,8,HSG 12 Cà Mau,7,HSG 12 Cần Thơ,7,HSG 12 Cao Bằng,5,HSG 12 Chuyên SPHN,11,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,14,HSG 12 Hà Nam,5,HSG 12 Hà Nội,17,HSG 12 Hà Tĩnh,16,HSG 12 Hải Dương,16,HSG 12 Hải Phòng,20,HSG 12 Hậu Giang,4,HSG 12 Hòa Bình,10,HSG 12 Hưng Yên,10,HSG 12 Khánh Hòa,4,HSG 12 KHTN,26,HSG 12 Kiên Giang,12,HSG 12 Kon Tum,3,HSG 12 Lai Châu,4,HSG 12 Lâm Đồng,11,HSG 12 Lạng Sơn,8,HSG 12 Lào Cai,17,HSG 12 Long An,18,HSG 12 Nam Định,7,HSG 12 Nghệ An,13,HSG 12 Ninh Bình,12,HSG 12 Ninh Thuận,7,HSG 12 Phú Thọ,18,HSG 12 Phú Yên,13,HSG 12 Quảng Bình,14,HSG 12 Quảng Nam,11,HSG 12 Quảng Ngãi,6,HSG 12 Quảng Ninh,20,HSG 12 Quảng Trị,10,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,17,HSG 12 Thừa Thiên Huế,19,HSG 12 Tiền Giang,3,HSG 12 TPHCM,13,HSG 12 Tuyên Quang,3,HSG 12 Vĩnh Long,7,HSG 12 Vĩnh Phúc,20,HSG 12 Yên Bái,6,HSG 9,573,HSG 9 2009-2010,1,HSG 9 2010-2011,21,HSG 9 2011-2012,42,HSG 9 2012-2013,41,HSG 9 2013-2014,35,HSG 9 2014-2015,41,HSG 9 2015-2016,38,HSG 9 2016-2017,42,HSG 9 2017-2018,45,HSG 9 2018-2019,41,HSG 9 2019-2020,18,HSG 9 2020-2021,50,HSG 9 2021-2022,53,HSG 9 2022-2023,55,HSG 9 2023-2024,15,HSG 9 An Giang,9,HSG 9 Bà Rịa Vũng Tàu,8,HSG 9 Bắc Giang,14,HSG 9 Bắc Kạn,1,HSG 9 Bạc Liêu,1,HSG 9 Bắc Ninh,12,HSG 9 Bến Tre,9,HSG 9 Bình Định,11,HSG 9 Bình Dương,7,HSG 9 Bình Phước,13,HSG 9 Bình Thuận,5,HSG 9 Cà Mau,2,HSG 9 Cần Thơ,4,HSG 9 Cao Bằng,2,HSG 9 Đà Nẵng,11,HSG 9 Đắk Lắk,12,HSG 9 Đắk Nông,3,HSG 9 Điện Biên,5,HSG 9 Đồng Nai,8,HSG 9 Đồng Tháp,10,HSG 9 Gia Lai,9,HSG 9 Hà Giang,4,HSG 9 Hà Nam,10,HSG 9 Hà Nội,15,HSG 9 Hà Tĩnh,13,HSG 9 Hải Dương,16,HSG 9 Hải Phòng,8,HSG 9 Hậu Giang,6,HSG 9 Hòa Bình,4,HSG 9 Hưng Yên,11,HSG 9 Khánh Hòa,6,HSG 9 Kiên Giang,16,HSG 9 Kon Tum,9,HSG 9 Lai Châu,2,HSG 9 Lâm Đồng,14,HSG 9 Lạng Sơn,10,HSG 9 Lào Cai,4,HSG 9 Long An,10,HSG 9 Nam Định,9,HSG 9 Nghệ An,21,HSG 9 Ninh Bình,14,HSG 9 Ninh Thuận,4,HSG 9 Phú Thọ,13,HSG 9 Phú Yên,9,HSG 9 Quảng Bình,14,HSG 9 Quảng Nam,12,HSG 9 Quảng Ngãi,13,HSG 9 Quảng Ninh,17,HSG 9 Quảng Trị,10,HSG 9 Sóc Trăng,9,HSG 9 Sơn La,5,HSG 9 Tây Ninh,16,HSG 9 Thái Bình,11,HSG 9 Thái Nguyên,5,HSG 9 Thanh Hóa,12,HSG 9 Thừa Thiên Huế,9,HSG 9 Tiền Giang,7,HSG 9 TPHCM,11,HSG 9 Trà Vinh,2,HSG 9 Tuyên Quang,6,HSG 9 Vĩnh Long,12,HSG 9 Vĩnh Phúc,12,HSG 9 Yên Bái,5,HSG Cấp Trường,80,HSG Quốc Gia,113,HSG Quốc Tế,16,Hứa Lâm Phong,1,Hứa Thuần Phỏng,1,Hùng Vương,2,Hưng Yên,43,Huỳnh Kim Linh,1,Hy Lạp,1,IMC,26,IMO,58,IMT,2,IMU,2,India - Ấn Độ,47,Inequality,13,InMC,1,International,349,Iran,13,Jakob,1,JBMO,41,Jewish,1,Journal,30,Junior,38,K2pi,1,Kazakhstan,1,Khánh Hòa,30,KHTN,64,Kiên Giang,74,Kon Tum,24,Korea - Hàn Quốc,5,Kvant,2,Kỷ Yếu,46,Lai Châu,12,Lâm Đồng,47,Lăng Hồng Nguyệt Anh,1,Lạng Sơn,37,Langlands,1,Lào Cai,35,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 Viết Hải,1,Lê Việt Hưng,2,Leibniz,1,Long An,52,Lớp 10 Chuyên,709,Lớp 10 Không Chuyên,355,Lớp 11,1,Lục Ngạn,1,Lượng giác,1,Lưu Giang Nam,2,Lưu Lý Tưởng,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,73,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,61,Ninh Thuận,26,Nội Suy Lagrange,2,Nội Suy Newton,1,Nordic,21,Olympiad Corner,1,Olympiad Preliminary,2,Olympic 10,134,Olympic 10/3,6,Olympic 10/3 Đắk Lắk,6,Olympic 11,122,Olympic 12,52,Olympic 23/3,2,Olympic 24/3,10,Olympic 24/3 Quảng Nam,10,Olympic 27/4,24,Olympic 30/4,61,Olympic KHTN,8,Olympic Sinh Viên,78,Olympic Tháng 4,12,Olympic Toán,344,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,32,Phú Yên,42,Phùng Hồ Hải,1,Phương Trình Hàm,11,Phương Trình Pythagoras,1,Pi,1,Polish,32,Problems,1,PT-HPT,14,PTNK,64,Putnam,27,Quảng Bình,64,Quảng Nam,57,Quảng Ngãi,49,Quảng Ninh,60,Quảng Trị,42,Quỹ Tích,1,Riemann,1,RMM,14,RMO,24,Romania,38,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,36,Sơn La,22,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,37,Thái Bình,45,Thái Nguyên,61,Thái Vân,2,Thanh Hóa,69,THCS,2,Thổ Nhĩ Kỳ,5,Thomas J. Mildorf,1,Thông Tin Toán Học,43,THPT Chuyên Lê Quý Đôn,1,THPT Chuyên Nguyễn Du,9,THPTQG,16,THTT,31,Thừa Thiên Huế,56,Tiền Giang,30,Tin Tức Toán Học,1,Titu Andreescu,2,Toán 12,7,Toán Cao Cấp,3,Toán Rời Rạc,5,Toán Tuổi Thơ,3,Tôn Ngọc Minh Quân,2,TOT,1,TPHCM,158,Trà Vinh,10,Trắc Nghiệm,1,Trắc Nghiệm Toán,2,Trại Hè,39,Trại Hè Hùng Vương,30,Trại Hè Phương Nam,7,Trần Đăng Phúc,1,Trần Minh Hiền,2,Trần Nam Dũng,12,Trần Phương,1,Trần Quang Hùng,1,Trần Quốc Anh,2,Trần Quốc Luật,1,Trần Quốc Nghĩa,1,Trần Tiến Tự,1,Trịnh Đào Chiến,2,Trường Đông,23,Trường Hè,10,Trường Thu,1,Trường Xuân,3,TST,544,TST 2008-2009,1,TST 2010-2011,22,TST 2011-2012,23,TST 2012-2013,32,TST 2013-2014,29,TST 2014-2015,27,TST 2015-2016,26,TST 2016-2017,41,TST 2017-2018,42,TST 2018-2019,30,TST 2019-2020,34,TST 2020-2021,30,TST 2021-2022,38,TST 2022-2023,42,TST 2023-2024,23,TST An Giang,8,TST Bà Rịa Vũng Tàu,11,TST Bắc Giang,5,TST Bắc Ninh,11,TST Bến Tre,10,TST Bình Định,5,TST Bình Dương,7,TST Bình Phước,9,TST Bình Thuận,9,TST Cà Mau,7,TST Cần Thơ,6,TST Cao Bằng,2,TST Đà Nẵng,8,TST Đắk Lắk,12,TST Đắk Nông,2,TST Điện Biên,2,TST Đồng Nai,13,TST Đồng Tháp,12,TST Gia Lai,4,TST Hà Nam,8,TST Hà Nội,12,TST Hà Tĩnh,15,TST Hải Dương,11,TST Hải Phòng,13,TST Hậu Giang,1,TST Hòa Bình,4,TST Hưng Yên,10,TST Khánh Hòa,8,TST Kiên Giang,11,TST Kon Tum,6,TST Lâm Đồng,12,TST Lạng Sơn,3,TST Lào Cai,4,TST Long An,6,TST Nam Định,8,TST Nghệ An,7,TST Ninh Bình,11,TST Ninh Thuận,4,TST Phú Thọ,13,TST Phú Yên,5,TST PTNK,15,TST Quảng Bình,12,TST Quảng Nam,7,TST Quảng Ngãi,8,TST Quảng Ninh,9,TST Quảng Trị,10,TST Sóc Trăng,5,TST Sơn La,7,TST Thái Bình,6,TST Thái Nguyên,8,TST Thanh Hóa,9,TST Thừa Thiên Huế,4,TST Tiền Giang,6,TST TPHCM,14,TST Trà Vinh,1,TST Tuyên Quang,1,TST Vĩnh Long,7,TST Vĩnh Phúc,7,TST Yên Bái,8,Tuyên Quang,14,Tuyển Sinh,4,Tuyển Sinh 10,1064,Tuyển Sinh 10 An Giang,18,Tuyển Sinh 10 Bà Rịa Vũng Tàu,22,Tuyển Sinh 10 Bắc Giang,19,Tuyển Sinh 10 Bắc Kạn,3,Tuyển Sinh 10 Bạc Liêu,9,Tuyển Sinh 10 Bắc Ninh,15,Tuyển Sinh 10 Bến Tre,34,Tuyển Sinh 10 Bình Định,19,Tuyển Sinh 10 Bình Dương,12,Tuyển Sinh 10 Bình Phước,21,Tuyển Sinh 10 Bình Thuận,15,Tuyển Sinh 10 Cà Mau,5,Tuyển Sinh 10 Cần Thơ,10,Tuyển Sinh 10 Cao Bằng,2,Tuyển Sinh 10 Chuyên SPHN,19,Tuyển Sinh 10 Đà Nẵng,18,Tuyển Sinh 10 Đại Học Vinh,13,Tuyển Sinh 10 Đắk Lắk,21,Tuyển Sinh 10 Đắk Nông,7,Tuyển Sinh 10 Điện Biên,5,Tuyển Sinh 10 Đồng Nai,18,Tuyển Sinh 10 Đồng Tháp,23,Tuyển Sinh 10 Gia Lai,10,Tuyển Sinh 10 Hà Giang,1,Tuyển Sinh 10 Hà Nam,16,Tuyển Sinh 10 Hà Nội,80,Tuyển Sinh 10 Hà Tĩnh,19,Tuyển Sinh 10 Hải Dương,17,Tuyển Sinh 10 Hải Phòng,15,Tuyển Sinh 10 Hậu Giang,3,Tuyển Sinh 10 Hòa Bình,15,Tuyển Sinh 10 Hưng Yên,12,Tuyển Sinh 10 Khánh Hòa,12,Tuyển Sinh 10 KHTN,21,Tuyển Sinh 10 Kiên Giang,31,Tuyển Sinh 10 Kon Tum,6,Tuyển Sinh 10 Lai Châu,6,Tuyển Sinh 10 Lâm Đồng,10,Tuyển Sinh 10 Lạng Sơn,6,Tuyển Sinh 10 Lào Cai,10,Tuyển Sinh 10 Long An,18,Tuyển Sinh 10 Nam Định,21,Tuyển Sinh 10 Nghệ An,23,Tuyển Sinh 10 Ninh Bình,20,Tuyển Sinh 10 Ninh Thuận,10,Tuyển Sinh 10 Phú Thọ,18,Tuyển Sinh 10 Phú Yên,12,Tuyển Sinh 10 PTNK,37,Tuyển Sinh 10 Quảng Bình,12,Tuyển Sinh 10 Quảng Nam,15,Tuyển Sinh 10 Quảng Ngãi,13,Tuyển Sinh 10 Quảng Ninh,12,Tuyển Sinh 10 Quảng Trị,7,Tuyển Sinh 10 Sóc Trăng,17,Tuyển Sinh 10 Sơn La,5,Tuyển Sinh 10 Tây Ninh,15,Tuyển Sinh 10 Thái Bình,17,Tuyển Sinh 10 Thái Nguyên,18,Tuyển Sinh 10 Thanh Hóa,27,Tuyển Sinh 10 Thừa Thiên Huế,24,Tuyển Sinh 10 Tiền Giang,14,Tuyển Sinh 10 TPHCM,23,Tuyển Sinh 10 Trà Vinh,6,Tuyển Sinh 10 Tuyên Quang,3,Tuyển Sinh 10 Vĩnh Long,12,Tuyển Sinh 10 Vĩnh Phúc,22,Tuyển Sinh 2008-2009,1,Tuyển Sinh 2009-2010,1,Tuyển Sinh 2010-2011,6,Tuyển Sinh 2011-2012,20,Tuyển Sinh 2012-2013,65,Tuyển Sinh 2013-2014,77,Tuyển Sinh 2013-2044,1,Tuyển Sinh 2014-2015,81,Tuyển Sinh 2015-2016,64,Tuyển Sinh 2016-2017,72,Tuyển Sinh 2017-2018,126,Tuyển Sinh 2018-2019,61,Tuyển Sinh 2019-2020,90,Tuyển Sinh 2020-2021,59,Tuyển Sinh 2021-202,1,Tuyển Sinh 2021-2022,69,Tuyển Sinh 2022-2023,113,Tuyển Sinh 2023-2024,49,Tuyển Sinh Chuyên SPHCM,7,Tuyển Sinh Yên Bái,6,Tuyển Tập,45,Tuymaada,6,UK - Anh,16,Undergraduate,69,USA - Mỹ,62,USA TSTST,6,USAJMO,12,USATST,8,USEMO,4,Uzbekistan,1,Vasile Cîrtoaje,4,Vật Lý,1,Viện Toán Học,6,Vietnam,4,Viktor Prasolov,1,VIMF,1,Vinh,32,Vĩnh Long,41,Vĩnh Phúc,86,Virginia Tech,1,VLTT,1,VMEO,4,VMF,12,VMO,58,VNTST,25,Võ Anh Khoa,1,Võ Quốc Bá Cẩn,26,Võ Thành Văn,1,Vojtěch Jarník,6,Vũ Hữu Bình,7,Vương Trung Dũng,1,WFNMC Journal,1,Wiles,1,Xác Suất,1,Yên Bái,25,Yên Thành,1,Zhautykov,14,Zhou Yuan Zhe,1,
      MOlympiad.NET: Mathematics and Youth Magazine Problems 2008
      Mathematics and Youth Magazine Problems 2008
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