Mathematics and Youth Magazine Problems 2013


Issue 427

  1. Let $a=123456789$. Which number is greater $2012^{9^{9^{a}}}$ or $2013^{a^{a^{9}}}$?.
  2. Let $ABC$ $(AB < AC)$ be a triangle, with two altitudes $BD,CE$ and $AB=c$, $AC=b$, $BD=h_{b}$, $CE=h_{c}$. Prove that \[c^{n}+h_{c^{n}} < b^{n}+h_{b}^{n},\quad\forall n\in\mathbb{N}^{*}.\]
  3. Find all positive integers $n$ such that \[A=\left[\frac{n^{2}+n-5}{2}\right]\] is a prime number, where $[a]$ is the largest integer not exceeding $a$.
  4. Find all postive integer soltuions of the equation \[\sqrt{x+y}-\sqrt{x}-\sqrt{y}+2=0.\]
  5. Let $ABC$ be a right triangle, right angle at $A$. The bisectors $BD$ and $CE$ intersect at $O$. The area of $BOC$ is $a$. Determine the product $BD\cdot CE$ in terms of $a$.
  6. Solve the system pf equations \[\begin{cases} 2\sqrt[4]{\frac{x^{4}}{3}+4} & =1+\sqrt{\frac{3}{2}}|y|\\ 2\sqrt[4]{\frac{y^{4}}{3}+4} & =1+\sqrt{\frac{3}{2}}|x| \end{cases}.\]
  7. The side lengths of a traingle $ABC$ are $AB=9$, $BC=\sqrt{39}$, $CA=\sqrt{201}$. Find a point $M$ on the circle $(C;\sqrt{3})$ such that the sum $MA+MB$ is the maximum. 
  8. Prove that in any traingle $ABC$, \begin{align*} & \sqrt{\left(\tan\frac{A}{2}+\tan\frac{B}{2}\right)\left(\tan\frac{B}{2}+\tan\frac{C}{2}\right)}\\  + &\sqrt{\left(\tan\frac{B}{2}+\tan\frac{C}{2}\right)\left(\tan\frac{C}{2}+\tan\frac{A}{2}\right)}\\ + &\sqrt{\left(\tan\frac{C}{2}+\tan\frac{A}{2}\right)\left(\tan\frac{A}{2}+\tan\frac{B}{2}\right)}\\ \leq & 2(\cot A+\cot B+\cot C).\end{align*}
  9. Let $N=1+10+10^{2}+\ldots+10^{4023}$. Find the 2013-th digit after the decimal comma of $\sqrt{N}$. 
  10. Find the maximum and minimum values of the expression \[P=\frac{a-b}{c}+\frac{b-c}{a}+\frac{c-a}{b},\] where $a,b,c$ are positive real numbers satisfying the condition \[\min\{a,b,c\}\geq\frac{1}{4}\max\{a,b,c\}.\] Let $\{S_{n}(x)\}$ be a sequence of real-valued functions defined by \[S_{n}(x)=\cos^{3}x-\frac{1}{3}\cos^{3}3x+\frac{1}{3^{2}}\cos^{3}3^{2}x-\ldots+\left(\frac{-1}{3}\right)^{n}\cos^{3}3^{n}x.\]
  11. Find all real values of $x$ such that \[\lim S_{n}(x)=\frac{3-3x}{4}.\]
  12. In a con-cyclic quadrilateral $ABCD$, let $A'$, $B'$, $C'$, $D'$ be the circumcenters of triangles $BCD$, $CDA$, $DAB$ and $ABC$ respectively. Let $A''$, $B''$, $C''$, $D''$ be the centers of the Euler circles of triangle $BCD$, $CDA$, $DAB$, $ABC$ respectively. Prove that the two quadrilateral $A'B'C'D'$, $A''B''C''D''$ are both convex and inversely similar.

Issue 428

  1. Determine all triple of prime numbers $a,b,c$ (not necessarily distinct) such that \[abc < ab+bc+ca.\]
  2. Let $ABC$ be a right triangle, with right angle at $A$ and $AH$ is the altitude from $A$, $\widehat{ACB}=30^{\circ}$. Construct an equilateral triangle $ACD$ ($D$ and $B$ are in opposite side $AC$). K is the foot of the perpendicular line from $H$ onto $AC$. The line through $H$ and parallel to $AD$ meets $AB$ at $M$. Prove the points $D,K,M$ are colinear.
  3. Consider a $6\times6$ board of squares with 4 corner squares being deleted. Find the smallest number of squares that can be painted black given that among the 5 squares in any figure, there is at lease one black.
  4. Let $a,b,c$ be real numbers in the interval $[1,2]$. Prove the inequality \[a^{2}+b^{2}+c^{2}+3\sqrt[3]{(abc)^{2}}\geq2(ab+bc+ca).\]
  5. Let $ABC$ be a non-right triangle $(AB < AC)$ with altitude $AH$. $E,F$ are the orthogonal projection of point $H$ onto $AB$ and $AC$ respectively. $EF$ meets $BC$ at $D$. Draw a semicircle with diameter $CD$ on the half-plane containing $A$ with edge $CD$. The line through $B$ and perpendicular to $CD$ meets the semicircle at $K$. Prove that $DK$ is tangent to the circumcircle of triangle $KEF$.
  6. Given that the equation \[ax^{3}-x^{2}+ax-b=0\quad(a\ne0,\,b\ne0)\] has three positive real roots. Determine the greatest value of the following expression \[P=\frac{11a^{2}-3\sqrt{3}ab-\frac{1}{3}}{9b-10(\sqrt{3}a-1)}.\]
  7. Solve the following system of equations \[\begin{cases} \sqrt{x-\frac{1}{4}}+\sqrt{y-\frac{1}{4}} & =\sqrt{3}\\ \sqrt{y-\frac{1}{16}}+\sqrt{z-\frac{1}{16}} & =\sqrt{3}\\ \sqrt{z-\frac{9}{16}}+\sqrt{x-\frac{9}{16}} & =\sqrt{3}\end{cases}.\]
  8. Let $a,b$ be real constants such that $ab>0$. Let $\{u_{n}\}$ be a sequence where $n=1,2,3,\ldots$ given by \[u_{1}=a,\quad u_{n+1}=u_{n}+bu_{n}^{2},\,\forall n\in\mathbb{N}^{*}.\] Determine the limit \[\lim_{n\to\infty}\left(\frac{u_{1}}{u_{2}}+\frac{u_{2}}{u_{3}}+\ldots+\frac{u_{n}}{u_{n+1}}\right).\]
  9. Find all positive integers $k$ with the property that there exists a polynomial $f(x)$ with integer coefficients of degree greater than 1 such that for all prime numbers $p$ and natural numbers $a,b$ if $p$ divides $(ab-k)$ then it also divides $(f(a)f(b)-k)$.
  10. Given $a_{i}\in[0,\alpha]$ ($i=\overline{1,n}$), ($\alpha>0$). Prove the inequality \[\prod_{i=1}^{n}(\alpha-a_{i})\leq\alpha^{n}\left(1-\sum_{i=1}^{n}\frac{a_{i}}{S_{i}+\alpha}\right)\] where ${\displaystyle S_{i}=\sum_{k=1}^{n}a_{k}-a_{i}}$ for all $i=\overline{1,n}$.
  11. Point $O$ is in the interior of triangle $ABC$. The ray $Ox$ parallel to $AB$ meets $BC$ at $D$, ray $Oy$ parallel to $BC$ meets $CA$ at $E$, ray $Oz$ parallel to $CA$ meets $AB$ at $F$. Prove that
    a) $3S_{DEF}\leq S_{ABC}$.
    b) $OD\cdot OE\cdot OF\leq27AB\cdot BC\cdot CA$.
  12. The circle $(O)$ and $(O')$ meet at points $A$, $B$. Point $C$ is fixed on $(O)$ and point $D$ is fixed on $(O')$. A moving point $P$ is on the opposite ray of ray $BA$. The circumcircles of traingles $PBC$, $PDB$ intersect $BD$, $BC$ at seconde points $E$, $F$ respectively. Prove that the midpoint of line segment $EF$ is always on a fixed segment $EF$ is always on a fixed straight line.

Issue 429

  1. Find an integer size square whose area is a 4-digit number such that the last rightmost three digits are idnentical.
  2. Determine the values of $a_{1},a_{2},a_{3},a_{4},a_{5},a_{6},a_{7}$. Given that $2a_{1}=3a_{2}$, $2a_{3}=4a_{4}$, $5a_{4}=2a_{5}$, $2a_{5}=5a_{6}$, $2a_{6}=3a_{7}$, $2a_{7}=3a_{1}$ and $$a_{1}+a_{2}+a_{3}+a_{4}+a_{5}+a_{6}+a_{7}=400.$$
  3. Find all pairs of integers $(x,y)$ such that $x^{2}(x^{2}+y^{2})=y^{p+1}$, where $p$ is a prime number.
  4. Find the maximum and minimum values of the expression \[P=xy+yz+zx-xyz\] where $x,y,z$ are non-negative real numbers satisfying \[x^{2}+y^{2}+z^{2}=3.\]
  5. Triangle $ABC$ inscribed in circle $(O)$ with $\widehat{BAC}=70^{\circ}$, $\widehat{ACB}=50^{\circ}$. The points $M,N,P,Q$ and $R$ on circle $(O)$ are such that $PA=AB=BR$, $QB=BC=CM$ and $NC=CA=AN$. Let $S$ be the intersection of arc $NQ$ and the diameter $PP'$ of $(O)$. Prove that $\Delta NRS\backsim\Delta NQR$.
  6. Solve the equation \[x^{3}-3x=\sqrt{x+2}.\]
  7. Find the measure of the angles of a triangle $ABC$ such that the expression \[T=-3\tan\frac{C}{2}+4(\sin^{2}A-\sin^{2}B)\] is greatest possible.
  8. Let $S.ABC$ be a triangular pyramid where the sides $SA,SB,SC$ are pairwise orthogonal, $SA=a$, $SB=b$, $Sc=c$. $H$ is the foot of the perpendicular from $S$ onto $ABC$. Prove the inequality \[aS_{HBC}+bS_{HAC}+cS_{HAB}\leq\frac{abc\sqrt{3}}{2}.\]
  9. Let $p$ be an odd prime number, and $x,y$ are two positive integers such that $\sqrt{x}+\sqrt{y}\leq\sqrt{2p}$. Find the minimum value of the following expression \[A=\sqrt{2p}-\sqrt{x}-\sqrt{y}.\]
  10. Does there exist a funtion $f:\mathbb{N}^{*}\to\mathbb{N}^{*}$ such that \[f(mf(n))=n+f(2013m),\quad\forall m,n\in\mathbb{N}^{*}?.\]
  11. The non-negative real numbers $a,b,c$ are such that $$\max\{a,b,c\}\leq4\min\{a,b,c\}.$$ Prove the inequality \[2(a+b+c)(ab+bc+ca)^{2}\geq9abc(a^{2}+b^{2}+c^{2}+ab+bc+ca).\]
  12. Let $ABC$ be a traingle inscribed in circle centered at $O$, and let $I$ be its incenter. $AI,BI,CI$ intersect $(O)$ at $A_{1},B_{1},C_{1}$; $A_{1}C_{1}$, $A_{1B_{1}}$ meet $BC$ at $M$, $N$; $B_{1}A_{1}$, $B_{1}C_{1}$ meet $CA$ at $P$, $Q$; $C_{1}B_{1}$, $C_{1}A_{1}$ meet $AB$ at $R$, $S$ respectively. Prove that \[S_{MNPQRS}\leq\frac{2}{3}S_{A_{1}B_{1}C_{1}}.\]

Issue 430

  1. Do there exist natural numbers $x,y,z$ such that \[5x^{2}+2016^{y+1}=2017^{z}?.\]
  2. Point $O$ is chosen in a right triangke $ABC$, right angle at $A$, such that $\widehat{ABO}=30^{0}$ and $OA=OC$. Point $E$ on side $BC$ such that $\widehat{EOB}=60^{0}.$ Determine the three angles of traingle $ABC$ given that the line $CO$ passes through the midpoint $I$ of the line segment $AE$.
  3. Find all pair of natural numbers $x,y$ such that $5^{x}=y^{4}+4y+1$.
  4. Solve the system of equations \[\begin{cases} x+\sqrt{y-2}+\sqrt{4-z} & =y^{2}-5z+11\\ y+\sqrt{z-2}+\sqrt{4-x} & =z^{2}-5x+11\\ z+\sqrt{x-2}+\sqrt{4-y} & =x^{2}-5y+11 \end{cases}.\]
  5. Let $AB=2a$ be a line segment with midpoint $O$. Two half-circles, one with center $O$ and diagonal $AB$, another with center $O'$ and diagonal $AO$ are drawn on the same half-plane divided by $AB$. Point $M$, different from $A$ and $O$, moves on the half-circle $(O')$. $OM$ meets the half-circle $(O)$ at $C$. Let $D$ be the second intersection point of $CA$ and half-circle $(O')$. The tangent line at $C$ of half-circle $(O)$ meets $OD$ at $E$. Find the position of point $M$ on $(O')$ such that $ME$ is parallel to $AB$.
  6. Let $ABCD$ be a quadrilateral where the diagonals $AC,BD$ are equal and perpendicular. The triangles $AMB$, $BNC$, $CPD$, $DQA$, similar in order, are constructed outside the given quadrilateral. $O_{1}$, $O_{2}$, $O_{3}$, $O_{4}$ are the midpoints of $MN$, $NP$, $PQ$, $QM$ respectively. Prove that the quadrilateral $O_{1}O_{2}O_{3}O_{4}$ is s square.
  7. Find a formula counting the number of all $2013$-digits natural numbers which are multiple of $3$ and all digits are taken from the set $X=\{3,5,7,9\}$.
  8. Solve for $x$, \[\log_{2}x=\log_{5-x}3.\]
  9. The positive integers $a_{1},a_{2},\ldots,a_{2013}$, $b_{1},b_{2},\ldots b_{2013}$ where $b_{k}>1$ for all $k$ are chosen from the set $X=\{1,2,\ldots,2013\}$. Prove that there exists a positive integer $n$ satisfying the following two conditions
    • ${\displaystyle n\leq\left(\prod_{i=1}^{2013}a_{i}\right)\left(\prod_{i=1}^{2013}b_{i}\right)+1}$.
    • $a_{k}b_{k}^{n}+1$ is a composite number for every $k\in X$.
  10. Let $a,b,c\in\left[0,\frac{1}{2}\right]$ be such that $a+b+c=1$. Prove the inequality \[a^{3}+b^{3}+c^{3}+4abc\leq\frac{9}{32}.\]
  11. Let $ap$ be a prime number, $p\equiv1\,(\text{mod }4)$. Determine the sum \[\sum_{k=1}^{p-1}\left[\frac{2k^{2}}{p}-2\left[\frac{k^{2}}{p}\right]\right],\] where $[a]$ denotes the largest integer not exceeding $a$.
  12. Let $ABC$ be a triangle inscribed inside circle $(O)$. Point $M$ not on lines $BC$, $CA$, $AB$ as weel as circle $(O)$; $AM$, $BM$, $CM$ intersect $(O)$ at $A_{1}$, $B_{1}$, $C_{1}$; $A_{2}$, $B_{2}$, $C_{2}$ are the circumcenters of triangles $MBC$, $MCA$, $MAB$ respectively. Prove that the lines $A_{1}A_{2}$, $B_{1}B_{2}$, $C_{1}C_{2}$ meet at a point on the circle $(O)$.

Issue 431

  1. Which number is greater \[A=\left(1+\frac{1}{2013}\right)\left(1+\frac{1}{2013^{2}}\right)\ldots\left(1+\frac{1}{2013^{n}}\right)\] where $n$ is a positive integer, or ${\displaystyle B=\frac{2013^{2}-1}{2012^{2}-1}}$?.
  2. Given four points in the plane such that no pair of points has distance less than $\sqrt{2}$ cm. Prove that there exists two of them having a distance greater than or equal to $2$ cm. 
  3. Find the last two digits of the number  \[2003^{2004^{\mathstrut^{.^{.^{.^{2013}}}}}}.\]
  4. Find the maximum and minimum value of the expression \[P=27\sqrt{x}+8\sqrt{y}\] where $x,y$ are non-negative real numbers satisfying \[x\sqrt{1-y^{2}}+y\sqrt{2-x^{2}}=x^{2}+y^{2}.\]
  5. Let $ABCD$ be a cyclic quadrilateral, inscribed in circle $(O)$. $I$ and $J$ are the midpoints of $BD$ and $AC$ respectively. Prove that $BD$ is the angle bisector of angle $AIC$ if and only if $AC$ is the angle bisector of angle $BJD$. 
  6. Solve the following system of equations \[\begin{cases} x^{3}(1-x)+y^{3}(1-y) & =12xy+18\\ |3x-2y+10|+|2x-3y| & =10 \end{cases}.\]
  7. Determine the greatest value of the expression \[E=a^{2013}+b^{2013}+c^{2013},\] where $a,b,c$ are real numbers satisfying \[a+b+c=0,\quad a^{2}+b^{2}+c^{2}=1.\]
  8. Let $S.ABC$ be a triangular pyramid, $G$ is the centroid of the base triangle $ABC$, $O$ is the midpoint of $SG$. A moving plane $(\alpha)$ through $O$ meets the edges $SA$, $SB$, $SC$ at $A',B'C'$ respectively. Prove that \[\frac{SA'^{2}}{AA'^{2}}+\frac{SAB'}{BB'^{2}}+\frac{SC'^{2}}{CC'^{2}}\geq\frac{AA'^{2}}{SA'^{2}}+\frac{BB'^{2}}{SB'^{2}}+\frac{CC'^{2}}{SC'^{2}}.\]
  9. Find all natural numbers $n$ such that \[A=\left[\frac{n+3}{4}\right]+\left[\frac{n+5}{4}\right]+\left[\frac{n}{2}\right]+n^{2}+3n-1\] is a prime number, where $[x]$ denotes the greatest integer not exceeding $x$.
  10. Consider the real-valued function \[y=a\sin(x+2013)+\cos2014x\] where $a$ is given real number. Let $M,N$ be the greatest and smallest values respectively of this function over $\mathbb{R}.$ Prove that $M^{2}+N^{2}\geq2$.
  11. Let $\{a_{n}\}$ be a sequence given by \[a_{1}=\frac{1}{2},\quad a_{n+1}=\frac{a_{n^{2}}}{a_{n^{2}-a_{n}+1}},\,n=1,2,\ldots\] a) Prove that the sequence $\{a_{n}\}$ converges to a finite limit and find this limit.
    b) Let $b_{n}=a_{1}+a_{2}+\ldots+a_{n}$ for each positive integer $n$. Determine the integer part $[b_{n}]$ and the limit $\lim_{n\to\infty}b_{n}$.
  12. Given four points $A,B,C,D$ on circle $(ABC)$ and $M$ is a point not on this circle. Let $T_{i}$ be the triangle whose three vertices are $3$ of $4$ given points, except point $i$ ($i=A,B,C,D$). Let $H_{i}$ be the triangle whose vertices are the feet of the perpendicular drawn from $M$ onto the edges (or extended edges) of triangles $T_{i}$ ($i=A,B,C,D$). Prove that
    a) The circumcenter of triangles $H_{i}$ ($i=A,B,C,D$) lie on the same circle, centered at $O'$.
    b) When $D$ moves on the circle $(ABC)$, $O'$ always lie on a fixed circle.

Issue 432

  1. The first $2013$ natural numbers from $1$ to $2013$ are writeen in a line in some order. Substract one from the first number, two from the second ... and 2013 from the $2013^{\text{th}}$ number. Is the product of the resulting $2013$ numbers odd or even?.
  2. Let $ABC$ be an acute triangle with orthocenter $O$. $AO$ meets $BC$ at $D$. Points $E$ and $F$ are on sides $AB$ and $AC$ respectively such that $DE=DB$, $DF=DC$. Prove that $DA$ is the angle bisector of angle $EDF$.
  3. Find all positive integers $a,b$ ($a\geq2$, $b\geq2$) so that $a+b$ is amultiple of $4$ and \[\frac{a(a-1)+b(b-1)}{(a+b)(a+b-1)}=\frac{1}{2}.\]
  4. Find $x,y$ such that \[\begin{cases} x\sqrt{x}+y\sqrt{y} & =2\\ x^{3}+2y^{2} & \leq y^{3}+2x\end{cases}.\]
  5. Given a circle centered at $O$, and diameter $AB$. Point $C$, different from $A$ and $B$, is chosen on circle $(O)$. Point $P$ on $AB$ such that $BP=AC$. The perpendicular from $P$ to $AC$ meet $AC$ at $H$. The intenal angle bisector of angle $CAB$ intersects circle $(O)$ at $E$ and intersects $PH$ at $F$. $CF$ meets circle $(O)$ at $N$. Prove that $CN$ passes through the midpoint of $AP$.
  6. Let $a,b,c$ the positive real number. Prove the following inequality \begin{align*} & \left(\frac{1}{a}+\frac{2}{b+c}+\frac{3}{a+b+c}\right)^{2}+\left(\frac{1}{b}+\frac{2}{c+a}+\frac{3}{a+b+c}\right)^{2}+\left(\frac{1}{c}+\frac{2}{a+b}+\frac{3}{a+b+c}\right)^{2}\\ & \geq\frac{81}{a^{2}+b^{2}+c^{2}}.\end{align*}
  7. Triangle $ABC$ is inscribed in circle $(O)$, another circle $(O')$ touches $AB$, $AC$ at $P,Q$ respectively and touches circle $(O)$ at other points $M$, $N$. Points $E$, $D$, $F$ are the perpendicular feet of point $S$ on $AM$, $MN$, $NA$ respectively. Prove that $DE=DF$.
  8. The real numbers $a,b,c$ satisfying the condition that the polynomial \[P(x)=x^{4}+ax^{3}+bx^{2}+cx+1\] has at least one real root. Determine all triple $(a,b,c)$ such that $s^{2}+b^{2}+c^{2}$ is smallest possible.
  9. Let $a$ and $B$ be two real numbers such that $a^{p}-b^{p}$ is a positive integers for all prime number $p$. Prove that $a$ and $b$ are integers. 
  10. The sequence $\{u_{n}\}$ is given recursively as follows \[u_{1}=\frac{1}{1+a},\quad\frac{1}{u_{n+1}}=\frac{1}{u_{n}^{2}}-\frac{1}{u_{n}}+1,\,\forall n\geq1\] where $a\in\mathbb{R},$ $a\ne-1$. Let $$S_{n}=u_{1}+u_{2}+\ldots+u_{n},\quad P_{n}=u_{1}u_{2}\ldots u_{n}.$$ Determine the value of the following expression $aS_{n}+P_{n}$.
  11. Determine all funtions $f:\mathbb{R}^{+}\to\mathbb{R}^{+}$ such that
    • $f$ is a decreasing function on $\mathbb{R}^{+}$.
    • $f(2x)=2012^{-x}f(x)$, $\forall x\in\mathbb{R}^{+}$
    where $\mathbb{R}^{+}=(0,+\infty)$.
  12. Let $ABC$ be a triangle inscribed in circle $(O)$. $AD$ is a diameter of $(O)$. Point $E$ belongs to the opposite ray of ray $DA$. The perpendicular through $E$ o $AD$ meets $BC$ at $T$. $TP$ is a tangent line to $(O)$ such that $P$ and $A$ are on opposite sides of $BC$; $AP$ meets $TE$ at $Q$. $M$ is the midpoint of $AQ$; $TM$ meets $AB$, $AC$ at $X$, $Y$ respectively. Prove that $M$ is the midpoint of $XY$.

Issue 433

  1. Find all positive integers $x,y,z$ such that \[x^{2}+y^{3}+z^{4}=90.\]
  2. Let $ABC$ be an equilateral triangle whose altitudes are $AD$, $BE$ and $CF$. Suppose $M$ is an arbitrary point inside triangle $ABC$. $I$, $K$, $L$ are the perpendicualr feet from $M$ to $AD$, $BE$, $CF$. Prove that the sum $AI+BK+CL$ does not depend on the position of $M$.
  3. The rational numbers $a$, $b$ satisfy the identity \[a^{2013}+b^{2013}=2a^{1006}b^{1006}.\] Prove that the equation $x^{2}+2x+ab=0$ has two rational solutions.
  4. Find the minimum value of the expression \[P=(x^{4}+y^{4}+z^{4})\left(\frac{1}{x^{4}}+\frac{1}{y^{4}}+\frac{1}{z^{4}}\right),\] where $x,y,z$ are positive real numbers that satisfy $x+y\leq z$.
  5. Let $AH$ be the altitude from $A$ of right triangle $ABC$, right angle at $A$. Point $D$ on the oppostite ray of $HA$ such that $HA=2HD$. Point $E$ is the reflection of $B$ through $D$; $I$ is the midpoint of $AC$; $DI$ and $EI$ meet $BC$ at $M$ and $K$ respectively. Prove that $\widehat{BDK}=\widehat{MCD}$.
  6. Solve the equation \[\sqrt{x+\sqrt{x^{2}-1}}=\frac{27\sqrt{2}}{8}(x-1)^{2}\sqrt{x-1}.\]
  7. A convex quadrilateral $ABCD$ with area $S$ is inscribed in a circle whose radius is $R$ and $AB=a$, $BC=b$, $CD=c$, $DA=d$, $AC=e$. If there exists a circle touching all the opposite rays of the rays $BA$, $DA$, $CD$ and $CB$. Prove that
    a) $R=\dfrac{S\cdot e}{p^{2}-e^{2}}$,
    b) $a^{2}+b^{2}+c^{2}+d^{2}+\dfrac{8SR}{e}=2p^{2}$,
    where $2p=a+b+c+d$.
  8. Find the maximum value of the expression \[\alpha(\sin^{2}A+\sin^{2}B+\sin^{2}C)-\beta(\cos^{3}A+\cos^{3}B+\cos^{3}C)\] where $A$, $B$, $C$ are three angles of an acute triangle and $\alpha$, $\beta$ are two given positive numbers.
  9. Find the maximum area of a convex pectagon in the coordinate plane $Oxy$ having the following properties: all interior angles are the same, all vertices have integer coordinates, there exists a side that is parallel to the axis $Ox$, there are exactly $16$ points, including the vertices, with integer coordinates on its boundary.
  10. Find all continuous functions $f$ such that \[(x+y)f(x+y)=xf(x)+yf(y)+2xy,\,\forall x,y\in\mathbb{R}.\]
  11. Let $(a_{n})$ be a sequence where $a_{1}\in\mathbb{R}$ and $a_{n+1}=|a_{n}-2^{1-n}|$, $\forall n\in\mathbb{N}^{*}$. Find ${\displaystyle \lim_{n\to\infty}a_{n}}$.
  12. A right triangle $ABC$ with right angle at $C$ is inscribed in circle $(O)$. $M$ is an arbitrary point moving on circle $(O)$, different from $A$, $B$, $C$. Point $N$ is the reflection of $M$ in $AB$, $P$ is the perpendicular foot of $N$ to $AC$, $MP$ meets $(O)$ at a second point $Q$. Prove that the circumcenter of triangle $APQ$ lies on a fixed circle.

Issue 434

  1. Find the largest possible perfect square of the form $4^{27}+4^{1020}+4^{x}$, where $x$ is a natural number.
  2. Let $ABC$ be a right triangle with right angle at vertex $A$ and $\widehat{ABC}=54^{0}$. The median $AM$ meets the internal angle-bisector $CD$ at $E$. Prove that $CE=AB$.
  3. Find all prime numbers $p$ such that $p-1$ and $p+1$ each has exactly $6$ divisors.
  4. Solve the system of equations \[ \begin{cases} 3\sqrt{x}+2\sqrt{y}+\sqrt{z} & =\dfrac{1}{6}\sqrt{xyz}\\ 6\sqrt{xy}+2\sqrt{yz}+3\sqrt{zx} & =108+18\sqrt{x+4}+12\sqrt{y+9}+6\sqrt{z+36} \end{cases}.\]
  5. Let $AB$ and $AC$ be the tangent lines to a circle $(O)$ through an external point $A$ ($B$ and $C$ are the poins of tangency). The median $BM$ of triangle $ABC$ intersects $(O)$ at $D$, the ray $AD$ meets $(O)$ at $E$. Prove that $BE||AC$.
  6. Given that $a,b,c$ are the sides of a triangle, prove the inequality \begin{align*} \sqrt{a^{2}-(b-c)^{2}}+\sqrt{b^{2}-(c-a)^{2}}+\sqrt{c^{2}-(a-b)^{2}} & \leq \sqrt{ab}+\sqrt{bc}+\sqrt{ca} \\ & \leq a+b+c \end{align*}
  7. The circle $(O)$ is inscribed in a triangle $ABC$. The tangents to $(O)$ which are parallel to the sides of the triangle are drawn, they intersect these sides at points $M,N,P,Q,R$ and $S$ ($M,S\in AB$, $N,P\in AC$, $Q,R\in BC$). Let $l_{1}$, $l_{2}$, $l_{3}$ be the lengths of the internal angle-bisectors from $A,B,C$ of triangles $AMN$, $BSR$ and $CPQ$ respectively. Prove that \[\frac{1}{l_{1}^{2}}+\frac{1}{l_{2}^{2}}+\frac{1}{l_{3}^{2}}\geq\frac{81}{p^{2}},\] where $p$ denotes the semiperimeter of triangle $ABC$.
  8. It is given that in a triangle $ABC$, $\tan\dfrac{B}{2}\tan\dfrac{C}{2}=\dfrac{1}{3}$. Solve for $x$ \[x^{2}+x-\cos A-\frac{1}{4}\cos(B-C)=0.\]
  9. Find all real numbers $x$ such that \[\left\{ \frac{x^{2}+1}{x^{2}+x+1}\right\} =\frac{1}{2}\] where $\{a\}$ denote te fractional part of $a$, that is $\{a\}=a-[a]$.
  10. Consider the sequence $(x_{n})$, \[x_{n+1}=x_{n}+\frac{1}{x_{n}}+\frac{2}{x_{n}^{2}}+\ldots+\frac{2012}{x_{n}^{2012}}\quad(n\in\mathbb{N}^{*})\] where $x_{1}>0$ is given. Determine all values of $\alpha$ such that the sequence $(nx_{n}^{\alpha})$ has a non-zero limit.
  11. Find all functions $f:\mathbb{R}\to\mathbb{R}$ such that \[f(xf(y)+3y^{2})+f(3xy+y)=f(3y^{2}+x)+4xy-x+y\] for all $x,y\in\mathbb{R}$.
  12. Let $G$ be the centroid of a tetrahedron $ABCD$. Points $X,Y<Z,T$ are chosen on the faces $(BCD)$, $(CDA)$, $(DAB)$, and $(ABC)$ respectively such that $XY$, $YZ$, $ZT$, $TX$ are parallel to $GA$, $GB$, $GC$ and $GD$. Determine the volume ratio of the two tetrahedrons $ABCD$ and $XYZT$.

Issue 435

  1. The number $s=\overline{3\ldots3}^{2}+\overline{5\ldots54\ldots4}^{2}$, written in decimal system, consist of $n+1$ digits $3$, $n-1$ digits $5$ and $n$ digits $4$. Given that $s=r^{2}$, find the value of $r$.
  2. Let $AM$ be the median of triangle $ABC$. On the half-plane containing $C$ created by the side $AB$, draw line segment $AE$ perpendicular to $AB$ such that $AB=AE$. On the half-plane containing $B$ created by $AC$, draw $AF$ perpendiclar to $AC$ such that $AF=AC$. Prove that $EF=2AM$ and $EF\perp AM$.
  3. Consider $n$ positive integers $a_{1},a_{2},\ldots a_{n}$ ($n>1$) satisfying \[a_{1}+a_{2}+\ldots+a_{n}=a_{1}a_{2}\ldots a_{n}.\] a) Prove that for any given value of $n$, the above equation always has solution.
    b) Determine all values of $n$ such that the equation $a_{1}<a_{2}<\ldots<a_{n}$.
  4. The numbers $a,b,c$ satisfy \[ab+bc+ca=2013abc,\quad2013(a+b+c)=1.\] Find the sum $A=a^{2013}+b^{2013}+c^{2013}$. 
  5. Prove that if in a trapezium $ABCD$ ($AB||CD$), $AC+CB=AD+DB$, then $ABCD$ is an isosceles trapezium.
  6. Solve the inequality on $\mathbb{R}$ \[(\sqrt{13}-\sqrt{2x^{2}-2x+5}-\sqrt{2x^{2}-4x+4})(x^{6}-x^{3}+x^{2}-x+1)\geq0.\]
  7. The three angles of an acute triangle $ABC$ are such that $A>\dfrac{\pi}{4}$, $B>\dfrac{\pi}{4}$, $C>\dfrac{\pi}{4}$. Determine the smallese value of the expression \[\frac{\tan A-2}{\tan^{2}C}+\frac{\tan B-2}{\tan^{2}A}+\frac{\tan C-2}{\tan^{2}B}.\]
  8. Let $S.ABCD$ be a pyramid inscribed in a sphere centred at $O$, and $AB=a$, $CD=b$. Draw parallellograms $ADKB$ and $SDHC$. Determine the ratio $\dfrac{HK}{EF}$ in terms of $a$ and $b$, where $E$ is the point of intersection of $AD$ and $BC$, and $F$ is the point of intersecion of $AC$ and $BD$.
  9. How many positive integers $n$ are there such that $n$ has $2013$ digits in decimal number system and $\dfrac{n}{7}$ is a positive integer with $2013$ odd digits?.
  10. Find all funtions $f:\mathbb{R}\to\mathbb{R}$, $g:\mathbb{R}\to\mathbb{R}$ such that the following two conditions are satisfied
    • $f(x)-2g(x)=g(y)+4y$, for all $x,y\in\mathbb{R}$;
    • $f(x)g(x)\leq33x^{2}$, for all $x\in\mathbb{R}$.
  11. Find all polynomials $T(x,y)$ such that \[T(x,y)T(z,t)=T(xz+yt,xt+yz)\] for all $x,y,z,t\in\mathbb{R}$. 
  12. Let $ABCD$ be a cyclic quadrilateral. The circle whose diameter is $AB$ meets $CA$, $CB$, $DA$ and $DB$ at $E,F,I$ and $J$ respectively (all differ from $A$ and $B$). Prove that the angle-bisector of an angle between $EF$ and $IJ$ is perpendicular to the line $CD.$

Issue 436

  1. Let \[n=1234567891011\ldots99100.\] Delete $100$ digits so that the remaining digits in the original order, is greatest possible.
  2. Find all integers $x,y,z$ such that \[(x-y)^{3}+3(y-z)^{2}+5|z-x|=35.\]
  3. For what values of $a$ is the numbers $a+\sqrt{15}$ and $\dfrac{1}{a}-\sqrt{15}$ are both intergers?.
  4. Solve the equation \[\sqrt{6-x}+\sqrt{2x+6}+\sqrt{6x-5}=x^{2}-2x-5.\]
  5. Let $(O)$ be a circle of diameter $AB$. Point $I$ is outside the circle, $IH$ is the perpendicular line to $AB$ through $I$ ($H$ lies between $O$ and $A$). $IA$, $IB$ meet $(O)$ at points $E$ and $F$ respectively; $EF$ meets $AB$ at $P$; $EH$ meets $(O)$ at the second point $M$; $PM$ intersects $(O)$ at the second point $N$. Let $K$ denote the midpoint of $EF$, $O'$ is the circumcenter of triangle $HMN$. Prove that $O'H||OK$.
  6. Prove that in any triangle $ABC$, the following inequality holds \[\left(\frac{h_{a}}{l_{a}}-\sin\frac{A}{2}\right)\left(\frac{h_{b}}{l_{b}}-\sin\frac{B}{2}\right)\left(\frac{h_{c}}{l_{c}}-\sin\frac{C}{2}\right)\leq\frac{r}{4R},\] where $h_{a},h_{b},h_{c}$ and $l_{a},l_{b},l_{c}$ are the length of the altitudes and the internal angle bisector from $A,B$ and $C$ respectively; any $R,r$ are its circumradius and inradius.
  7. Let $S.ABC$ be a triangular pyramid where $\widehat{BAC}=90^{0}$, $BC=2a$, $\widehat{ACB}=\alpha$. The plane $(SAB)$ is perpendicular to $(ABC)$. Find the volume of this pyramid, given that $SAB$ is an isosceles triangle at vertex $S$ and $SBC$ is a right triangle.
  8. Solve the system of equations \[\begin{cases}x^{8}y^{8}+y^{4} & =2x\\ 1+x & =x(1+y)\sqrt{xy} \end{cases}.\]
  9. Let $A=2013^{30n^{2}+4n+2013}$ where $n\in\mathbb{N}$. Let $X$ be the set of remainders when dividing $A$ by $21$ for some natural number $n$. Determine the set $X$.
  10. Let $(x_{n})$ be a sequence of natural numbers with \[x_{1}=2,\quad x_{n+1}=\left[\frac{3}{2}x_{n}\right],\forall n=1,2,3,\ldots,\] where $[x]$ denotes the greates integer not exceeding $x$. Prove that in the sequence $(x_{n})$, there are infinitely many odd numbers and infinitely many even numbers.
  11. Given a real number $x\geq1$. Find the limit \[\lim_{n\to\infty}(2\sqrt[n]{x}-1)^{n}.\]
  12. Let $S$ denote the area of a given triangle, and let $P$ be an arbitrary point. $A',B',C'$ are the midpoints of $BC$, $CA$ and $AB$ respectively; $h_{a},h_{b},h_{c}$ denote the corresponding altitudes. Prove that \[PA^{2}+PB^{2}+PC^{2}\geq\frac{4}{\sqrt{3}}S\max\left\{ \frac{PA+PA'}{h_{a}},\frac{PB+PB'}{h_{b}},\frac{PC+PC'}{h_{c}}\right\}.\]

Issue 437

  1. Let $A$ be the sum (where $n!$ denote the product $1.2.3.\ldots n$) \[A=\frac{1}{1.1!}+\frac{1}{2.2!}+\ldots+\frac{1}{n.n!}+\ldots+\frac{1}{2013.2013!}.\] Prove that $A<\dfrac{3}{2}$.
  2. In a right triangle $ABC$ with right angle at $A$, let $D$ be the midpoint of $AC$, $E$ be the point on side $BC$ such that $BE=2CE$. Prove that $BD=3ED$.
  3. Determine all possible triples of integers $x,y,z$ satisfying the equation \[6(y^{2}-1)+3(x^{2}+y^{2}z^{2})+2(z^{2}-9x)=0.\]
  4. Let $a,b,c$ be three real numbers satisfying the conditions $a\ne0$ and $2a+3b+6c=0$. Find the smallest possible distance between the two roots of equation $ax^{2}+bc+c=0$.
  5. Given that in a triangle $ABC$, $AB=2$, $AC=3$, $BC=4$. Prove that \[\widehat{BAC}=\widehat{ABC}+2\widehat{ACB}.\]
  6. Let $a,b,c$ be positive real numbers satisfuing \[a+b+c=\frac{1}{a}+\frac{1}{b}+\frac{1}{c}.\] Prove the inequality \[3(a+b+c)\geq\sqrt{8a^{2}+1}+\sqrt{8b^{2}+1}+\sqrt{8c^{2}+1}.\]
  7. A circle centered at $O$ and radius $R$ circumscribes about a triangle $ABC$ whose sides are $AB=c$, $BC=a$, $CA=b$. Let $E$ be the center of the Euler circle of triangle $ABC$. Prove that if $a^{2}+b^{2}+c^{2}=5R^{2}$ then $E$ lies on the circle $(O;R)$.
  8. Solve the equation \[3^{x}-x-1=\log_{3}\frac{(2x+1)\log_{3}(2x+1)}{x}.\]
  9. For which positive integers $n$ will the following equation has positive integer solutions \[\frac{1}{x_{1}^{2}}+\frac{1}{x_{n}^{2}}+\ldots+\frac{1}{x_{n}^{2}}=4.\]
  10. Prove that in any triangle $ABC$, the following inequality holds \[\frac{\cos\left(\frac{B}{2}-\frac{C}{2}\right)}{\sin\frac{A}{2}}+\frac{\cos\left(\frac{C}{2}-\frac{A}{2}\right)}{\sin\frac{B}{2}}+\frac{\cos\left(\frac{A}{2}-\frac{B}{2}\right)}{\sin\frac{C}{2}} \leq 2\left(\frac{\tan\frac{A}{2}}{\tan\frac{B}{2}}+\frac{\tan\frac{B}{2}}{\tan\frac{C}{2}}+\frac{\tan\frac{C}{2}}{\tan\frac{A}{2}}\right).\]
  11. Let $(a_{n})$ be the sequence of real numbers such that \[a_{1}=34,\quad a_{n+1}=4a_{n}^{3}-104a_{n}^{2}-107a_{n}\] for all positive integers $n$. Find all prime numbers $p$ satisfying the following two conditions $p\equiv3(\text{mod }4)$ and $a_{2013}+1$ is divisible by $p$.
  12. Given five points $A,B,C,D,E$ on the same circle. Let $M$ be the midpoint of $DE$. The Euler circles of triangles $ADE$ and $BDE$ meet at $C'$ (different from $M$); the Euler circles of triangles $BDE$ and $CDE$ meet at $A'$ (different from $M$); the Euler circles of triangles $CDE$ and $ADE$ meet at $B'$ (different from $M$). Prove that the lines $AA'$, $BB'$ and $CC'$ are concurrent.

Issue 438

  1. Determine all triples of (not necessarily distinct) prome numbers $a,b,c$ such that \[a(a+1)+b(b+1)=c(c+1).\]
  2. In an isosceles right triangle $ABC$, right angle at vertex $A$, let $E$ be the midpoint of side $AC$. The line perpendicular to $BE$ through $A$ meets $BC$ at $D$. Prove that $AD=2ED$. 
  3. Find all positive integers $x,y$ and $t$ such that $t\leq6$ and the following equation is satisfied \[x^{2}+y^{2}-4x-2y-7t-2=0.\]
  4. Find the greatest integer not exceeding $A$, where \[A=\frac{1}{\sqrt{1}}+\frac{1}{\sqrt{2}}+\ldots+\frac{1}{\sqrt{2013}}.\]
  5. Let $ABC$ be an acute triangle with altitudes $BE$ and $CF$. The line segments $FI$ and $EJ$ are perpendicular to $BC$ ($I,J$ belong to $BC$). Point $K,L$ on $AB$, $AC$ respectively such that $KI||AC$, $LJ||AB$. Prove that the lines $EI$, $FJ$ and $KL$ are concurrent. 
  6. Solve for $x$ \[\sqrt{13}\sqrt{2x^{2}-x^{4}}+9\sqrt{2x^{2}+x^{4}}=32.\]
  7. Given that $x,y$ satisfy the conditions \[x-y\geq0,\,x+y\geq0,\,\sqrt{\left(\frac{x+y}{2}\right)^{3}}+\sqrt{\left(\frac{x-y}{2}\right)^{3}}=27.\] Find the smallest possible value of $x$.
  8. In a triangle $ABC$, let $m_{a},l_{b},l_{c}$ and $p$ denote the median length from $A$, the lengths of the internal angle bisectors from $B,C$ and its semiperimeter. Prove that \[m_{a}+l_{b}+l_{c}\leq p\sqrt{3}.\]
  9. a) Let $a_{1},a_{2},\ldots,a_{n}$ be $n$ rational numbers. Prove that if $a_{1}^{m}+a_{2}^{m}+\ldots+a_{n}^{m}$ is an integer for all positive integers $m$, then $a_{1},a_{2},\ldots a_{n}$ are integer numbers.
    b) Does the conclusion above remain valid if one only assumes that $a_{1},a_{2},\ldots a_{n}$ are real numbers?. 
  10. Find all continuous funtions $f:\mathbb{R}\to\mathbb{R}$ such that the following identity is satisfied for all $x,y$ \[ f(x)+f(y)+2=2f\left(\frac{x+y}{2}\right)+2f\left(\frac{x-y}{2}\right).\]
  11. Let $ABC$ be a triangle with side lengths $BC=a$, $AC=b$, $AB=c$ and median lengths $m_{a},m_{b},m_{c}$ drawn from vertex $A,B$ and $C$ respectively. Prove that for any point $M$ \begin{align*}\left(\frac{m_{b}}{b}+\frac{m_{c}}{c}\right)\frac{MA}{a}+\left(\frac{m_{c}}{c}+\frac{m_{a}}{a}\right)\frac{MB}{a}+\left(\frac{m_{a}}{a}+\frac{m_{b}}{b}\right)\frac{MC}{c} & \geq3,\\ \left(\frac{b}{m_{b}}+\frac{c}{m_{c}}\right)\frac{MA}{a}+\left(\frac{c}{m_{c}}+\frac{a}{m_{a}}\right)\frac{MB}{a}+\left(\frac{a}{m_{a}}+\frac{b}{m_{b}}\right)\frac{MC}{c} & \geq4.\end{align*}
  12. Let $ABC$ be an isosceles triangle at vertex $A$. A circle $\omega$ which touches the sides $AB,AC$ meets $BC$ at $K,L$. $AK$ meets $\omega$ at $M$. Let $P,Q$ be the reflection points of $K$ through points $B,C$ respectively. Let $O$ be the circumcenter of triangle $MPQ$. Prove that $M$, $O$ and the center of circle $\omega$ are collinear.

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Abel Albania AMM Amsterdam An Giang Andrew Wiles Anh APMO Austria (Áo) Ba Đình Ba Lan Bà Rịa Vũng Tàu Bắc Bộ Bắc Giang Bắc Kạn Bạc Liêu Bắc Ninh Bắc Trung Bộ Bài Toán Hay Balkan Baltic Way BAMO Bất Đẳng Thức Bến Tre Benelux Bình Định Bình Dương Bình Phước Bình Thuận Birch BMO Booklet Bosnia Herzegovina BoxMath Brazil British Bùi Đắc Hiên Bùi Thị Thiện Mỹ Bùi Văn Tuyên Bùi Xuân Diệu Bulgaria Buôn Ma Thuột BxMO Cà Mau Cần Thơ Canada Cao Bằng Cao Quang Minh Câu Chuyện Toán Học Caucasus CGMO China - Trung Quốc Chọn Đội Tuyển Chu Tuấn Anh Chuyên Đề Chuyên Sư Phạm Chuyên Trần Hưng Đạo Collection College Mathematic Concours Cono Sur Contest Correspondence Cosmin Poahata Crux Czech-Polish-Slovak Đà Nẵng Đa Thức Đại Số Đắk Lắk Đắk Nông Đan Phượng Danube Đào Thái Hiệp ĐBSCL Đề Thi Đề Thi HSG Đề Thi JMO Điện Biên Định Lý Định Lý Beaty Đỗ Hữu Đức Thịnh Do Thái Doãn Quang Tiến Đoàn Quỳnh Đoàn Văn Trung Đống Đa Đồng Nai Đồng Tháp Du Hiền Vinh Đức Duyên Hải Bắc Bộ E-Book EGMO ELMO EMC Epsilon Estonian Euler Evan Chen Fermat Finland Forum Of Geometry Furstenberg G. Polya Gặp Gỡ Toán Học Gauss GDTX Geometry Gia Lai Gia Viễn Giải Tích Hàm Giảng Võ Giới hạn Goldbach Hà Giang Hà Lan Hà Nam Hà Nội Hà Tĩnh Hà Trung Kiên Hải Dương Hải Phòng Hậu Giang Hậu Lộc Hilbert Hình Học HKUST Hòa Bình Hoài Nhơn Hoàng Bá Minh Hoàng Minh Quân Hodge Hojoo Lee HOMC HongKong HSG 10 HSG 10 Bắc Giang HSG 10 Thái Nguyên HSG 10 Vĩnh Phúc HSG 11 HSG 11 Bắc Giang HSG 11 Lạng Sơn HSG 11 Thái Nguyên HSG 11 Vĩnh Phúc HSG 12 HSG 12 2010-2011 HSG 12 2011-2012 HSG 12 2012-2013 HSG 12 2013-2014 HSG 12 2014-2015 HSG 12 2015-2016 HSG 12 2016-2017 HSG 12 2017-2018 HSG 12 2018-2019 HSG 12 2019-2020 HSG 12 2020-2021 HSG 12 2021-2022 HSG 12 Bắc Giang HSG 12 Bình Phước HSG 12 Đồng Tháp HSG 12 Lạng Sơn HSG 12 Long An HSG 12 Quảng Nam HSG 12 Quảng Ninh HSG 12 Thái Nguyên HSG 12 Vĩnh Phúc HSG 9 HSG 9 2010-2011 HSG 9 2011-2012 HSG 9 2012-2013 HSG 9 2013-2014 HSG 9 2014-2015 HSG 9 2015-2016 HSG 9 2016-2017 HSG 9 2017-2018 HSG 9 2018-2019 HSG 9 2019-2020 HSG 9 2020-2021 HSG 9 2021-202 HSG 9 2021-2022 HSG 9 Bắc Giang HSG 9 Bình Phước HSG 9 Đồng Tháp HSG 9 Lạng Sơn HSG 9 Long An HSG 9 Quảng Nam HSG 9 Quảng Ninh HSG 9 Vĩnh Phúc HSG Cấp Trường HSG Quốc Gia HSG Quốc Tế Hứa Lâm Phong Hứa Thuần Phỏng Hùng Vương Hưng Yên Hương Sơn Huỳnh Kim Linh Hy Lạp IMC IMO IMT India - Ấn Độ Inequality InMC International Iran Jakob JBMO Jewish Journal Junior K2pi Kazakhstan Khánh Hòa KHTN Kiên Giang Kim Liên Kon Tum Korea - Hàn Quốc Kvant Kỷ Yếu Lai Châu Lâm Đồng Lạng Sơn Langlands Lào Cai Lê Hải Châu Lê Hải Khôi Lê Hoành Phò Lê Khánh Sỹ Lê Minh Cường Lê Phúc Lữ Lê Phương Lê Quý Đôn Lê Viết Hải Lê Việt Hưng Leibniz Long An Lớp 10 Lớp 10 Chuyên Lớp 10 Không Chuyên Lớp 11 Lục Ngạn Lượng giác Lương Tài Lưu Giang Nam Lý Thánh Tông Macedonian Malaysia Margulis Mark Levi Mathematical Excalibur Mathematical Reflections Mathematics Magazine Mathematics Today Mathley MathLinks MathProblems Journal Mathscope MathsVN MathVN MEMO Metropolises Mexico MIC Michael Guillen Mochizuki Moldova Moscow MYM MYTS Nam Định Nam Phi National Nesbitt Newton Nghệ An Ngô Bảo Châu Ngô Việt Hải Ngọc Huyền Nguyễn Anh Tuyến Nguyễn Bá Đang Nguyễn Đình Thi Nguyễn Đức Tấn Nguyễn Đức Thắng Nguyễn Duy Khương Nguyễn Duy Tùng Nguyễn Hữu Điển Nguyễn Mình Hà Nguyễn Minh Tuấn Nguyễn Phan Tài Vương Nguyễn Phú Khánh Nguyễn Phúc Tăng Nguyễn Quản Bá Hồng Nguyễn Quang Sơn Nguyễn Tài Chung Nguyễn Tăng Vũ Nguyễn Tất Thu Nguyễn Thúc Vũ Hoàng Nguyễn Trung Tuấn Nguyễn Tuấn Anh Nguyễn Văn Huyện Nguyễn Văn Mậu Nguyễn Văn Nho Nguyễn Văn Quý Nguyễn Văn Thông Nguyễn Việt Anh Nguyễn Vũ Lương Nhật Bản Nhóm $\LaTeX$ Nhóm Toán Ninh Bình Ninh Thuận Nội Suy Lagrange Nội Suy Newton Nordic Olympiad Corner Olympiad Preliminary Olympic 10 Olympic 10/3 Olympic 11 Olympic 12 Olympic 24/3 Olympic 24/3 Quảng Nam Olympic 27/4 Olympic 30/4 Olympic KHTN Olympic Sinh Viên Olympic Tháng 4 Olympic Toán Olympic Toán Sơ Cấp PAMO Phạm Đình Đồng Phạm Đức Tài Phạm Huy Hoàng Pham Kim Hung Phạm Quốc Sang Phan Huy Khải Phan Thành Nam Pháp Philippines Phú Thọ Phú Yên Phùng Hồ Hải Phương Trình Hàm Phương Trình Pythagoras Pi Polish Problems PT-HPT PTNK Putnam Quảng Bình Quảng Nam Quảng Ngãi Quảng Ninh Quảng Trị Quỹ Tích Riemann RMM RMO Romania Romanian Mathematical Russia Sách Thường Thức Toán Sách Toán Sách Toán Cao Học Sách Toán THCS Saudi Arabia - Ả Rập Xê Út Scholze Serbia Sharygin Shortlists Simon Singh Singapore Số Học - Tổ Hợp Sóc Trăng Sơn La Spain Star Education Stars of Mathematics Swinnerton-Dyer Talent Search Tăng Hải Tuân Tạp Chí Tập San Tây Ban Nha Tây Ninh Thạch Hà Thái Bình Thái Nguyên Thái Vân Thanh Hóa THCS Thổ Nhĩ Kỳ Thomas J. Mildorf THPT Chuyên Lê Quý Đôn THPTQG THTT Thừa Thiên Huế Tiền Giang Tin Tức Toán Học Titu Andreescu Toán 12 Toán Cao Cấp Toán Chuyên Toán Rời Rạc Toán Tuổi Thơ Tôn Ngọc Minh Quân TOT TPHCM Trà Vinh Trắc Nghiệm Trắc Nghiệm Toán Trại Hè Trại Hè Hùng Vương Trại Hè Phương Nam Trần Đăng Phúc Trần Minh Hiền Trần Nam Dũng Trần Phương Trần Quang Hùng Trần Quốc Anh Trần Quốc Luật Trần Quốc Nghĩa Trần Tiến Tự Trịnh Đào Chiến Trường Đông Trường Hè Trường Thu Trường Xuân TST TST 2010-2011 TST 2011-2012 TST 2012-2013 TST 2013-2014 TST 2014-2015 TST 2015-2016 TST 2016-2017 TST 2017-2018 TST 2018-2019 TST 2019-2020 TST 2020-2021 TST 2021-2022 TST Bắc Giang TST Bình Phước TST Đồng Tháp TST Lạng Sơn TST Long An TST Quảng Nam TST Quảng Ninh TST Thái Nguyên TST Vĩnh Phúc Tuyên Quang Tuyển Sinh Tuyển Sinh 10 Tuyển Sinh 10 Bắc Giang Tuyển Sinh 10 Bình Phước Tuyển Sinh 10 Đồng Tháp Tuyển Sinh 10 Lạng Sơn Tuyển Sinh 10 Long An Tuyển Sinh 10 Quảng Nam Tuyển Sinh 10 Quảng Ninh Tuyển Sinh 10 Thái Nguyên Tuyển Sinh 10 Vĩnh Phúc Tuyển Sinh 2010-2011 Tuyển Sinh 2011-2012 Tuyển Sinh 2011-2022 Tuyển Sinh 2012-2013 Tuyển Sinh 2013-2014 Tuyển Sinh 2014-2015 Tuyển Sinh 2015-2016 Tuyển Sinh 2016-2017 Tuyển Sinh 2017-2018 Tuyển Sinh 2018-2019 Tuyển Sinh 2019-2020 Tuyển Sinh 2020-2021 Tuyển Sinh 2021-202 Tuyển Sinh 2021-2022 Tuyển Tập Tuymaada UK - Anh Undergraduate USA - Mỹ USA TSTST USAJMO USATST USEMO Uzbekistan Vasile Cîrtoaje Vật Lý Viện Toán Học Vietnam Viktor Prasolov VIMF Vinh Vĩnh Long Vĩnh Phúc Virginia Tech VLTT VMEO VMF VMO VNTST Võ Anh Khoa Võ Quốc Bá Cẩn Võ Thành Văn Vojtěch Jarník Vũ Hữu Bình Vương Trung Dũng WFNMC Journal Wiles Yên Bái Yên Định Yên Thành Zhautykov Zhou Yuan Zhe
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MOlympiad.NET: Mathematics and Youth Magazine Problems 2013
Mathematics and Youth Magazine Problems 2013
MOlympiad.NET
https://www.molympiad.net/2022/03/mym-2013.html
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BƯỚC 1: CHIA SẺ BÀI VIẾT NÀY LÊN MẠNG XÃ HỘI
STEP 2: CLICK THE LINK ON YOUR SOCIAL NETWORK
BƯỚC 2: BẤM VÀO ĐƯỜNG DẪN TRÊN MẠNG XÃ HỘI CỦA BẠN