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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MOlympiad.NET: Mathematics and Youth Magazine Problems 2014
Mathematics and Youth Magazine Problems 2014
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