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Mathematics and Youth Magazine Problems 2014

Issue 439

  1. Find all possible ways of inserting three distinct digits into the positions represented by a star in $\overline{155*710*4*16}$ so that the resulting number is divisible by 396.
  2. The triangles $XBC$, $YCA$ and $ZAB$ are constructed externally on the sides of a triangle $ABC$ such that triangle $XBC$ is isosceles with angle $BXC$ equals $120^{0}$ and $YCA$, $ZAB$ are both equilateral. Prove that $XA$ is perpendicular to $YZ$.
  3. Find all positive integer solutions $x,y$ of the equation \[(x^{2}-9y^{2})^{2}=33y+16.\]
  4. Solve the following system of equations \[\begin{cases}6(1-x)^{2} & =\frac{1}{y}\\ 6(1-y)^{2} & =\frac{1}{x} \end{cases}.\]
  5. Point $C$ lies on a half-circle $(O)$ with diameter $AB=2R$, $CH$ is the altitude from $C$ to $AB$ ($H$ differs from $O$). The points $E,F$ move on the half-circle such that $\widehat{CHE}=\widehat{CHF}$. Prove that the line $EF$ always passes through a fixed point.
  6. Solve for $x$ \[\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{2+x}}}}=\sqrt[3]{6+\sqrt[3]{6+\sqrt[3]{6+\sqrt[3]{6+}x}}}.\]
  7. Solve the following system of equations \[\begin{cases} 4x^{2} & =(\sqrt{x^{2}+1}+1)(x^{2}-y^{3}+3y-2)\\ (x^{2}+y^{2})^{2}+1 & =x^{2}+2y \end{cases}.\]
  8. Let $BC=a$, $CA=b$, $AB=c$ be the side lengths of a triangle $ABC$; $R$ and $r$ denote its circumradius and inradius respectively. If $S$ is the area of triangle $ABC$, prove that \[\frac{R}{r}\geq\max\left\{ \frac{1}{2};\sqrt{\frac{ab^{3}+bc^{3}+ca^{3}}{3S^{2}}};\sqrt{\frac{ab^{3}+bc^{3}+ca^{3}}{3S^{2}}}\right\} .\]
  9. Find all odd positive integers $n$ such that $15^{n}+1$ is divisible by $n$.
  10. Determine all possible pairs of functions $f:\mathbb{R}\to\mathbb{R}$; $g:\mathbb{R}\to\mathbb{R}$ such that for any $x,y\in\mathbb{R}$, the following identity holds \[f(x+g(y))=xf(y)-yg(y)+g(x).\]
  11. $n$ students ($n\geq2$) are standing in a straigh line. Each time the teacher blow a whistle, exactly two students exchange their positions.  Can it be possible that after an odd number of such whistles, all students returned to their original positions?.
  12. Let $AH$ ($H\in BC$) be the altitude of an acute triangle $ABC$. Point $P$ moves on the segment $AH$. Let $E,F$ denote the feet of the perpendicular from $P$ to $AB,AC$ respectively.
    a) Prove that the points $B,R,F,C$ are concyclic.
    b) Let $O'$ denote the center of the circle containing $B,E,F,C$. Prove that $PO'$ always passes through a fix point, independent of the position of point $P$ chosen on $AH$.

Issue 440

  1. Which number is greater? $P$ or $Q$, given that $$\begin{align*} P & =\frac{20}{30}+\frac{20}{70}+\frac{20}{126}+\ldots+\frac{20}{798};\\ Q & =\left(\frac{31}{2}\cdot\frac{32}{2}\cdot\frac{33}{2}\ldots\frac{60}{2}\right):(1.3.5\ldots59). \end{align*}$$
  2. Given $2014$ points $A_{1},A_{2},\ldots,A_{2014}$ and a circle with radius $1$ on the plane, prove that there always exists a point $M$ on the circle such that \[MA_{1}+MA_{2}+\ldots+MA_{2014}\geq2014.\]
  3. Prove that for any natural number $n$, \[n^{4}-5n^{3}-2n^{2}-10n+4\] is not divisible by $49$.
  4. Let $R$ denote the radius of the circumcircle of a given triangle $ABC$. The internal and external angle bisector of angle $\widehat{ACB}$ meet $AB$ at $E$ and $F$ respectively. Prove that if $CE=CF$, then $AC^{2}+BC^{2}=4R^{2}$.
  5. Solve the following system of equations \[\begin{cases} 2x\left(1+\frac{1}{x^{2}-y^{2}}\right) & =5\\ 2(x^{2}+y^{2})\left(1+\frac{1}{(x^{2}-y^{2})^{2}}\right) & =\frac{17}{2}\end{cases}.\]
  6. Determine the funtion \[f(x)=ax^{2}+bx+c \] where $a,b,c$ are integers such that $f(0)=2014$, $f(2014)=0$ and $f(2^{n})$ is a multiple of $3$ for any natural number $n$.
  7. The positive real numbers $x,y,z$ satisfy the equation $xy=1+z(x+y)$. Find the greatest value of \[P=\frac{2xy(xy+1)}{(1+x^{2})(1+y^{2})}+\frac{z}{1+z^{2}}.\]
  8. In an acute triangle $ABC$, the three altitudes $AA_{1},BB_{1},CC_{1}$ meet at $H$. Prove that $ABC$ is an equilateral triangle if and only if \[ HA^{2}+HB^{2}+HC^{2}=4(HA_{1}^{2}+HB_{1}^{2}+HC_{1}^{2}).\]

Issue 441

  1. How many triples of positive integers $(a,b,c)$ are there such that $$\text{lcm}(a,b)=1000,\quad \text{lcm}(b,c)=2000,\quad \text{lcm}(a,c)=2000?.$$
  2. Let $ABC$ be an isosceles triangle $A$ with $\widehat{BAC}=100^{0}$, point $D$ on segment $BC$ such that $\widehat{CAD}=20^{0}$, point $E$ on the ray $AD$ such that triangle $ACE$ is isosceles at vertex $C$. Determine the measure of all angles of triangle $BDE$. 
  3. The sum of $m$ distinct even positive integers and $n$ distinct odd positive integers equal $2014$. Find the greatest possible value of $3m+4n$.
  4. Triangle $ABC$ is inscribed in circle center at $O$. Parallel lines are drawn through vertices $A,B,C$ such that they are not parallel to any of the sides of triangle $ABC$. These parallel lines intersect $(O)$ at $A_{1},B_{1},C_{1}$ respectively. Prove that the orthocenters of triangles $A_{1}BC$, $B_{1}CA$, $C_{1}AB$ are collinear. 
  5. Solve the system of equations \[ \begin{cases} (1+x)(1+x^{2})(1+x^{4}) & =1+y^{7}\\ (1+y)(1+y^{2})(1+y^{4}) & =1+x^{7} \end{cases}.\]
  6. Find all polynomials with real coefficients $P(x)$ such that the following conditions are satisfied \[\begin{cases} P(x)-10 & =\sqrt{P(x^{2}+3)}-13\quad(x\geq0)\\ P(2014) & =2024 \end{cases}.\]
  7. Let $h_{a},h_{b},h_{c}$ and $l_{a},l_{b},l_{c}$ denote the altitudes and inner angle-bisectors of a triangle $ABC$. Prove that \[\frac{1}{h_{a}h_{b}}+\frac{1}{h_{b}h_{c}}+\frac{1}{h_{c}h_{a}}\geq\frac{1}{l_{a}^{2}}+\frac{1}{l_{b}^{2}}+\frac{1}{l_{c}^{2}}.\]
  8. Given that $0<x<\frac{\pi}{2}$. Prove that at least one of the two numbers $\left(\frac{1}{\sin x}\right)^{\frac{1}{\cos^{2}x}}$, $\left(\frac{1}{\cos x}\right)^{\frac{1}{\sin^{2}x}}$ is greater than $\sqrt{3}$.

Issue 442

  1. Find two whole numbers of the form $\overline{ab}$ and $\overline{ba}$ ($a\ne b$) such that \[\frac{\overline{ab}}{\overline{ba}}=\frac{\underset{2014\text{ digits}}{\overline{a\underbrace{3\ldots3}b}}}{\underset{2014\text{ digits}}{\overline{b\underbrace{3\ldots3}a}}}.\]
  2. The sum $A$ below consists of 2014 summands \[A=\frac{1}{19^{1}}+\frac{2}{19^{2}}+\frac{3}{19^{3}}+\ldots+\frac{2014}{19^{2014}}.\] Compare the number $A^{2013}$ with $A^{2014}$.
  3. Let $ABCD$ be a quadriteral whose diagonals $AC$ and $BD$ are perpendicular. $M$ and $N$ are the midpoints of line segments $AB,AD$ respectively. Points $E,F$ are the feet of perpendicular lines from $M$ and $N$ onto $CD,BC$ respectively. Prove that $MNEF$ is a cyclic quadrilateral.
  4. Solve for $x$ \[4x^{3}+4x^{2}-5x+9=4\sqrt[4]{16x+8}.\]
  5. The real numbers $x,y,z$ satisfy $x+y+z=1$. Prove the inequality \[44(xy+yz+zx)\leq(3x+4y+5z)^{2}.\] 
  6. Prove that the following equation has no real solutions \[9x^{4}+x(12x^{2}+6x-1)+(x+1)(9x^{2}+12x+5)+1=0.\]
  7. Triangle $ABC$ inscribed in a circle centerd at $O$ and radius $R$, where $CA\ne CB$, $\widehat{ACB}=90^{0}$. The circumcircle centered at $S$ of triangle $AOB$ meets $CA,CB$ at points $M,N$ respectively. Let $K$ be the reflection of $S$ in the line $MN$. Prove that $SK=R$. 
  8. The real numbers $x,y,z$ satisfy $x^{2}+y^{2}+z^{2}=8$. Determine the largest and smallest values of the following expression \[P=(x-y)^{5}+(y-z)^{5}+(z-x)^{5}.\]

Issue 443

  1. $21$ distinct integers are chosen so that the sum of any subset of $11$ numbers among them is always greater than the sum of the remaining $10$. If one of them is $101$, and the largest number is $2014$, find the other $19$ numbers.
  2. In a triangle $ABC$ where $\widehat{BAC}=40^{0}$ and $\widehat{ABC}=60^{0}$, point $D$ and $E$ are chosen on the sides $AC$ and $AB$ respectively such that $\widehat{CBD}=40^{0}$ and $\widehat{BCE}=70^{0}$. $BD$ and $CE$ intersect at point $F$. Prove that $AF$ is perpendicular to $BC$.
  3. Solve the following system of equations \[\begin{cases} 2\sqrt{2x}-\sqrt{y} & =1\\ \sqrt[3]{8x^{3}+y^{3}} & =\sqrt[3]{2}(\sqrt{x}+\sqrt{y}-1) \end{cases}.\]
  4. In a triangle $ABC$, points $E,D$ on the sides $AB$ and $AC$ respectively such that $\widehat{ABD}=\widehat{ACE}$. The circumcircle of triangle $ADB$ meets $CE$ at $M$ and $N$. The circumcircle of triangle $AEC$ meets $BD$ at $I$ and $K$. Prove that the points $M,I,N,K$ lie on a circle.
  5. Prove that for all positive real numbers $a,b,c$ the following inequality holds \[\frac{a^{2}}{a+b}+\frac{b^{2}}{b+c}+\frac{c^{2}}{c+a}\geq\frac{\sqrt{2}}{4}(\sqrt{a^{2}+b^{2}}+\sqrt{b^{2}+c^{2}}+\sqrt{c^{2}+a^{2}}).\]
  6. Determine all real solutions of the equation \[(x^{5}+x-1)^{5}+x^{5}=2.\]
  7. Let $M$ be a point inside a given triangle $ABC$ and let $x,y,z$ denote the distance from $M$ onto $BC,CA,AB$ respectively. Prove that $\widehat{BAM}=\widehat{CBM}=\widehat{ACM}$ if and only if \[\frac{bx}{c}=\frac{cy}{a}=\frac{az}{b}\] where $BC=a$, $CA=b$, $AB=c$.
  8. Let $x,y,z$ be theree arbitrary numbers from the interval $[0,1]$. Determine the maximum value of $P$, where \[P=\frac{x}{y+z+1}+\frac{y}{z+x+1}+\frac{z}{z+y+1}+(1-x)(1-y)(1-z).\]

Issue 444

  1. Find the maximum calue of positive integer $n$ such that $2013$ can be written as the sum of $n$ compound numbers. How does the answer change if $2013$ is replaced by $2014$.
  2. Let $ABC$ be a right triangle, right angle at $A$, $\widehat{B}=60^{0}$. Point $E$ on side $AC$ such that $\widehat{ABE}=20^{0}$. Point $K$ on the half line $BE$ such that $EK=BC$. Find the measure of the angle $\widehat{BCK}$.
  3. Solve the inequality \[\frac{x^{2}+8}{x+1}+\frac{x^{3}+8}{x^{2}+1}+\frac{x^{4}+8}{x^{3}+8}+\ldots+\frac{x^{101}+8}{x^{100}+1}\geq800.\]
  4. The quadrilateral $ABCD$ is inscribed in circle $(O)$ where angle $\widehat{BAD}$ is obtuse. The rays through $A$ and perpendicular to $AD,AB$ meet $CB,CD$ at $P$ and $Q$ respectively. $PQ$ intersects $BD$ at $M$. Prove that $\widehat{MAC}=90^{0}$.
  5. Solve the system of equations \[\begin{cases} \sqrt{2x-3}-\sqrt{y} & =2x-6\\ x^{3}+y^{3}+7(x+y)xy & =8xy\sqrt{2(x^{2}+y^{2})} \end{cases}.\]
  6. The positive real numbers $a,b,c$ satisfy the equation $abc=1$. Prove the inequality \[a^{3}+b^{3}+c^{3}+\frac{ab}{a^{2}+b^{2}}+\frac{bc}{b^{2}+c^{2}}+\frac{ca}{c^{2}+a^{2}}\geq\frac{9}{2}.\]
  7. Let $ABC$ be a triangle. $D$ is the midpoint of side $BC$ and $M$ is an arbitrary point on segment $BD$. $MEAF$ is a parallellogram where vertex $E$ lies on $AB$, $F$ lies on $AC$, $MF$ and $AD$ intersect at $H$. The line through $B$ and parallel to $EH$ intersects $MF$ at $K$; $AK$ meets $BC$ at $I$. Find the ratio $\dfrac{IB}{ID}$.
  8. The sequence $\{v_{n}\}_{n}$ satisfies \[v_{1}=5,\quad v_{n+1}=v_{n}^{4}-4v_{n}^{2}+2.\] Find a closed formular for $v_{n}$.

Issue 445

  1. Prove that \[\overline{\underset{2014\text{ digits}}{\underbrace{111\ldots111}}\underset{2014\text{ digits}}{\underbrace{222\ldots222}}}-\overline{\underset{2014\text{ digits}}{\underbrace{333\ldots333}}}\] is a perfect square.
  2. Given a triangle $ABC$ with $\widehat{BAC}>90^{0}$ and the lengths of its sides are three consecutive even numbers. Find these lengths. 
  3. Let $a,b$ be two positive real numbers such that $a+b$, $ab$ are positive integers and $[a^{2}+ab]+[b^{2}+ab]$ is a perfect square, where $[x]$ is the greatest integer not exceeding $x$. Prove that $a,b$ are positive integers.
  4. Let $ABC$ be an acute triangle with altitudes $AD$, $BE$, $CF$. On the opposite rays of the rays $DA$, $EB$, $FC$ choose three points $M,N,P$ respectively such that $\widehat{BMC}=\widehat{CNA}=\widehat{APB}=90^{0}$. Prove that the lines containing the sides of the hexagon $APBMCN$ are both tangent to a circle.
  5. Find all integers $m$ such that the equation \[x^{3}+(m+1)x^{2}-(2m-1)x-(2m^{2}+m+4)=0\] has an integer solution.
  6. Given any triple of real numbers $a,b,c>1$. Prove the following inequality \[(\log_{b}a+\log_{c}a-1)(\log_{c}b+\log_{a}b-1)(\log_{a}c+\log_{b}c-1)\leq1.\]
  7. Let $ABC$ ($AB<AC$) be an acute triangle inscribed in a circle $(O)$. The altitudes $AD$, $BE$, $CF$ intersect at $H$. Let $K$ be the midpoint of $BC$. The tangent lines to the circle $(O)$ at $B$ and $C$ meets at $J$. Prove that $HK$, $JD$, $EF$ are concurrent.
  8. Find all functions $f:\mathbb{R}\to\mathbb{R}$ such that $f$ is bounded on a certain interval containing $0$ and $f$ satisfies \[2f(2x)=x+f(x)\] for every $x\in\mathbb{R}$.
  9. Let \[f(x)=x^{3}-3x^{2}+9x+1964\] be a polynomial. Prove that there exists an integer $a$ such that $f(a)$ is divisible by $3^{2014}$. 
  10. Does there exist a continuous funtion $f:\mathbb{R}\to\mathbb{R}$ satisfying the following property: for any $x\in\mathbb{R}$, among $f(x)$, $f(x+1)$, $f(x+2)$ there are exactly two rational numbers and one irrational number?.
  11. Given a sequence $\{a_{n}\}_{1}^{\infty}$ where \[a_{1}=1,\,a_{2}=2014,\quad a_{n+1}=\frac{2013a_{n}}{n}+\left(1+\frac{2013}{n-1}\right)a_{n-1}.\] Find \[\lim_{n\to\infty}\left(\frac{1}{a_{1}}+\frac{1}{a_{2}}+\ldots+\frac{1}{a_{n}}\right). \]
  12. Let $ABCD$ be a quadrilateral circumscribing a circle $(I)$. The sides $AB$ and $BC$ are tangent to $(I)$ at $M$ and $N$ respectively. Let $E$ be the intersection of $AC$ and $MN$, and $F$ be the intersection of $BC$ and $DE$. $DM$ intersects $(I)$ at another point, say $T$. Prove that $FT$ is tangent to $(I)$.

Issue 446

  1. Find all prome numbers $p,q,r$ satisfying \[(p+1)(q+2)(r+3)=4pqr.\]
  2. Given a triangle $ABC$ with $\widehat{A}=75^{0}$, $\widehat{B}=45^{0}$. On the side $AB$, choose a point $D$ such that $\widehat{ACD}=45^{0}$. Prove that $DA=2DB$.
  3. Solve the following system of equations \[\begin{cases} \sqrt{x+y+2}+x+y & =2(x^{2}+y^{2})\\ \frac{1}{x}+\frac{1}{y} & =\frac{1}{x^{2}}+\frac{1}{y^{2}} \end{cases}.\]
  4. Given a triangle $ABC$. Let $(I)$ be the inscribed circle and $(J)$ the escribed circle corresponding to the angle $A$. Suppose that $(J)$ is tangent to the lines $BC$, $CA$ and $AB$ at $D,E$ and $F$ respectively. The line $JD$ meets the line $EF$ at $N$. The line which contains $I$ and is perpendicular to the line $BC$ intersects the line $AN$ at $P$. Let $M$ be the midpoint of $BC$. Prove that $MN=MP$.
  5. Find all the integer solutions of the following equation \[x^{3}=4y^{3}+x^{2}y+y+13.\]
  6. Let $$f(x)=\frac{4^{x+2}}{4^{x}+2}.$$ Find \[f(0)+f\left(\frac{1}{2014}\right)+f\left(\frac{2}{2014}\right)+\ldots+f\left(\frac{2013}{2014}\right)+f(1).\]
  7. Given a tetrahedron $ABCD$. Let $d_{1},d_{2},d_{3}$ be the distances between the pairs of opposite sides $AB$ and $CD$, $AC$ and $BD$, $AD$ and $BC$. Prove that \[V_{ABCD}\geq\frac{1}{3}d_{1}d_{2}d_{3}.\]
  8. Given an integer $n$ which is greater than $1$. Let $a_{1},a_{2},\ldots,a_{n}$ be arbitrary positive real numbers satisfying \[\frac{1}{a_{1}}+\frac{1}{a_{2}}+\ldots+\frac{1}{a_{n}}=1.\] Prove that \[a_{1}^{a_{2}}+a_{2}^{a_{3}}+\ldots+a_{n-1}^{a_{n}}+a_{1}+a_{2}+\ldots+a_{n}>n^{3}+n.\]
  9. Let $T$ be a set of $n$ elements. What is the maximal number of subsets of $T$ which can be picked so that each subset has exactly 3 elements and any two subsets has nonempty intersection?.
  10. Let $p$ be a prime number. Find all the polynomials $f(x)$ with integer coefficients such that for every positive integer $n$, $f(n)$ is a divisor of $p^{n}-1$.
  11. Let $x,y$ be the positive real numbers satisfying $[x]\cdot[y]=30^{4}$, where $[a]$ is the greatest integer not wxceeding $a$. Find the minimum and maximum values of \[P=[x[x]]+[y[y]].\]
  12. Given a triangle $ABC$. Let $E,F$ be points on $CA$, $AB$ respectively such that $EF\parallel BC$. The perpendicular bisector of $BC$ intersects $AC$ at $M$ and the perpendicular bisector of $EF$ intersects $AB$ at $N$. The circle circumscribing the triangle $BCM$ meets $CF$ at $P$ which is different from $C$. The circle circumscribing the triangle $EFN$ meets $CF$ at $Q$ which is different from $F$. Prove that the perpecdicular bisector of $PQ$ contains the midpoint of $MN$.

Issue 447

  1. Find all the integer solutions of the following equation \[1+x+x^{2}+x^{3}=y^{2}.\]
  2. Let $x,y,z$ be three coprime positive integers satisfying \[(x-z)(y-z)=z^{2}.\] Prove that $xyz$ is a perfect square. 
  3. Solve the following equation \[\frac{1}{\sqrt{3x}}+\frac{1}{\sqrt{9x-3}}=\frac{1}{\sqrt{5x-1}}+\frac{1}{\sqrt{7x-2}}.\]
  4. Given a circle $(O,R)$ and a chord $AB$ with the distance from $O$ is $d$ ($0<d<R$). Two circles $(I)$, $(K)$ are externally tangent at $C$, are both tangent to $AB$ and are internally tangent with $(O)$ ($I$ and $K$ are in the same half-plane determined by the line through $AB$). Find the locus of the points $C$ which vary when $(I)$ and $(K)$ vary.
  5. Find all positive integers $a$ and $b$ so that both equations $x^{2}-2ax-3b=0$ and $x^{2}-2bx-3a=0$ have positive integer solution.
  6. Find all positive real numbers $x,y,z$ satisfying system of equations \[\begin{cases} \dfrac{1}{x+1}+\dfrac{1}{y+1}+\dfrac{1}{z+1} & =1\\ xyz(x+y+z)(x+1)(y+1)(z+1 & =1296 \end{cases}.\]
  7. Given a tetrahedron $ABCD$ and the lengths of its sides $AB=BD=DC=x$, $BC=CA=AD=y$. Prove that \[ \frac{3}{5}<\frac{x}{y}<\frac{5}{3}.\]
  8. Find the maximum value of the expression \[P=|(a^{2}-b^{2})(b^{2}-c^{2})(c^{2}-a^{2})|,\] in which $a,b,c$ are nonnegative numbers satisfying $a+b+c=\sqrt{5}$.
  9. Solve equation \[x^{4}+ax^{3}+bx^{2}+2ax+4\] given $9(a^{2}+b^{2})=16$.
  10. Find all pairs of positive integers $(a,b)$ satisfying the following properties: $4a+1$ and $4b-1$ are coprime and $a+b$ is a divisor of $16ab+1$.
  11. Given two sequences \[a_{1}=0,\,a_{2}=16,\,a_{3}=18,\,a_{n+2}=8a_{n}+6a_{n-1}\] and \[b_{1}=3,\,b_{2}=19,\,b_{3}=69,\,b_{n+2}=3b_{n+1}+5b_{n}-b_{n-1}\] for $n\geq2$. Prove that \begin{align*} b_{n} & =C_{n}^{0}a_{n}+C_{n}^{1}a_{n-1}+\ldots+C_{n}^{n-1}a_{1}+3C_{n}^{n},\\ a_{n} & =C_{n}^{0}b_{n}-C_{n}^{1}b_{n-1}+C_{n}^{2}b_{n-2}-\ldots+(-1)^{n-1}C_{n}^{n-1}b_{1}+(-1)^{n}3C_{n}^{n}. \end{align*}
  12. Given a triangle $ABC$ and its circumscribed circle $(O)$. The points $A_{1},B_{1}$and $C_{1}$ are on the sides $BC,CA$ and $AB$ respectively. The circumscribed circles $(AB_{1}C_{1})$, $(BC_{1}A_{1})$, and $(CA_{1}B_{1})$ intersect $(O)$ at $A_{2},B_{2}$ and $C_{2}$ respectively. Find the positions of $A_{1},B_{1}$ and $C_{1}$ so that $\dfrac{S_{A_{1}B_{1}C_{1}}}{S_{A_{2}B_{2}C_{2}}}$ is minimal.

Issue 448

  1. Let $m,n$ be two positive integers such that $3^{m}+5^{n}$ is divisible by $8$. Prove that $3^{n}+5^{m}$ is also divisible by $8$.
  2. Given a triangle $ABC$ with $A$ is an obtuse angle. Let $M$ be the midpoint of $BC$. Inside $\widehat{BAC}$, draw two rays $Ax$ and $Ay$ such that $\widehat{BAx}=\widehat{CAy}=22^{0}$. Let $H$ be the projection of $B$ on $Ax$, and $I$ the projection of $C$ on $Ay$. Find the angle $HMI$.
  3. Solve the following equation \[\sqrt[3]{x^{2}+3x+3}+\sqrt[3]{2x^{2}+3x+2}=6x^{2}+12x+8.\]
  4. Let $ABC$ be a right triangle with the right angle $A$ and let $AB=a$, $AC=b$. Two internal angle bisectors $BB_{1}$ and $CC_{1}$ intersect at $R$, $AR$ intersects $B_{1}C_{1}$ at $M$. Compute the distance from $M$ to $BC$ in terms of $a$ and $b$.
  5. Let $a,b,c$ be positive real numbers satisfying $a^{3}+b^{3}+c^{3}=1$. Prove that \[\frac{a^{2}+b^{2}}{ab(a+b)^{3}}+\frac{b^{2}+c^{2}}{bc(b+c)^{3}}+\frac{c^{2}+a^{2}}{ca(c+a)^{3}}\geq\frac{9}{4}.\]
  6. Express 2015 as a sum of integers $a_{1},a_{2},\ldots,a_{n}$ which are greater than $1$ such that ${\displaystyle \sum_{i=1}^{n}\sqrt[a_{i}]{a_{i}}}$ is maximal.
  7. Given a quadrilateral $ABCD$ and $a,b,c,d$ respectively are external angle bisectors of $\widehat{DAB}$, $\widehat{ABC}$, $\widehat{BCD}$, $\widehat{CDA}$. Denote $K=a\cap b$, $L=b\cap c$, $M=c\cap d$, $N=d\cap a$. Prove that the quadrilateral $KLMN$ inscribes a circle whose radius is \[\frac{KM\cdot LN}{AB+BC+CD+DA}.\]
  8. Suppose that the polynomial \[f(x)=x^{3}+ax^{2}+bx+c\] has three non-negative solutions. Find the maximal real number $\alpha$ such that \[f(x)\geq\alpha(x-a)^{2},\quad\forall x\geq0.\]
  9. Let $[x]$ be the greatest integer not exceeding $x$ and let $\{x\}=x-[x]$. Find $$\left\{ \frac{p^{2012}+q^{2016}}{120}\right\}$$ where $p,q$ are primes numbers which are greater than 5.
  10. Let $x,y,z$ be positive real numbers satisfying $x^{3}+y^{2}+z=2\sqrt{3}+1$. Find the minimum value of the expression \[P=\frac{1}{x}+\frac{1}{y^{2}}+\frac{1}{z^{3}}.\]
  11. Given a sequence $\{a_{n}\}$ whose terms are greater than 1 and satisfy \[\lim_{n\to\infty}\frac{\ln(\ln a_{n})}{n}=\frac{1}{2014}. \] Let $b_{n}=\sqrt{a_{1}+\sqrt{a_{2}+\ldots+\sqrt{a_{n}}}}$ ($n\in\mathbb{N}^{*}$). Prove that $\lim_{n\to\infty}b_{n}$ is a finite number.
  12. Given a triangle $ABC$ and $O$ is any point inside the triangle. Let $P,Q$ and $R$ respectively be the projections of $O$ on $BC$, $CA$ and $AB$ respectively. Let $A_{1},B_{1}$ and $C_{1}$ be arbitrary points other than $A,B,C$ on the lines $BC,CA$ and $AB$ respectively. Let $A_{2},B_{2}$and $C_{2}$ are the reflections of $A_{1},B_{1}$ and $C_{1}$ through the points $P,Q$ and $R$. Let \begin{align*} Z_{1} & \equiv(AB_{1}C_{1})\cap(BC_{1}A_{1})\cap(CA_{1}B_{1}),\\ Z_{2} & \equiv(AB_{2}C_{2})\cap(BC_{2}A_{2})\cap(CA_{2}B_{2}). \end{align*} Prove that $O$ is equidistant from $Z_{1}$ and $Z_{2}$.

Issue 449

  1. Find the minimum value of the products of $5$ different integers among which the sum of any $3$ arbitrary numbers is always greater than the sum of the remains.
  2. Let $ABC$ be a triangle with $AB>AC$ and $AB>BC$. On the side $AB$ choose $D$ and $E$ such that $BC=BD$ and $AC=AE$. Choose $K$ on $CA$ and $I$ on $CB$ such that $DK$ is parallel to $BC$ and $EI$ is parallel to $CA$. Prove that $CK=CI$.
  3. Solve the follwowing equation \[\frac{1}{\sqrt{x+3}}+\frac{1}{\sqrt{3x+1}}=\frac{2}{1+\sqrt{x}}.\]
  4. Given an acute triangle $ABC$ with the orthocenter $H$. Let $M$ be a point inside the triangle such that $\widehat{MAB}=\widehat{MCA}$. Let $E$ and $F$ respectively be the orthogonal projections of $M$ on $AB$ and $AC$. Let $I$ and $J$ respectively be the midpoints of $BC$ and $MA$. Prove that 3 lines $MH$, $EF$ and $IJ$ are concurrent.
  5. Find all pairs of integers $(x,y)$ satisfying \[x^{4}+y^{3}=xy^{3}+1.\]
  6. Solve the following equation \[ 8^{x}-9|x|=2-3^{x}.\]
  7. Given a triangle $ABC$ with the sides $AB=c$, $CA=b$, $BC=a$. Assume that the radius of the circumscribed circle is $R$ and the radius of the inscribed circle is $r$. Show that \[ \frac{r}{R}\leq\frac{3(ab+bc+ca)}{2(a+b+c)^{2}}.\]
  8. Let $x,y,z$ be 3 positive real numbers with $x\geq z$. Find the minimum value of the expression \[P=\frac{xz}{y^{2}+yz}+\frac{y^{2}}{xz+yz}+\frac{x+2z}{x+z}.\]
  9. Find the integer part of the expression \[B=\frac{1}{3}+\frac{5}{7}+\frac{9}{13}+\ldots+\frac{2013}{2015}.\]
  10. Find all polynomials $f(x)$ with integer coefficients such that $f(n)$ is a divisor of $3^{n}-1$ for every positive integer $n$. 
  11. Let $\{x_{n}\}$ be a sequence satisfying \[x_{0}=4,\,x_{1}=34,\,x_{n+2}\cdot x_{n}=x_{n+1}^{2}+18\cdot10^{n+1},\,\forall n\in\mathbb{N}.\] Let ${\displaystyle S_{n}=\sum_{k=0}^{26}x_{n+k}}$, $n\in\mathbb{N}^{*}$. Prove that, for every odd natural number $n$, $66|S_{n}$.
  12. Given a triangle $ABC$. The point $E$ and $F$ respectively vary on the sides $CA$ and $AB$ such that $BF=CE$. Let $D$ be the intersection of $BE$ and $CF$. Let $H$ and $K$ respectively be the orthocenters of $DEF$ and $DBC$. Prove that, when $E$ and $F$ change, the line $HK$ always passes through a fixed point.

Issue 450

  1. Find all positive integers $a$ and $b$ such that $b|a+2$ and $a|b+3$.
  2. Given a right triangle $ABC$ with the right angle $A$. Choose $E$ on the side $BC$ such that $EC=2EB$. Prove that $AC^{2}=3(EC^{2}-EA^{2})$.
  3. Solve the following equation \[\frac{1}{x+\sqrt{x^{2}-1}}=\frac{1}{4x}+\frac{3x}{2x^{2}+2}.\]
  4. Let $BC$ be a chord of a circle with center $O$ and radius $R$. Assume that $BC=R$. Let $A$ be apoint on the major arc $BC$ ($A\ne B$, $A\ne C$), and $M,N$ points on the chord $AC$ such that $AC=2AN=\frac{3}{2}AM$. Choose $P$ on $AB$ such that $MP$ is perpendicular to $AB$. Prove that three points $P,O$ and $N$ are collinear.
  5. Assume that equation \[ax^{3}-x^{2}+bx-1=0,\quad(a\ne0)\] has three positive real solutions. Find the minimum value of the expression \[M=(1-2ab)\frac{b}{a^{2}}.\]
  6. Let $x$ and $y$ be two positive real numbers satisfying $32x^{6}+4y^{3}=1$. Find the maximum value of the expression \[P=\frac{(2x^{2}+y+3)^{3}}{3(x^{2}+y^{2})-3(x+y)+2}.\]
  7. Given an acute triangle $ABC$ ($AB>AC$). The heights $BB'$ and $CC'$ intersect at $H$. Let $M,N$ respectively be the midpoints of the sides $AB,AC$ and $O$ the circumcenter. $AH$ intersects $B'C'$ at $E$, and $AO$ intersects $MN$ at $F$. Prove that $EF\parallel OH$.
  8. Given three positive numbers $a,b,c$. Find the maximum value of $k$ so that the following inequality holds \[\frac{a}{b}+\frac{b}{c}+\frac{c}{a}-3\geq3\left(\frac{a^{2}+b^{2}+c^{2}}{ab+bc+ca}-1\right).\]
  9. Find all positive integers $x,y,z$ which form an airthmetic progression and satisfy the following equation \[\frac{x^{2}(x+y)(x+z)}{(x-y)(x-z)}+\frac{y^{2}(y+z)(y+x)}{(y-z)(y-x)}+\frac{z^{2}(z+x)(z+y)}{(z-x)(z-y)}=2016+(x+y-z)^{2}.\]
  10. Given a $999\times999$ table of squares. Each square is colored by white or red. Consider a set of triples of squares $(C_{1},C_{2},C_{3})$ which satisfy the following properties: the first two squares $C_{1},C_{2}$ are in the same row, the last two squares $C_{2},C_{3}$ are in the same column, $C_{1},C_{3}$ are white, and $C_{2}$ is red. Find the maximum number of elements in such a set.
  11. Find all positive integers $n>1$ and all primes $p$ such that the polynomial $f(x)=x^{n}-px+p^{2}$ ca be factorized as a product of two non-constant polynomials with integer coefficients. 
  12. Assume that $ABC$ is an equilateral triangle and $M$ is a point which is not on the lines through $BC$, $CA$ and $AB$. Prove that the Euler lines of the triangles $MBC$, $MCA$, and $MAB$ are either concurrent or parallel.

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Name

Abel,5,Albania,2,AMM,2,Amsterdam,5,An Giang,40,Andrew Wiles,1,Anh,2,APMO,21,Austria (Áo),1,Ba Đình,2,Ba Lan,1,Bà Rịa Vũng Tàu,72,Bắc Bộ,2,Bắc Giang,59,Bắc Kạn,3,Bạc Liêu,15,Bắc Ninh,58,Bắc Trung Bộ,3,Bài Toán Hay,5,Balkan,40,Baltic Way,32,BAMO,1,Bất Đẳng Thức,69,Bến Tre,68,Benelux,15,Bình Định,62,Bình Dương,36,Bình Phước,48,Bình Thuận,41,Birch,1,BMO,40,Booklet,12,Bosnia Herzegovina,3,BoxMath,3,Brazil,2,British,16,Bùi Đắc Hiên,1,Bùi Thị Thiện Mỹ,1,Bùi Văn Tuyên,1,Bùi Xuân Diệu,1,Bulgaria,6,Buôn Ma Thuột,2,BxMO,14,Cà Mau,21,Cần Thơ,25,Canada,40,Cao Bằng,11,Cao Quang Minh,1,Câu Chuyện Toán Học,43,Caucasus,3,CGMO,11,China - Trung Quốc,25,Chọn Đội Tuyển,491,Chu Tuấn Anh,1,Chuyên Đề,125,Chuyên SPHCM,7,Chuyên SPHN,26,Chuyên Trần Hưng Đạo,2,Collection,8,College Mathematic,1,Concours,1,Cono Sur,1,Contest,666,Correspondence,1,Cosmin Poahata,1,Crux,2,Czech-Polish-Slovak,26,Đà Nẵng,48,Đa Thức,2,Đại Số,20,Đắk Lắk,72,Đắk Nông,13,Đan Phượng,1,Danube,7,Đào Thái Hiệp,1,ĐBSCL,2,Đề Thi,1,Đề Thi HSG,2126,Đề Thi JMO,1,DHBB,28,Điện Biên,12,Định Lý,1,Định Lý Beaty,1,Đỗ Hữu Đức Thịnh,1,Do Thái,3,Doãn Quang Tiến,5,Đoàn Quỳnh,1,Đoàn Văn Trung,1,Đống Đa,4,Đồng Nai,62,Đồng Tháp,62,Du Hiền Vinh,1,Đức,1,Dương Quỳnh Châu,1,Dương Tú,1,Duyên Hải Bắc Bộ,28,E-Book,31,EGMO,29,ELMO,19,EMC,10,Epsilon,1,Estonian,5,Euler,1,Evan Chen,1,Fermat,3,Finland,4,Forum Of Geometry,2,Furstenberg,1,G. Polya,3,Gặp Gỡ Toán Học,30,Gauss,1,GDTX,3,Geometry,14,GGTH,30,Gia Lai,38,Gia Viễn,2,Giải Tích Hàm,1,Giảng Võ,1,Giới hạn,2,Goldbach,1,Hà Giang,4,Hà Lan,1,Hà Nam,38,Hà Nội,258,Hà Tĩnh,87,Hà Trung Kiên,1,Hải Dương,64,Hải Phòng,54,Hậu Giang,11,Hậu Lộc,1,Hélènne Esnault,1,Hilbert,2,Hình Học,33,HKUST,7,Hòa Bình,31,Hoài Nhơn,1,Hoàng Bá Minh,1,Hoàng Minh Quân,1,Hodge,1,Hojoo Lee,2,HOMC,5,HongKong,8,HSG 10,116,HSG 10 2010-2011,4,HSG 10 2011-2012,6,HSG 10 2012-2013,5,HSG 10 2013-2014,4,HSG 10 2014-2015,5,HSG 10 2015-2016,2,HSG 10 2016-2017,5,HSG 10 2017-2018,3,HSG 10 2018-2019,3,HSG 10 2019-2020,8,HSG 10 2020-2021,2,HSG 10 2021-2022,2,HSG 10 2022-2023,3,HSG 10 Bà Rịa Vũng Tàu,2,HSG 10 Bắc Giang,1,HSG 10 Bạc Liêu,2,HSG 10 Bắc Ninh,3,HSG 10 Bình Định,1,HSG 10 Bình Dương,1,HSG 10 Bình Thuận,3,HSG 10 Chuyên SPHN,5,HSG 10 Đắk Lắk,2,HSG 10 Đồng Nai,4,HSG 10 Gia Lai,2,HSG 10 Hà Nam,3,HSG 10 Hà Tĩnh,13,HSG 10 Hải Dương,9,HSG 10 KHTN,9,HSG 10 Kon Tum,1,HSG 10 Nghệ An,1,HSG 10 Ninh Thuận,1,HSG 10 Phú Yên,2,HSG 10 Quảng Trị,2,HSG 10 Thái Nguyên,8,HSG 10 Thanh Hóa,1,HSG 10 Trà Vinh,5,HSG 10 Vĩnh Phúc,14,HSG 1015-2016,3,HSG 11,117,HSG 11 2010-2011,4,HSG 11 2011-2012,5,HSG 11 2012-2013,7,HSG 11 2013-2014,4,HSG 11 2014-2015,8,HSG 11 2015-2016,2,HSG 11 2016-2017,5,HSG 11 2017-2018,4,HSG 11 2018-2019,5,HSG 11 2019-2020,5,HSG 11 2020-2021,5,HSG 11 2021-2022,1,HSG 11 An Giang,1,HSG 11 Bà Rịa Vũng Tàu,1,HSG 11 Bắc Giang,4,HSG 11 Bạc Liêu,2,HSG 11 Bắc Ninh,4,HSG 11 Bình Định,11,HSG 11 Bình Dương,3,HSG 11 Bình Thuận,1,HSG 11 Cà Mau,1,HSG 11 Đà Nẵng,9,HSG 11 Đồng Nai,1,HSG 11 Hà Nam,1,HSG 11 Hà Tĩnh,10,HSG 11 Hải Phòng,1,HSG 11 Kiên Giang,4,HSG 11 Lạng Sơn,11,HSG 11 Nghệ An,6,HSG 11 Ninh Bình,2,HSG 11 Quảng Bình,9,HSG 11 Quảng Ngãi,8,HSG 11 Quảng Trị,3,HSG 11 Sóc Trăng,1,HSG 11 Thái Nguyên,8,HSG 11 Thanh Hóa,4,HSG 11 Trà Vinh,1,HSG 11 Tuyên Quang,1,HSG 11 Vĩnh Long,2,HSG 11 Vĩnh Phúc,10,HSG 12,623,HSG 12 2009-2010,2,HSG 12 2010-2011,39,HSG 12 2011-2012,44,HSG 12 2012-2013,58,HSG 12 2013-2014,53,HSG 12 2014-2015,44,HSG 12 2015-2016,36,HSG 12 2016-2017,47,HSG 12 2017-2018,58,HSG 12 2018-2019,44,HSG 12 2019-2020,43,HSG 12 2020-2021,51,HSG 12 2021-2022,34,HSG 12 2022-2023,25,HSG 12 An Giang,7,HSG 12 Bà Rịa Vũng Tàu,11,HSG 12 Bắc Giang,17,HSG 12 Bạc Liêu,3,HSG 12 Bắc Ninh,13,HSG 12 Bến Tre,18,HSG 12 Bình Định,16,HSG 12 Bình Dương,8,HSG 12 Bình Phước,8,HSG 12 Bình Thuận,8,HSG 12 Cà Mau,8,HSG 12 Cần Thơ,7,HSG 12 Cao Bằng,5,HSG 12 Chuyên SPHN,9,HSG 12 Đà Nẵng,3,HSG 12 Đắk Lắk,20,HSG 12 Đắk Nông,1,HSG 12 Điện Biên,3,HSG 12 Đồng Nai,20,HSG 12 Đồng Tháp,18,HSG 12 Gia Lai,13,HSG 12 Hà Nam,4,HSG 12 Hà Nội,15,HSG 12 Hà Tĩnh,15,HSG 12 Hải Dương,14,HSG 12 Hải Phòng,19,HSG 12 Hậu Giang,3,HSG 12 Hòa Bình,10,HSG 12 Hưng Yên,9,HSG 12 Khánh Hòa,2,HSG 12 KHTN,26,HSG 12 Kiên Giang,11,HSG 12 Kon Tum,2,HSG 12 Lai Châu,4,HSG 12 Lâm Đồng,10,HSG 12 Lạng Sơn,8,HSG 12 Lào Cai,16,HSG 12 Long An,17,HSG 12 Nam Định,7,HSG 12 Nghệ An,12,HSG 12 Ninh Bình,11,HSG 12 Ninh Thuận,6,HSG 12 Phú Thọ,16,HSG 12 Phú Yên,12,HSG 12 Quảng Bình,12,HSG 12 Quảng Nam,9,HSG 12 Quảng Ngãi,5,HSG 12 Quảng Ninh,20,HSG 12 Quảng Trị,9,HSG 12 Sóc Trăng,4,HSG 12 Sơn La,5,HSG 12 Tây Ninh,6,HSG 12 Thái Bình,11,HSG 12 Thái Nguyên,13,HSG 12 Thanh Hóa,18,HSG 12 Thừa Thiên Huế,18,HSG 12 Tiền Giang,3,HSG 12 TPHCM,12,HSG 12 Tuyên Quang,2,HSG 12 Vĩnh Long,6,HSG 12 Vĩnh Phúc,22,HSG 12 Yên Bái,6,HSG 9,533,HSG 9 2009-2010,1,HSG 9 2010-2011,21,HSG 9 2011-2012,44,HSG 9 2012-2013,44,HSG 9 2013-2014,36,HSG 9 2014-2015,40,HSG 9 2015-2016,39,HSG 9 2016-2017,42,HSG 9 2017-2018,47,HSG 9 2018-2019,50,HSG 9 2019-2020,20,HSG 9 2020-2021,53,HSG 9 2021-2022,57,HSG 9 2022-2023,1,HSG 9 An Giang,8,HSG 9 Bà Rịa Vũng Tàu,7,HSG 9 Bắc Giang,12,HSG 9 Bạc Liêu,1,HSG 9 Bắc Ninh,12,HSG 9 Bến Tre,9,HSG 9 Bình Định,10,HSG 9 Bình Dương,6,HSG 9 Bình Phước,13,HSG 9 Bình Thuận,5,HSG 9 Cà Mau,1,HSG 9 Cần Thơ,4,HSG 9 Cao Bằng,1,HSG 9 Chuyên SPHN,2,HSG 9 Đà Nẵng,10,HSG 9 Đắk Lắk,11,HSG 9 Đắk Nông,2,HSG 9 Điện Biên,3,HSG 9 Đồng Nai,7,HSG 9 Đồng Tháp,10,HSG 9 Gia Lai,8,HSG 9 Hà Giang,3,HSG 9 Hà Nam,9,HSG 9 Hà Nội,25,HSG 9 Hà Tĩnh,16,HSG 9 Hải Dương,14,HSG 9 Hải Phòng,7,HSG 9 Hậu Giang,4,HSG 9 Hòa Bình,3,HSG 9 Hưng Yên,9,HSG 9 Khánh Hòa,4,HSG 9 Kiên Giang,15,HSG 9 Kon Tum,8,HSG 9 Lai Châu,1,HSG 9 Lâm Đồng,13,HSG 9 Lạng Sơn,9,HSG 9 Lào Cai,3,HSG 9 Long An,9,HSG 9 Nam Định,8,HSG 9 Nghệ An,19,HSG 9 Ninh Bình,13,HSG 9 Ninh Thuận,3,HSG 9 Phú Thọ,12,HSG 9 Phú Yên,8,HSG 9 Quảng Bình,13,HSG 9 Quảng Nam,11,HSG 9 Quảng Ngãi,12,HSG 9 Quảng Ninh,15,HSG 9 Quảng Trị,9,HSG 9 Sóc Trăng,8,HSG 9 Sơn La,4,HSG 9 Tây Ninh,16,HSG 9 Thái Bình,9,HSG 9 Thái Nguyên,5,HSG 9 Thanh Hóa,17,HSG 9 Thừa Thiên Huế,8,HSG 9 Tiền Giang,6,HSG 9 TPHCM,10,HSG 9 Trà Vinh,2,HSG 9 Tuyên Quang,5,HSG 9 Vĩnh Long,11,HSG 9 Vĩnh Phúc,12,HSG 9 Yên Bái,4,HSG Cấp Trường,89,HSG Quốc Gia,109,HSG Quốc Tế,16,HSG11 2021-2022,3,Hứa Lâm Phong,1,Hứa Thuần Phỏng,1,Hùng Vương,2,Hưng Yên,39,Hương Sơn,2,Huỳnh Kim Linh,1,Hy Lạp,1,IMC,26,IMO,57,IMT,2,IMU,2,India - 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MOlympiad.NET: Mathematics and Youth Magazine Problems 2014
Mathematics and Youth Magazine Problems 2014
MOlympiad.NET
https://www.molympiad.net/2022/03/mym-2014.html
https://www.molympiad.net/
https://www.molympiad.net/
https://www.molympiad.net/2022/03/mym-2014.html
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