Mathematics and Youth Magazine Problems 2011


Issue 403

  1. Given the sum $$S=\frac{5}{1.2 .3}+\frac{8}{2.3 .4}+\frac{11}{3.4 .5}+\ldots+\frac{6026}{2008.2009 .2010}.$$ Compare $S$ with $2$.
  2. Let $A B C$ be a triangle with $\widehat{B A C}=50^{\circ}$, $\widehat{A B C}=72^{\circ}$. Outside of the triangle $A B C,$ draw a triangle $B D C$ such that $\widehat{C B D}=28^{\circ}$; $\widehat{B C D}=22^{\circ}$. Find the measure of the angle $A D B$.
  3. Find all possible pair of integers $x$, $y$ satisfying the following condition $$x^{2}+y^{2}=(x-y)(x y+2)+9$$
  4. Given $a, b, c, d \in[0 ; 1]$ satisfies the following condition $$a+b+c+d=x+y+z+t=1.$$ Prove the inequality $$a x+b y+c z+d t \geq 54 a b c d.$$
  5. Let $A B C$ be a triangle with $\widehat{B A C}=45^{\circ}$. The attitudes $B D$ and $C E$ intersect at $H .$ Let $I$ be a midpoint of $D E .$ Prove that the line $H I$ goes through the centroid of the triangle $A B C$.
  6. Solve the equation $$\sqrt{x}+\sqrt[4]{x}+4 \sqrt{17-x}+8 \sqrt[4]{17-x}=34.$$
  7. A number is said to be an interesting number if it has $10$ digits, all are distinct, and is a multiple of $11111$. How many interesting numbers are there?
  8. Let $A_{1} A_{2} A_{3} \ldots A_{n}$ be a convex polygon $(n \geq 3)$ on the plane $(P)$ and let $S$ be a point outside $(P)$. Another plane $(\alpha)$ intersects the sides $S A_{1}, S A_{2}, \ldots, S A_{n}$ at $B_{1}, B_{2},\ldots, B_{n}$ respectively such that $$\frac{S A_{1}}{S B_{1}}+\frac{S A_{2}}{S B_{2}}+\ldots+\frac{S A_{n}}{S B_{n}}=a$$ where $a$ is a given positive number. Prove that such a plane $(\alpha)$ always contains a fixed point.
  9. Two circles $\omega_{1}$, $\omega_{2}$ intersect at points $A$, $B$. $C D$ is a common tangent line of $\omega_{1}$, $\omega_{2}$ $(C \in \omega_{1}, D \in \omega_{2}$) where point $B$ is closer to $C D$ than point $A$. $C B$ cuts $A D$ at $E$, $D B$ cuts $CA$ at $F$ and $E F$ cuts $A B$ at $N$. $K$ is the orthogonal projection of $N$ onto $C D$.
    a) Prove that $\widehat{C A B}=\widehat{D A K}$.
    b) Let $O$ be the circumcenter of the triangle $A C D$ and $H$ is the orthocenter of the triangle $K E F .$ Prove that $O$, $B$, $H$ are collinear
  10. Let $\left(x_{n}\right)$ be the sequence where $$x_{1}=5,\quad x_{n+1}=\frac{x_{n}^{2010}+3 x_{n}+16}{x_{n}^{2009}-x_{n}+11},\,n=1,2, \ldots$$ For each positive number $n,$ put $\displaystyle y_{n}=\sum_{i=1}^{n} \frac{1}{x_{i}^{2009}+7}$. Determine $\displaystyle \lim_{n\to\infty}y_{n}$.
  11. Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ satisfy the following equation $$f(f(x-y))=f(x) \cdot f(y)+f(x)-f(y)-x y,\,\forall x, y \in \mathbb{R}.$$
  12. For each $n \in \mathbb{N}$, let $a_{n}$ be a number of bijections $f:\{1,2, \ldots, n\} \rightarrow\{1,2, \ldots, n\}$ such that $f(f(k))=k$ for all $k \in\{1,2, \ldots, n\}$.
    a) Prove that $a_{n}$ is an even number for every $n \geq 2$.
    b) Prove that for every $n \geq 10$ and $n$ is divisible by $3$ then $a_{n}-a_{n-9}$ is divisible by $3$.

Issue 404

  1. Given seven distinct prime numbers $a$, $b$, $c$, $a+b+c$, $a+b-c$, $a-b+c$, $-a+b+c$ in which the sum of two of three numbers $a$, $b$, $c$ equals $800$. Let $d$ be the difference between the largest and the smallest number among these seven integers. What is the maximum value of $d ?$
  2. A triangle $A B C$ has sides $A B=2cm$, $A C=4cm$ and median $A M=\sqrt{3}cm$. Find the measure of the angle $B A C$, the length of side $B C$ and the area of triangle $A B C$.
  3. Find the largest natural number $k$ so that $n^{5}-2011 n$ is divisible by $k$ for all natural number $n$.
  4. Solve the equation $$\left(x^{4}-625\right)^{2}-100 x^{2}-1=0$$
  5. Let $A B C$ be an acute triangle $(A B \neq A C)$ inscribed in the circle $(O),$ and $H$ is its orthocenter. Let $d$ be an arbitrary line passing through $H$. Draw the line $d^{\prime}$ symmetric to $d$ through $B C$. Find the position of the line $d$ so that $d^{\prime}$ touches the circumcircle $(O)$.
  6. Find the largest constant $k$ such that $$\frac{a^{2}}{b}+\frac{b^{2}}{c}+\frac{c^{2}}{a} \geq k\left(a^{2}+b^{2}+c^{2}\right)$$ for all positive real numbers $a$, $b$, and $c$ whose sum equals $1 .$
  7. Let $A B C$ be a triangle inscribed in the circle $(O)$, and let $G$ be its centroid; $D$, $E$, and $F$ are the circumcenters of triangles $G B C$, $G C A$, $G A B$ respectively. Prove that $O$ is the centroid of $D E F$.
  8. Solve the equation $$\sqrt[3]{\cos 5 x+2 \cos x}-\sqrt[3]{2 \cos 5 x+\cos x}=2 \sqrt[3]{\cos x}(\cos 4 x-\cos 2 x).$$
  9. Let $x$, $y$, $z$ be real numbers such that $x \geq 1$, $y \geq 2$, $z \geq 3$ and $$\frac{1}{x+\sqrt{x-1}}+\frac{2}{y+\sqrt{y-2}}+\frac{3}{z+\sqrt{z-3}}=12$$ Find the maximum and minimum value of the function $f(x, y, z)=x+y+z$.
  10. Let $\left(x_{n}\right)$ be a sequence given by $$x_{1}=\frac{5}{2},\quad x_{n+1}=\sqrt{x_{n}^{3}-12 x_{n}+\frac{20 n+21}{n+1}},\,\forall n \in \mathbb{N}^{*}.$$ Prove that the sequence $\left(x_{n}\right)$ converges and find its limit.
  11. Find all functions $f: \mathbb{R} \rightarrow(0 ; 2011]$ such that $$f(x) \leq 2011\left(2-\frac{2011}{f(y)}\right),\,\forall x>y.$$
  12. Given four points $A_{I}$ $(i=1,2,3,4),$ no three of them are colinear and a point $M$ so that $A_{i}$ $(i=1,2,3,4)$ and $M$ do not lie on the same circle. Let $T_{i}$ be a triangle having $A_{j}$ $(j=1,2,3,4 ; j \neq i)$ as its vertices, $C_{i}$ is the circle (or the line) passing through the feet of the projections through $M$ onto three sides (or extended sides) of triangle $T_{i}$. Prove that $C_{I}$ $(i=1,2,3,4)$ have a common point.

Issue 405

  1. Which of the following two numbers is greater? $$A=\frac{326}{1955}+\frac{988}{1975}+\frac{662}{1985},\quad B=\dfrac{3951}{3950}+\dfrac{1}{5955}+\dfrac{1}{11730}.$$
  2. Let $A B C$ be an isosceles triangle with $\widehat{B A C}=96^{\circ} .$ A point $M$ is inside the triangle such that $\widehat{M B C}=12^{\circ}$, $\widehat{M C B}=24^{\circ} .$ Prove that $M A=M C$.
  3. Find the maximum value of the expression $P=\max \{a, b, c\}-\min \{a, b, c\}$ where $a, b, c$ are real numbers satisfying the condition $$a+b+c=a^{3}+b^{3}+c^{3}-3 a b c=2.$$
  4. Solve the equation $$21 x-25+2 \sqrt{x-2}=19 \sqrt{x^{2}-x+2}+\sqrt{x+1}$$
  5. $A B C$ is a triangle inscribed the circle $(O)$ with $\widehat{B A C}=60^{\circ}$, $A K$ is the angle-bisector of $\widehat{B A C}$ ($K$ is on the circle $(O)$). Let $F$ be the midpoint of $A K,$ the ray $O F$ meets the altitude $C E$ of triangle $A B C$ at $H .$ Prove that $B H$ is perpendicular to $A C$.
  6. Find the minimum value of the expression $$P=\left(5 a+\frac{2}{b+c}\right)^{3}+\left(5 b+\frac{2}{c+a}\right)^{3}+\left(5 c+\frac{2}{a+b}\right)^{3}$$ where $a, b, c$ are positive real numbers satisfying $a^{2}+b^{2}+c^{2}=3$
  7. $OABC$ is a trirectangular tetrahedron at vertex $O$. $O H$ is the altitude from $O$ of tetrahedron. Let $R$ be the circumradius of triangle $A B C .$ Prove that $O H \leq \dfrac{R \sqrt{2}}{2} .$ When does equality occur?
  8. a) Find all distinct permutations of the word $TOANHOCTUOITRE$.
    b) How many permutations are there that has three consecutive $T - TTT$?
    c) How many permutations are there without adjacent $T$s?
  9. Let $k$ be a positive integer, $\alpha$ is an arbitrary real number. Find the limit of sequence $\left(a_{n}\right)$ where $$a_{n}=\frac{\left[1^{k} \cdot \alpha\right]+\left[2^{k} \cdot \alpha\right]+\ldots+\left[n^{k} \cdot \alpha\right]}{n^{k+1}},\, n=1,2, \ldots$$ here the notation $[x]$ is the largest integer that does not exceed $x .$
  10. Find all functions $f: \mathbb{N} \rightarrow \mathbb{N}$ which satisfy the following conditions
    • $f$ is strictly increasing;
    • $f(f(n))=4 n+9$ for all $n \in \mathbb{N}^{*}$
    • $f(f(n)-n)=2 n+9$ for all $n \in \mathbb{N}^{*}$
  11. Does there exist a positive integer $n \geq 2$ so that $$f(x)=1+4 x+4 x^{2}+\ldots+4 x^{2 n}$$ is a perfect square polynomial?
  12. Let $A B C$ be a triangle inscribed the circle $(O)$ and $A^{\prime}$ is a fixed point on $(O)$. $P$ moves on $B C$, $K$ belongs to $A C$ so that $P K$ is always parallel to a fixed line $d$. The circumcircle of triangle $A P K$ cuts the circle $(O)$ at a second point $E$. $A E$ cuts $B C$ at $M$. $A^{\prime} P$ cuts the circle $(O)$ at a second point $N .$ Prove that the line $M N$ passes through a fixed point.

Issue 406


Issue 407

  1. Denote $T(a)$ the number of digits of the natural number $a$. If $T\left(5^{n}\right)-T\left(2^{n}\right)$ is even, is $n$ necessarily odd or even?
  2. A triangle $A B C$ has $\widehat{B A C}<90^{\circ}$ and $H A=B C$ where $H$ is its orthocenter. Find the measure of angle $B A C$.
  3. Find all pair of integers $x, y$ such that $$7^{x}+24^{x}=y^{2}$$
  4. Let $x, y, z$ be arbitrary positive real numbers, prove the inequality $$\frac{x^{2}-z^{2}}{y+z}+\frac{z^{2}-y^{2}}{x+y}+\frac{y^{2}-x^{2}}{z+x} \geq 0.$$ When does equality occur?
  5. From point $M$ outside circle $(O),$ draw tangent $M A$ and secant $M B C$ ($B$ is between $M$ and $C$). Let $H$ be the projection of $A$ onto $M O, K$ is the intersection of segment $M O$ with $(O)$. Prove that
    a) The quadrilateral $O H B C$ is cyclic.
    b) $B K$ is the internal angle-bisector of angle $H B M$
  6. Solve the system of equations $$\begin{cases}\sqrt{\dfrac{x^{2}+y^{2}}{2}}+\sqrt{\dfrac{x^{2}+x y+y^{2}}{3}} &=x+y \\ x \sqrt{2 x y+5 x+3} &=4 x y-5 x-3 \end{cases}$$
  7. Let $\left(x_{n}\right)$ be a sequence defined by $$3 x_{n+1}=x_{n}^{3}-2,\,n=1,2, \ldots.$$ For what values of $x_{1}$ does the sequence $\left(x_{n}\right)$ converge? Determine this limit when it converges.
  8. $S . A B C$ is a tetrahedron with isosceles perpendicular to plane $(A B C)$. $D$ is the midpoint of $B C .$ Let $\alpha$ is the angle between edge $S B$ and plane $(A B C)$; $\beta$ is the angle between edge $S B$ and plane $(S A D)$. Prove that $$\cos ^{2} \alpha+\cos ^{2} \beta>1$$
  9. Let $x, y, z$ be positive real numbers such that $x^{2}+y^{2}+z^{2}+2 x y z=1 .$ Prove the inequality $$8(x+y+z)^{3} \leq 10\left(x^{3}+y^{3}+z^{3}\right) + 11(x+y+z)(1+4 x y z)-12 x y z.$$
  10. Let $p$ be an odd prime, $n$ is a positive integer so that $(n, p)=1$. Find the number of tuples $\left(a_{1}, a_{2}, \ldots, a_{p-1}\right)$ such that the sum $\displaystyle\sum_{k=1}^{p-1} k a_{k}$ is divisible by $p,$ and $a_{1}, a_{2}, \ldots, a_{p-1}$ are natural numbers which do not exceed $n-1$.
  11. Find all the functions $f$ which is defined on $\mathbb{R},$ take value on $\mathbb{R}$ and satisfying the equation $f(x+y+f(y))=f(f(x))+2 y,$ for all real numbers $x$, $y$.
  12. Let $p, r, r_{a}, r_{b}, r_{c}$ be semiperimeter, inradius, and exradius opposite angles $A$, $B$, $C$ of triangle $A B C$ having side lengths $B C=a$, $C A=b$, $A B=c$. Prove the inequality $$\sqrt{a b}+\sqrt{b c}+\sqrt{c a} \geq p+\sqrt{r r_{a}}+\sqrt{r r_{b}}+\sqrt{r r_{c}}$$ When does equality hold?

Issue 408

  1. How many integers $n$ are there such that $-1964 \leq n \leq 2011$ and the fraction $\dfrac{n^{2}+2}{n+9}$ is reducible?
  2. Let $a_{1}, a_{2}, a_{3}, \ldots$ be a sequence satisfying the conditions $$a_{2}=3,\, a_{50}=300,\quad a_{n}+a_{n+1}=a_{n+2},\quad \forall n \geq 1 .$$ Find the sum of the first $48$ terms $S=a_{1}+a_{2}+\ldots+a_{48}$.
  3. Let $p$ be a prime number. Let $x, y$ be nonzero natural numbers such that $\dfrac{x^{2}+p y^{2}}{x y}$ is also a natural number. Prove that $$\frac{x^{2}+p y^{2}}{x y}=p+1$$
  4. Solve the system of equations $$\begin{cases}x-2 \sqrt{y+1} &=3 \\ x^{3}-4 x^{2} \sqrt{y+1}-9 x-8 y &=-52-4 x y \end{cases}$$
  5. Let $A B$ be a fixed line segment. Point $M$ is such that $M A B$ is an acute triangle. Let $H$ be the orthocenter of $M A B$, $I$ is the midpoint of $A B$, $D$ is the projection of $H$ onto $MI$. Prove that the product $M I$. $D I$ does not depend on the position of $M$.
  6. Let $a, b, c$ be real numbers such that $\sin a+\sin b+\sin c \geq \dfrac{3}{2}$. Prove the inequality $$\sin \left(a-\frac{\pi}{6}\right)+\sin \left(b-\frac{\pi}{6}\right)+\sin \left(c-\frac{\pi}{6}\right) \geq 0$$
  7. Denote by $[x]$ the largest integer not exceeding $x$. Solve the equation $$x^{2}-(1+[x]) x+2011=0.$$
  8. Let $E$ be the center of the nine-point circle (the Euler's circle) of triangle $A B C$ with edge-lengths $B C=a$, $A C=b$, $A B=c$; $E_{1}$, $E_{2}$, $E_{3}$ are respectively the projections of $E$ onto $B C$, $C A$, $A B$ and let $R$ be the circumradius of triangle $A B C$. Prove that $$\frac{S_{E_{1} E_{2} E_{3}}}{S_{A B C}}=\frac{a^{2}+b^{2}+c^{2}}{16 R^{2}}-\frac{5}{16}$$
  9. Find the minimum value of the expression $$\tan B+\tan C-\tan A \tan A+\tan C-\tan B \quad \tan A+\tan B-\tan C$$ where $A$, $B$, $C$ are three angles of an acute triangle $A B C$ and $C \geq A$
  10. a) Prove that for each positive integer $n,$ the equation $$x+x^{2}+x^{3}+\ldots+2011 x^{2 n+1}=2009$$ has a unique real root.
    b) Let $x_{n}$ be denote the real solution in part a). Prove that $0<x_{n}<\dfrac{2010}{2011}$.
  11. Let $\left(u_{n}\right)$ be a sequence given by $$u_{1}=2011,\quad u_{n+1}=\frac{\pi}{8}\left(\cos u_{n}+\frac{\cos 2 u_{n}}{2}+\frac{\cos 3 u_{n}}{3}\right),\, \forall n \geq 1.$$ Prove that the sequence $\left(u_{n}\right)$ has a finite limit.
  12. Let $A_{1} B_{1} C_{1} D_{1}$ and $A_{2} B_{2} C_{2} D_{2}$ be two squares in opposite direction (that is, if the vertices $A_{1}$, $B_{1}$, $C_{1}$, $D_{1}$ are in clockwise order, then $A_{2}$, $B_{2}$, $C_{2}$, $D_{2}$ are ordered counterclockwise) with centers $O_{1}$, $O_{2}$ suppose that $D_{2}$, $D_{1}$ are respectively in $A_{1} B_{1}$, $A_{2} B_{2}$. Prove that the lines $B_{1} B_{2}$, $C_{1} C_{2}$ and $O_{1} O_{2}$ are concurrent.

Issue 409

  1. Without taking common denominator, find the integer $x$ given that $$\left(\frac{2009}{2010}+\frac{2010}{2011}+\frac{2011}{2009}\right) \cdot(x-2011)>3 x-6033$$
  2. In a triangle $A B C$, $M$, $N$, $P$ are midpoints of sides $B C$, $C A$ and $A B$ respectively. Choose the points $A_{1}$, $A_{2}$, $B_{1}$, $B_{2}$, $C_{1}$, $C_{2}$ on the opposite rays of rays $B A$, $C A$, $C B$, $A B$, $A C$ and $B C$ respectively so that $B A_{1}=C A_{2}$, $C B_{1}=A B_{2}$, $A C_{1}=B C_{2} . A_{0}, B_{0}, C_{0}$ are midpoints of $A_{1} A_{2}, B_{1} B_{2}, C_{1} C_{2}$ respectively. Prove that the lines $A_{0} M, B_{0} N, C_{0} P$ meet at a common point.
  3. Let $b$ be a positive integer with the following properties
    • $b$ equals a sum of three squares.
    • $b$ possess a divisor of the form $a=3 k^{2}+3 k+1$ $(k \in \mathbb{N})$.
    Prove that $b^{n}$ is also a sum of three squares for any positive integer $n$.
  4. Given three non-negative numbers $a, b, c$, prove the inequality $$a+b+c \geq \frac{a-b}{b+2}+\frac{b-c}{c+2}+\frac{c-a}{a+2}.$$ When does equality hold?
  5. Let $A B C$ be a triangle with circumcircle $(O)$, $I$ is the midpoint of side $B C$. $M$ is chosen on $I C$ (differ from both $C$ and $I$). $A M$ meets $(O)$ at $D$. Point $E$ is on $B D$ such that $\widehat{B M E}=\widehat{M A I}$. $E M$ and $D C$ intersect at $F$. Prove that $$\frac{C F}{C D}=\frac{B E}{B D}$$
  6. Solve for $x$ $$\sqrt{\frac{x+2}{2}}-1=\sqrt[3]{3(x-3)^{2}}+\sqrt[3]{9(x-3)}$$
  7. Triangle $A B C$ is inscribed in a fixed circle $(O)$. The medians from $A$, $B$, $C$ meets $(O)$ at $A_{1}$, $B_{1}$, $C_{1}$ respectively. Which triangle makes the value of the expression $$p=\frac{A A_{1}^{2}+B B_{1}^{2}+C C_{1}^{2}}{A B^{2}+B C^{2}+C A^{2}}$$ minimum possible?
  8. In a triangle $A B C$, prove that $$\frac{\sin A \cdot \sin B}{\sin ^{2} \frac{C}{2}}+\frac{\sin B \cdot \sin C}{\sin ^{2} \frac{A}{2}}+\frac{\sin C \cdot \sin A}{\sin ^{2} \frac{B}{2}} \geq 9$$
  9. Let $A B C$ be a triangle with points $A'$, $B'$, $C'$ on sides $B C$, $C A$ and $A B$ respectively such that $$\frac{A^{\prime} B}{A^{\prime} C}=\frac{B^{\prime} C}{B^{\prime} A}=\frac{C^{\prime} A}{C^{\prime} B}.$$ $A A^{\prime}$ and $B B^{\prime}$ meet at $D$, $B B$ ' meets $C C$ ' at $E$ and $F$ is the intersection of $C C'$ and $A A '$. Parallel lines to $A A'$, $B B'$, $C C'$ through point $O$ in the interior of $A B C$ meet $B C$, $C A$, $A B$ at $A_{1}$, $B_{1}$, $C_{1}$ respectively. Prove that for any point $M$ $$A D\left(M A_{1}-O A_{1}\right)+B E\left(M B_{1}-O B_{1}\right)+C F\left(M C_{1}-O C_{1}\right) \geq 0$$
  10. Given $P=(n+1)^{7}-n^{7}-1$ $(n \in \mathbb{N})$. Prove that there are infinitely many natural numbers $n$ so that $P$ is a perfect square.
  11. Determine all functions $f: \mathbb{R} \rightarrow \mathbb{R}$, continuous on $\mathbb{R}$ such that $$f(x y)+f(x+y)=f(x y+x)+f(y),\, \forall x, y \in \mathbb{R}.$$
  12. $f: \mathbb{R} \rightarrow \mathbb{R}$ is a function with the following properties
    • $f(1)=2011$,
    • $f(x+1) f(x)=(f(x))^{2}+f(x)-1, \forall x \in \mathbb{R}$.
    Let $\displaystyle S_{1}=\sum_{i=1}^{n} \frac{1}{f(i)-1}$, $\displaystyle S_{2}=\sum_{i=1}^{n} \frac{1}{f(i)+1}$. Find $\displaystyle\lim_{n \rightarrow+\infty}\left(S_{1}+S_{2}\right)$.

Issue 410

  1. Find all pairs of coprime positive integers $x, y$ so that $$\frac{x+y}{x^{2}+y^{2}}=\frac{7}{25}$$
  2. Let $A B C$ be an equilateral triangle whose altitudes $A H$, $B K$ intersect at $G .$ The angle-bisector of angle $B K H$ meets $C G$, $A H$, $B C$ at $M$, $N$, $P$ respectively. Prove that $K M=N P$.
  3. Find the minimum value of the expression $S=2011 c a-a b-b c$ where $a, b, c$ satisfy $a^{2}+b^{2}+c^{2} \leq 2$.
  4. Let $A B C$ be an isosceles right triangle with right angle at $A .$ Let $M$, $N$, $O$ be respectively the midpoints of $A B$, $A C$, $B C$. The line perpendicular to $C M$ from $O$ cuts $M N$ at $G .$ Compare the lengths of the two segments $G M$ and $G N$.
  5. Solve the equation $$\sqrt{7 x^{2}+25 x+19}-\sqrt{x^{2}-2 x-35}=7 \sqrt{x+2}$$
  6. Let $A B C$ be a triangle. Let $A M$, $B N$, $C P$ be its internal angle-bisectors ($M \in B C$, $N \in C A$, $P \in A B$). Find the measure of angle $B A C$ so that $P M$ is perpendicular to $N M$
  7. Solve the equation $$(\sin x-2)\left(\sin ^{2} x-\sin x+1\right)=3 \sqrt[3]{3 \sin x-1}+1.$$
  8. Find all values of $a, b$ so that the equation $$x^{4}+a x^{3}+b x^{2}+a x+1=0$$ has at least one solution and the sum $a^{2}+b^{2}$ is smallest possible.
  9. Let $a, b, c$ be positive numbers. Prove that $$\left(a^{2012}-a^{2010}+3\right)\left(b^{2012}-b^{2010}+3\right)\left(c^{2012}-c^{2010}+3\right) \geq 9(a b+b c+c a).$$ When does equality occur?
  10. Let $A B C$ be a triangle. An arbitrary line cuts the lines $B C$, $C A$, $A B$ at $M$, $N$, $P$ respectively. Let $X$, $Y$, $Z$ be respectively the centroids of triangles $A N P$, $B P M$, $C M N$. Prove that $$S_{X Y Z}=\frac{2}{9} S_{A B C}$$
  11. Let $\left(a_{n}\right)$ $\left(n \in \mathbb{N}^{*}\right)$ be a sequence given by $$a_{1}=0 ,\, a_{2}=38,\, a_{3}=-90,\quad a_{n+1}=19 a_{n-1}-30 a_{n-2},\, \forall n \geq 3.$$ Prove that $a_{2011}$ is divisible by 2011 .
  12. For all positive integers $n$ greater than $2$. Find the number of functions $$f:\{1,2,3, \ldots, n\} \rightarrow\{1,2,3,4,5\}$$ satisfying $|f(k+1)-f(k)| \geq 3$ where $k \in\{1,2, \ldots, n-1\}$.

Issue 411

  1. The natural numbers $1,2 \ldots, 2011^{2}$ are arranged in some order in a $2011 \times 2011$ square table, each square contains one number. Prove that there exists two adjacent squares (that is two squares having a common edge or a common vertex) such that the difference between the corresponding assigned numbers is not smaller than $2012$.
  2. Find the value of the following $2009$-terms sum $$S=\left(1+\frac{1}{1.3}\right)\left(1+\frac{1}{2.4}\right)\left(1+\frac{1}{3.5}\right) \ldots\left(1+\frac{1}{2009.2011}\right)$$
  3. Find the integers $x$, $y$ satisfying the expression $$x^{3}+x^{2} y+x y^{2}+y^{3}=4\left(x^{2}+y^{2}+x y+3\right)$$
  4. $M$ is a point in the interior of a triangle $A B C .$ Let $P$, $Q$, $R$, $H$, $G$ be respectively the centroid of triangles $M B C$, $M A C$, $M A B$, $P Q R$, $A B C$. Prove that points $M$, $H$ and $G$ are colinear.
  5. $a$, $b$ and $c$ are positive real numbers whose sum is $3$. Prove the inequality $$\frac{4}{(a+b)^{3}}+\frac{4}{(b+c)^{3}}+\frac{4}{(c+a)^{3}} \geq \frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}$$
  6. The incircle $(I)$ of a triangle $A B C$ touches $B C$, $C A$, $A B$ at $D$, $E$, $F$ respectively. The line passing through $A$ and parallel to $B C$ meets $E F$ at $K$. $M$ is the midpoint of $B C$. Prove that $I M$ is perpendicular to $D K$.
  7. Solve the system of equations $$\begin{cases}\sqrt{\dfrac{x^{2}+y^{2}}{2}}+\sqrt{\dfrac{x^{2}+x y+y^{2}}{3}}&=x+y \\ x \sqrt{2 x y+5 x+3} &=4 x y-5 x-3\end{cases}$$
  8. Let $a, b, c$ be real numbers such that the equation $a x^{2}+b x+c=0$ has two real solutions, both are in the closed interval $[0 ; 1] .$ Find the maximum and mininum values of the expression $$M=\frac{(a-b)(2 a-c)}{a(a-b+c)}.$$
  9. Let $P(x)$ and $Q(x)$ be two polynomials with real coefficients, each has at least one real solution, so that $$P\left(1+x+Q(x)+(Q(x))^{2}\right)=Q\left(1+x+P(x)+(P(x))^{2}\right).$$ For any $x \in \mathbb{R}$. Prove that $P(x) \equiv Q(x)$.
  10. Let $a, b, c, d$ be positive numbers such that $a \geq b \geq c \geq d$ and $a b c d=1 .$ Find the smallest constant $k$ such that the following inequality holds $$\frac{1}{a+1}+\frac{1}{b+1}+\frac{1}{c+1}+\frac{k}{d+1} \geq \frac{3+k}{2}.$$
  11. Find all continuous functions $f: \mathbb{R} \rightarrow \mathbb{R}$ satisfying $$\{f(x+y)\}=\{f(x)+f(y)\}$$ for every $x, y \in \mathbb{R}$ ($[t]$ is the largest integer not exceed $t$ and $\{t\}=t-[t]$.)
  12. Let $A B C$ be a triangle, $P$ is an arbitrary point inside the triangle. Let $d_{a}$, $d_{b}$, $d_{c}$ be respectively the distances from $P$ to $B C$, $C A$, $A B$; $R_{a}$, $R_{b}$, $R_{c}$ are the circumradii of triangles $P B C$, $P C A$, $P A B$ respectively. Prove that $$\frac{\left(d_{a}+d_{b}+d_{c}\right)^{2}}{P A \cdot P B \cdot P C} \geq \frac{\sqrt{3}}{2}\left(\frac{\sin A}{R_{a}}+\frac{\sin B}{R_{b}}+\frac{\sin C}{R_{c}}\right)$$

Issue 412

  1. Pick $n$ numbers $(n \geq 2)$ from the first hundred natural numbers (from $1$ to $100$) so that the sum of any two distinct numbers is a multiple of $6 .$ What is the largest possible number $n$ so that this can be done?
  2. Given $A=\dfrac{5^{a}}{5^{b+c}}$ and $B=\dfrac{5^{a}+2011}{5^{b+c}+2011}$ where $a, b, c$ are the side lengths of a triangle. Compare $A$ and $B$.
  3. Do there exists three integers $x$, $y$ and $z$ such that $$|x-2005 y|+|y-2007 z|+|z-2009 x|=2011^{x}+2013^{y}+2015^{z} ?$$
  4. Determine the following sum of $2011$ terms $$S=\frac{1}{1^{4}+1^{2}+1}+\ldots+\frac{2011}{2011^{4}+2011^{2}+1}$$
  5. Given a circle $(O),$ a chord $B C$ ($B C$ is not a diameter) and point $A$ moving on the major arc $B C$. Draw a circle $\left(O_{1}\right)$ passing through $B$ and touches $A C$ at $A$, another circle $\left(O_{2}\right)$ passing through $C$ and touches $A B$ at $A .\left(O_{1}\right)$ meets $\left(O_{2}\right)$ at a second point $D$, different from $A$. Prove that line $A D$ always passes through a fixed point.
  6. A quadrilateral $A B C D$ with $A C \perp B D$ is inscribed in a fixed circle $(O ; R)$. Let $p$ be the perimeter of $A B C D$. Prove that $$\frac{A B^{2}}{p-A B}+\frac{B C^{2}}{p-B C}+\frac{C D^{2}}{p-C D}+\frac{D A^{2}}{p-D A} \geq \frac{4 R \sqrt{2}}{3}$$
  7. Solve the system of equations $$\begin{cases}(17-3 x) \sqrt{5-x}+(3 y-14) \sqrt{4-y} &=0 \\ 2 \sqrt{2 x+y+5}+3 \sqrt{3 x+2 y+11} &=x^{2}+6 x+13\end{cases}$$
  8. Prove that the following inequality holds for any triangles $A B C$ $$\cos ^{2} \frac{A-B}{2}+\cos ^{2} \frac{B-C}{2}+\cos ^{2} \frac{C-A}{2} \geq 24 \sin \frac{A}{2} \sin \frac{B}{2} \sin \frac{C}{2}.$$
  9. Two circles $\left(C_{1}\right),\left(C_{2}\right)$ are given such that the center $O$ of $\left(C_{2}\right)$ lies on $\left(C_{1}\right) .$ Let $C$, $D$ be their intersection points. Points $A$ and $B$ on $\left(C_{1}\right)$ and $\left(C_{2}\right)$ respectively such that $A C$ touches $\left(C_{2}\right)$ at $C$ and $B C$ touches $\left(C_{1}\right)$ at $C$. The line $A B$ intersects $\left(C_{2}\right)$ at $E$ and $\left(C_{1}\right)$ at $F$. $C E$ meets $\left(C_{1}\right)$ at $G, C F$ meets $G D$ at $H$. Prove that $G O$ intersects $E H$ at the circumcenter of triangle $D E F$.
  10. Let $a_{1}, a_{2} \ldots, a_{n}$ be $n$ positive real numbers such that $$a_{1}^{2}+a_{2}^{2}+\ldots+a_{k}^{2} \leq \frac{k(2 k-1)(2 k+1)}{3}$$ where any $k=\overline{1, n}$. Find the largest possible value of the expression $$P=a_{1}+2 a_{2}+\ldots+n a_{n}.$$
  11. Given a sequence $\left(x_{n}\right)$ such that $$x_{n}=2 n+a \sqrt[3]{8 n^{3}+1}, \forall n=1,2, \ldots$$ where $a$ is any real number.
    a) For what values of $a$ does the sequence has finite limit?
    b) Find $a$ such that $\left(x_{n}\right)$ is eventually increasing.
  12. Find all function $f: \mathbb{R} \rightarrow \mathbb{R}$ such that $$f\left(x^{3}+y^{3}\right)=x^{2} f(x)+y^{2} f(y)$$ where $x, y \in \mathbb{R}$

Issue 413

  1. On the cardboards, Write each five-digit numbers, from $11111$ to $99999$, on a cardboard. After mixing the cardboards, place them in a sequence in certain order. Prove that the resulting number is not a power of $2 .$
  2. Find all triple of pairwise distinct prime numbers $a, b, c$ such that $$20 a b c<30(a b+b c+c a)<21 a b c.$$
  3. Point $O$ on the median $A D$ of triangle $A B C$ is chosen such that $\dfrac{A O}{A D}=k$ $(0<k<1)$. The rays $B O$, $C O$ cut $A C$, $A B$ at $E$, $F$ respectively. Determine the value of $k$ so that $$S_{A E O F}=\frac{1}{15} S_{A B C}.$$
  4. Solve the equation $$\sqrt{x^{2}+x+19}+\sqrt{7 x^{2}+22 x+28}+\sqrt{13 x^{2}+43 x+37}=3 \sqrt{3}(x+3).$$
  5. Let $A B C$ be a right triangle with right angle at $A$. $D$ is a point within the triangle so that $C D=C A$. Choose point $M$ on the edge $A B$ so that $\widehat{B D M}=\dfrac{1}{2} \widehat{A C D}$; $N$ is the intersection of $M D$ and the altitude $A H$ of triangle $A B C$. Prove that $D M=D N$.
  6. Let $A B C$ be a triangle inscribed the circle $(O)$. $F$ is an arbitrary point on the arc $\widehat{A B}$ (not containing $C$) $(F$ differs from $A$ and $B$). $M$ is the midpoint of the arc $\widehat{B C}$ (not containing $A$); $N$ is the midpoint of the arc $\widehat{A C}$ (not containing $B$). The line passing through $C$ and parallel to $M N$ cuts the circle $(O)$ at another point $P .$ Let $I$, $I_{1}$, $I_{2}$ be the incenters of triangles $A B C$, $F A C$, $F B C$. $P I$ cuts the circle $(O)$ at $G$. Prove that the four points $I_{1}$, $F$, $G$, $I_{2}$ are concyclic.
  7. Solve the equation $$(2 \sin x-3)\left(4 \sin ^{2} x-6 \sin x+3\right)=1+3 \sqrt[3]{6 \sin x-4}.$$
  8. $x, y, z,$ and $t$ are four real numbers longing to the interval $\left[\frac{1}{2} ; \frac{2}{3}\right]$. Find the least and greatest values of the expression $$P=9\left(\frac{x+z}{x+t}\right)^{2}+16\left(\frac{x+t}{x+y}\right)^{2}$$
  9. Prove that given any prime number $p,$ there exist natural numbers $x$, $y$, $z$, $t$ so that $x^{2}+y^{2}+z^{2}-t p=0$ and $0<t<p$.
  10. Let $\left(u_{n}\right)$ be a sequence given by $$u_{1}=a ,\quad u_{n+1}=\frac{(\sqrt{2}+1) u_{n}-1}{\sqrt{2}+1+u_{n}},\,\forall n \geq 1.$$ a) Find the condition of $a$ so that all terms in the sequence are well-defined.
    b) Find the value of $a$ such that $u_{2011}=2011$.
  11. Find all functions $f: \mathbb{N}^{*} \rightarrow \mathbb{N}^{*}$ satisfying the following condition $$\frac{f(x+y)+f(x)}{2 x+f(y)}=\frac{2 y+f(x)}{f(x+y)+f(y)}, \forall x, y \in \mathbb{N}^{*}$$
  12. Let $A B C$ be a triangle, $\widehat{B A C} \neq 90^{\circ}$. $D$ is a fixed point on the edge $B C$. $P$ is a point inside the triangle $A B C$. Let $B_{1}$, $C_{1}$ be respectively the projections of $P$ onto $A C$, $A B$. $D B_{1}$ cuts $A B$ at $C_{2}$, $D C_{1}$ cut $A C$ at $B_{2}$. $Q$ is the intersection differs from $A$ of the circumcircles of triangles $A B_{1} C_{1}$ and $A B_{2} C_{2}$. Prove that the line $P Q$ always go through a fixed point when $P$ is moving.

Issue 414

  1. Do there exist two natural numbers $a, b$ such that $$(3 a+2 b)(7 a+3 b)-4=\overline{22} * 12 * 2011 ?$$
  2. Equilateral triangles $A B E$ and $B C F$ are constructed outside triangle $A B C .$ Let $G$ be the centroid of triangle $A B E$ and $I$ be the midpoint of $A C .$ Find the measure of angle $G I F$.
  3. Find the smallest positive integer $n$ such that $2^{n}-1$ is divisible by $2011$.
  4. Prove that for all integers $k$, the equation $$x^{4}-2010 x^{3}+(2009+k) x^{2}-2007 x+k=0$$ does not have two distinct integer roots.
  5. From a point $M$ outside the cycle $(O),$ draw the tangents $M A, M B$ and the secant $M C D$ to $(O), C$ lies between $M$ and $D .$ $A B$ cuts $C D$ at $N .$ Prove that $$\frac{1}{M D}+\frac{1}{N D}=\frac{2}{C D}$$
  6. Let $A B C$ be a right triangle, right angle at $A,$ satisfying $A B+\sqrt{3} A C=2 B C$. Find the position of point $M$ such that $$4 \sqrt{3} \cdot M A+3 \sqrt{7} \cdot M B+\sqrt{39} \cdot M C$$ is smallest possible.
  7. Solve the equation $$\log _{3}\left(7^{x}+2\right)=\log _{5}\left(6^{x}+19\right)$$
  8. Let $A B C$ be a triangle satisfying $$\tan \frac{A}{2} \tan \frac{B}{2}=\frac{1}{2}.$$ Prove that $A B C$ is a right triangle iff $$\sin \frac{A}{2} \sin \frac{B}{2} \sin \frac{C}{2}=\frac{1}{10}.$$
  9. Let $A B C$ be a triangle inscribed the circle $(O ; R), M$ is a point not on the circle respectively. Let $r$, $r_{1}$ be respectively the radii of the incircles of triangles $A B C$ and $A_{1} B_{1} C_{1}$. Prove that $$\left|R^{2}-O M^{2}\right| \geq 4 r \cdot r_{1}$$
  10. Find the greatest positive constant $k$ satisfying the inequality $$ \frac{k}{a^{3}+b^{3}}+\frac{1}{a^{3}}+\frac{1}{b^{3}} \geq \frac{16+4 k}{(a+b)^{3}}.$$
  11. Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that $$f(x f(y)+y)+f(x y+x)=f(x+y)+2 x y.$$
  12. For each positive integer $n$ consider a function $f_{n}$ in $\mathbb{R}$ defined by $$f_{n}(x)=\sum_{i=1}^{2 n} x^{i}+1.$$ Prove the following statements
    a) $f_{n}$ obtains its minimum value at a unique point $x_{n},$ for each positive integer $n .$ Put $S_{n}=f_{n}\left(x_{n}\right)$.
    b) $S_{n}>\dfrac{1}{2}$ for all $n \in \mathbb{N}^{*} .$ Moreover, $\dfrac{1}{2}$ is the best constant possible in the sense that there does not exist any real number $a>\dfrac{1}{2}$ such that $S_{n}>a$ for all $n \in \mathbb{N}^{*}$.
    c) The sequence $\left(S_{n}\right)$ $(n=1,2, \ldots)$ is decreasing and $\displaystyle\lim_{n\to\infty} S_{n}=\dfrac{1}{2}$.
    d) $\displaystyle\lim_{n\to\infty} x_{n}=-1$.
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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 2011
Mathematics and Youth Magazine Problems 2011
MOlympiad.NET
https://www.molympiad.net/2022/04/mym-2011.html
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