Mathematics and Youth Magazine Problems 2003


Issue 307

  1. Find the measures of the sides of all triangles satisfying the conditions : these measures are whole numbers, the perimeter and the area are expressed by equal numbers.
  2. Find the least value of the expression $$(\sqrt{1+a}+\sqrt{1+b})(\sqrt{1+c}+\sqrt{1+d})$$ where $a, b, c, d$ are positive numbers satisfying the condition $a b c d=1$.
  3. Solve the system of equations $$\begin{cases}\sqrt{x}+\sqrt{y}+\sqrt{z} &=3 \\ (1+x)(1+y)(1+z) &=(1+\sqrt[3]{x y z})^{3}\end{cases}$$
  4. Let $A B C D$ be a parallelogram. Take a point $M$ on the side $A B$, a point $N$ on the side $C D$. Let $P$ be the point of intersection of $A N$ and $D M$, $Q$ be the point of intersection of $B N$ and $C M$. Prove that $P Q$ passes through a fixed point when $M$ and $N$ move respectively on $A B$ and $C D$.
  5. Let $A B C D$ be a convex quadrilateral. Prove that $$ \min \{A B, B C, C D, D A\} \leq \frac{\sqrt{A C^{2}+B D^{2}}}{2} \leq  \max \{A B, B C, C D, D A\}$$
  6. Find all natural numbers $n \geq 3$ such that in the coordinate plane, there exists a regular $n$-polygon all vertices of which have integral coordinates.
  7. The sequence of numbers $\left(u_{n}\right)$ $(n=1,2,3, \ldots)$ is defined by $$u_{n}=\sum_{k=1}^{n} \frac{(-1)^{k-1}}{k},\,\forall n=1,2,3 \ldots.$$ Prove that the sequence has limit and find the limit.
  8. Find the greatest value of the expression $$x_{1}^{3}+x_{2}^{3}+\ldots+x_{n}^{3}-x_{1}^{4}-x_{2}^{4}-\ldots-x_{1}^{4}$$ ($n$ is a given number), where the numbers $x_{i}$ $(i=1,2, \ldots, n)$ satisfy the conditions $0 \leq x_{i} \leq 1$ $(i=1,2, \ldots, n)$ and $x_{1}+x_{2}+\ldots+x_{n}=1$.
  9. In a triangle $A B C$, let $m_{u}$, $m_{b}$, $m_{c}$ be respectively the measures of the medians issued from $A$, $B$, $C$, let $r_{a}$, $r_{b}$, $r_{c}$ be respectively the radii of the escribed circles in the angles $A$, $B$, $C$. Prove that $$r_{a}^{2}+r_{b}^{2}+r_{c}^{2} \geq m_{a}^{2}+m_{b}^{2}+m_{c}^{2}.$$ When does equality occur?
  10. Let $H$ and $O$ be respectively the orthocenter and the circumcenter of the triangle $A B C$ of a tetrahedron $S A B C$ such that $S A$, $S B$, $S C$ are orthogonal each to the others. Prove that $$\frac{O H^{2}}{S H^{2}}+2=\frac{1}{4 \cos A \cdot \cos B \cdot \cos C}$$ where $\cos A$, $\cos B$, $\cos C$ are cosinus of the angles of triangle $A B C$.

Issue 308

  1. There are three bells in the laboratory. The first bell rings every $4$ minutes, the second every $12$ minutes, the third every $16$ minutes. The three bells ring simultaneously at $7^{\text {h }} 30$ in the morning.
    a) At what next time do the three bells ring simultaneously?
    b) From $7^{\mathrm{h}} 30$ to $11^{\mathrm{h}} 30$ p.m., how many times do only two bells ring simultaneously?
  2. a) In the figure let $\widehat{B E C}=30^{\circ}$, $\widehat{A C D}=70^{\circ}$, $\widehat{C D E}=110^{\circ}$ and $\widehat{B A C}=\widehat{C E D}=\widehat{A B E}$ and justify the answer.
    b) How many triangles are there in the figure? Write down these triangles.
  3. Find the integer-solutions of the equation $$4(a-x)(x-b)+b-a=y^{2}$$ where $a, b$ are given integers, $a>b$.
  4. Prove that the equation $$(n+1) x^{n+2}-3(n+2) x^{n+1}+a^{n+2}=0$$ (where $n$ is a given even positive integer and $a>3$) has no solution.
  5. Prove that $$\sqrt{\frac{a}{b}}+\sqrt{\frac{b}{a}}+\frac{3(\sqrt{a}+\sqrt{b})}{\sqrt{a+b}}>6$$
  6. Let $A B C$ be a triangle ; on the opposite rays of the rays $B A, C A$ take respectively the points $E, F$ (distinct from $B, C)$ $B F$ cuts $C E$ at $M$. Prove that $$\frac{M B}{M F}+\frac{M C}{M E} \geq 2 \sqrt{\frac{A B \cdot A C}{A F \cdot A E}}$$ When does equality occur?
  7. Let be given a convex quadrilateral $A B C D$. $A B$ cuts $C D$ at $E$, $A D$ cuts $B C$ at $F$. The diagonals $A C$, $B D$ intersect at $O$. Let $M$, $N$, $P$, $Q$ be respectively the midpoints of $A B$, $B C$, $C D$, $DA$. $CF$ cuts $M P$ at $H$, $O E$ cuts $N Q$ at $K$. Prove that $H K$ is parallel to $E F$.
  8. Let $x, y, p$ be integers such that $p>1$ and $x^{2002}$, $y^{2003}$ are divisible by $p$. Prove that $1+x+y$ is not divisible by $p$.
  9. Let $a_{1}, a_{2}, \ldots a_{n}$ and $b_{1}, b_{2}, \ldots, b_{n}$ be positive numbers. Prove that
    a) $\displaystyle \frac{d_{1}^{r}}{b_{1}^{r-1}}+\frac{c_{2}}{b_{2}^{r-1}}+\ldots+\frac{a_{n}^{r}}{b_{n}^{r-1}} \geq \frac{\left(a_{1}+a_{2}+\ldots+a_{n}\right)^{r}}{\left(b_{1}+b_{2}+\ldots+b_{n}\right)^{r-1}}$ where $r$ is a rational number, $r>1$;
    b) $\displaystyle \frac{a_{1}^{s}}{b_{1}}+\frac{a_{2}^{s}}{b_{2}}+\ldots+\frac{a_{n}^{s}}{b_{n}} \geq \frac{\left(a_{1}+a_{2}+\ldots+a_{n}\right)^{s}}{n^{s-2}\left(b_{1}+b_{2}+\ldots+b_{n}\right)}$ where $s$ is a rational number, $s \geq 2$.
  10. The sequence of numbers $\left(v_{n}\right)$ is defined by $$v_{0}=1,\quad v_{n}=\frac{-1}{3+v_{n-1}},\,\forall n=1,2,3, \ldots$$ Prove that the sequence has a limit and find this limit.
  11. $M$ is a point inside an acute triangle $A B C$ with $B C=a$, $C A=b$, $A B=c$. Let $D$, $E$, $F$ be respectively the orthogonal projections of $M$ on the sides $B C$, $C A$, $A B$. Find the greatest value of the expression $$P=a \cdot M E \cdot M F + b \cdot M F \cdot M D+c \cdot M D \cdot M E$$ and determine the position of $M$ where this expression assumes its greatest value. 
  12. Let $S_{A}$, $S_{B}$, $S_{C}$, $S_{D}$ be respectively the areas of the faces $B C D$, $C D A$, $D A B$, $A B C$ of a tetrahedron $A B C D$ and $R$ be the radius of the circumscribed sphere of $A B C D$. Prove that $$R \geq \frac{T}{S_{A}+S_{B}+S_{C}+S_{D}}$$ where $$T^{2}=A B^{2} S_{A} S_{B}+A C^{2} S_{A} S_{C}+A D^{2} S_{A} S_{D}+B C^{2} S_{B} S_{C}+B D^{2} S_{B} S_{D}+C D^{2} S_{C} S_{D}.$$ When does equality occur?

Issue 309

  1. Compare the value of the expression $$A=\frac{3}{4}+\frac{8}{9}+\frac{15}{16}+\ldots+\frac{9999}{10000}$$ with the numbers $98$ and $99$.
  2. Let $A B C$ be a triangle with $\widehat{A B C}=30^{\circ}$, $\widehat{A C B}=20^{\circ}$. The perpendicular bisector of $A C$ cuts $B C$ at $E$ and cuts the ray $B A$ at $F$. Prove that $A F=E F$ and $A C=B E$.
  3. Find the intergers $x, y$ satisfying $$\frac{x+2 y}{x^{2}+y^{2}}=\frac{7}{20}.$$
  4. Prove that $$\frac{a+b}{2} \geq \sqrt{a b}+\frac{(a-b)^{2}(3 a+b)(a+3 b)}{8(a+b)\left(a^{2}+6 a b+b^{2}\right)}$$ for positive numbers $a, b$.
  5. Solve the inequation $$(x-1) \sqrt{x^{2}-2 x+5}-4 x \sqrt{x^{2}+1} \geq 2(x+1)$$
  6. On a line, take three distinct points $A$, $B$, $C$ in this order. Draw the tangents $A D$, $A E$ to the circle with diameter $B C$ ($D$ and $E$ are the touching points). Draw $D H \perp C E$ at $H$. Let $P$ be the midpoint of $D H$. The line $C P$ cuts again the circle at $Q$. Prove that the circle passing through the three points $A$, $D$, $Q$ is tangent to the line $A C$.
  7. $M$ is a point on the incircle of triangle $A B C$. Let $K$, $H$, $J$ be respectively the orthogonal projections of $M$ on the lines $A B$, $B C$, $C A$. Determine the position of $M$ so that the sum $M K+M H+M J$.
    a) attains its greatest value
    b) attains its least value
  8. Determine all sequences of positive integers $\left(x_{n}\right)$ $(n=1,2,3, \ldots)$ satisfying $$x_{1}=1,\, x_{2}>1,\quad x_{n+2}=\frac{1+x_{n+1}^{4}}{x_{n}},\,\forall n=1,2,3, \ldots$$
  9. Solve the system of equations $$\begin{cases}\log _{2}(1+3 \cos x) &=\log _{3}(\sin y)+2 \\ \log _{2}(1+3 \sin y) &=\log _{3}(\cos x)+2\end{cases}$$
  10. Let $n$ be a given positive integer. Find the least number $t=t(n)$ such that for all real numbers $x_{1}, x_{2}, \ldots, x_{n}$, holds the following inequality $$\sum_{k=1}^{n}\left(x_{1}+x_{2}+\ldots+x_{k}\right)^{2} \leq t\left(x_{1}^{2}+x_{2}^{2}+\ldots+x_{n}^{2}\right)$$
  11. Prove that for arbitrary triangle $A B C$, $$\sin A \cos B+\sin B \cos C+\sin C \cos A \leq \frac{3 \sqrt{3}}{4}.$$ When does equality occur?
  12. In space, let be given $n$ distinct points $A_{1}, A_{2}, \ldots, A_{n}$ $(n \geq 2)$ and $n$ points $K_{1}, K_{2}, \ldots, K_{n}$ ($K_{i}$ does not coincide with $A_{i}$ for every $i=1,2, \ldots, n)$. Prove that there exist $n$ spheres (with positive radii) $S_{i}$ $(i=1,2, \ldots, n)$ satisfying simultaneously the following two conditions
    a) the spheres do not intersect each others,
    b) the products $P_{A_{i} / S_{i}} \cdot P_{K_{i} / S_{i}}$ are negative numbers for all $i=1,2, \ldots, n$, where $P$ denotes the power of a point with respect to a sphere.

Issue 310

  1. Do there exist three positive integers $a, b, c$ such that the sum $$\frac{a}{a+b}+\frac{b}{b+c}+\frac{c}{c+a}$$ is an integer?
  2. Let $A B C$ be a triangle with $\widehat{B A C}=90^{\circ}$, $\widehat{A B C}=50^{\circ}$. Construct the triangle $B C D$ such that $\widehat{B C D}=\widehat{C B D}=30^{\circ}$ and $D$ is outside of triangle $A B C$. The lines $A B$ and $C D$ intersect at $E$. The lines $A C$ and $B D$ intersect $F$. Calculate the angles $\widehat{A E F}$ and $\widehat{A F E}$.
  3. Find integral solutions of the equation $$x^{3}-y^{3}=x y+2003.$$
  4. Solve the system of equations $$\begin{cases} x+y+z &=-1 \\ x y z &=1 \\ \dfrac{x}{y^{2}}+\dfrac{y}{z^{2}}+\dfrac{z}{x^{2}} &=\dfrac{y^{2}}{x}+\dfrac{z^{2}}{y}+\dfrac{x^{2}}{z}\end{cases}$$
  5. Find the least value of the expression $$\frac{a^{3}}{1+b}+\frac{b^{3}}{1+a},$$ where $a$ and $b$ are positive numbers satisfying the condition $a b=1$.
  6. Let $A B C D$ a convex quadrilateral such that the diagonals $C A$ and $D B$ are the angled-bisectors of $\widehat{B C D}$ and $\widehat{A D C}$. Let $E$ be the point of intersection of $C A$ and $D B$. Prove that $$E C \cdot E D=E A \cdot E B+E A \cdot E D+E B \cdot E C$$ when and only when $\widehat{A E D}=45^{\circ}$.
  7. Let be given a semi-circle with diameter $A B$ and a fixed point $C$ on the segment $A B$ ($C$ distinct from $A$, $B$). $M$ is a point on the semi-circle. The line passing through $M$, perpendicular to $M C$, cuts the tangents of the semi-circle at $A$ and $B$ respectively at $E$ and $F$. Find the least value of the area of triangle $C E F$ when $M$ moves on the semi-circle.
  8. Prove that $$(a+b)(b+c)(c+a) \geq 2(1+a+b+c)$$ where $a, b, c$ are positive numbers satisfying the condition $a b c=1$.
  9. Solve the following equation with parameter $m$ $$x^{3}+5 x^{2}+(5 m+1) x+m^{2}=\left(x^{2}-x+1\right)^{2}$$
  10. Prove the inequalities $$\frac{2 x^{n}}{1+x^{n+1}} \leq\left(\frac{1+x}{2}\right)^{n-1} \leq \frac{x^{n}-1}{n(x-1)}$$ for positive number $x \neq 1$ and positive integer $n$.
  11. Prove that for arbitrary acute triangle $A B C$ $$\tan A+\tan B+\tan C+6(\sin A+\sin B+\sin C) \geq 12 \sqrt{3}.$$
  12. Let $S A B C$ be a tetrahedron with $S A=B C$, $S B=C A$, $S C=A B$. A plane passing through the incenter of triangle $A B C$ cuts the rays $S A$, $S B$, $S C$ respectively at $M$, $N$, $P$. Prove that $$S M+S N+S P \geq S A+S B+S C.$$

Issue 311

  1. Compare the fractions $$A=\frac{2003^{2003}+1}{2003^{2004}+1}, \quad B=\frac{2003^{2002}+1}{2003^{2003}+1}$$ Can you do it by distinct methods?
  2. Let $A B C$ be a triangle. Construct the segment $B D$ so that $\widehat{A B D}=60^{\circ}$, $B D=B A$ and the ray $B A$ lies between the rays $B C$, $B D$. Construct the segment $B E$ so that $\widehat{C B E}=60^{\circ}$, $B E=B C$ and the ray $B C$ lies between the rays $B A$, $B E$. Let $M$ be the midpoint of $D E$, $P$ be the point of intersection of the perpendicular bisectors of the segments $B A$ and $B D$. Calculate the angles of triangle $C M P$.
  3. Let $x_{1}$, $x_{2}$ be the solutions of the equation $x^{2}-4 x+1=0$. Prove that for every positive integer $n$, $x_{1}^{n}+x_{2}^{n}$ can be written as the sum of the squares of three consecutive integers.
  4. Find all triples of numbers $a, b, c$ satisfying $$\left(a^{2}+1\right)\left(b^{2}+2\right)\left(c^{2}+8\right)=32 a b c.$$
  5. Find the least value of the expression $$\frac{a^{8}}{\left(a^{2}+b^{2}\right)^{2}}+\frac{b^{8}}{\left(b^{2}+c^{2}\right)^{2}}+\frac{c^{8}}{\left(c^{2}+a^{2}\right)^{2}}$$ where $a, b, c$ are positive numbers satisfying the condition $a b+b c+c a=1$.
  6. Let $A B C$ be a triangle, right at $A$. Construct a square $E F G D$ so that $E$, $F$ lie on the side $B C$ and $G$, $D$ lie respectively on the sides $A C$, $A B$. Let $R_{1}$, $R_{2}$, $R_{3}$ be respectively the inradii of the triangles $B D E$, $C G F$, $A D G$. Prove that the area of $E F G D$ assumes its greatest value when and only when $$R_{1}^{2}+R_{2}^{2}=R_{3}^{2}.$$
  7. A chord $D E$ of the circumcircle of triangle $A B C$ cuts the incircle of $A B C$ at $M$ and $N$. Prove that $D E \geq 2 M N$.
  8. Each domino consists of two squares having a common side, on its first and second squares are marked $x$ and respectively $y$ dots. Thid momino is denoted by $(x, y)$ or $(y, x) .$ For every non ordered pair of number $x, y$, there's only one domino $(x, y)$. Divide the set of dominoes with $1 \leq x \leq 5,1 \leq y \leq 5$ into three groups, each group is presented as a closed circuit where the numbers $a, b, c, d$, $a \mid b] b \mid c$ are not necessarily distinct. $a|e| e \mid d\rfloor d$ How many such dividings are there?
  9. Prove that for every $x \in\left[0 ; \frac{\pi}{2}\right]$, holds $$\sin x \leq \frac{4}{\pi} x-\frac{4}{\pi^{2}} x^{2}.$$
  10. Prove that for every triangle $A B C$, hold $$\sqrt{3} \leq \frac{\cos (A / 2)}{1+\sin (A / 2)}+\frac{\cos (B / 2)}{1+\sin (B / 2)}+\frac{\cos (C / 2)}{1+\sin (C / 2)}<2$$
  11. Le $A B C D$ be a convex a quadrilateral, the diagonals $A C$ and $B D$ of which are orthogonal. The lines $B C$ and $A D$ intersect at $I$, the lines $A B$ and $C D$ intersect at $J$. Prove that the quadrilateral $B D I J$ is inscribable when and only when $A B \cdot C D=A D \cdot B C$. 
  12. The tetrahedron $A B C D$ has its four altitudes concurrent at a point $H$ inside the tetrahedron. Let $M$, $N$, $P$ be respectively the midpoints of $B C$, $C D$, $D B$. Let $R$ and $R_{1}$ be respectively the circumradii of the tetrahedra $A B C D$ and $H M N P$. Prove that $R=2 R_{1}$.

Issue 312

  1. Find the the first three digits from the left side of the number $$1^{1}+2^{2}+3^{3}+\ldots+999^{999}+1000^{1000} .$$
  2. Prove that if the numbers $a, b, c, x, y, z$ satisfy the condition $$\frac{b z+c y}{x(-a x+b y+c z)}=\frac{c x+a z}{y(a x-b y+c z)}=\frac{a y+b x}{z(a x+b y-c z)}$$ then $$\frac{x}{a\left(b^{2}+c^{2}-a^{2}\right)}=\frac{y}{b\left(a^{2}+c^{2}-b^{2}\right)}=\frac{z}{c\left(a^{2}+b^{2}-c^{2}\right)}$$
  3. Given an integer $n>1$ such that $3^{n}-1$ is divisible by $n$, show that $n$ is even.
  4. Find all positive solutions of the system of equations $$\begin{cases}\dfrac{3 x}{x+1}+\dfrac{4 y}{y+1}+\dfrac{2 z}{z+1} &=1 \\ 8^{9} \cdot x^{3} \cdot y^{4} \cdot z^{2} &=1\end{cases}$$
  5. Prove that $$\left(a^{3}+b^{3}+c^{3}\right)\left(\frac{1}{a^{3}}+\frac{1}{b^{3}}+\frac{1}{c^{3}}\right) \geq \frac{3}{2}\left(\frac{b+c}{a}+\frac{c+a}{b}+\frac{a+b}{c}\right)$$ where $a, b, c$ are positive real numbers.
  6. Prove that $$(a+b+c)\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)+\frac{3(a-b)(b-c)(c-a)}{a b c} \geq 9$$ where $a, b, c$ are the lengths of the sides of a triangle.
  7. The circle inscribed a triangle $A B C$ meets the sides $B C$, $C A$, $A B$ at $D$, $E$, $F$, respectively. Let $H$ be the projection of $D$ on $E F$. Show that $\widehat{B H D}=\widehat{C H D}$.
  8. Let $p_{1}, p_{2}, \ldots, p_{n}$ and $q_{1}, q_{2}, \ldots, q_{m}$ be $m+n$ different prime numbers such that $p_{i} \geq n+1$ $(i=1,2, \ldots, n)$, $q_{j} \geq m+1$ $(j=1,2, \ldots, m)$. Put $$P=p_{1} p_{2} \ldots p_{n} \quad \text{and} \quad Q=q_{1} q_{2} \ldots q_{m}.$$ Prove that the equation $P^{s} x+Q^{t} y=1$, where $s$, $t$ are fixed positive integers, have infinitely many integral solutions $\left(x_{0}, y_{0}\right)$ such that $\left(P, x_{0}\right)=1$ and $\left(Q, y_{0}\right)=1$.
  9. Solve the equation $$\left(16 \cos ^{4} x+3\right)^{4}=2048 \cos x-768.$$
  10. Prove that $$\frac{x_{1}^{m}}{x_{2}^{n}+x_{3}^{n}+\ldots+x_{n}^{n}}+\ldots+\frac{x_{n}^{m}}{x_{1}^{n}+x_{2}^{n}+\ldots+x_{n-1}^{n}} \geq \frac{n}{n-1} \sqrt[n]{\left(\frac{2003}{n}\right)^{m-n}}$$ where $m \geq n>1$ are integers and $x_{i}$ $(i=1,2, \ldots, n)$ are positive numbers such that $$x_{1}^{n}+x_{2}^{n}+\ldots+x_{n}^{n}\geq 2003 .$$
  11. Show that for a triangle $A B C$ with three acute angles, $$\frac{\cos A \cos B}{\sin 2 C}+\frac{\cos B \cos C}{\sin 2 A}+\frac{\cos C \cos A}{\sin 2 B} \geq \frac{\sqrt{3}}{2}$$
  12. Construct in the base plane $(A B C)$ of a tetrahedron $S A B C$ the triangles $A_{1} B C$, $B_{1} C A$, $C_{1} A B$ on the same side as $\triangle A B C$ and $A_{2} B C$, $B_{2} C A$, $C_{2} A B$ on the opposite side as $A B C$ such that $$\Delta A_{1} B C=\Delta A_{2} B C=\Delta S B C,\,\Delta B_{1} C A=\Delta B_{2} C A=\triangle S C A,\,\Delta C_{1} A B=\Delta C_{2} A B=\Delta S A B.$$ Prove that $\dfrac{R_{1}}{R_{2}}=\dfrac{r}{r_{S}}$, where $R_{1}$, $R_{2}$ are the radii of the circumcircles of $A_{1} B_{1} C_{1}$, $A_{2} B_{2} C_{2}$ and $r$, $r_{S}$ are the radii of the spheres inscribed and escribed the tetrahedron $S A B C$ opposite to $S$, respectively.

Issue 313

  1. Let $$A=\frac{1}{14}+\frac{1}{29}+\ldots+\frac{1}{n^{2}+(n+1)^{2}+(n+2)^{2}}+\ldots+\frac{1}{1877}.$$ Prove that $0,15<A<0,25$.
  2. The median $B M$ and the angled-bisector $C D$ of a triangle $A B C$ intersect at $K$ such that $K B=K C$. Calculate the measures of the angles $\widehat{A B C}$, $\widehat{A C B}$, knowing that the measure of $\widehat{B A C}$ is $105^{\circ}$.
  3. Prove that the number $A=2^{n}+6^{n}+8^{n}+9^{n}$ ($n$ is a positive integer) is divisible by $5$ when and only when $n$ is not divisible by $4$.
  4. Solve the equation $$2 \sqrt[3]{x+2}+\sqrt[3]{3 x+1} \geq \sqrt[3]{2 x-1}$$
  5. Prove that if the polynomial $$x^{3}+a x^{2}+b x+c$$ has $3$ distinct roots then the polynomial $$x^{3}-b x^{2}+a c x-c^{2}$$ has also $3$ distinct roots.
  6. The measures of the sides $B C$, $C A$, $A B$ of a triangle $A B C$ are respectively $a$, $b$, $c$ and the measures of its altitudes $A A^{\prime}$, $B B^{c}$, $C C^{\prime}$ are respectively $h_{a}$, $h_{b}$, $h_{c}$. Prove that the triangle $A B C$ is equilateral when and only when $$\sqrt{a+h_{a}}+\sqrt{b+h_{b}}+\sqrt{c+h_{c}}=\sqrt{a+h_{b}}+\sqrt{b+h_{c}}+\sqrt{c+h_{a}}$$
  7. A triangle $A B C$ has $\dfrac{1}{4} A C<A B<4 A C$. A line passing through its centroid $G$ cuts the sides $A B$, $A C$ respectively at $E$, $F$. Determine the position of $E$ so that $A E+A F$ attains its least value.
  8. Find the least value of the expression $$P =\frac{x_{1}^{3}}{x_{2}+x_{3}+\ldots+x_{n}}+\frac{x_{2}^{3}}{x_{1}+x_{3}+\ldots+x_{n}}+\ldots +\frac{x_{n}^{3}}{x_{1}+x_{2}+\ldots+x_{n-1}}$$ where $x_{1}, x_{2}, \ldots, x_{n}$ are real positive numbers satisfying the condition $$x_{1} x_{2}+x_{2} x_{3}+\ldots+x_{n-1} x_{n}+x_{n} x_{1}=s$$ ($s$ is a given constant).
  9. The polynomial $P(x)$ satisfies the conditions $$P(2003)=2003 !,\quad x \cdot P(x-1) \equiv (x-2003) \cdot P(x).$$ Prove that the polynomial $f(x)=$ $P^{2}(x)+1$ is irreducible.
  10. The sequence of real numbers $\left(x_{n}\right)$ $(n=0,1,2, \ldots)$ is defined by $$x_{0}=a,\quad x_{n+1}=2 x_{n}^{2}-1,\,\forall n=0,1,2, \ldots$$ Find all values of $a$ so that $x_{n}<0$ for all $n=0,1,2, \ldots$.
  11. Let $M$ be a point inside the triangle $A B C$. Let $d_{1}$, $d_{2}$, $d_{3}$ be respectively the distances from $M$ to the lines $B C$, $C A$, $A B$, let $R$ and $r$ be respectively the circumradius and the inradius of $\triangle A B C$. Prove that $$\frac{M A \cdot M B \cdot M C}{d_{1} \cdot d_{2} \cdot d_{3}} \geq \frac{4 R}{r}$$
  12. The altitudes of a tetrahedron $A B C D$ are concurrent at a point $H$. Prove that $$(H A+H B+H C+H D)^{2} \leq A B^{2}+A C^{2}+A D^{2}+B C^{2}+B D^{2}+C D^{2}$$

Issue 314

  1. Calculate $\frac{A}{B}$, where $$\begin{align}A&=\frac{1}{2.32}+\frac{1}{3.33}+\ldots+\frac{1}{n(n+30)}+\ldots+\frac{1}{1973.2003},\\ B&=\frac{1}{2.1974}+\frac{1}{3.1975}+\ldots+\frac{1}{n(n+1972)}+\ldots+\frac{1}{31.2003}.\end{align}$$
  2. Find the value of the expression $S=x^{4}+y^{4}+z^{4}$, knowing that $x+y+z=0$ and $x^{2}+y^{2}+z^{2}=1$.
  3. Prove that for every integer $n>1$ $$\frac{1}{3 \sqrt{2}}+\frac{1}{4 \sqrt{3}}+\ldots+\frac{1}{(n+1) \sqrt{n}}<\sqrt{2}$$
  4. Solve the equation $$x^{2}-2 x+3=\sqrt{2 x^{2}-x}+\sqrt{1+3 x-3 x^{2}}$$
  5. Find the least value of the expression $$\frac{a}{1+b-a}+\frac{b}{1+c-b}+\frac{c}{1+a-c}$$ where $a, b, c$ are positive numbers satisfying the condition $a+b+c=1$.
  6. A tangent to the incircle with center $I$ and with radius $r$ of a triangle $A B C$ cuts the circumcircle of $\triangle A B C$ at $M$ and $N$. Prove that $M N>2 r$.
  7. Let be given a rhombus $A B C D$ with $\widehat{B A D}=60^{\circ}$. $P$ is a point on the line $A B$, dinstinct from $A$ and $B$. The lines $C P$ and $D A$ intersect at $E$. The lines $D P$ and $B E$ intersect at $M$. Find the locus of $M$ when $P$ moves on the line $A B$.
  8. Let be given an integer $k \geq 2$. Prove that $\left(k^{2003}\right) !$ is divisible by $$(k !)\left(1+k+k^{2}+\ldots+k^{2002}\right)$$ where $n !$ denotes $1.2.3...n$.
  9. Find the greatest value of the expression $x^{3}+y^{3}+(9-x-y)^{3}$, where $x, y$ are positive integers satisfying the conditions $y \leq x \leq 5$, $x+y \leq 8$ and $y+\dfrac{x+1}{2} \geq 5$.
  10. The periodic functions $f(x): \mathbb R \rightarrow \mathbb R$ and $g(x): \mathbb R \rightarrow \mathbb R$ satisfy the condition $$\lim_{x \rightarrow+\infty}(f(x)-g(x))=0.$$ Prove that $f(x)=g(x)$ for every real number $x$.
  11. Let $A D$, $B E$, $C F$ be the altitudes of an acute triangle $A B C$. Let $R$ and $r$ be respectively the circumradii of the triangles $A B C$ and $D E F$. Prove that $$\sin ^{2} A+\sin ^{2} B+\sin ^{2} C=2+\frac{r}{R}$$
  12. Let $S_{A}$, $S_{B}$, $S_{C}$, $S_{D}$ be respectively the areas of the faces opposite to the vertices $A$, $B$, $C$, $D$ of a tetrahedron $A B C D$. Denote the measures of the dihedral angles with sides $B C$, $D A$; $A C$, $B D$; $A B$ $C D$ respectively by $\alpha$, $\alpha^{\prime}$; $\beta$, $\beta^{\prime}$; $\gamma$, $\gamma^{\prime}$. Prove the relation $$S_{A}^{2}+S_{B}^{2}+S_{C}^{2}+S_{D}^{2}=2 T$$ where $$T=S_{A}S_{D} \cos \alpha+S_{B} S_{D} \cos \beta+S_{C} S_{D} \cos \gamma+S_{B} S_{C} \cos \alpha^{\prime}+S_{A} S_{C} \cos \beta^{\prime}+S_{A} S_{B} \cos \gamma^{\prime}.$$

Issue 315

  1. Let $A$ be the product of the consecutive integers from $1$ to $1001$ and $B$ be the product of the consecutive integers from $1002$ to $2002$. Is $A+B$ divisible by $2003$? Can you do it by distinct methods?
  2. Prove that the sum $$A=\frac{1}{3^{2}}-\frac{1}{3^{4}}+\ldots+\frac{1}{3^{4 n-2}}-\frac{1}{3^{4 n}}+\ldots-\frac{1}{3^{100}}$$ is less than $0$, $1$. Give a generalization of the problem.
  3. Find the greatest value of the sum $$T=\frac{1}{a}+\frac{1}{b}+\frac{1}{c},$$ where $a, b, c$ are positive integers satisfying the condition $$\frac{1}{a}+\frac{1}{b}+\frac{1}{c}<1.$$
  4. Solve the equation $$\sqrt{\frac{42}{5-x}}+\sqrt{\frac{60}{7-x}}=6.$$
  5. Find the greatest value of the expression $x y+y z+z x$, where $x, y, z$ are positive real numbers satisfying the conditions $x \geq y \geq z$ and $32-3 x^{2}=z^{2}=16-4 y^{2}$.
  6. Let be given an angle $\widehat{x A y}=90^{\circ}$ and a point $M$ inside the angle. Let $H$ and $K$ be respectively the projections of $M$ on $A x$ and $A y$. On the line passing through $M$ perpendicular to $H K$ take a point $P$ such that $P M=H K$. Find the locus of $P$ when $M$ moves inside the angle $x A y$.
  7. Let be given an acute triangle $A B C$ inscribed in a circle with center $O$ and radius $R$. Let $D$, $E$, $F$ be respectively the points of intersection of the lines $A O$ and $B C$, $B O$ and $A C$, $C O$ and $A B$. Prove that $$A D+B E+C F \geq \frac{9 R}{2}.$$
  8. Find all natural number a so that there exists a natural number $n>1$ such that $a^{n}+1$ is divisible by $n^{2}$.
  9. Prove the inequality $$\frac{\sin ^{n+2} x}{\cos ^{n} x}+\frac{\cos ^{n+2} x}{\sin ^{n} x} \geq 1$$ where $0<x<\dfrac{\pi}{2}$ and $n$ is a positive integer.
  10. The sequences of numbers $\left(u_{n}\right)$ and $\left(v_{n}\right)$ $(n=0,1,2, \ldots)$  are defined by $$u_{0}=2001,\, u_{1}=2002,\, v_{0}=v_{1}=1,\quad u_{n+2}=2002 \sqrt{\frac{u_{n}}{v_{n+1}^{2001}}},\, v_{n+2}=2002 \sqrt{\frac{v_{n}}{u_{n+1}^{2001}}}$$ for every $n=0,1,2, \ldots$. Prove that these sequences have finite limits and find these limits.
  11. Let be given a triangle $A B C$ with $B C=a$, $C A=b$, $A B=c$ and $\widehat{B}>\widehat{C}$. Prove that a necessary and sufficient condition for $\widehat{A}=2(\hat{B}-\widehat{C})$ is $$(b-c)(b+c)^{2}=a^{2} b.$$
  12. Let $R$ be the radius of the circumscribed sphere of the tetrahedron $A_{1} A_{2} A_{3} A_{4}$ and let $S_{i}$ $(i=1,2,3,4)$ be the area of the face opposite to the vertex $A_{i}$ $(i=1,2,3,4)$. $M$ is an arbitrary point in space. Prove that $$\frac{M A_{1}}{S_{1}}+\frac{M A_{2}}{S_{2}}+\frac{M A_{3}}{S_{3}}+\frac{M A_{4}}{S_{4}} \geq \frac{2 \sqrt{3}}{R}.$$

Issue 316

  1. Consider all $7$-digit numbers, each of which is formed by seven distinct digits belonging to $\{1,2,3,4,5,6,7\}$.
    a) Are there three numbers $a, b, c$ among them such that $a+b=c$?
    b) Are there two distinct numbers $a$, $b$ among them so that $a$ is divisible by $b$?
  2. Let $A B C$ be a triangle with $\widehat{A B C}=30^{\circ}$, $\widehat{B A C}=130^{\circ}$ and let $A x$ be the opposite ray of the ray $A B$. The angled-bisector of $\widehat{A B C}$ cuts the angled-bisector of $\widehat{C A x}$ at $D$. The line $B A$ cuts the line $C D$ at $E$. Compare the measures of $A C$ and $C E$.
  3. Find all pairs of prime numbers $p$, $q$ so that $$p^{3}-q^{5}=(p+q)^{2}.$$
  4. Solve the system of equations $$\begin{cases} 20 \cdot \dfrac{y}{x^{2}}+11 y &=2003 \\ 20\cdot \dfrac{z}{y^{2}} + 11 z&=2003 \\ 20 \cdot \dfrac{x}{z^{2}}+11 x &=2003\end{cases}$$
  5. Find the least value of the expression $$\frac{a}{\sqrt{b}}+\frac{b}{\sqrt{c}}+\frac{c}{\sqrt{a}},$$ where $a, b, c$ are positive numbers satisfying the condition $a+b+c \geq 3$.
  6. Let $A B C$ be a triangle with $B C=a$, $A B=A C=b$ and suppose that the angledbisector of $\widehat{A C B}$ cuts the side $A B$ at $D$ so that $C D+D A=a$. Prove that $$a^{3}+b^{3}=3 a b^{2}.$$
  7. Let $D$ be a point on the the side $B C$ of triangle $A B C$ ($D$ distinct from $B$, $C)$ and let $E$ and $F$ be respectively the incenters of triangle $A B D$ and triangle $A C D$. Prove that if the four points $B$, $C$, $E$, $F$ lie on a circle then $$\frac{A D+D B}{A D+D C}=\frac{A B}{A C}.$$
  8. Prove that for every natural number $k>0$, the number $(\sqrt{2}+\sqrt{3})^{2 k}$ can be written in the form $a_{k}+b_{k} \sqrt{6}$ where $a_{k}$, $b_{k}$ are positive integers. Find the relations defining the sequences $\left(a_{k}\right)$, $\left(b_{k}\right)$, $k=1,2,3, \ldots$. Prove that for every $k \geq 2$, $a_{k-1} \cdot a_{k+1}-6 b_{k}^{2}$ is a constant not depending on $k$.
  9. Find the greatest value of the expression $$\left(x^{a_{1}}+x^{a_{2}}+\ldots+x^{a_{n}}\right)\left(\frac{1}{x^{a_{1}}}+\frac{1}{x^{a_{2}}}+\ldots+\frac{1}{x^{a_{n}}}\right)$$ where $x$ is an arbitrary positive number, $x \neq 1$, $n$ is a given even natural number, $a_{1}, a_{2}, \ldots, a_{n}$ are given numbers belonging to the segment $[m, m+1]$, where $m$ is a given natural number.
  10. Consider the equation $$x^{2}-2 x+\cos \alpha=0$$ where the parameter $\alpha$ belongs to the interval $(0 ; \pi / 2)$.
    a) Prove that the equation has two positive roots $x_{1}$, $x_{2}$.
    b) Let $f(x)$ be the trinomial with leading coefficient $1$, the two roots of which are $t_{1}=x_{1}^{x_{1}}+x_{2}^{x_{1}}$ and $t_{2}=x_{1}^{x_{2}}+x_{2}^{x_{2}}$. Find all value $c$ such that $f(c)<0$ for all $\alpha$ in the interval $(0 ; \pi / 2)$
  11. Let $A B C$ be a triangle with $B C=a$, $C A=b$, $A B=c$, let $O$ and $R$ be respectively its circumcenter and circumradius, let $I_{a}$, $I_{b}$, $I_{c}$ be respectively the centers of its escribed circles in the angles $A$, $B$, $C$ and let $r$ be the inradius of $\triangle A B C$. Prove that $$\frac{1}{2 R} \leq \frac{O I_{a}}{(a+b)(a+c)}+\frac{O I_{b}}{(b+c)(b+a)}+\frac{O I_{c}}{(c+a)(c+b)} \leq \frac{1}{4 r}.$$
  12. Consider the tetrahedra $A B C D$ with opposite congruent sides ($A B=C D$, $A C=B D$, $A D=B C$) and the dihedral angles $\alpha$, $\beta$, $\gamma$ respectively of sides $B C$, $C A$, $A B$ are acute. Find the least value of the expression $$T=\sqrt{\cos ^{2} \alpha+\frac{1}{\cos ^{2} \alpha}}+\sqrt{\cos ^{2} \beta+\frac{1}{\cos ^{2} \beta}}+\sqrt{\cos ^{2} \gamma+\frac{1}{\cos ^{2} \gamma}}.$$

Issue 317

  1. Calculate the following sums
    a) $A=1.2+2.3+\ldots+n(n+1)+\ldots+98.99$.
    b) $B=1.99+2.98+\ldots+n(100-n)+\ldots+98.2+99.1$.
  2. Find all pairs of rational numbers $x, y$ such that both numbers $x+y$ and $\dfrac{1}{x}+\dfrac{1}{y}$ are integers.
  3. Find the remainder of the division of $13376^{2003 !}$ by $2000$, where $n!$ is the product of the $n$ integers from $1$ to $n$.
  4. Prove that $$\frac{1}{a^{2}+2 b^{2}+3}+\frac{1}{b^{2}+2 c^{2}+3}+\frac{1}{c^{2}+2 a^{2}+3} \leq \frac{1}{2}$$ where $a, b, c$ are real numbers satisfying the condition $a b c=1$. When does equality occur?
  5. Let be given the real numbers $a, b, c, x, y$, $z$ satisfying the conditions $a x^{2003}=b y^{2003}=c z^{2003}$ and $x y+y z+z x=x y z \neq 0$. Prove that $$\sqrt[2003]{a x^{2012}+b y^{2002}+c z^{2012}}=\sqrt[2003]{a}+\sqrt[2003]{b}+\sqrt[2003]{c}.$$
  6. Let be given a triangle $A B C$. Construct outside of $A B C$ the parallelograms $A B E F$ and $A C P Q$ so that $A F=A C$, $A Q=A B$. Let $D$ be the point of intersection of $B P$ and $C E$. The lines $Q D$ and $F D$ cut $B C$ respectively at $M$ and $N$. Calculate the ratio $\dfrac{MN}{BC}$.
  7. A quadrilateral $A B C D$ with $A B>A C$ circumscribes about a circle with center $O$. Let $E$ and $F$ be the points of intersection of $B D$ with the circle. The line $O H$ passing through $O$ cuts orthogonally $A C$ at $H$. Prove that $\widehat{B H E}=\widehat{D H F}$.
  8. Let be given positive integers $m, n, k$ with $n>m$. Prove that the number of positive integral solutions of the system of equations $$\begin{cases}x_{1}+x_{2}+\ldots+x_{n} &=y_{1}+y_{2}+\ldots+y_{m}+1 \\ x_{1}+x_{2}+\ldots+x_{n} & \leq n k\end{cases}$$ is equal to $\displaystyle \sum_{i=0}^{n(k-1)} C_{n-1+i}^{n-1} \cdot C_{n-2+i}^{m-1}$.
  9. For every positive integer $k$, consider the sequence of numbers $\left(x_{n}^{k}\right)(n=1,2, \ldots)$ defined by $$x_{1}^{k}=1,\quad x_{n}^{k}=\sum_{i=1}^{n} \frac{i^{k}}{i !},\,\forall n=2,3, \ldots$$ a) Prove that the sequence $\left(x_{n}^{k}\right)(n=1,2, \ldots)$ has a finite limit for every positive integer $k$. b) Put $\displaystyle E_{k}=\lim _{n \rightarrow \infty} x_{n}^{k}$. Prove that $y_{k}=\dfrac{E_{k}}{E_{1}}$ is a positive integer for every positive integer $k$.
  10. Let $a_{1}, a_{2}, \ldots, a_{n}$ be real numbers distinct from 0 and $u_{1}, u_{2}, \ldots, u_{n}$ be positive real numbers such that $u_{1}<u_{2}<\ldots<u_{n}$, and let $f(x)=\sum_{i=1}^{n} a_{i} \cos \left(u_{i} x\right)$ be a periodic function defined on $R$. Prove that $\dfrac{u_{i}}{u_{1}}$ is a rational number for every $i=2,3, \ldots, n$.
  11. Let be given a convex polygon $A_{1} A_{2} \ldots A_{n}$ $(n \geq 3)$ and $M$ be a point inside the polygon. Let $\alpha_{i}=\widehat{M A_{i} A_{i+1}}$ ($i=1,2, \ldots, n$ and $A_{n+1}$ is considered as $A_{1}$). Prove that $$\min _{1 \leq i \leq n}\left\{\alpha_{i}\right\} \leq \frac{(n-2) \pi}{2 n}.$$
  12. The sides $O A$, $O B$, $O C$ of a tetrahedron. $O A B C$ are orthogonal each to the others. The incircle of triangle $A B C$ touches the sides $B C$, $C A$, $A B$ respectively at $D$, $E$, $F$. Let $\alpha$, $\beta$, $\gamma$ be respectively the measures of the dihedral angles with sides $B C$, $C A$, $A B$ of the tetrahedron. Prove that
    a) $V_{O D E F}=\dfrac{1}{3} O A \cdot O B \cdot O C \cdot \sin \dfrac{A}{2} \cdot \sin \dfrac{B}{2} \cdot \sin \dfrac{C}{2}$ where $A$, $B$, $C$ denote the angles of triangle $A B C$.
    b) $\tan\alpha \cdot \tan\beta \cdot \tan\gamma \geq 2 \sqrt{2}$. When does equality occur ?

Issue 318

  1. Can the sum of the digits of a perfect square be equal to one of the following numbers
    a) $2003$;
    b) $2004$;
    c) $2007$?
  2. Let $A B C$ be a triangle with $A B=A C$, $\widehat{B A C}=$ $90^{\circ}$ and $M$ be the midpoint of $B C$. $D$ is a point on the ray $B C$ ($D$ distinct from $B$, $M)$. Draw the line $B K$ perpendicular to $A D$ at $K$. Prove that $K M$ is the interior or exterior angled bisector of $\Delta B K D$ issued from the vertex $K$.
  3. Prove that if $2 n$ is the sum of two perfect squares (greater than $1$) then $n^{2}+2 n$ can be written as the sum of four distinct perfect squares (greater than $1$).
  4. Solve the system of equations $$\begin{cases} x^{2}(y+z)^{2} &=\left(3 x^{2}+x+1\right) y^{2} z^{2} \\ y^{2}(z+x)^{2} &=\left(4 y^{2}+y+1\right) z^{2} x^{2} \\ z^{2}(x+y)^{2} &=\left(5 z^{2}+z+1\right) x^{2} y^{2} \end{cases}$$
  5. Prove the inequality $$\left(1+a^{n+1}\right)\left(1+b^{n+1}\right)\left(1+c^{n+1}\right) \geq\left(1+a b^{n}\right)\left(1+b c^{n}\right)\left(1+c a^{n}\right)$$ where $a, b, c$ are positive numbers and $n$ is a positive integer, $n \leq 4$.
  6. Let be given a trapezoid $A B C D$ with $A B || C D$ and $A B \perp B D$. The diagonals $A C$ and $B D$ intersect at $G$. On the line $C E$ perpendicular to $A C$, take the point $E$ so that $C E=A G$ and the line $C D$ does not cut the segment $G E$. On the ray $D C$, take the point $F$ such that $D F=G B$. Prove that $G F$ is perpendicular to $E F$.
  7. The circle with center $O$, radius $R$ is tangent to the circle with center $O^{\prime}$, radius $R^{\prime}$ $(R>R^{\prime})$ at the point $A$. The ray $A x$ of a right angle $x A y$ cuts the circle with center $O$ again at $B$ and the ray $A y$ cuts the circle with center $O'$ again at $C$. Let $H$ be the projection of $A$ on $B C$. Prove that when the right angle $x A y$ rotates around the point $A$, the point $H$ moves on a circle.
  8. Find all positive integers $k$ so that the equation $$x^{3}+y^{3}+z^{3}=k x^{2} y^{2} z^{2}$$ has positive integral solution and solve this equation.
  9. Prove that the sequence of numbers $$s_{n}=\sum_{k=1}^{n} k \sin \frac{k}{n^{3}}(n=1,2,3, \ldots)$$ has finite limit and find this limit.
  10. Find all functions $f: \mathbb R \rightarrow \mathbb R$ satisfying the condition $$f(x)=\max _{y \in \mathbb R}\left\{x y^{2003}+y x^{2003}-f(y)\right\},\,\forall x \in \mathbb R.$$
  11. Let $A B C D$ be an inscribable quadrilateral such that the circle with diameter $C D$ cuts the segments $A C$, $A D$, $B C$, $B D$ respectively at $A_{1}$, $A_{2}$, $B_{1}$, $B_{2}$ and the circle with diameter $A B$ cuts the segments $C A$, $C B$, $D A$, $D B$ respectively at $C_{1}$, $C_{2}$, $D_{1}$, $D_{2}$. Prove that there exists a circle touching the four lines $A_{1} A_{2}$, $B_{1} B_{2}$, $C_{1} C_{2}$, $D_{1} D_{2}$.
  12. In space, consider four rays $P a$, $P b$, $P c$, $P d$ so that no three of them are coplanar and $\widehat{a P b}=\widehat{c P d}$, $\widehat{b P c}=\widehat{d P a}$, $\widehat{c P a}=\widehat{b P d}$.
    a) Prove that there exist such four rays.
    b) Let $\alpha$, $\beta$, $\gamma$, $\delta$ be respectively the angles formed by the rays $P a$, $P b$, $P c$, $P d$ with a fifth given ray $P t$. Find the set of all rays $P x$ such that $$\cos \widehat{a P x}+\cos \widehat{b P x}+\cos \widehat{c P x}+\cos \widehat{d P x}=\cos \alpha+\cos \beta+\cos \gamma+\cos \delta.$$

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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 2003
Mathematics and Youth Magazine Problems 2003
MOlympiad.NET
https://www.molympiad.net/2022/04/mym-2003.html
https://www.molympiad.net/
https://www.molympiad.net/
https://www.molympiad.net/2022/04/mym-2003.html
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