Mathematics and Youth Magazine Problems 2007


Issue 355

  1. Let $a, b$ be two natural numbers satisfying $$2006 a^{2}+a=2007 b^{2}+b.$$ Prove that $a-b$ is a perfect square.
  2. Let $A B C$ be a triangle with $\widehat{B A C}=90^{\circ}, \widehat{A B C}$ $=60^{\circ} .$ Take the point $M$ on the side $B C$ such that $A B+B M=A C+C M$. Caculate the measure of $\widehat{C A M}$
  3. Find all positive integers $x, y$ greater than 1 so that $2 x y-1$ divisible by $(x-1)(y-1)$.
  4. Prove that $$\frac{a^{4} b}{2 a+b}+\frac{b^{4} c}{2 b+c}+\frac{c^{4}}{2 c+a} \geq 1$$ where $a, b, c$ are positive numbers satisfying the condition $a b+b c+c a \leq 3 a b c$. When does equality occur?
  5. Let be given two circles $\left(O_{1}\right)$, $\left(O_{2}\right)$ with centers $O_{1}$, $O_{2}$ with distinct radii, externally touching each other at a point $T$. Let $O_{1} A$ be a tangent to $\left(O_{2}\right)$ at a point $A,$ let $O_{2} B$ be a tangent to $\left(O_{1}\right)$ at a point $B$ so that the points $A, B$ are on the same side with respect to the line $O_{1} O_{2} .$ Let $H$ be the point on $O_{1} A, K$ be the point on $O_{2} B$ so that the lines $B H, A K$ are perpendicular to $O_{1} O_{2}$. The line $T H$ cuts $\left(O_{1}\right)$ again at $E,$ the line $T K$ cuts $\left(O_{2}\right)$ againt at $F$. The line $E F$ cuts $A B$ at $S$. Prove that the lines $O_{1} A$, $O_{2} B$ and $T S$ are concurrent.
  6. Let $S$ be a set consisting of 43 distinct positive integers not exceeding $100 .$ For each subset $X$ of $S$ let $t_{X}$ be the product of its elements. Prove that there exist two disjoint substs $A$ and $B$ of $S$ such that $t_{A} t_{B}^{2}$ is the cube of a natural numbers.
  7. Find the greast value of the expression $$\frac{a}{c}+\frac{b}{d}+\frac{c}{a}+\frac{d}{b}-\frac{a b c d}{(a b+c d)^{2}}$$ where $a, b, c d$ are distinct real numbers satisfying the conditions $\dfrac{a}{b}+\dfrac{b}{c}+\dfrac{c}{d}+\dfrac{d}{a}=4$ and $a c=b d$
  8. a) Let $f(x)$ be a polynomial of degree $n$ with leading coefficient $a$. Suppose that $f(x)$ has $n$ distinct roots $x_{1}, x_{2}, \ldots, x_{n}$ all not equal to zero. Prove that $$\frac{(-1)^{n-1}}{a x_{1} x_{2} \ldots x_{n}} \sum_{k=1}^{n} \frac{1}{x_{k}}=\sum_{k=1}^{n} \frac{1}{x_{k}^{2} f^{\prime}\left(x_{k}\right)}.$$ b) Does there exist a polynomial $f(x)$ of degree $n,$ with leading coefficient $a=1,$ such that $f(x)$ has $n$ distinct roots $x_{1}, x_{2}, \ldots, x_{n},$ all not equal to zero, satisfying the condition $$\frac{1}{x_{1} f^{\prime}\left(x_{1}\right)}+\frac{1}{x_{2} f^{\prime}\left(x_{2}\right)}+\ldots+\frac{1}{x_{n} f^{\prime}\left(x_{n}\right)}+\frac{1}{x_{1} x_{2} \ldots x_{n}}=0 ?$$
  9. Let $A D$, $B E$, $C F$ be the altitudes and $H$ be the orthocenter of an acute triangle $A B C .$ Let $M$, $N$ be respectively the points of intersection of $D E$ and $C F$ and of $D E$ and $B E$. Prove that the line passing through $A$ perpendicular to the line $M N$ passes through the circumcenter of triangle $B H C$.

Issue 356

  1. Caculate the following sum $S$ of 1002 terms $$S=\frac{1.3}{3.5}+\frac{2.4}{5.7}+\ldots+\frac{(n-1)(n+1)}{(2 n-1)(2 n+1)}+\ldots+\frac{1002.1004}{2005.2007}.$$
  2. Let $B E$ and $C F$ be two altitudes of a triangle $A B C .$ Prove that $A B=A C$ when and only when $A B+B E=A C+C F$.
  3. Let $A$ be a natural number greater than $9$, written in decimal system with digits $1,3,7,9$. Prove that $A$ has at least a prime divisor not less than $11 .$
  4. Find the least value of the expression $$P=\frac{b_{1}+b_{2}+b_{3}+b_{4}+b_{5}}{a_{1}+a_{2}+a_{3}+a_{4}+a_{5}}$$ where $a_{1}, a_{2}, a_{3}, a_{4}, a_{5}, b_{1}, b_{2}, b_{3}, b_{4}, b_{5}$ are non negative real numbers satisfying the conditions $a_{i}^{2}+b_{i}^{2}=1$ $(i=1,2,3,4,5)$ and $a_{1}^{2}+a_{2}^{2}+a_{3}^{2}+a_{4}^{2}+a_{5}^{2}=1$
  5. Let be given a nonright-angled triangle $A B C$ with its altitudes $A A^{\prime}$, $B B^{\prime}$, $C C^{\prime}$. Let $D$, $E$, $F$ be the centers of the escribed circles of the trangles $A B^{\prime} C^{\prime}$, $B C^{\prime} A^{\prime}$, $C A^{\prime} B^{\prime}$ opposite $A$, $A^{\prime}$, $A^{\prime}$ respectively. The escribed circle of triangle $A B C$ opposite $A$ touches the lines $B C$, $C A$, $A B$ at $M$, $N$, $P$ respectively. Prove that the circumcenter of triangle $D E F$ is the orthocenter of triangle $M N P$.
  6. Ten teams participated in a football competition where each team played against every other team exactly once. When the competition was over, it turned out that for every three teams $A$, $B$, $C$ if $A$ defeated $B$ and $B$ defeated $C$ then $A$ defeated $C$. Prove that there were four teams $A$, $B$, $C$, $D$ such that $A$ defeated $B$, $B$ defeated $C$, $C$ defeated $D$ or such that each match between them was a draw.
  7. Prove that for abitrary positive numbers $a, b, c$ such that $a b c \geq 1,$ we have $$a+b+c \geq \frac{1+a}{1+b}+\frac{1+b}{1+c}+\frac{1+c}{1+a}.$$ When does equality occur?
  8. The sequence of numbers $\left(u_{n}\right) ; n=1$, $2, \ldots,$ is defined by : $u_{0}=a$ and $$u_{n+1}=\sin ^{2}\left(u_{n}+11\right)-2007$$ for all natural number $n,$ where $a$ is a given real number. Prove that
    a) The equation $\sin ^{2}(x+11)-x=2007$ has a unique solution. Denote it by $b$.
    b) $\lim u_{n}=b$
  9. Let $A B C$ be a nonregular triangle. Take three points $A_{1}$, $B_{1}$, $C_{1}$ lying on the sides $B C$. $C A$, $A B$ respectively such that $\dfrac{B A_{1}}{B C}=\dfrac{C B_{1}}{C A}=\dfrac{A C_{1}}{A B}$. Prove that if the triangles $A B_{1} C_{1}$, $B C_{1} A_{1}$, $C A_{1} B_{1}$ have equal circumradii then $A_{1}$, $B_{1}$, $C_{1}$ are the midpoints of the sides $B C$, $C A$, $A B$ respectively.

Issue 357

  1. Let be given two bottles, the first bottle contains 1 liter of water, the second bottle is empty. One pours $\frac{1}{2}$ of the quantity of water contained in the first bottle into the second one, then one pours $\frac{1}{3}$ of the quantity of water contained in the second bottle into the first one, then one pours $\frac{1}{4}$ of the quantity of water contained in the first bottle into the second one, and so on, one pours $\frac{1}{5},$ then $\frac{1}{6},$ then $\frac{1}{7}, \ldots$ After the $2007^{\mathrm{th}}$ turn of such pouring, what are the quantities of water left in each bottle?
  2. Let $A B C$ be a right-angled triangle with right angle at $A$ and let $I$ be the point of intersection of its innner angled-bisectors. Take the orthogonal projection $E$ of $A$ on the line $B I$ then the orthogonal projection $F$ of $A$ on the line $C E .$ Prove that $2 E F^{2}=A I^{2}$
  3. Find all finite subset $A \subset \mathbb{N}^{*}$ such that there exists a finite subset $B \subset \mathbb N^{*}$ containning $A$ so that the sum of the numbers in $B$ is equal to the sum of the squares of the numbers in $A$.
  4. Prove that for every natural number $n \geq 2,$ we have $$1+\sqrt{1+\frac{4}{3 !}}+\sqrt[3]{1+\frac{9}{4 !}}+\ldots+\sqrt[n]{1+\frac{n^{2}}{(n+1) !}}<n+\frac{1}{2}$$ where $n !$ denotes $1.2 .3 \ldots n$
  5. Let $A B C D$ be a square inscribed in the circle $(O) .$ On the minor arc $\widehat{B C},$ take an arbitrary point $M$ distinct from $B, C .$ The lines $C M$ and $D B$ intersect at a point $E,$ the lines $D M$ and $A B$ intersect at a point $F$. Prove that the triangles $A B E$ and $D O F$ have equal areas.
  6. Prove that for every couple of positive integers $n, k$ the number $(\sqrt{n}-1)^{k}$ can be written in the form $\sqrt{a_{k}}-\sqrt{a_{k}-(n-1)^{k}}$ with $a_{k} \in \mathbb{N}^{*}$
  7. Prove that for every triangle $A B C$, we have $$\frac{3 \sqrt{3}}{2 \cos \frac{A}{2} \cos \frac{B}{2} \cos \frac{C}{2}}+8 \sin \frac{A}{2} \sin \frac{B}{2} \sin \frac{C}{2} \geq 5.$$ When does equality occur?
  8. Consider the sequence of numbers $\left(x_{n}\right)(n=0,1,2, \ldots)$ defined as follows $x_{0}$, $x_{1}$, $x_{2}$ are given positive numbers; $$x_{n+2}=\sqrt{x_{n+1}}+\sqrt{x_{n}}+\sqrt{x_{n-1}},\, \forall n \geq 1 .$$ Prove that the sequence $\left(x_{n}\right)(n=0,1,2, \ldots)$ has a finite limit and find its limit.
  9. Let be given a tetrahedron $A_{1} A_{2} A_{3} A_{4}$ Its inscribed sphere has center $I$, has radius $r$ and touches the faces opposite the vertices $A_{1}$, $A_{2}$, $A_{3}$, $A_{4}$ at $B_{1}$, $B_{2}$, $B_{3}$, $B_{4}$ respectively. Let $h_{1}$, $h_{2}$, $h_{3}$, $h_{4}$ be the measures of the altitudes of $A_{1} A_{2} A_{3} A_{4}$ issued from $A_{1}$, $A_{2}$, $A_{3}$, $A_{4}$ respectively. Prove that for every point $M$ in space, we have $$\frac{M B_{1}^{2}}{h_{1}}+\frac{M B_{2}^{2}}{h_{2}}+\frac{M B_{3}^{2}}{h_{3}}+\frac{M B_{4}^{2}}{h_{4}} \geq r.$$

Issue 358

  1. Let $a, b, c$ be positive numbers such that $a^{3}+b^{3}=c^{3}$. Compare $a^{2007}+b^{2007}$ and $c^{2007}$
  2. Let $A B C$ be an isosceles triangle at $A$. Let $E$ be an arbitrary point on the side $B C$. The line through $E$ perpendicular to $A B$ meets with the line through $C$ perpendicular to $A C$ at a point denoted by $D$. Let $K$ be the midpoint of $B E$ Find the measure angle $\widehat{A K D}$.
  3. Find the least positive integer $n$ $(n>1)$ such that $\dfrac{1^{2}+2^{2}+\ldots+n^{2}}{n}$ is a perfect square number.
  4. Prove the following inequality $$\left(1+\frac{2 a}{b}\right)^{2}+\left(1+\frac{2 b}{c}\right)^{2}+\left(1+\frac{2 c}{a}\right)^{2} \geq \frac{9(a+b+c)^{2}}{a b+b c+c a}$$ where $a, b, c$ are arbitrary positive real numbers. When does equality occur?
  5. Solve the equation $$x^{3}-\sqrt[3]{6+\sqrt[3]{x+6}}=6.$$
  6. Let $A B C$ be a right triangle at $A$. Draw the circle $(B)$ with center at $B$ and radius $B A$ and draw a diameter $A D .$ Choose two points $E$ and $F$ on the line $B C$ such that $B$ is their midpoint. $D E$ and $D F$ meets with $(B)$ at $M$ and $N$ respectively. Prove that $M, N$ and $C$ are colinear.
  7. Let $A B C$ be an equilateral triangle with circumcircle $(O) .$ Draw a circle $\left(O_{1}\right)$ which is tangent to both side $B C$ and arc $\widehat{B C}$. Construct the circles $\left(O_{2}\right)$, $\left(O_{3}\right)$ similarly for the remaining sides $C A$ and $A B .$ Prove that $A O_{1}=B O_{2}=C O_{3}$ iff (ie. if and only if) the circles $\left(O_{1}\right)$, $\left(O_{2}\right)$ and $\left(O_{3}\right)$ all have the same radius. (Notation: $(X)$ is a circle with center at $X$.)
  8. Let $\left(a_{n}\right)$ $(n=0,1,2, \ldots)$ be a sequence given by $$a_{0}=29,\, a_{1}=105,\,a_{2}=381,\, a_{n+3}=3 a_{n+2}+2 a_{n+1}+a_{n},\,\forall n=0,1,2, \ldots .$$ Prove that for each positive integer $m,$ there exists a number $n$ such that $a_{n}$, $a_{n+1}-1$, $a_{n+2}-2$ are all divisible by $m$
  9. Let $f$ be a continuous function on the interval [0,1] and satisfies the following properties $$f(0)=0,\,f(1)=1,\quad 6 f\left(\frac{2 x+y}{3}\right)=5 f(x)+f(y),\,\forall x \geq y \in [0,1].$$ Find $f\left(\dfrac{8}{23}\right)$.
  10. Let $a, b, c$ be non-negative real numbers with sum equal to $1 .$ Prove that $$\sqrt{a+(b-c)^{2}}+\sqrt{b+(c-a)^{2}}+\sqrt{c+(a-b)^{2}} \geq \sqrt{3}$$
  11. Let $H$ and $I$ be, respectively, the orthorcenter and the incenter of an acute triangle $A B C$. The lines $A H$, $B H$ and $C H$ meet the circumcircles of triangles $HBC$, $HCA$ and $HAB$ at $A_{1}$, $B_{1}$, $C_{1}$ respectively. $A I$, $B I$ and $C I$ meet the circumcircles of triangles $I B C$, $I C A$ and $I A B$ at $A_{2}$, $B_{2}$, $C_{2}$ respectively. Prove that $$H A_{1} \cdot H B_{1} \cdot H C_{1}+64 R^{3} \geq 9 \cdot I A_{2} \cdot I B_{2} \cdot I C_{2}.$$
  12. Let $G$ be the centroid of a tetrahedron $A B C D$. Let $A_{1}$, $B_{1}$, $C_{1}$, $D_{1}$ be arbitrary points on $G A$, $G B$, $G C$ and $G D$ respectively. GA meets the plane $\left(B_{1} C_{1} D_{1}\right)$ at $A_{2} .$ Construct the points $B_{2}$, $C_{2}$, $D_{2}$ similarly. Prove that $$3\left(\frac{G A}{G A_{1}}+\frac{G B}{G B_{1}}+\frac{G C}{G C_{1}}+\frac{G D}{G D_{1}}\right) = \frac{G A}{G A_{2}}+\frac{G B}{G B_{2}}+\frac{G C}{G C_{2}}+\frac{G D}{G D_{2}}$$

Issue 359

  1. Determine the sum of 2005 terms $$S=\frac{2^{2}}{1.3}+\frac{3^{2}}{2.4}+\frac{4^{2}}{3.5}+\ldots+\frac{2006^{2}}{2005.2007}$$
  2. Let $ABC$ be an isosceles right triangle with right angle at $A .$ Pick an arbitrary point $M$ on $A C$. The foot of the altitude with $B C$ through $M$ is $H .$ Let $I$ be the midpoint of $BM$. Find the value of the angle $\angle H A I .$
  3. Prove that there exist infinitely many positive integers $n$ such that $n !$ is a multiple of $n^{2}+1$
  4. Let $a, b, c$ be positive numbers such that $a+b+c=a b c .$ Prove that $$\frac{a}{b^{3}}+\frac{b}{c^{3}}+\frac{c}{a^{3}} \geq 1.$$
  5. Solve the equation $$\frac{4}{x}+\sqrt{x-\frac{1}{x}}=x+\sqrt{2 x-\frac{5}{x}}$$
  6. Let $A B C$ be a right triangle at $A$ with $A B<A C$, $B C=2+2 \sqrt{3}$ and its incircle has radius is $1 .$ Determine the value of the angles $\angle B$ and $\angle C$.
  7. Let $A B C$ be an acute triangle with orthocenter $H .$ Let $A G$ be the altitude through $A$ and let $D$ be the midpoint of $B C .$ The circles, whose diameters are $B C$ and $A D$ respectively, meet at $E$ and $F$. Prove that a) The circumcircles of $H, E$ and $G$ and of $H$, $F$ and $G$ are both tangent to the circle whose diameter is the side $B C$. b) $H, E$, and $F$ lie on the same line.
  8. Find the maximum value of the following expression $$x_{1}^{3} x_{2}^{2}+x_{2}^{3} x_{3}^{2}+\ldots+x_{n}^{3} x_{1}^{2}+n^{2(n-1)} x_{1}^{3} x_{2}^{3} \ldots x_{n}^{3}$$ where $x_{1}, x_{2}, \ldots, x_{n}$ are non-negative numbers whose sum is $1$ $(n \geq 2)$.
  9. Solve the system of equations $$\begin{cases}20\left(x+\dfrac{1}{x}\right) &=11\left(y+\dfrac{1}{y}\right) &=2007\left(z+\dfrac{1}{z}\right) \\ x y+y z+z x &=1 \end{cases}$$
  10. Find the limit at $x=0$ of the following function $$\frac{\sqrt{\frac{\cos 2 x+\sqrt[3]{1+3 x}}{2}}-\sqrt[3]{\frac{\cos 3 x+3 \cos x-\ln (1+x)^{4}}{4}}}{x}$$
  11. Let $\left(O_{1}\right)$ be a circle inside a triangle $A B C$ and touches both sides $A B$ and $A C$. Construct a second circle $\left(O_{2}\right)$ through $B$, $C$ and touches outside of $\left(O_{1}\right)$ at $T .$ Prove that the angle bisector of $\widehat{B T C}$ passes through the incenter of the triangle $A B C$.
  12. Let $A_{1} A_{2} A_{3} A_{4}$ be a tetrahedron with perpendicular opposite edges. Denote by $S_{i}$ the area of the opposing faces of the vertices $A_{I}$ $(i=1,2,3,4) .$ Find all possible points $M$ where the sum $$S_{1} \cdot M A_{1}+S_{2} \cdot M A_{2}+S_{3} \cdot M A_{3}+S_{4} \cdot M A_{4}$$ is smallest possible.

Issue 360

  1. Consider the following sequence $$a_{1}=3,\, a_{2}=4,\, a_{3}=6, \ldots, a_{n+1}=a_{n}+n, \ldots$$ a) Is $2006$ a number in the above sequence? b) What is the $2007^{\text {th }}$ number in this sequence? c) Calculate the sum of the first $100$ numbers in the above sequence?
  2. Let $A B C$ be a right triangle with right angle at $A,$ and $\widehat{A C B}=54^{\circ} .$ Choose a point $E$ on the open ray in opposite direction to $C A$ such that $\widehat{A B E}=54^{\circ} .$ Prove that $B C<A E$
  3. Consider the following sum of $2006$ terms $$S=\sqrt{\frac{2+1}{2}}+\sqrt[3]{\frac{3+1}{3}}+\sqrt[4]{\frac{4+1}{4}}+\ldots+200 \sqrt[2]{\frac{2007+1}{2007}}.$$ Find $[S]$. (Here $[a]$ denote the largest integer which does not exceed $a$.)
  4. Solve the following system of equations $$\begin{cases} x-\dfrac{4}{x} &=2 y-\dfrac{2}{y} \\ 2 x &=y^{3}+3\end{cases}$$
  5. Let $a, b, c$ be numbers, all greater than or equal $-\dfrac{3}{2},$ such that $$a b c+a b+b c+c a+a+b+c \geq 0.$$ Prove that $a+b+c \geq 0$.
  6. Let $M$, $N$ and $P$ be three points outside a given triangle $A B C$ such that $$\angle C A N=\angle C B M=30^{\circ},\,\angle A C N=\angle B C M=20^{\circ},\,\angle P A B=\angle P B A=40^{\circ} .$$ Show that $M N P$ is an equilateral triangle.
  7. Let $A B C$ be a right triangle with right angle at $A$ and let $A H$ be the altitute through $A$. Denote by $I$, $I_{1}$, $I_{2}$ the incenters of the triangles $A B C$, $A H B$ and $A H C$. respectively. Prove that the circumcircle of the triangle $I I_{1} I_{2}$ is exactly the incircle of the triangle $A B C$.
  8. Find all functions $f: \mathbb{R}^{+} \rightarrow \mathbb{R}^{+}$ which satisfies the following identity $$ f(x) \cdot f(y)=f(x+y f(x)), \forall x, y \in \mathbb{R}^{+}$$
  9. Find the largest real number $m$ such that there exists a number $k$ in the interval $[1 ; 2]$ such that the following inequality $$\left(\frac{a}{b}+\frac{b}{c}+\frac{c}{a}\right)^{2 k} \geq(k+1)(a+b+c)\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)+m$$ is satisfied for all positive real numbers $a, b, c$.
  10. Let $\left(u_{k}\right)$ $(k=1,2, \ldots, n,)$ be an increasing sequence and let $A_{n}$ be the set of all positive numbers of the form $u_{i}-u_{j}$ $(1 \leq j<i \leq n)$. Prove that if $A_{n}$ has fewer that $n$ elements, then $\left(u_{k}\right)$ $(k=1,2, \ldots, n)$ forms an arithmetic sequence.
  11. Let $A B C D$ be a quadrilateral with nonparallel opposite sides and let $O$ denote the intersection of the diagonals $A C$ and $B D .$ The circumcircles of the triangles $O A B$ and $O C D$ meet at $X$ and $O$. The circumcircles of the triangles $O A B$ and $O C B$ meet at $Y$ and $O .$ The circles with diameters $A C$ and $B D$ intersect at $Z$ and $T$. Prove that either $X$, $Y$, $Z$, $T$ are colinear or they lie in a circle.
  12. Consider a closed polygon $A_{1} A_{2} \ldots A_{n}$ where each of its $n$ sides is tangent to a sphere $\mathscr{C}$ at center $O$ and radius $R .$ Let $G$ be the centroid of the system of points $A_{i}$, $i=1,2, \ldots, n$ (that is, $\overline{G A_{1}}+\overline{G A_{2}}+\ldots+\overline{G A_{n}}=\overrightarrow{0}$) and denote $\angle A_{i}=\angle A_{i-1} A_{i} A_{i+1}$ (with the convention that $A_{0} \equiv A_{n}$ and $A_{n+1} \equiv A_{1}$.) Prove the inequality $$ \sum_{i=1}^{n} \frac{G A_{i}}{\sin \frac{A_{i}}{2}} \geq \frac{1}{n R} \sum_{1 \leq i<j \leq n} A_{i} A_{j}^{2}$$

Issue 361

  1. The number $(\overline{9 x})^{8}$, where $x \in\{0,1,2, \ldots, 9\}$ contains how many digits when written in decimal form?
  2. Let $H$ be the orthocenter of an acute triangle $A B C$ and denote by $M$ the midpoint of $B C$. The perpendicular line to $M H$ through $H$ meets $A B$ and $A C$ at $P$ and $Q$ respectively. Prove that $H P=H Q$.
  3. Prove that if $a, b, c$ and $d$ are distinct positive integers such that the sum $$\frac{a}{a+b}+\frac{b}{b+c}+\frac{c}{c+d}+\frac{d}{d+a}$$ is also an integer then the product $a b c d$ is a perfect square.
  4. Compare the values of the following two numbers $$A=\min _{|y| \leq 1} \max _{|x| \leq 1}\left(x^{2}+y x\right) ,\quad B=\max _{|x| \leq 1} \min _{|y| \leq 1}\left(x^{2}+y x\right).$$
  5. Solve the system of equations $$\begin{cases}\sqrt{x}+\sqrt{y}+\sqrt{z}-\dfrac{1}{\sqrt{x}}-\dfrac{1}{\sqrt{y}}-\dfrac{1}{\sqrt{z}} &=\dfrac{8}{3} \\ x+y+z+\dfrac{1}{x}+\dfrac{1}{y}+\dfrac{1}{z} &=\dfrac{118}{9} \\ x \sqrt{x}+y \sqrt{y}+z \sqrt{z}-\dfrac{1}{x \sqrt{x}}-\dfrac{1}{y \sqrt{y}}-\dfrac{1}{z \sqrt{z}} &=\dfrac{728}{27}\end{cases}.$$
  6. Let $A B C D$ be a parallelogram. Let $M$ be a point in the plane spanned by the parallelogram $A B C D$ such that $\widehat{M D A}=\widehat{M B A}$. Prove that the two triangles $M A B$ and $M C D$ share a common orthocenter.
  7. Let $AD$, $BE$, $CF$ be the three bisectors of a triangle $A B C$ ($D \in B C$,$ E \in C A$, $F \in A B)$. Denote by $O$ the circumcenter of this triangle and by $R$ its. Let $O_{1}$, $O_{2}$ and $O_{3}$ be the circumcenters of the triangles $A B D$. $B C E$ and $A C F$ respectively. Prove that $$\frac{3}{2} R \leq O O_{1}+O O_{2}+O O_{3}<2 R.$$
  8. Let $a, b, c$ be positive real numbers such that $\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c} \leq 1 .$ Find the smallest possible value of $$P=[a+b]+[b+c]+[c+a]$$ where $[x]$ is the largest integers which is smaller than $x$.
  9. Find all funtions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that the following equality holds for all $x, y \in \mathbb{R}$ $$f\left(x^{3}-y\right)+2 y\left(3 f^{2}(x)+y^{2}\right)=f(y+f(x)).$$
  10. Prove that in any triangle $A B C$, the following inequality holds $$-1<6 \cos A+3 \cos B+2 \cos C<7$$
  11. Let ABC be a triangle with circumcircle $(O ; R)$. Let $E$ be the midpoint of $A B$ and denote by $F$ the point on $A C$ such that $\dfrac{A F}{A C}=\dfrac{1}{3} .$ Construct a parallelogram $A E M F$ Prove that $$M A+M B+M C \leq \sqrt{11\left(R^{2}-O M^{2}\right)}.$$ Now suppose that $(O ; R)$ is fixed. Construct a triangle $A B C,$ inscribed in $(O, R)$ such that $$M A+M B+M C=\sqrt{11\left(R^{2}-O M^{2}\right)}.$$
  12. Let $A B C D$ be a tetrahedron whose sides $D A$, $D B$ and $D C$ are pairwise perpendicular. Denote by $x, y, z$ the angles formed from sides $A B$, $B C$ and $C A$ respectively. Prove that $$\left(2+\tan ^{2} x\right)\left(2+\tan ^{2} y\right)\left(2+\tan ^{2} z\right) \geq 64.$$

Issue 362

  1. Let $$S=\frac{1}{2^{2}}+\frac{2}{2^{3}}+\ldots+\frac{n}{2^{n+1}}+\ldots+\frac{2006}{2^{2007}}.$$ Compare $S$ with $1 .$
  2. Let $A H$ be the altitude of a right triangle $A B C,$ right angle at $A .$ Let $P$ and $Q$ be the incenters of $A B H$ and $A C H$ respectively. $P Q$ meets with $A B$ at $E$ and with $A C$ at $F .$ Prove that $A E=A F$.
  3. Find all possible pairs of positive integers $(x ; y)$ such that $4 x^{2}+6 x+3$ is a multiple of $2 x y-1$.
  4. Solve the following equation $$\sqrt{2 x^{2}+4 x+7}=x^{4}+4 x^{3}+3 x^{2}-2 x-7.$$
  5. Find the maximum value of the following expression $$A=\frac{x}{x^{2}+y z}+\frac{y}{y^{2}+2 x}+\frac{z}{z^{2}+x y},$$ where $x, y, z$ are positive real numbers such that $x^{2}+y^{2}+z^{2}=x y z$.
  6. $(O)$ is a point chosen arbitrarily inside a triangle $A B C$. $A O$, $B O$ and $C O$ meet $B C$, $C A$ and $A B$ at $M$, $N$ and $P$ respectively. Prove that the value of ratio $$\left(\frac{O A \cdot A P}{O P}\right)\left(\frac{O B \cdot B M}{O M}\right)\left(\frac{O C \cdot C N}{O N}\right)$$ does not depend on the position of the point $O$.
  7. Let $M$ be a point inside a circle $(O)$ with center at $O$ and radius $R .$ Draw two chords $C D$ and $E F$ through $M$ but not passing through $O$. The tangent lines with $(O)$ at $C$ and $D$ intersects at $A$, and the tangent lines at $E$ and $F$ meets at $B$. Prove that $O M$ and $A B$ are orthogonal.
  8. Let $p>2007$ be a prime number and $n$ is an integer exceeding $2006 p .$ Prove that $\mathrm{C}_{n}^{2006 \mathrm{p}}-\mathrm{C}_{k}^{2006}$ is a multiple of $p,$ where $k=\left[\dfrac{n}{p}\right]$ is the largest integer which is not larger than $\dfrac{n}{p}$
  9. Let $\left(u_{n}\right)$ be a sequence given by $u_{1}$ and the formula $u_{n+1}=\dfrac{k+u_{n}}{1-u_{n}},$ where $k>0$ where $n=1,2, \ldots .$ Given that $u_{13}=u_{1},$ find the value of $k$.
  10. Suppose $a, b,$ and $c$ are the three lengths of sides of a triangle. Prove that $$\frac{a}{\sqrt{a^{2}+3 b c}}+\frac{b}{\sqrt{b^{2}+3 c a}}+\frac{c}{\sqrt{c^{2}+3 a b}} \geq \frac{3}{2}$$
  11. Let $P$ be a arbitrary point in a quadrilateral $A B C D$ such that $\widehat{P A B}$, $\widehat{P B A}$, $\widehat{P B C}$, $\widehat{P C B}$, $\widehat{P C D}$, $\widehat{P D C}$, $\widehat{P A D}$ and $\widehat{P D A}$ are all acute angles. Let $M$, $N$, $K$, $L$ denote the feet of the altitude from $P$ on $A B$, $B C$, $C D$ and $D A$ respectively. Find the smallest value of the sum $$\frac{A B}{P M}+\frac{B C}{P N}+\frac{C D}{P K}+\frac{D A}{P L}$$
  12. Let $A B C . A^{\prime} B^{\prime} C^{\prime}$ be a triangular prism. $M$ and $N$ are two points, chosen on $A A^{\prime}$ and $C B$ respectively such that $$\frac{A M}{A A^{\prime}}=\frac{C N}{C B} .$$ Prove that the length of $M N$ is no less than the distance from $A^{\prime}$ to the straight line $C^{\prime} B$.

Issue 363

  1. The first $100$ positive integer numbers are written consecutively in a certain order. Call the resulting number $A$. Is $A$ a multiple of $2007$?
  2. Let $ABC$ be a nonisosceles triangle, where $AB$ is the shortest side. Choose a point $D$ in the opposite ray of $BA$ such that $BD=BC$. Prove that $\angle ACD<90^\circ$.
  3. Let $a$, $b$, $c$ be positive reals such that $a+b+c=1$. Prove that $$\left(a+\dfrac{1}{b}\right)\left(b+\dfrac{1}{c}\right)\left(c+\dfrac{1}{a}\right)\geq \left(\dfrac{10}{3}\right)^3.$$
  4. Solve the equation in $\mathbb{R}$ $$(x^4+5x^3+8x^2+7x+5)^4+(x^4+5x^3+8x^2+7x+3)^4=16$$
  5. Let $AH$ denote the altitude of a right triangle $ABC$, right angle at $A$ and suppose that $AH^2=4AM\cdot AN$ where $M$, $N$ are the feet of the altitude from $H$ to $AB$ and $AC$, respectively. Find the measures of the angles of triangle $ABC$.
  6. Find all $(x,y)\in\mathbb{Z}^2$ such that $$x^{2007}=y^{2007}-y^{1338}-y^{669}+2.$$
  7. Let $(x_n)$ be a sequence given by $$x_1=5,\quad x_{n+1}=x_n^2-2,\,\forall n\geq 1.$$ Calculate $\displaystyle\lim_{n\to\infty}\dfrac{x_{n+1}}{x_1x_2\cdots x_n}.$
  8. Let $a$, $b$, $c$ and denote the three sides of a triangle $ABC$. Its altitudes are $h_a$, $h_b$, $h_c$ and the radius of its three escribed circles are $r_a$, $r_b$, $r_c$. Prove that $$\dfrac{a}{h_a+r_a}+\dfrac{b}{h_b+r_b}+\dfrac{c}{h_c+r_c}\geq \sqrt{3}.$$
  9. In a quadrilateral $ABCD$, where $AD=BC$ meets at $O$, and the angle bisector of the angles $DAB$, $CBA$ meets at $I$. Prove that the midpoints of $AB$, $CD$, $OI$ are colinear.
  10. Prove that for all $a,b,c\in [1,+\infty)$ we have $$\left(a-\dfrac{1}{b}\right)\left(b-\dfrac{1}{c}\right)\left(c-\dfrac{1}{a}\right)\geq \left(a-\dfrac{1}{a}\right)\left(b-\dfrac{1}{b}\right)\left(c-\dfrac{1}{c}\right).$$
  11. Find the number of binary strings of length $n$ $(n>3)$ in which the substring $01$ occurs exactly twice.
  12. Let $f:\mathbb{N}\to\mathbb{R}$ be a function such that $$f(1)=\dfrac{2007}{6},\quad \dfrac{f(1)}{1}+\dfrac{f(2)}{2}+\cdots+\dfrac{f(n)}{n}=\dfrac{n+1}{2}\cdot f(n)\forall n\in\mathbb{N}.$$ Find the limit $\displaystyle\lim_{n\to\infty} (2008+n)f(n)$.

Issue 364

  1. Find the last two digit numbers of $3^{9999}-2^{9999}$.
  2. Compare the sum (consisting of $n+1$ terms) $$S_{n}=\frac{2}{101+1}+\frac{2^{2}}{101^{2}+1}+\ldots+\frac{2^{n+1}}{101^{2 \prime}+1}$$ with $0,02$.
  3. Let $a$, $b$ and $c$ be positive numbers such that $a b+b c+c a=1$. Prove the inequality $$\frac{1}{a b}+\frac{1}{b c}+\frac{1}{c a} \geq 3+\sqrt{\frac{1}{a^{2}}+1}+\sqrt{\frac{1}{b^{2}}+1}+\sqrt{\frac{1}{c^{2}}+1}.$$ When does equality occur?
  4. Solve the equation $$\sqrt{7 x^{2}-22 x+28}+\sqrt{7 x^{2}+8 x+13}+\sqrt{31 x^{2}+14 x+4}=3 \sqrt{3}(x+2).$$
  5. Let $A B C D$ be a rectangular with $A B<B C$. Let $M$ be a point, different from $A$ and $B$, on the half-circle with $A B$ as its diameter and on the same side with $CD$. $MA$ and $MB$ meet $CD$ at $P$ and $Q$ respectively. $M C$ and $M D$ meet $A B$ at $E$ and $F$ respectively. Find the position of the point $M$ on the half-circle such that the sum $P Q+E F$ is smallest possible. Calculate this smallest value.
  6. Find all pairs of positive integers $a$, $b$ such that $q^{2}-r=2007,$ where $q$ and $r$ are respectively the quotient and the remainder obtained when dividing $a^{2}+b^{2}$ by $a+b$
  7. Consider the equation $$a x^{3}-x^{2}+b x-1=0$$ where $a, b$ are real numbers, $a \neq 0$ and $a \neq b$ such that all of its roots are positive real numbers. Find the smallest value of $$P=\frac{5 a^{2}-3 a b+2}{a^{2}(b-a)}.$$
  8. Choose five points $A$, $B$, $C$, $D$ and $E$ on a sphere with radius $R$ such that $$\widehat{B A C}=\widehat{C A D} =\widehat{D A E}=\widehat{E A B}=\frac{2}{3} \widehat{B A D}=\frac{2}{3} \widehat{C A E}.$$ Prove the inequality $$A B+A C+A D+A E \leq 4 \sqrt{2} R.$$
  9. Suppose that $M$, $N$ and $P$ are three points lying respectively on the edges $A B$, $B C$. $C A$ of a triangle $A B C$ such that $$A M+B N+C P=M R+N C+P A.$$ Prove the inequality $S_{M N P} \leq \dfrac{1}{4} S_{A B C}$.
  10. Find the limit $$\lim_{n \rightarrow+\infty}\sqrt{2-\sqrt{2}} \cdot \sqrt{2-\sqrt{2+\sqrt{2}}} \cdots \underbrace{\sqrt{2-\sqrt{2+\sqrt{2+\ldots+\sqrt{2}}}}}_{n \text { signsor railiad }}$$
  11. Denote by $[x]$ the largest integer not exceeding $x$ and write $\{x\}=x-[x] .$ Find the limit $\displaystyle \lim _{n \rightarrow+\infty}(7+4 \sqrt{3})^{n}$.
  12. Let $n$ be a positive integer and $2 n+2$ real numbers $a, b, a_{1}, a_{2}, \ldots, a_{n}, b_{1}, b_{2}, \ldots, b_{n}$ such that $a_{i} \neq 0(i=1,2, \ldots, n)$ and the function $$F(x)=\sum_{i=1}^{n} \sqrt{a_{i} x+b_{i}}-(a x+b)$$ satisfies the following property: There exists distinct real numbers $\alpha, \beta$ such that $F(\alpha)=F(\beta)=0$ Prove that $\alpha$ and $\beta$ are the only real solutions of the equation $F(x)=0$

Issue 365

  1. Write $2005^{2006}$ as a sum of natural numbers, then calculate the sum of all the digits occurred in these summands. Can one obtain either $2006$ or $2007$ in this way? Why?
  2. Let $A B C$ be an isosceles right triangle, right angle at $A$. Choose a point $D$ in the half-plane on the side of $A B$ that does not contain $C$ such that $D A B$ is also an isosceles right triangle, with right angle at $D$. Let $E$ be a point (differs from $A$) on $A D$. The perpendicular line with $B E$ through $E$ intersects with $A C$ at $F .$ Prove that $E F=E B$.
  3. Find the maximum and minimum values of the following expression $$\frac{x-y}{x^{4}+y^{4}+6}.$$
  4. Solve the equation $$\sqrt{2}\left(x^{2}+8\right)=5 \sqrt{x^{3}+8}.$$
  5. The two diagonals $A C$ and $B D$ of an inscribed quadrilateral $A B C D$ intersect at $O$. The circumcircles $\left(S_{1}\right)$ of $A B O$ and $\left(S_{2}\right)$ of $CDO$ meet at $O$ and $K$. Through $O$, draw parallel lines with $A B$ and $C D$; they meet with $\left(S_{1}\right)$ and $\left(S_{2}\right)$ at $N$ and $M$, respectively. Let $P$ and $Q$ be two points on $O N$ and $O M$ respectively such that $\dfrac{O P}{P N}=\dfrac{M Q}{Q}$. Prove that $O$, $K$, $P$ and $Q$ lie on the same circle.
  6. There are nine cards in a box, labelled from $1$ to $9$. How many cards should be taken from the box so that the probability of getting a card whose label is a multiple of 4 will be greater than $\dfrac{5}{6} ?$
  7. Let $x_{1}, x_{2}, \ldots, x_{n}$ be $n$ arbitrary real numbers chosen in a given closed interval $[a ; b]$. Prove the inequality $$\sum_{i=1}^{n}\left(x_{i}-\frac{a+b}{2}\right)^{2} \geq \sum_{i=1}^{n}\left(x_{i}-\frac{1}{n} \sum_{i=1}^{n} x_{i}\right)^{2}.$$ When does equality occur?
  8. Let $H$ be the orthocenter of an acute triangle $A B C$ whose edges are $B C=a, C A=b$, and $A B=c .$ Prove that $$\frac{H A^{2}+H B^{2}+H C^{2}}{a^{2}+b^{2}+c^{2}} \leq(\cot A \cdot \cot B)^{2}+(\cot B \cdot \cot C)^{2}+(\cot C \cdot \cot A)^{2}.$$
  9. Let $p$ be a prime number, greater than $3$ and $k=\left[\dfrac{2 p}{3}\right]$. (The notation $[x]$ denote the largest integer which is not exceeding $x$.) Prove that $\displaystyle \sum_{i=1}^{k} C_{p}^{i}$ is a multiple of $p^{2}$.
  10. Find the measures of the angles of triangle $A B C$, such that $$\cos \frac{5 A}{2}+\cos \frac{5 B}{2}+\cos \frac{5 C}{2}=\frac{3 \sqrt{3}}{2}.$$
  11. Let $\left(x_{n}\right)(n=1,2, \ldots)$ be a boundedabove sequence such that $$x_{n+2} \geq \frac{1}{4} x_{n+1}+\frac{3}{4} x_{n},\,\forall n=1,2, \ldots$$ Prove that this sequence has limit.
  12. Let $\alpha$, $\alpha^{\prime}$; $\beta$, $\beta^{\prime}$ and $\gamma$, $\gamma^{\prime}$ be the dihedral angles, opposite to the edges $B C$, $DA$, $CA$, $DB$ and $A B$, $D C$ respectively, of a tetrahedron $A B C D$. Prove the inequality $$\sin \alpha+\sin \alpha^{\prime}+\sin \beta+\sin \beta^{\prime}+\sin \gamma+\sin \gamma^{\prime} \leq 4 \sqrt{2}.$$ When does equality occur?

Issue 366

  1. Let $\left(a_{n}\right)$ be $a_{1}=3$, $a_{2}=8$, $a_{3}=13$, $a_{4}=24$, $a_{5}=31$, $a_{6}=48, \ldots,$ in general, $$a_{n+2}=\begin{cases}a_{n}+4 n+8 & \text { if } n \text { is odd } \\ a_{n}+4 n+6 & \text { if } n \text { is cven }\end{cases}.$$ a) Do the numbers $2007$ and $2024$ appear in this sequence?.
    b) Find the $2007$-th number in this sequence.
  2. Let $ABC$ be a triangle with $\widehat{B A C}=40^{\circ}$ and $\widehat{A B C}=60^{\circ} .$ Denote $D$ and $E$ are two points on $A B$ and $A C$ respectively such that $\widehat{D C B}=70^{\circ}$ and $\widehat{E B C}=40^{\circ}$; $D C$ and $E B$ meets at a point $F$. Prove that $A F$ and $B C$ are orthogonal.
  3. Let $x,y,z,t$ and $u$ be positive real numbers such that $x+y+z+t+u=4$. Find the smallest possible value of the following expression $$P=\frac{(x+y+z+t)(x+y+z)(x+y)}{x y z t u}.$$
  4. Solve for $x$ $$2 \sqrt{x+1}+6 \sqrt{9-x^{2}}+6 \sqrt{(x+1)\left(9-x^{2}\right)}=38+10 x-2 x^{2}-x^{3}.$$
  5. Let $P A$ and $P B$ be two tangent lines through a point $P$ outside a circle with center $O$. $OP$ and $A B$ meet at $M$. Draw a secant $C D$ through $M$ ($C D$ does not contain $O$). The tangent lines at $C$ and $D$ meet at $Q$. Find the measure of the angle $O P Q$.
  6. Find all pairs of positive integers $(x; y)$ such that $$x^{y}+y=y^{x}+x.$$
  7. Prove that if the equation $x^{3}+a x^{2}+b x+c=0$ has three distinct real roots, then so is $$x^{3}+a x^{2}+\left(-a^{2}+4 b\right) x+a^{3}-4 a b+8 c=0.$$
  8. Let $I$ denote the incenter of an acute triangle $A B C$. The incircle $(I)$ touches $B C$, $C A$ and $A B$ at $D$, $E$ and $F$ respectively. The angle bisector of $B I C$ meets $B C$ at $M$. $A M$ meets $E F$ at $P$. Prove that $$P D \geq \frac{1}{2} \sqrt{4 D E \cdot D F-E F^{2}}.$$
  9. Let $a_{1}, a_{2}, \ldots, a_{2007}$ be pairwise distinct integers, all greater than $1$ such that $\displaystyle \sum_{i=1}^{20} a_{i}=2017035$. Could it be possible that the sum $\displaystyle \sum_{i=1}^{2007} a_{i}^{a_{i} a_{i}}$ is a perfect square?
  10. Find all polynornial with real coefficients $P(x)$ such that $$P(P(x)+x)=P(x) P(x+1),\,\forall x \in \mathbb{R}.$$
  11. Let $a_{1}, a_{2}, a_{3}, a_{4}$ and $a_{5}$ be nonnegative real numbers whose sum equal $1$. Prove the inequality $$a_{2} a_{3} a_{4} a_{5}+a_{1} a_{3} a_{4} a_{5}+a_{1} a_{2} a_{3} a_{5}+a_{1} a_{2} a_{3} a_{4} \leq \frac{1}{256}+\frac{3275}{256} a_{1} a_{2} a_{3} a_{4} a_{5}.$$ When does equality occur?
  12. Let $P$ be the intersection of the diagonals of an inscribed quadrilateral $A B C D$. Prove that all four Euler lines of the triangles $PAB$, $PBC$, $PCD$ and $PDA$ intersect in a single point. (The Euler line of a triangle is the line connecting its centroid and its orthocenter).
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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 2007
Mathematics and Youth Magazine Problems 2007
MOlympiad.NET
https://www.molympiad.net/2022/04/mym-2007.html
https://www.molympiad.net/
https://www.molympiad.net/
https://www.molympiad.net/2022/04/mym-2007.html
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