Mathematics and Youth Magazine Problems 2013

This article has
views, Facebook comments and 0 Blogger comments. Leave a comment.

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.



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 2013
Mathematics and Youth Magazine Problems 2013
Loaded All Posts Not found any posts VIEW ALL Readmore Reply Cancel reply Delete By Home PAGES POSTS View All RECOMMENDED FOR YOU Tag ARCHIVE SEARCH ALL POSTS Not found any post match with your request Back Home Sunday Monday Tuesday Wednesday Thursday Friday Saturday Sun Mon Tue Wed Thu Fri Sat January February March April May June July August September October November December Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec just now 1 minute ago $$1$$ minutes ago 1 hour ago $$1$$ hours ago Yesterday $$1$$ days ago $$1$$ weeks ago more than 5 weeks ago Followers Follow THIS PREMIUM CONTENT IS LOCKED
Copy All Code Select All Code All codes were copied to your clipboard Can not copy the codes / texts, please press [CTRL]+[C] (or CMD+C with Mac) to copy Table of Content