Mathematics and Youth Magazine Problems 2012

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Issue 415

  1. Let $$A=\dfrac{2011^{2011}}{2012^{2012}},\quad B=\dfrac{2011^{2011}+2011}{2012^{2012}+2012}.$$ Which number is greater, $A$ or $B$?.
  2. Given \[A=\sqrt{6+\sqrt{6+\ldots+\sqrt{6}}},\:B=\sqrt[3]{6+\sqrt[3]{6+\ldots+\sqrt[3]{6}}},\] where there are exactly $n$ square roots in $A$ and $n$ cube roots in $B$. Write $[x]$ for the greatest integer not exceeding $x$. Determine the value of $\left[\dfrac{A-B}{A+B}\right]$.
  3. Find all pairs of natural numbers $x,y$ such that \[x^{2}-5x+7=3^{y}.\]
  4. Prove the inequality \[\left(1+\frac{1}{2}\right)\left(1+\frac{1}{2^{2}}\right)\ldots\left(1+\frac{1}{2^{n}}\right)<3.\]
  5. Let $ABCD$ be aparallelogram. Points $H$ and $K$ are chosen on lines $AB$ and $BC$ such that triangles $KAB$ and $HCB$ are isosceles ($KA=AB$, $HC=CB$). Prove that
    a) Triangle $KDH$ is also isosceles.
    b) Triangle $KAB$, $BCH$ and $KDH$ are similar.
  6. In a triangle $ABC$ with $a=BC$, $b=CA$, $c=AB$, $A_{1}$ is the midpoint of $BC$; $O$ and $I$ are its circumcenter and incenter respectively. Prove that if $AA_{1}$ isperpendicular to $OI$ then \[\min\{b,c\}\leq a\leq\max\{b,c\}.\]
  7. The real numbers $x,y$ and $z$ are such that \[\begin{cases}\sqrt{x}\sin\alpha+\sqrt{y}\cos\alpha-\sqrt{z} & =-\sqrt{2(x+y+z)}\\ 2x+2y-13\sqrt{z} & =7 \end{cases},\quad\pi\leq\alpha\leq\frac{3\pi}{2}.\]Determine the value of $(x+y)z$.
  8. Solve the following system of equations in two variables \[\begin{cases}\log_{2}x & =2^{y+2}\\ 2\sqrt{1+x}+xy\sqrt{4+y^{2}} & =0 \end{cases}.\]
  9. A collection of prime numbers (each prime can be repeated) is said to be beautiful if their product is exactly ten times their sum. Find all beautiful collections. 
  10. Points $A,B,C,D,E$ in clockwise order, lie on the same circle. $M,N,P,Q$ are the feet of perpendicular lines from $E$ onto $AB$, $BC$, $CD$, $DA$. Prove that $MN$, $NP$, $PQ$, $QM$ are tangent lines to a certain parabole whose focus point if $E$. 
  11. The sequence $(a_{n})$ is defined recursively by the following rules \[a_{1}=1,\quad a_{n+1}=\frac{1}{a_{1}+\ldots+a_{n}}-\sqrt{2},\:n=1,2,\ldots.\] Find the limit of the sequence $(b_{n})$ where \[b_{n}=a_{1}+\ldots+a_{n}.\]
  12. Let $\alpha$ and $\beta$ be two real roots of the equation \[4x^{2}-4tx-1=0\] where $t$ is a parameter. Let $f(x)=\dfrac{2x-t}{x^{2}+1}$ be a funtion defined on the interval $[\alpha;\beta]$, and let \[g(t)=\max_{x\in[\alpha;\beta]}f(x)-\min_{x\in[\alpha;\beta]}f(x).\] Prove that if a triple $a,b,c\in\left(0;\frac{\pi}{2}\right)$ are such that $\sin a+\sin b+\sin c=1$, then \[\frac{1}{g(\tan a)}+\frac{1}{g(\tan b)}+\frac{1}{g(\tan c)}<\frac{3\sqrt{6}}{4}.\]

Issue 416

  1. Find all natural numbers $x,y,z$ such that \[2010^{x}+2011^{y}=2012^{z}.\]
  2. The natural numbers $a_{1},a_{2},\ldots,a_{100}$ satisfy the equation \[\frac{1}{a_{1}}+\frac{1}{a_{2}}+\ldots+\frac{1}{a_{100}}=\frac{101}{2}.\]Prove that there are at least two equal numbers.
  3. Let $a,b,c$ be positive real numbers. Prove the inequality \[\frac{(a+b)^{2}}{ab}+\frac{(b+c)^{2}}{bc}+\frac{(c+a)^{2}}{ca}\geq9+2\left(\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}\right).\]
  4. Solve the equation \[4x^{2}+14x+11=4\sqrt{6x+10}.\]
  5. In a triange $ABC$, te incircle $(I)$ meets $BC$, $CA$ at $D$, $E$ respectively. Let $K$ be the point of reglection of $D$ through the midpoint of $BC$, the line through $K$ and perpendicular to $BC$ meets $DE$ at $L$, $N$ is the midpoint of $KL$. Prove that $BN$ and $AK$ are orthogonal.
  6. Determine the maximum value of the expression \[A=\frac{mn}{(m+1)(n+1)(m+n+1)}\] where $m,n$ are natural numbers.
  7. Triangle $ABC$ ($AB>AC$) is inscribed in circle $(O)$. The exterior angle bisector of $BAC$ meets $(O)$ at another point $E$; $M,N$ are the midpoints of $BC$, $CA$ respectively; $F$ os the perpendicular foot of $E$ on $AB$, $K$ is the intersection of $MN$ and $AE$. Prove that $KF$ and $BC$ are parallel.
  8. Solve the equation \[\sin^{2n+1}x+\sin^{n}2x+(\sin^{n}x-\cos^{n}x)^{2}-2=0\] where $n$ is a given positive integer.
  9. Find all polynomials $P(x)$ such that \[P(2)=12,\quad P(x^{2})=x^{2}(x^{2}+1)P(x),\:\forall x\in\mathbb{R}.\]
  10. Let $r_{1},r_{2},\ldots,r_{n}$ be $n$ rational numbers such that $0<r_{i}\leq\dfrac{1}{2}$, ${\displaystyle \sum_{i=1}^{n}r_{i}=1}$ ($n>1$), and let $f(x)=[x]+\left[x+\dfrac{1}{2}\right]$. Find the greatest value of the expression ${\displaystyle P(k)=2k-\sum_{i=1}^{n}f(kr_{i})}$ where $k$ runs over the integers $\mathbb{Z}$ (the notation $[x]$ means the greatest integer not exceeding $x$).
  11. Suppose that $f:\mathbb{R}\to\mathbb{R}$ is a continuous funtion such that $f(x)+f(x+1006)$ is a rational number if and only if $x\in\mathbb{R}$, \[f(x+20)+f(x+12)+f(x+2012)\] is itrational. Prove that $f(x)=f(x+2012)$ for all $x\in\mathbb{R}$.
  12. Prove the following inequality \[\frac{m_{a}}{h_{a}}+\frac{m_{b}}{h_{b}}=\frac{m_{c}}{h_{c}}\leq1+\frac{R}{r},\] where $m_{a},b_{b},m_{c}$ are medians; $h_{a},h_{b},h_{c}$ are the altitudes from $A$, $B$, $C$ and $R$, $r$ are the circumradius and inradius, respectively.

Issue 417

  1. Which number is bigger, $2^{3100}$ or $3^{2100}$?.
  2. Let $ABC$ be an isosceles triangle with $AB=AC$. $BM$ is the median from $B$. $N$ is a point on $BC$ such that $\widehat{CAN}=\widehat{ABM}$. Prove that $CM\geq CN$.
  3. Let $a,b,c$ be positive numbers such that \[|a+b+c|\leq1,\,|a-b+c|\leq1,\,|4a+2b+c|\leq8,\,|4a-2b+c|\leq8.\] Prove the inequality \[|a|+3|b|+|c|\leq7.\]
  4. Solve the equation \[(x-2)(x^{2}+6x-11)^{2}=(5x^{2}-10x+1)^{2}.\]
  5. Let $ABC$ be a right triangle, with right angle at$A$, $AH$ is the altitude from $A$ and $I,J$ ae the incenters of triangles $HAB$ and $HAC$, respectively. $IJ$ cuts $AB$ at $M$ and meets $AC$ at $N$. Let $X$ and $Y$ be the intersections of $HI$ with $AB$ and $HJ$ with $AC$; $BY$, $CX$ cuts $MN$ at $P$ and $Q$ respectively. Prove that \[\frac{AI}{AJ}=\frac{HP}{HQ}.\]
  6. Let $x,y,z$ be real numbers such that $x^{2}+y^{2}+z^{2}=3$. Find the minimum and maximum value of the expression \[P=(x+2)(y+2)(z+2).\]
  7. In a triangle $ABC$, let $m_{a},m_{b},m_{c}$ be its median lengths, and $l_{a},l_{b},l_{c}$ be the lengths of its inner bisectors, $p$ is half of its perimeter. Prove the inequality \[m_{a}+m_{b}+m_{c}+l_{a}+l_{b}+l_{c}\leq2\sqrt{3}p.\]
  8. Let $S.ABC$ be a pyramid where surface $SAB$ is a isosceles triangle at $S$ and $\widehat{BSA}=120^{0}$, the plane $(SAB)$ is perpendicular to $(ABC)$. Prove that $\dfrac{S_{ABC}}{S_{SAC}}\leq\sqrt{3}$, when does the equality occur?. (Denote by $S_{DEF}$ the area of triangle $DEF$)
  9. A natural number $n$ is a good number if it is possible to partition any square into $n$ smaller squares such that at least two of them are not equal.
    a) Prove that if $n$ is a good number, then $n\geq4$.
    b) Prove that both $4$ and $5$ are not good.
    c) Find all good numbers.
  10. A sequence $a_{0},a_{1},\ldots,a_{n}$ ($n\geq2$) is defined by \[a_{0}=0,\quad a_{k}=\frac{1}{n+1}+\frac{1}{n+2}+\ldots+\frac{1}{n+k},\,k=1,2,\ldots,n.\] Prove the inequality \[\sum_{k=0}^{n-1}\frac{e^{a_{k}}}{n+k+1}+(\ln2-a_{n})e^{a_{n}}<1\] where ${\displaystyle e=\lim_{n\to\infty}\left(1+\frac{1}{n}\right)^{n}}$.
  11. Find all functions $f:\mathbb{R}^{+}\to\mathbb{R}^{+}$ satisfying \[f(x)f(yf(x))=f(y+f(x)),\quad x,y\in\mathbb{R}^{+}.\]
  12. Given a triangle $ABC$ inscribed in a circle $(O,R)$, with center $G$ and area $S$. Prove that \[a^{2}+b^{2}+c^{2}\geq\left(4\sqrt{3}+\frac{OG^{2}}{R^{2}}\right)S+(a-b)^{2}+(b-c)^{2}+(c-a)^{2}.\]

Issue 418

  1. Given \[A=1^{5}+2^{5}+3^{5}+\ldots+2011^{5}.\] Find the last digit of $A$.
  2. Let $ABC$ be an isosceles right triangle with right angle at $A$. On the half-plane defined by $AB$ containing $C$ draw an isosceles right triangle $ABD$ with right angle at $B$. Let $E$ be the midpoint of segment $BD$. Draw $CM$ perpendicular to $AE$ at $M$. Let $N$ be the midpoint of segment $CM$, $K$ is the intersection of $BM$ and $DN$. Find the measure of the angle $BKD$.
  3. Find all positive integer solutions of the equation \[3^{x}-32=y^{2}.\]
  4. Find all minimal value of the expression \[A=\frac{1}{x^{3}+xy+y^{3}}+\frac{4x^{2}y^{2}+2}{xy}\] where $x$ and $y$ are positive real numbers satisfying $x+y=1$.
  5. Let $ABC$ be an acute triangle with orthocenter $H$. Prove that $ABC$ is an equilateral triangle if and only if \[\frac{AH}{BC}=\frac{BH}{CA}=\frac{CH}{AB}.\]
  6. Let $ABC$ be a triangle with circumcenter $O$, and incenter $I$. $BC$ touches the circle $(I)$ at $D$. The circle whose diameter is $AI$ meets $(O)$ at $M$ ($M\ne A$) and cuts the line passing through $A$ parallel to $BC$ at $N$. Prove that $MO$ passes through the midpoint of $DN$.
  7. Solve the system of equations \[\begin{cases}\sqrt{xy+(x-y)(\sqrt{xy}+2)}+\sqrt{x} & =y+\sqrt{y}\\(x+1)(y+\sqrt{xy}+x(1-x)) & =4\end{cases}.\]
  8. Let $ABC$ be an acute triangle. Prove the inequaltiy \[\cos^{3}A+\cos^{3}B+\cos^{3}C+\cos A.\cos B.\cos C\geq\frac{1}{2}.\]
  9. For each natural number $n$, let $(S_{n})$ be the sum of all digits of $n$ (in the decimal system). Put $S_{k}(n)=S(S(\ldots(S(n))\ldots))$ ($k$ times). Find all natural numbers $n$ such that \[S_{1}(n)+S_{2}(n)+\ldots S_{k}(n)+\ldots+S_{223}(n)=n.\]
  10. Does there exist a set $X$ satisfying the following two conditions
    • $X$ contains $2012$ natural numbers.
    • The sum of any arbitrary elements in $X$ is the $k$-th power of a positive integer ($k\geq2$).
  11. Find all functions $f:\mathbb{R}\to\mathbb{R}$ satisfing \[f\left(\frac{xf(y)}{2}\right)+f\left(\frac{yf(x)}{2}\right)=4xy,\:\forall x,y\in\mathbb{R}.\]
  12. Fix two circles $(K)$ and $(O)$, where $(K)$ is inside $(O)$. Two circles $(O_{1})$, $(O_{2})$ are moving so that they always externally touch each other at $M$. Both also internally touch $(O)$, and externally touch $(K)$. Prove that $M$ belongs to a fixed circle.

Issue 419

  1. Let $$A=\frac{1}{1^{2}}+\frac{1}{2^{2}}+\frac{1}{3^{2}}+\ldots+\frac{1}{50^{2}}$$ and $B=\dfrac{165}{101}$. Compare $A$ and $B$.
  2. Let $A B C$ be a right isosceles triangle with right angle at $A .$ If there exists a point $M$ inside the triangle with $\widehat{M B A}=$ $\widehat{M A C}=\widehat{M C B}$. Find the ratio $M A: M B: M C$.
  3. Find the minimum values of the natural numbers $a, b, c$ satisfying $$\begin{align} & a+(a+1)+(a+2)+\ldots+(a+6) \\ =& b+(b+1)+(b+2)+\ldots+(b+8) \\ = &c+(c+1)+(c+2)+\ldots+(c+10).\end{align}$$
  4. Solve the following equation $$6(x-1) \sqrt{x+1}+\left(x^{2}+2\right)(\sqrt{x-1}-3)=x\left(x^{2}+2\right).$$
  5. Let $M$ be the midpoint of the arc $A B$ of a semicircle with center $O$ and diameter $A B$. $A C$ meets $M O$ at $D$. Prove that the circumcenter of triangle $M D C$ always lies on a fixed line when $C$ moves on the semicircle.
  6. Let $a, b, c$ be positive real numbers. Prove that $$6\left(a^{3}+b^{3}+c^{3}\right) \geq 18 a b c+\left(\sqrt[3]{a(b-c)^{2}}+\sqrt[3]{b(c-a)^{2}}+\sqrt[3]{c(a-b)^{2}}\right)^{3}.$$
  7. Let $A B C$ be an acute triangle which is not isosceles; and $H$, $O$ be its orthocenter and circumcenter respectively; let $D$, $E$ be respectively the foot of the altitude from $A$, $B$. The lines $O D$ and $B E$ intersect at $K$, $O E$ and $A D$ intersect at $L$. Let $M$ be the midpoint of edge $A B$. Prove that $K$, $L$, $M$ are collinear if and only if $C$, $D$, $O$, $H$ lies on the same circle.
  8. Find all pairs of positive integers $(n, k)$ satisfying $C_{3 n}^{n}=3^{n} n^{k},$ where $$C_{p}^{m}=\frac{p !}{m !(p-m) !} ; 0 \leq m \leq p, p \neq 0, m, p \in \mathbb{N}.$$
  9. Let $a, b, c$ be three positive real numbers satisfying $$15\left(\frac{1}{a^{2}}+\frac{1}{b^{2}}+\frac{1}{c^{2}}\right)=10\left(\frac{1}{a b}+\frac{1}{b c}+\frac{1}{c a}\right)+2012.$$ Find the largest possible value of the expression $$P=\frac{1}{\sqrt{5 a^{2}+2 a b+2 b^{2}}}+\frac{1}{\sqrt{5 b^{2}+2 b c+2 c^{2}}}+\frac{1}{\sqrt{5 c^{2}+2 c a+2 a^{2}}}.$$
  10. Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that for all $x, y \in \mathbb{R}$ we have $$f(x f(y))+f(f(x)+f(y))=y f(x)+f(x+f(y)).$$
  11. On the interval $[a ; b],$ pick $k$ distinct points $x_{1}, x_{2}, \ldots, x_{k}$. Let $d_{n}$ be the product of the distances from $x_{n}$ to the $k-1$ remaining points; $n=1,2,3 \ldots, k .$ Find the smallest value of $\displaystyle \sum_{n=1}^{k} \frac{1}{d_{n}}$.
  12. Given a triangle $A B C$ and an arbitrary point $M$. Prove that $$\frac{M A}{B C}+\frac{M B}{C A}+\frac{M C}{A B} \geq \frac{B C+C A+A B}{M A+M B+M C}.$$

Issue 420

  1. Find the integer value of the expression $f(x ; y)=\dfrac{x^{2}+x+2}{x y-1}$ where $x, y$ are positive integers.
  2. Let $A B C$ be an acute triangle which is not isosceles at $A$. The perpendicular bisectors of $A B$, $A C$ cut the median $A M$ at $E$, $F$ respectively. $B E$ and $C F$ meet at $K .$ Prove that $\widehat{A K B}=\widehat{A K C}$ and $\widehat{M A B}=\widehat{K A C}$.
  3. Find all triples of integers $(x ; y ; z)$ such that $$2 x y+6 y z+3 z x-|x-2 y-z|=x^{2}+4 y^{2}+9 z^{2}-1.$$
  4. For each positive integer $n(n=1,2, \ldots),$ put $a_{n}=\dfrac{4 n}{n^{4}+4} .$ Prove that $$a_{1}+a_{2}+\ldots+a_{n}<\frac{3}{2}$$
  5. Let $A B C$ be an acute triangle. The internal angle-bisector of angle $B A C$ cuts $B C$ at $D$. $E$, $F$ are the orthogonal projections of point $D$ on $A B$ and $A C$ respectively, $K$ is the intersection of $C E$ and $B F, H$ is the intersection of $B F$ with the circumcircle of triangle $A E K$. Prove that $D H$ is perpendicular to $B F$
  6. Solve the system of equations $$\begin{cases} x+6 \sqrt{x y}-y &=6 \\ x+\dfrac{6\left(x^{3}+y^{3}\right)}{x^{2}+x y+y^{2}}-\sqrt{2\left(x^{2}+y^{2}\right)} &=3 \end{cases}.$$
  7. Let $a, b, c$ be non-negative real numbers whose sum equals $1$. Prove that $$\left(1+a^{2}\right)\left(1+b^{2}\right)\left(1+c^{2}\right) \geq\left(\frac{10}{9}\right)^{3}$$
  8. Point $M$ inside the triangle $A B C$ with area $S$. Let $x, y, z$ be distances of $M$ to $A$, $B$, $C$ respectively. Prove that $$(x+y+z)^{2} \geq 4 \sqrt{3} S.$$ When does the equality hold?
  9. A nonempty set $S \subseteq \mathbb{Z}$ posesses the following properties
    • There exist $a, b \in S$ such that $(a, b)=(a-2 b-2)=1$,
    • If $x, y \in S$ then $x^{2}-y \in S$ ($x$, $y$ may be identical).
    Prove that $S=\mathbb{Z}$. ($(a, b)$ is the greatest common divisor of two integers $a$ and $b$.)
  10. Find the greatest number $k$ such that the inequality $$\sqrt{a+2 b+3 c}+\sqrt{b+2 c+3 a}+\sqrt{c+2 a+3 b} \geq k(\sqrt{a}+\sqrt{b}+\sqrt{c})$$ holds for all positive numbers $a, b, c$
  11. Let $\left(x_{n}\right)$ be a sequence defined by $$x_{1}=\frac{1001}{1003} ,\quad x_{n+1}=x_{n}-x_{n}^{2}+x_{n}^{3}-x_{n}^{4}+\ldots+x_{n}^{2011}-x_{n}^{2012},\, \forall n \in \mathbb{N}.$$ Find $\displaystyle \lim _{n \rightarrow+\infty}\left(n x_{n}\right)$.
  12. Given four distinct points $A$, $B$, $C$, $D$ lying on a circle with center $O$. Let $I$, $J$ be the feet of the perpendicular to $A B$ and $A D$ through $C$; $K$, $L$ are the feet of the perpendicular to $B C$ and $B A$ through $D$; $N$ is the midpoint of $C D$; $M$ is the intersection of $I J$ and $K L$. $I J$ meets $O D$ at $E$ and $K L$ meets $O C$ at $F$. Prove that the five points $M$, $N$, $O$, $E$ and $F$ lie on the same circle.

Issue 421

  1. Given the sum of $2012$ terms $$S=\frac{1}{5}+\frac{2}{5^{2}}+\frac{3}{5^{3}}+\frac{4}{5^{4}}+\ldots+\frac{2012}{5^{2012}}$$ Compare $S$ with $\dfrac{1}{3}$.
  2. Let $A B C$ be a triangle with $\widehat{A B C}=40^{\circ}, \widehat{A C B}=30^{\circ} .$ Outside this triangle, construct triangle $A D C$ with $\widehat{A C D}=\widehat{C A D}=50^{\circ} .$ Prove that the triangle $B A D$ is isosceles.
  3. Find all natural numbers $a, b, c$ such that $c < 20$ and $a^{2}+a b+b^{2}=70 c$.
  4. Find the largest possible value of the expression $$P=\sqrt{1-\frac{x}{y+z}}+\sqrt{1-\frac{y}{z+x}}+\sqrt{1-\frac{z}{x+y}}$$ where $x, y, z$ are side lengths of a triangle.
  5. Given a circle $(O),$ with a fixed chord $B C$. $A$ is a point moving on the line $B C$, outside the circle $(O)$. $AM$ and $AN$ are the tangent lines to circle $(O)$ $(M, N \in (O))$. The line through $B$ and parallel to $A M$ meets $M N$ at $E .$ Prove that the circumcircle of triangle $B E N$ always passes through two fixed points when point $A$ moves on the line $B C$.
  6. Given that $\dfrac{1}{3} < x \leq \dfrac{1}{2}$ and $y \geq 1$. Find the minimum value of $$P=x^{2}+y^{2}+\frac{x^{2} y^{2}}{((4 x-1) y-x)^{2}}.$$
  7. Let $\left(a_{n}\right)$ be a sequence of positive real numbers, given by
    • $a_{0}=1$,
    • $a_{m}<a_{n}$, for all $m, n \in \mathbb{N}$, $m<n$.
    • $a_{n}=\sqrt{a_{n+1} \cdot a_{n-1}}+1$ and $4 \sqrt{a_{n}}=a_{n+1}-a_{n-1}$ for all $n \in \mathbb{N}^{*}$.
    Determine the sum $T=a_{0}+a_{1}+a_{2}+\ldots+a_{2012}$.
  8. The base of a triangular prism $A B C \cdot A^{\prime} B^{\prime} C^{\prime}$ is an equilateral triangle with side lengths $a$ and the lengths of its adjacent sides also equal $a$. Let $I$ be the midpoint of $A B$ and $B^{\prime} I \perp(A B C)$. Find the distance from $B^{\prime}$ to the plane $\left(A C C^{\prime} A^{\prime}\right)$ in term of $a$.
  9. Find all polynomials $P(x)$ with real coefficients satisfying $$P^{2}(x)-1=4 P\left(x^{2}-4 x+1\right)$$
  10. Find $\alpha, \beta$ so that the largest value of $$y=|\cos x+\alpha \cos 2 x+\beta \cos 3 x|$$ is smallest possible.
  11. Let $A B C$ be a triangle with side lengths $a$, $b$ and $c$. Let $S$ and $p$ be respectively the area and the semiperimeter of this triangle. Prove the inequality $$\frac{1}{a^{2}(p-a)^{2}}+\frac{1}{b^{2}(p-b)^{2}}+\frac{1}{c^{2}(p-c)^{2}} \geq \frac{9}{4 S^{2}}$$
  12. Given an acute triangle $A B C$ inscribed the circle $(O)$ with $B C>C A>A B$. On the circle $(O)$, select six distinct points $M$, $N$, $P$, $Q$, $R$ and $S$ (which are also distinct from the vertices of triangle $A B C$) so that $Q B=B C=C R$, $S C=C A=A M$ and $N A=A B=B P$. Let $I_{A}$, $I_{B}$ and $I_{C}$ be the incenters of triangles $A P S$, $B N R$ and $C M Q$ respectively. Prove that $\Delta I_{A} I_{B} I_{C} \sim \Delta A B C$.

Issue 422

  1. Let $$A=1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\ldots+\frac{1}{2011}-\frac{1}{2012},\quad B=\frac{1}{1007}+\frac{1}{1008}+\ldots+\frac{1}{2012}.$$ Compute the value of $\left(\dfrac{A}{B}\right)^{2012}$.
  2. Let $f(x)$ be a polynomial with integer coefficients such that $f(3) \cdot f(4)=5 .$ Prove that $f(x)-6$ does not have any integer solution.
  3. Find all triple of integers $a, b, c$ such that $$2^{a}+8 b^{2}-3^{c}=283.$$
  4. Given a triangle $A B C$, $B C=a$, $C A=b$, $A B=c$, $\widehat{A B C}=45^{\circ}$ and $\widehat{A C B}=120^{\circ}$. Point $I$ is taken on the opposite ray of $C B$ such that $\widehat{A I B}=75^{\circ} .$ Find the length of $A I$ in term of $a$, $b$ and $c$
  5. Point $K$ lies on side $B C$ of a triangle $A B C$. Prove that $$A K^{2}=A B \cdot A C - K B \cdot K C$$ if and only if $A B=A C$ or $\widehat{B A K}=\widehat{C A K}$.
  6. A non-isosceles triangle $A B C$ has $B C=a$, $C A=b$, $A B=c$. Let $\left(A A_{1}, A A_{2}\right)$, $\left(B B_{1}, B B_{2}\right)$, $\left(C C_{1}, C C_{2}\right)$ be the median and the altitude from vertices $A$, $B$ and respectively. Prove that $$\frac{a^{2}}{b^{2}-c^{2}} \overline{A_{1} A_{2}}+\frac{b^{2}}{c^{2}-a^{2}} \overrightarrow{B_{1} B_{2}}+\frac{c^{2}}{a^{2}-b^{2}} \overrightarrow{C_{1} C_{2}}=\overrightarrow{0}$$
  7. Let $a, b, c \in(0 ; 1)$ and $$a b+b c+c a+a+b+c=1+a b c.$$ Prove that $$\frac{1+a}{1+a^{2}}+\frac{1+b}{1+b^{2}}+\frac{1+c}{1+c^{2}} \leq \frac{3}{4}(3+\sqrt{3})$$
  8. Let $A B C$ be an acute triangle with all angles greater than $45^{\circ}$. Prove that $$\frac{2}{1+\tan A}+\frac{2}{1+\tan B}+\frac{2}{1+\tan C} \leq 3(\sqrt{3}-1).$$ When does equality occur?
  9. Two sequences $\left(a_{n}\right)$ and $\left(b_{n}\right)$ are defined inductively as follows $$a_{0}=3, b_{0}=-3,\quad a_{n}=3 a_{n-1}+2 b_{n-1},\,b_{n}=4_{n-1}+3 b_{n-1},\,\forall n \geq 1.$$ Find all natural numbers $n$ such that $\displaystyle\prod_{k=0}^{n}\left(b_{k}^{2}+9\right)$ is a perfect square.
  10. Let $n$ be a positive integer. How many strings of length $n: a_{1} a_{2} \ldots a_{n}$ where $a_{i}$ is chosen from $\{0,1,2, \ldots, 9\}(i=1,2, \ldots, n)$ are there such that the number of occurrences of 0 is even?
  11. Let $\left(u_{n}\right)$ be a sequence defined by $u_{0}=a \in[0 ; 2), u_{n}=\dfrac{u_{n-1}^{2}-1}{n}$ for all $n=1,2,$ $3, \ldots$ Find $\displaystyle\lim _{n \rightarrow+\infty}\left(u_{n} \sqrt{n}\right)$.
  12. Let $A B C$ be a triangle, inscribed in the circle $(O)$ with altitudes $A D$, $B E$ and $C F$. $A A^{\prime}$ is a diameter of $(O)$. $A^{\prime} B$, $A^{\prime} C$ intersect $A C$, $A B$ at $M$, $N$ respectively. Points $P$, $Q$ are in $E F$, such that $P B$, $Q C$ are perpendicular to $B C$. The line passing through $A$ and orthogonal to $Q N$, $P M$ cuts $(O)$ at $X$, $Y$ respectively. The tangents to circle $(O)$ at $X$ and $Y$ meet at $J$. Prove that $J A^{\prime}$ is perpendicular to $B C$.

Issue 423

  1. Find all numbers abcde, where all five digits are distinct and $\overline{a b c d}=(5 e+1)^{2}$
  2. Find all positive integers $x, y, z$ such that $x+3=2^{y}$ và $3 x+1=4^{z}$
  3. Find the last digit of the sum $$S=1^{2}+2^{2}+3^{3}+\ldots+n^{n}+\ldots+2012^{2012}.$$
  4. Given a function $f$ such that $$f\left(1+\frac{\sqrt{2}}{x}\right)=\frac{(1+2011) x^{2}+2 \sqrt{2 x}+2}{x^{2}}$$ for all nonzero $x$. Determine $f(\sqrt{2012-\sqrt{2011}})$
  5. Let $A B C$ be a triangle inscribed in the circle $(O)$. The tangents of $(O)$ at $B$ and $C$ meet at $T$. The line passing through $T$ and parallel to $B C$ cuts $A B$ and $A C$ respectively at $B_{1}$ and $C_{1}$ Prove that $\widehat{B_{1} O C_{1}}$ is an acute angle.
  6. On the outside of triangle $A B C$, construct equilateral triangles $A B C_{1}$, $B C A_{1}$, $CAB_{1}$ and inside of $A B C$ construct equilateral triangles $A B C_{2}$, $B C A_{2}$, $C A B_{2}$. Let $G_{1}$, $G_{2}$, $G_{3}$ be respectively the centroids of $A B C_{1}$, $B C A_{1}$, $C A B_{1}$ and let $G_{4}$, $G_{5}$, $G_{6}$ be respectively the centroids of triangles $A B C_{2}$, $BCA_{2}$ and $CAB_{2}$. Prove that the centroids of triangle $G_{1} G_{2} G_{3}$ and of triangle $G_{4} G_{5} G_{6}$ coincide.
  7. Solve the equation $$3^{3 x}+3^{x}=\log _{3}\left(2^{x}+x\right)+2^{x}+3^{2^{x}+x}.$$
  8. Let $A$, $B$, $C$ be the three angles of an acute triangle. Prove the inequality $$\sqrt{\frac{\cos A \cos B}{\cos C}}+\sqrt{\frac{\cos B \cos C}{\cos A}}+\sqrt{\frac{\cos C \cos A}{\cos B}}>2.$$
  9. Find the largest positive integer $n$ $(n \geq 3)$ such that there exists a sequence of positive integers $a_{1}, a_{2}, \ldots, a_{n}$ satisfying the condition $$a_{k+1}+1=\frac{a_{k}^{2}+1}{a_{k-1}+1},\, k \in\{2,3, \ldots, n-1\}.$$
  10. Let $p$ be an odd prime number, $n$ is a positive integer so that $p-1$, $p$, $n$ and $n+1$ are pairwise coprime. Find all positive integers $x$, $y$ satisfying $$x^{p-1}+x^{p-2}+\ldots+x+2=y^{n+1}.$$
  11. Solve the system of equations $$\begin{cases}\sqrt{5 x^{2}+2 x y+2 y^{2}}+\sqrt{2 x^{2}+2 x y+5 y^{2}} &=3(x+y) \\ \sqrt{2 x+y+1}+2 \sqrt[3]{7 x+12 y+8} &=2 x y+y+5\end{cases}.$$
  12. Let $A B C$ be a triangle inscribed in the circle $(O)$ and let $I$ be its incenter. $A I$, $B I$, $Cl$ cut the circle $(O)$ at $A^{\prime}$, $B^{\prime}$ and $C^{\prime}$ respectively; $A^{\prime} C^{\prime}$, $A^{\prime} B^{\prime}$ cut $B C$ at $M$, $N$; $B^{\prime} A^{\prime}$; $B^{\prime} C^{\prime}$ cut $C A$ at $P$, $Q$; $C^{\prime} B^{\prime}$, $C^{\prime} A$ cut $A B$ at $R$, $S$. Prove that $$\frac{2}{3} S_{A B C} \leq S_{M N P Q R S} \leq \frac{2}{3} S_{A^{\prime} B^{\prime} C^{\prime}}.$$

Issue 424

  1. Find all $2$-digit numbers such that when multiplied by $2,3,4,$ $5,6,7,8,9,$ the sum of the digits of the resulting numbers are equal.
  2. Let $$S=\frac{2}{2013+1}+\frac{2^{2}}{2012^{2}+1}+\frac{2^{3}}{2013^{2^{2}}+1}+\ldots+\frac{2^{2014}}{2013^{2^{2013}}+1}.$$ Which number is greater? $S$ or $\dfrac{1}{1006}$?.
  3. Find all integer solutions of the equation $$(y-2) x^{2}+\left(y^{2}-6 y+8\right) x=y^{2}-5 y+62$$
  4. Let $x$, $y$ be two rational numbers such that $$x^{2}+y^{2}+\left(\frac{x y+1}{x+y}\right)^{2}=2 .$$ Prove that $\sqrt{1+x y}$. is also a rational number.
  5. Let $O$ denote the point of intersection of the two diagonals $A C$ and $B D$ of a convex quadrilateral $A B C D$. Let $E$, $F$, $H$ be the feet of the altitudes from $B$, $C$ and $O$ respectively onto $A D$. Prove that $$ A D \cdot B E \cdot C F \leq A C \cdot B D \cdot O H.$$ When does equality holds?
  6. $a, b, c$ are positive real numbers satisfying $a b c=1$. Prove that $$\frac{a^{3}+5}{a^{3}(b+c)}+\frac{b^{3}+5}{b^{3}(c+a)}+\frac{c^{3}+5}{c^{3}(a+b)} \geq 9$$
  7. Solve the equation $$\left(x^{3}+\frac{1}{x^{3}}+1\right)^{4}=3\left(x^{4}+\frac{1}{x^{4}}+1\right)^{3}$$
  8. Let $A B C$ be a triangle with acute angle $A$. Point $P$ inside the triangle $A B C$ such that $\widehat{B A P}=\widehat{A C P}$ and $\widehat{C A P}=\widehat{A B P}$. Let $M$ and $N$ be the incenters of triangles $A B P$ and $A C P$ respectively, $R$ is the circumradius of triangle $A M N$. Prove that $$\frac{1}{R}=\frac{1}{A B}+\frac{1}{A C}+\frac{1}{A P}.$$
  9. Solve the equation $$[x]^{3}+2 x^{2}=x^{3}+2[x]^{2}$$ where $[t]$ denotes the largest integer not exceeding $t$.
  10. In the interior of a unit square, there are $n\left(n \in \mathbb{N}^{*}\right)$ circles whose sum of areas is greater than $n-1$. Prove that the circles has at least a common point of intersection.
  11. Given that the following equation $$a_{0} x^{n}+a_{1} x^{n-1}+\ldots+a_{n-1} x+a_{n}=0 $$ has $n$ distinct roots. Prove that $$\frac{n-1}{n}>\frac{2 a_{0} a_{2}}{a_{1}^{2}}.$$
  12. Let $O$, $I$ and $I_{a}$ denote the circumcenter, incenter and excenter in the angle $A$ of a triangle $A B C$. $A I$ meets $B C$ at $D$. BI meets $C A$ at $E$. The line through $I$ and perpendicular to $O I_{a}$ intersects $A C$ at $M$. Prove that $D E$ passes through the midpoint of line segment $I M$.

Issue 425

  1. Find all natural numbers $N$ such that $N$ decreases by a factor of $1997$ after truncating the last several digits.
  2. Let $A B C$ be a right triangle with right angle at $A$ and $\widehat{A C B}=15^{\circ}$, Point $D$ on edge $A C$ such that the line passing through $D$ and perpendicular to $B D$ cuts $B C$ at $E$ and $D E=2 D A$. Find the measure of angle $A D B$.
  3. Find all positive integers $n$ such that $[A]=4951$ where $A$ is the sum of $n$ terms $$A=\left(1+\frac{1}{2}\right)+\left(2+\frac{2}{2^{2}}\right)+\left(3+\frac{3}{2^{3}}\right)+\ldots+\left(n+\frac{n}{2^{n}}\right).$$ Here $[x]$ denotes the largest integer not exceeding $x$
  4. Find the minimum value of the expression $$P=\frac{1+\sqrt[3]{x}+\sqrt[3]{y}+\sqrt[3]{z}}{x y+y z+z x},$$ where $x, y, z$ are positive numbers satisfying $x+y+z=3$
  5. Solve the equation $$x^{2}-2 x+7+\sqrt{x+3}=2 \sqrt{1+8 x}+\sqrt{1+\sqrt{1+8 x}}.$$
  6. Let $A B C$ be a non-isosceles triangle with medians $A A^{\prime}$, $B B^{\prime}$ and $C C^{\prime}$; and altitudes $A H$, $B F$ and CK. Given that $C K=B B^{\prime}$, $B F=A A^{\prime}$. Determine the ratio $\dfrac{C C^{\prime}}{A H}$.
  7. $a_{1}, a_{2}, \ldots, a_{n}$ $(n \geq 3)$ are positive numbers that $$\left(a_{1}+a_{2}+\ldots+a_{n}\right)^{2}>\frac{3 n-1}{3}\left(a_{1}^{2}+a_{2}^{2}+\ldots+a_{n}^{2}\right).$$ Prove that for any triple $a_{i}, a_{j}, a_{k}$ are three edge lengths of some triangle, where natural numbers $i, j,$ $k$ satisfying $0<i<j<k \leq n$.
  8. The volume of a given parallelogrambased pyramid $S.ABCD$ is $V$. Assume that plane $(P)$ cuts$S A$, $S B$, $S C$, $S D$ at $A^{\prime}$, $B^{\prime}$, $C^{\prime}$, $D^{\prime}$ respectively such that $$\frac{S A}{S A^{\prime}}+\frac{S B}{S B^{\prime}}+\frac{S C}{S C^{\prime}}+\frac{S D}{S D^{\prime}}=8.$$ Denote the volume of the pyramid $S . A^{\prime} B^{\prime} C^{\prime}$ by $V_{1}$ and that of $S . A^{\prime} C^{\prime} D^{\prime}$ by $V_{2}$. Prove the inequality $$\frac{1}{\sqrt[3]{V_{1}}}+\frac{1}{\sqrt[3]{V_{2}}} \leq \frac{4 \sqrt[3]{2}}{\sqrt[3]{V}}.$$
  9. Write $2012^{2013}$ as a sum of $2013$ positive interger $a_{1}, a_{2}, a_{3}, \ldots, a_{2013} ;$ and let $$T=a_{1}^{13}+a_{2}^{13}+a_{3}^{13}+\ldots+a_{2013}^{13}.$$ Prove that $T+2012^{2013}$ is not a perfect square.
  10. The incircle $(I)$ of a triangle $A B C$ touches the edges $B C$, $C A$, $A B$ at $D$, $E$, $F$, respectively. $M$ is the intersection of $B C$ and the internal angle bisector of angle $B I C$, $N$ is the intersection of $E F$ and the internal angle bisector of angle $E D F$. Prove that $A$, $M$, $N$ are collinear.
  11. If $p(x)$ and $q(x)$ are polynomials with integer coefficients, write $p(x) \equiv q(x) \pmod 2$ if the coefficients of $p(x)-q(x)$ are all even. A sequence of polynomials $p_{n}(x)$ is such that $p_{1}(x)=p_{2}(x)=1$ and $$p_{n+2}(x)=p_{n+1}(x)+x p_{n}(x),\,\forall n \geq 1.$$ Prove that $p_{2^{n}}(x) \equiv 1\pmod 2, \forall n \in \mathbb{N}$.
  12. Let $A B C$ be an acute triangle. Prove the inequality $$\frac{\cos B \cos C}{\cos \frac{B-C}{2}}+\frac{\cos C \cos A}{\cos \frac{C-A}{2}}+\frac{\cos A \cos B}{\cos \frac{A-B}{2}} \leq \frac{3}{4}$$

Issue 426

  1. Prove that for any natural number $n>4$ there exists a pair of natural numbers $x, y$ with $\dfrac{n}{2} \leq x<n$ and $\dfrac{n}{2} \leq y<n,$ such that $x^{2}-y$ is divisible by $n$ 
  2. Let $A B C$ be an isosceles right triangle with right angle at $A$. On the ray $A C$ choose two points $E$ and $F$ such that $\widehat{A B E}=15^{\circ}$ and $C E=C F$. What is the measure of the angle $C B F$?.
  3. Find all positive integer solutions of the equation $$65\left(x^{3} y^{3}+x^{2}+y^{2}\right)=81\left(x y^{3}+1\right).$$
  4. Solve the system of equations $$\begin{cases} 9 x^{2}+9 x y+5 x-4 y+9 \sqrt{y} &=7 \\ \sqrt{x-y+2}+1 &=9(x-y)^{2}+\sqrt{7 x-7 y} \end{cases}.$$
  5. Let $A B C$ be an isosceles triangle where the angle $B A C$ is obtuse. Suppose $D$ is a point on edge $A B$ such that $B C=C D \sqrt{2}$. The line through $D$ and perpendicular to $A B$ meets $C A$ at $E$. Prove that $C D$ passes through the midpoint of $B E$.
  6. Let $$S=\sqrt{2}+\sqrt[3]{\frac{3}{2}}+\sqrt[4]{\frac{4}{3}}+\sqrt[5]{\frac{5}{4}}+\ldots+2013 \sqrt[2]{\frac{2013}{2012}}.$$ Find the integer part of $S$.
  7. Given that $a, b, c$ are edge lengths of a triangle. Prove the following inequality $$\sqrt{(a+b-c)(b+c-a)(c+a-b)} \leq \frac{3 \sqrt{3} a b c}{(a+b+c) \sqrt{a+b+c}}.$$
  8. Let $A B C D$ be a trirectangular tetrahedron where the edges $A B$, $A C$, $A D$ are pairwise perpendicular. $M$ is an arbitrary point in the space. Given that $A B=4$, $A C=8$, $A D=12$. Find the minimum value of the expression $$P=\sqrt{7} M A+\sqrt{11} M B+\sqrt{23} M C+\sqrt{43} M D.$$
  9. Let $\left(a_{n}\right)$ be a sequence of positive integers where $$a_{1}=1,\, a_{2}=2,\quad a_{n+2}=4 a_{n+1}+a_{n},\,\forall n \geq 1 .$$ Prove that
    a) $a_{n} a_{n+2}+(-1)^{n} .5$ is a perfect square for all $n \geq 1$.
    b) The equation $x^{2}-4 x y-y^{2}=5$ has infinitely many positive integer solutions.
  10. Let $a$ be a real number from $(0,1)$ and $b$ is a complex number, $|b|<1$. Prove that $$|b|+\left|\frac{a-b}{1-a b}\right| \geq a.$$
  11. Let $0<\alpha<\dfrac{\pi}{2}$. Prove that $$(\cot \alpha)^{\cos 2 \alpha} \geq \frac{1}{\sin 2 \alpha}$$
  12. A tetrahedron $A B C D$ is inscribed in a sphere centered at $O$. Point $G$ does not belong to planes $(B C D)$, $(C D A)$, $(D A B)$, $(A B C)$ and the sphere $(O)$. $X$, $Y$, $Z$, $T$ are the centers of the circumscribed spheres of the tetrahedron $G B C D$, $G C D A$, $G D A B$, $G A B C$ respectively. Prove that $G$ is the centroid of tetrahedron $A B C D$ if and only if $O$ is the centroid of tetrahedron $X Y Z T$.



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 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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 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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 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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 2012
Mathematics and Youth Magazine Problems 2012
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