Mathematics and Youth Magazine Problems 2015

Issue 451

1. Suppose that $a,b$ and $c$ are positive integers such that \[\frac{1}{a}+\frac{1}{b}+\frac{1}{c}<1.\] Find the maximum value of the expression \[S=\frac{1}{a}+\frac{1}{b}+\frac{1}{c}.\]
2. Assume that the triangle $ABC$ is isosceles at $A$. Draw the height $AH$. Let $D$ be the midpoint of $AH$. Choose $E$ on $CD$ such that $HE$ is perpendicular to $CD$. Prove that $\widehat{AEB}=90^{0}$.
3. Solve the system of equations \[\begin{cases} \dfrac{x^{2}-1}{y}+\dfrac{y^{2}-1}{x} & =3\\ x^{2}-y^{2}+\dfrac{12}{x} & =9 \end{cases}.\]
4. Two circles $(O,R)$ and $(O',R')$ intersect at $A$ and $B$. A point $M$ varies arbitrary on the opposite ray of the ray $AB$. From $M$, draw two tangent lines to the circle $(O',R')$ at $C$ and $D$ with $D$ is inside the circle $(O,R)$. The lines $AD$ and $AC$ intersect $(O,R)$ respectively at $P$ and $Q$ ($P$, $Q$ are different from $A$). Prove that the line $PQ$ always goes through a fixed point when $M$ varies. 
5. Find the minimum value of the expression \[A=\frac{2}{3xy}+\sqrt{\frac{3}{y+1}}\] where $x,y$ are positive real numbers satisfying $x+y\leq3$.
6. Consider the quadratic equation $ax^{2}+bx+c=0$ ($a\ne0$), where $a,b$ and $c$ are real numbers satisfying $145a+144b+144c=0$. Prove that this equation cannot have two solutions which are inverses of two nonzero consecutive perfect squares.
7. Given a tetrahedron $ABCD$ inscribed in a sphere $(S)$. Let $A_{1},B_{1},C_{1},D_{1}$ resprectively the centroids of the triangles $BCD$, $ACD$, $ABD$, $ABC$. The lines $AA_{1}$, $BB_{1}$, $CC_{1}$, $DD_{1}$ intersect $(S)$ respectively at $A_{2},B_{2},C_{2},D_{2}$. Find the minimum value of the expression \[P=\frac{AA_{2}^{2}+BB_{2}^{2}+CC_{2}^{2}+DD_{2}^{2}}{AB\cdot CD+AC\cdot BD+AD\cdot BC}.\]
8. Let $a,b,c$ be three positive real numbers. Prove that \[\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\geq\frac{a^{2}+nb^{2}}{a^{3}+nb^{3}}+\frac{b^{2}+nc^{2}}{b^{3}+nc^{3}}+\frac{c^{2}+na^{2}}{c^{3}+na^{3}}\] for all $n\in\mathbb{N}$, $n\geq2$.
9. Find all quadruples of posotive integers $(x,y,z,p)$, $p$ is a prime, such that \[p^{x}+(p-1)^{2y}=(2p-1)^{z}.\]
10. Find the maximal $K$ such that the following inequality \[\sqrt{a+K|b-c|^{\alpha}}+\sqrt{b+K|c-a|^{\alpha}}+\sqrt{c+K|a-b|^{\alpha}}\leq2\] always holds for all $\alpha\geq1$ and $a,b,c$ are arbitrary nonnegative real numbers satisfying $a+b+c=1$.
11. Find all functions $f:\mathbb{R}\to\mathbb{R}$ such that for all $x,y\in\mathbb{R}$ \[f(x^{2}+f(y))=4y+\frac{1}{2}f^{2}(x).\]
12. Given a circle $(O)$, $ABC$ is an acute triangle inscribed in $(O)$. The tangent line to $(O)$ at $A$ intersects $BC$ at $S$; $K$ is the perpendicular projection of $A$ on $OS$; $BK$ and $CK$ intersect $CA$, $AB$ respectively at $E$ and $F$. Prove that the line which contains $A$ and is perpendicular to $EF$ always goes through a fixed point when $A$ varies on $(O)$ ($B$ and $C$ are fixed).

Issue 452

1. Let $x,y$ be positive natural numbers. Find the minimum value of the expression $A=|36^{x}-5^{y}|$.
2. Let $O$ be the midpoint of the interval $AB$. On a half plane determined by the lines through $AB$, draw two rays $Ox$, $Oy$ which are perpendicular to each other. On the $Ox$, $Oy$, respectively choose two points $M,N$ which are different from $O$. Prove that $AM+BN\geq MN$. 
3. Solve the system of inequalities \[\begin{cases} 2\sqrt{x^{2}-xy+y^{2}} & \leq(x+y)^{2}\\ \sqrt{1-(x+y)^{2}} & =1-x \end{cases}. \]
4. Given a triangle $ABC$ which is isosceles at $A$ and is inscribed in a circle $(O)$. Let $AK$ be a diameter. Let $I$ be any point on the minor are $AB$ ($I$ is different from $A$ and $B$). $KI$ intersects $BC$ at $M$. The perpendicular bisector of $MI$ intersects the sides $AB$,$AC$ respectively at $D,E$. Let $N$ be the midpoint of $DE$. Prove that $A,M$ and $N$ are colinear.
5. Find integers $m$ so that the following equation \[x^{3}+(m+1)x^{2}-(2m-1)x-(2m^{2}+m+4)=0\] has integer solutions.
6. Suppose that $a,b,c$ are three nonnegative numbers. Prove that \[(a+bc)^{2}+(b+ca)^{2}+(c+ab)^{2}\geq\sqrt{2}(a+b)(b+c)(c+a).\]
7. Given a triangle $ABC$ which is isosceles at $A$ and is inscribed in a circle $(O)$. Let $D$ be the midpoint of $AB$. The ray $CD$ intersects $(O)$ at $E$. Let $F$ be the point on $(O)$ so that $CF\parallel AE$. The ray $EF$ intersects $AC$ at $G$. Prove that $BG$ is tangent to the circle $(O)$.
8. Determine the minimum value of the funtion \[f(x)=\sqrt{\sin x+\tan x}+\sqrt{\cos x+\cot x}.\]
9. On the plane $Oxy$, consider the set $M$ consisting of the points $(x,y)$ such that $x,y\in\mathbb{N}^{*}$ and $x\leq12,y\leq12$. Each point in $M$ is colored be red, white or blue. Prove that there exists a rectangle satisfying the following properties: its sides are parallel to coordinate axes and its vertives are in $M$ and are colored by the same color.
10. Find the minimum positive integer $t$ so that there exist $t$ integers $x_{1},x_{2},\ldots,x_{t}$ satisfying \[x_{1}^{3}-x_{2}^{3}+x_{3}^{3}-\ldots+(-1)^{t+1}x_{t}^{3}=2065^{2014}.\]
11. Let $a$ be a positive integer and $(x_{n})$ a sequence given by $x_{1}=1$ and \[x_{n+1}=\sqrt{x_{n}^{2}+2ax_{n}+2a+1}-\sqrt{x_{n}^{2}-2ax_{n}+2a+1},\forall n\in\mathbb{N}^{*}.\] Find $a$ so that the sequence $(x_{n})$ has a finite limit.
12. Given a triangle $ABC$. A line $\Delta$ which does not contain $A,B,C$ intersects $BC,CA,AB$ respectively at $A_{1},B_{1},C_{1}$. Let $A_{b},A_{c}$ respectively be the symmetric points of $A_{1}$ through $AB,AC$. Let $A_{a}$ be the midpoint of $A_{b}A_{c}$. The points $B_{b},C_{c}$ are determined similarly to the way we construct the point $A_{a}$. Prove that the points $A_{a}$, $B_{b}$, $C_{c}$ are collinear

Issue 453

1. Find all natural numbers $x,y,z$ such that $x!+y!+z!=\overline{xyz}$ and $\overline{xyz}$ is a three-digit number. Recall that $n!=1.2.3\ldots n$ and $0!=1$.
2. Given a triangle $ABC$ with $\widehat{BAC}=50^{0}$, $\widehat{ABC}=60^{0}$. On the sides $AB$ and $BC$, choose $D$ and $E$ respectively such that $\widehat{ACD}=\widehat{CAE}=30^{0}$. Find the angle $\widehat{CDE}$.
3. Given three positive numbers $a,b,c$. Find the maximum value of the expression \[T=\frac{a+b+c}{(4a^{2}+2b^{2}+1)(4c^{2}+3)}.\]
4. Suppose that $ABC$ is an acute triangle. Its altitudes $AD$, $BE$ and $CF$ are concurrent at the point $H$. Let $K$ be a point on the side $DC$ and choose $S$ on $HK$ such that $AS\perp HK$. Let $I$ be the intersection between $EF$ and $AH$. Prove that $SH$ is the angle bisector of the angle $\widehat{DSI}$.
5. Let $a,b,c$ be positive numbers satisfying \[a^{4}+b^{4}+c^{4}\leq2(a^{2}b^{2}+b^{2}c^{2}+c^{2}a^{2}).\] Prove that at lease one of the following equations has no solution \[ax^{2}+2bx+2c=0,\] \[bx^{2}+2cx+2a=0,\] \[cx^{2}+2ax+2b=0.\]
6. Solve the system of equations \[\begin{cases} x^{3}+y^{3} & =1\\ x^{5}+y^{5} & =1 \end{cases}.\]
7. Given a quadrilateral pyramid $S.ABCD$ whose base $ABCD$ is a parallelogram. A point $M$ varies on the side $AB$ ($M$ is different from $A$ and $B$) Let $(\alpha)$ be a plane which goes through $M$ and is parallel to $SA$ and $BD$. Determine the cross section of $S.ABCD$ determined by $(\alpha)$ and find the position for $M$ so that the cross section has maximal area.
8. Solve the equation \[3\sin^{3}x+2\cos^{3}x=2\sin x+\cos x.\]
9. Find all pairs of positive integers $(a,b)$ so that $4a+1$ and $4b-1$ are coprime and $a+b$ is a divisor of $16ab+1$.
10. Given a polynomial \[P(x)=a_{n}x^{n}+\ldots+a_{1}x+a_{0}\quad(a_{n}\ne0,\,n\geq2)\] with integer coefficients. Prove that there are infinitely many integers $k$ such that $P(x)+k$ cannot be factorized as a product of two positive degree polynomials with integral coefficients.
11. Suppose that the funtion $f:\mathbb{N\to\mathbb{N}}$ is onto and the funtion $g:\mathbb{N}\to\mathbb{N}$ is one-to-one. Assume furthermore that $f(x)\geq g(n)$ for all $n\in\mathbb{N}$. Show that \[f(n)=g(n),\,\forall n\in\mathbb{N}.\]
12. Given a triangle $ABC$ inscribed in a circle $(O)$. $M$ and $N$ are two fixed points on $(O)$ so that $MN\parallel BC$. Let $P$ vary on the line $AM$. The line through $P$ which is parallel to $BC$ intersects $CA$, $AB$ at $E,F$ respectively. The circumscribed circle of the triangle $NEF$ intersects $(O)$ at the second point $Q$ (different from $N$). Prove that the line $PQ$ always goes through a fixed point when $P$ moves.

Issue 454

1. Given $n$ numbers $a_{1},a_{2},\ldots,a_{n}$ and $n$ distinct primes $p_{1},p_{2},\ldots,p_{n}$ ($n\geq2$) satisfying \[ p_{1}|a_{1}-a_{2}|=p_{2}|a_{2}-a_{3}|=\ldots=p_{n}|a_{n}-a_{1}|.\] Prove that $a_{1}=a_{2}=\ldots=a_{n}$.
2. Choose 100 different natural numbers so that each of them is less than or equal to 2015 and has remainder 10 when divided by 17. Prove that, among these ones, we can always choose three numbers of which the sum is less than or equal to 999.
3. Prove that, for every positive integer $n$, the value of the expression \[\sqrt{\frac{(1^{4}+4)(2^{4}+4)\ldots(n^{4}+4)}{2}}\]is an irrational number.
4. Given a triangle $ABC$ and $D$ is any point on the side $BC$ ($D$is different from $B$ and $C$). The perpendicular bisector of $BD$and $CD$ intersect $AB$ and $AC$ respectively at $M$ and $N$. Let $H$ be the orthogonal projection of $D$ on the line $MN$ and $E,F$ respectively the midpoints of $BD$ and $CD$. Prove that $\widehat{EHF}=\widehat{BAC}$.
5. Given the equation $ax^{2}+bx+c=0$ where the coefficients $a,b,c$are integers and $a>0$. Suppose that the equation has two distinctpositive roots which are less than $1$. Find the smallese possible value for the coefficient $a$.
6. Solve the following system of equations \[\begin{cases} \sqrt{x-\sqrt{y}} & =\sqrt{z}-1\\ \sqrt{y-\sqrt{z}} & =\sqrt{x}-1\\ \sqrt{z-\sqrt{x}} & =\sqrt{y}-1 \end{cases}.\]
7. Let $P$ be a point on the plane containing a triangle $ABC$. Suppose that $A_{1}=BC\cap AP$, $B_{1}=AC\cap BP$, $C_{1}=AB\cap CP$, $A_{2}=BC\cap B_{1}C_{1}$, $B_{2}=AC\cap A_{1}C_{1}$, $C_{2}=AB\cap A_{1}B_{1}$; $A_{3}=B_{1}C_{1}\cap AP$, $B_{3}=BP\cap A_{1}C_{1}$, $C_{3}=A_{1}B_{1}\cap CP$. Prove that $A_{2}$, $B_{2}$ and $C_{2}$ respectively are on the lines going through $B_{3}C_{3}$, $A_{3}C_{3}$ and $A_{3}B_{3}$.
8. Given three positive numbers $a,b,c$. Prove that \[\left(\frac{a}{a+b}\right)^{2}+\left(\frac{b}{b+c}\right)^{2}+\left(\frac{c}{c+a}\right)^{2}+3 \geq \frac{5}{2}\left(\frac{a}{a+b}+\frac{b}{b+c}+\frac{c}{c+a}\right).\]
9. Let $d$ be a line in the coordinate plane with the equation $y=\frac{3}{2}x+\frac{1}{3}$. Let $a_{1}$ and $a_{2}$ be two distinct lines which are parallel to $d$. Suppose furthermore that the distances from $a_{1}$ and $a_{2}$ to $d$ both are equal to $\frac{1}{12}$. Is there any integer point, i.e. point with both coordinates are integers, between or on two lines $a_{1}$ and $a_{2}$?.
10. Prove that, for each positive integer $n$, the equation $2014^{x}+nx=2013$ has a unique solution, say $x_{n}$. Find $\lim_{n\to\infty}x_{n}$.
11. Find all continuous funtions $f:\mathbb{R}\to\mathbb{R}$ which satisfy \[ (x+y)f(x+y)=xf(x)+yf(y)+2xy,\quad\forall x,y\in\mathbb{R}.\]
12. Given a triangle $ABC$. A point $M$ varies on the side $BC$. Let $(I_{1})$ and $(I_{2})$ be the inscribed circles of the triangles $ABM$ and $ACM$ respectively. A common tangent line $XY$ of $(I_{1})$ and $(I_{2})$, which is different from $BC$, intersects $AM$ at $N$ ($X\in(I_{1})$ and $Y\in(I_{2})$). Let $Z$ and $T$ respectively be the tangent points between $AM$ and $(I_{1})$, $(I_{2})$. $XT$ cuts $YZ$ at $K$. Prove that $NK$ always goes through a fixed point.

Issue 455

1. Find a natural number with more than $3$ digits knowing that if we delete its the last $3$ digits, we will get a new number whose cube is exactly equal to that wanted number.
2. Given two positive real numbers $a$ and $b$ satisfying the following conditions $a^{2015}-a-1=0$ and $b^{4030}-b-3a=0$. Compare $a$ and $b$.
3. Solve the equation \[x+y+x\sqrt{1-y^{2}}+y\sqrt{1-x^{2}}=\frac{3\sqrt{3}}{2}.\]
4. On a circle centered at the point $I$, fix two points $B$ and $C$. A point $A$ varies on the circle such that the triangle $ABC$ is always acute. On the side $AC$, choose $M$ so that $MA=3MC$. Let $H$ be the perpendicular projection of $M$ on the side $AB$. Prove that $H$ always lies on a fixed circle.
5. Find all prime numbers $x$ and $y$ such that \[(x^{2}+2)^{2}=2y^{4}+11y^{2}+x^{2}y^{2}+9.\]
6. Solve the following system of equations \[\begin{cases} x^{3}+y^{3} & =4y^{2}-5y+4x+4\\ 2y^{3}+z^{3} & =4z^{2}-5z+6y+6\\ 3z^{3}+x^{3} & =4y^{2}-5x+9z+8 \end{cases}.\]
7. Given a triangle $ABC$. Suppose that the length of the sides are given by $BC=a$, $CA=b$, $AB=c$ and $m_{a}$, $m_{b}$ and $m_{c}$ are the length of the corresponding medians. Prove that \[(a^{2}+b^{2}+c^{2})\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)\geq2\sqrt{3}(m_{a}+m_{b}+m_{c}).\] When does the equality happen?.
8. Consider an acute triangle $ABC$ with the angles $A,B$ and $C$. Find the maximum value of the expression \[M=\frac{\tan^{2}A+\tan^{2}B}{\tan^{4}A+\tan^{4}B}+\frac{\tan^{2}B+\tan^{2}C}{\tan^{4}B+\tan^{4}C}+\frac{\tan^{2}C+\tan^{2}A}{\tan^{4}C+\tan^{4}A}.\]
9. Find that coefficient of $x^{2}$ in the expansion of the following expression \[(1+x)(1+2x)(1+4x)\ldots(1+2^{2013}x).\]
10. Let $a_{1},a_{2},\ldots,a_{n}$ be positive numbers such that \[a_{1}+a_{2}+\ldots+a_{n}=\frac{1}{a_{1}}+\frac{1}{a_{2}}+\ldots+\frac{1}{a_{n}}.\]Find the minimum value of the expression \[A=a_{1}+\frac{a_{2}^{2}}{2}+\ldots+\frac{a_{n}^{2}}{n}.\]
11. Find the biggest real number $k$ satisfying the condition: for any 3 real numbers $a,b,c$ such that $|a|+|b|+|c|<k$, the following system of inequalities has no solution \[\begin{cases} x^{16}+ax^{9}+bx^{4}+cx+15 & \leq0\\ |x^{16}-x^{9}+1|+|x^{4}-x+1| & \leq2 \end{cases}.\]
12. Suppose that $ABC$ is an acute triangle inscribed in the circle $(O)$ and $AD$ is an altitude. The tangent lines at $B$, $C$ of $(O)$ intersect at $T$. On the line segment $AD$, choose $K$ such that $\widehat{BKC}=90^{0}$. Let $G$ be the centroid of $ABC$. Suppose that $KG$ intersects $OT$ at $L$. Choose the point $P$, $Q$ on the side $BC$ so that $LP\parallel OB$, $LQ\parallel OC$. Choose the points $E$ and $F$ respectively on the sides $CA$ and $AB$ such that $QE$, $PF$ are both perpendicular to $BC$. Let $(T)$ be the circle centerd at $T$ and containing $B$ and $C$. Prove that the circle circumscribing the triangle $AEF$ is tangent to $(T)$.

Issue 456

1. Find all finite sets of primes such that for each set, the product of its elements is 10 times the sum of its element.
2. Let $ABC$ be an isosceles triangle with the vertex angle $\widehat{BAC}=80^{0}$. Choose $D$ and $E$ on the sides $BC$ and $CA$ respectively such that $\widehat{BAD}=\widehat{ABE}=30^{0}$. Find the angle $\widehat{BED}$.
3. Solve the equation \[\frac{1}{\sqrt{x}}+\frac{1}{\sqrt{2x-1}}=\sqrt{5}\left(\frac{1}{\sqrt{6x-1}}+\frac{1}{\sqrt{9x-4}}\right).\]
4. Let $ABCD$ be a square and let $a$ be the length each side. On the sides $AB$ and $BC$, choose $M$ and $N$ respectively such that $\widehat{MDN}=45^{0}$. Find the positions of $M$ and $N$ so that the length $MN$ is minimal.
5. Find all positive integers $x$ and $y$ such that \[x^{4}+y^{2}+13y+1\leq(y-2)x^{2}+8xy.\]
6. Solve the following system of equations \[ \begin{cases} x+y+z & =3\\ x^{2}y+y^{2}z+z^{2}x & =4\\ x^{2}+y^{2}+z^{2} & =5 \end{cases}.\]
7. Given a quadrilateral pyramid $S.ABCD$ with the following properties: the base $ABCD$ is a rectangle and $SA$ is perpendicular to the plane $(ABCD)$. Suppose that $G$ is the centroid of the triangle $SBC$ and let $d$ be the distance from $G$ to the plane $(SBD)$. Let $SB=a$, $BD=b$ and $SD=c$. Prove that \[a^{2}+b^{2}+c^{2}\geq162d^{2}.\]
8. Prove that the following equation \[(x+1)^{\frac{1}{x+1}}=x^{\frac{1}{x}}\] has a unique solution.
9. Given positive integers $a_{1},a_{2},\ldots a_{15}$ satisfying
   a) $a_{1}<a_{2}<\ldots<a_{15}$,
   b) for each $k$ ($k=1,\ldots,15$), if we denote $b_{k}$ the largest divisor of $a_{k}$ such that $b_{k}<a_{k}$, then $b_{1}>b_{2}>\ldots>b_{15}$.
   Prove that $a_{15}>2015$.
10. Given the following polynomial \[f(x)=x^{3}+3x^{2}+6x+1975.\] In the interval $[1,3^{2015}]$, how many are there integers $a$ such that $f(a)$ is divisible by $3^{2015}$?.
11. Find all injections $f:\mathbb{R\to\mathbb{R}}$ satisfying $$ f(x^{5})+f(y^{5}) = (x+y)[f^{4}(x)-f^{3}(x)f(y)+f^{2}(x)f^{2}(y)-f(x)f^{3}(y)+f^{4}(y)]$$ for all $x,y\in\mathbb{R}$.
12. Given a triangle $ABC$ and let $G$ be its centroid. Choose a point $M$, which is different from $G$, inside the triangle. Suppose that $AM$, $BM$, and $CM$ intersect $BC$, $CA$ and $AB$ at $A_{0},B_{0},C_{0}$ respectively. Choose $A_{1},A_{2}$ on $B_{0}C_{0}$ such that $A_{0}A_{1}\parallel CA$ and $A_{0}A_{2}\parallel AB$. We choose four points $B_{1},B_{2},C_{1},C_{2}$ similarly. Let $G_{1},G_{2}$ be the centroids of the triangles $A_{1}B_{1}C_{1}$, $A_{2}B_{2}C_{2}$ respectively. Prove that
    a) $A_{1}B_{2}\parallel B_{1}C_{2}\parallel C_{1}A_{2}$,
    b) $MG$ goes through the midpoint of $G_{1}G_{2}$.

Issue 457

1. Find all natural numbers $n$ satisfying \[2.2^{2}+3.2^{3}+4.2^{4}+\ldots+n.2^{n}=2^{n+34}.\]
2. Find integers $a,b,c$ such that \[|a-b|+|b-c|+|c-a|=2014^{a}+2015^{a}.\]
3. Suppose that $f(x)$ is a polynomial with integral coefficients and $f(1)=2$. Show that $f(7)$ is not a perfect square.
4. Given an acute triangle $ABC$ with altitudes $AH,BK$. Let $M$ be the midpoint of $AB$. The line through $CM$ intersect $HK$ at $D$. Draw $AL$ perpendicular to $BD$ at $L$. Prove that the circle containning $C,K$ and $L$ is tangent to the line going through $BC$.
5. Solve the following system of equations \[\begin{cases} 9x^{3}+2x+(y-1)\sqrt{1-3y} & =0\\ 9x^{2}+y^{2}+\sqrt{5-6x} & =6 \end{cases}\] for $x,y\in\mathbb{R}$.
6. Suppose that $f(x)$ is a polynomial of degree $3$ and its leading coefficient is equal to $2$. Also assume that $f(2014)=2015$, $f(2015)=2016$. Find $f(2016)-f(2013)$.
7. Let $S_{tp}$ and $V$ respectively be the surface area and the volume of the tetrahedron $ABCD.$ Prove that \[\left(\frac{1}{6}S_{tp}\right)^{3}\geq\sqrt{3}V^{2}.\]
8. Given an $n$-sided convex polygon ($n\geq4$) $A_{1}A_{2}\ldots A_{n}$. Prove that $$\begin{align*} & n+\sin A_{1}+\sin A_{2}+\ldots+\sin A_{n}\\ \leq & 2\left(\cos\frac{A_{1}-A_{2}}{4}+\cos\frac{A_{2}-A_{3}}{4}+\ldots+\cos\frac{A_{n}-A_{1}}{4}\right). \end{align*}$$ When does the equality happen?.
9. Find all triples $(x,y,p)$ where $x$ and $y$ are positive integers and $p$ is a prime number satisfying $p^{x}-y^{p}=1$.
10. Let $k$ be a real number which is greater than $1$. Consider the following sequence \[x_{1}=\frac{1}{2}\sqrt{k^{2}-1},\quad x_{2}=\sqrt{\frac{k^{2}-1}{4}+\frac{1}{2}\sqrt{k^{2}-1}},\ldots,\] \[x_{n}=\underset{n\text{ square root symbols}}{\underbrace{\sqrt{\frac{k^{2}-1}{4}+\sqrt{\frac{k^{2}-1}{4}+\ldots+\sqrt{\frac{k^{2}-1}{4}+\frac{1}{2}\sqrt{k^{2}-1}}}}}}.\] Prove that $\left\{ x_{n}\right\} $ converges and find $\lim_{n\to\infty}x_{n}.$ 
11. For each positive integer $n$, put $\displaystyle\psi(n)=\sum_{d|n}d^{2}$.
    a) Prove that $\psi(n)$ is multiplicative, i.e. \[\psi(ab)=\psi(a)\psi(b)\text{ if }(a,b)=1.\] b) Suppose the $l$ is an odd positive integer. Prove that there are only finitely many positive integers $n$ such that $\psi(n)=\psi(n+l)$.
12. Given a triangle $ABC$ with the circumscribed circle $(O)$ and the inscribed circle $(I)$. The tangent lines to $(O)$ at $B$ and $C$ intersect at $T$. Let $M$ be the midpoint of $BC$ and $D$ be the midpoint of the the arc $BC$ which does not contain $A$. Suppose that $AM$ intersects $(O)$ at $E$ and $AT$ intersects the side $BC$ at $F$. Let $J$ be the midpoint of $IF$. Prove that $\widehat{AEI}=\widehat{ADJ}$.

Issue 458

1. For a given prime number $p$, find positive integers $x,y$ such that  \[\frac{1}{x}+\frac{1}{y}=\frac{1}{p}.\]
2. Given an acute triangle $ABC$ with the orthocenter $H$. Let $M$ be the midpoint of $BC$. The line through $A$ parallel to $MH$ meets the line through $H$ parallel to $MA$ at $N$. Prove that \[AH^{2}+BC^{2}=MN^{2}.\]
3. Suppose that \[\begin{cases} a^{3}-a^{2}+a-5 & =0\\ b^{3}-2b^{2}+2b+4 & =0 \end{cases}.\] Find $a+b$.
4. From a point $M$ outside the circle $(O)$ draw to tangents $MA,MB$ to $(O)$ ($A,B$ are points of tangency). $C$ is an arbitrary point on the minor arc $AB$ of $(O)$. The rays $AC$ and $BC$ intersect $MB$ and $MA$ at $D$ and $E$ respectively. Prove that the circumcircles of the triangles $ACE$, $BCD$ and $OCM$ meet at another point which is different from $C$.
5. Find all triples of positive integers $(a,b,c)$ such that  \[(a^{5}+b)(a+b^{5})=2^{c}.\]
6. Solve the following system of equations \[\begin{cases} x+y+z+\sqrt{xyz} & =4\\ \sqrt{2x}+\sqrt{3y}+\sqrt{3z} & =\dfrac{7\sqrt{2}}{2}\\ x & =\min\{x,y,z\}\end{cases}. \]
7. Given a diamond $ABCD$ with $\widehat{BAD}=120^{0}$. Let $M$ vary on the side $BD$. Assume that $H$ and $K$ are the orthogonal projections of $M$ on the lines through $AB$ and $AD$ respectively. Let $N$ be the midpoint of $HK$. Prove that the line through $MN$ always passes through a fixed point.
8. Given a triangle $ABC$. Find the maximum value and the minimum value of the expression \[P=\cos^{2}2A\cdot\cos^{2}2B\cdot\cos^{2}2C+\frac{\cos4A\cdot\cos4B\cdot\cos4C}{8}.\]
9. Find the smallest $k$ such that \[S=a^{3}+b^{3}+c^{3}+kabc-\frac{k+3}{6}\left[a^{2}(b+c)+b^{2}(c+a)+c^{2}(a+b)\right]\leq0\] for all triples $(a,b,c)$ which are the lengths of the sides of a triangle.
10. Given a sequence of polynomials $\{P_{n}(x)\}$ satisfying the following conditions $P_{1}(x)=2x$, $P_{2}(x)=2(x^{2}+1)$ and \[P_{n}(x)=2xP_{n-1}(x)-(x^{2}-1)P_{n-2}(x),\quad n\in\mathbb{N},n\geq3.\] Prove that $P_{n}(x)$ is divisible by $Q(x)=x^{2}+1$ if and only if $n=4k+2$, $k\in\mathbb{N}$.
11. Consider the funtion \[f(n)=1+2n+3n^{2}+\ldots+2016n^{2015}.\] Let $(t_{0},t_{1},\ldots,t_{2016})$ and $(s_{0},s_{1},\ldots,s_{2016})$ be two permutations of $(0,1,\ldots,2016)$. Prove that there exist two different numbers in the following set $$A=\left\{s_{0}f(t_{0}),s_{1}f(t_{1}),\ldots,s_{2016}f(t_{2016})\right\}$$ such that their difference is divisible by $2017$.
12. Given a triangle $ABC$ and an arbitrary point $M$. Prove that \[\frac{1}{BC^{2}}+\frac{1}{CA^{2}}+\frac{1}{AB^{2}}\geq\frac{9}{(MA+MB+MC)^{2}}.\]

Issue 459

1. Let $$\begin{align*} A & =\frac{1}{1.2^{2}}+\frac{2}{2.3^{2}}+\frac{1}{3.4^{2}}+\ldots+\frac{1}{49.50^{2}},\\ B & =\frac{1}{2^{2}}+\frac{1}{3^{2}}+\frac{1}{4^{2}}+\ldots+\frac{1}{50^{2}}.\end{align*}$$ Compare $A$ and $B$ with $\dfrac{1}{2}$.
2. For all pairs of real numbers $(a,b)$ such that the polynomial \[A(x)=x^{2}-2ax+2a^{2}+b^{2}-5\] has solutions. Find the minimum value of the expression \[P=(a+1)(b+1).\]
3. Suppose that $a_{1},a_{2},\ldots,a_{n}$ are different positive integers such that \[\frac{1}{a_{1}}+\frac{1}{a_{2}}+\ldots+\frac{1}{a_{n}}=1\] and also assume that the biggest number among them is equal to $2p$, where $p$ is some prime number. Determine $a_{i}$ ($i=\overline{1,n}$).
4. Let $ABC$ be a isosceles right triangle ($AB=BC$) and let $O$ be the midpoint of $AC$. Through $C$ draw the line $d$ perpendicular to $BC$. Let $Cx$ be the opposite ray of the ray $CB$ and $M$ be an arbitrary point on $Cx$. Assume that $E$ is the intersection between $BE$ and $OM$. Prove that when $M$ varies on $Cx$, $I$ always belongs to a fixed curve.
5. Solve the following equation \[x^{2}-2x=2\sqrt{2x-1}.\]
6. Solve the following inequation \[\frac{2x^{3}+3x}{7-2x}>\sqrt{2-x}.\]
7. Given an acute and non-isosceles triangle $ABC$ with the altitudes $AH$, $BE$, $CF$. Let $I$ be the incenter of $ABC$ (the center of the inscribed circle) and $R$ be the circumradius of $ABC$ (the radius of the circumscribed circle). Let $M,N$ and $P$ respectively be the midpoint of $BC,CA$ and $AB$. Assume that $K,J$ and $L$ respectively are the intersections between $MI$ and $AH$, $NI$ and $BE$, and $PI$ and $CF$. Prove that \[\frac{1}{HK}+\frac{1}{EJ}+\frac{1}{FL}>\frac{3}{R}.\]
8. Let $a,b,c$ be the lengths of three sides of a triangle whose perimeter is equal to $3$. Find the minimum value of the expression \[T=a^{3}+b^{3}+c^{3}+\sqrt{5}abc.\]
9. For every positive integer $n$ prove that the following numbers are perfect square \[ 10([(4+\sqrt{15})^{n}+1])([(4+\sqrt{15})^{n+1}]+1)-60,\] \[6([(4+\sqrt{15})^{n}+1])([(4+\sqrt{15})^{n+1}]+1)-60,\] \[15([(4+\sqrt{15})^{n}+1])^{2}-60,\] where $[x]$ is the integerpart of $x$.
10. Let $f(x)=x^{3}-3x^{2}+1$.
    a) Find the number of different real solutions of the equation $f(f(x))=0$.
    b) Let $\alpha be$the maximal positive solution of $f(x)$.
    Prove that $[\alpha^{2020}]$ is divisible by $17$ (notice that $[x]$ is the integer part of $x$).
11. Give the sequence $(u_{n})$ where $u_{1}=2$, $u_{2}=20$, $u_{3}=56$ and \[u_{n+3}=7u_{n+2}-11u_{n+1}+5u_{n}-3.2^{n},\quad\forall n\in\mathbb{N}^{*}.\] Find the remainder of the division $u_{2011}$ by $2011$.
12. Consider any triangle $ABC$ and any line $d$. Let $A_{1},B_{1},C_{1}$ are the projections of $A,B,C$ onto $d$. It is a fact that the line through $A_{1}$ and perpendicular to $BC$, the line through $B_{1}$ and perpendicular to $AC$, and the line through $C_{1}$ and perpendicular to $AB$ are concurrent ar a point which is called the orthogonal pole of $d$ with respect to $ABC$. Prove that for any triangle and any point $P$ on its circumcircle, the Simon line corresponding to $P$ and the orthogonal pole of the Simon line with respect to the given triangle.

Issue 460

1. Let $a$ be a natural number with all different digits and $b$ is another number obtained by using the all the digits of $a$ but in a different order. Given \[a-b=\underset{n}{\underbrace{111\ldots1}}\] ($n$ digit $1$ where $n$ is a positive integer). Find the maximum value of $n$.
2. Find positive integers $x,y,z$ such that \[(x-y)^{3}+(y-z)^{3}+4|z-x|=27.\]
3. Let $a,b,c$ be the lengths of three sides of a triangle. Prove that \[2\left(\frac{a}{b}+\frac{b}{c}+\frac{c}{a}\right)\geq\frac{a}{c}+\frac{b}{a}+\frac{c}{b}+3.\]
4. Let $ABC$ be an isosceles triangle ($AB=AC$) inscribed in a given circle $(O,R)$. Draw $BH$ perpendicular to $AC$ at $H$. Find the maximum length of $BH$.
5. Solve the system of equations \[\begin{cases} \dfrac{7}{2}+\dfrac{3y}{x+y} & =\sqrt{x}+4\sqrt{y}\\ (x^{2}+y^{2})(x+1) & =4+2xy(x-1) \end{cases}.\]
6. For any $m>1$, prove that the following equation has a unique solution \[x^{3}-3\sqrt[3]{3x+2m}=2m.\]
7. Find all values of the parameters $p$ and $q$ such that the corresponding system of equations \[ \begin{cases} x^{2}+y^{2}+5 & =q^{2}+2x-4y\\ x^{2}+(12-2p)x+y^{2} & =2py+12p-2p^{2}-27 \end{cases}\] has two solutions $(x_{1},y_{1})$ and $(x_{2},y_{2})$ satisfying \[ x_{1}^{2}+y_{1}^{2}=x_{2}^{2}+y_{2}^{2}. \]
8. Given a triangle $ABC$ with $BC=a$, $CA=b$ and $AB=c$. Let $R,r$ and $p$ respectively be the circumradius, the inradius, and the semiperimeter of $ABC$. Prove that \[\frac{ab+bc+ca}{p^{2}+9Rr}\geq\frac{4}{5}.\] When does the equality occur?.
9. Given three positive real numbers $a,b,c$ such that $abc\geq1$. Prove that \[\frac{a^{4}b^{2}c^{2}}{bc+1}+\frac{b^{4}c^{2}a^{2}}{ca+1}+\frac{c^{4}a^{2}b^{2}}{ab+1}\geq\frac{3}{2}.\]
10. Find all positive integers $k$ such that there exist 2015 different positive integers whose sum is divisible by the sum of any $k$ numbers among them.
11. Find the maximum $k$ such that the inequality \[\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+1+2k\geq(k+2)\sqrt{a+b+c+1}\] holds for any positive real numbers $a,b,c$ satisfying $abc=1$.
12. Given a triangle $ABC$ and $D$ varies on the side $BC$. Let $(I_{1})$, $(I_{2})$ respectively be the incircles of the triangles $ABD$, $ACD$. Suppose that $(I_{1})$ is tangent to $AB$, $BD$ at $E,X$ and $(I_{2})$ is tangent to $AC$, $CD$ at $F,Y$. Assume that $AI_{1}$, $AI_{2}$ respectively intersects $EX$, $FY$ at $Z,T$. Show that
    a) $X,Y,Z,T$ both belong to a circle with center $K$.
    b) $K$ belongs to a fixed line.

Issue 461

1. Let $(2n-1)!!$ and $(2n)!!$ donote the products $1.3.5\ldots(2n-1)$ and $2.4.6\ldots(2n)$ respectively. Prove that the number \[A=(2013)!!+(2014)!!\] is divisible by $2015.$
2. Given an isolates triangle $ABC$ with $AB=AC$ and $\widehat{A}=3\widehat{B}$. On the half-plane determined by $BC$ that contians $A$, draw the array $Cy$ such that $\widehat{BCy}=132^{0}$. The array $Cy$ intersects the bisector $Bx$ of the angle $B$ at $D$. Calculate $\widehat{ADB}$.
3. Solve the equation \[\frac{1}{\sqrt{x^{2}+3}}+\frac{1}{\sqrt{1+3x^{2}}}=\frac{2}{x+1}.\]
4. Given an equilateral triangle $ABC$ and a point $)$ inside the triangle. Let $M,N,P$ respectively be the intersections between $AO,BO,CO$ and the sides of the triangle. Prove that
   a) ${\displaystyle \frac{1}{AM}+\frac{1}{BN}+\frac{1}{CP}\leq\frac{1}{3}\left(\frac{1}{OM}+\frac{1}{ON}+\frac{1}{OP}\right)}$,
   b) ${\displaystyle \frac{1}{AM}+\frac{1}{BN}+\frac{1}{CP}\leq\frac{2}{3}\left(\frac{1}{OA}+\frac{1}{OB}+\frac{1}{OC}\right)}$.
5. Solve the system of equations \[\begin{cases} \sqrt[3]{9(x\sqrt{x}+y^{3}+z^{3})} & =x+y+z\\ x^{2}+\sqrt{y} & =2z\\ \sqrt{y}+z^{2} & =\sqrt{1-x}+2 \end{cases}.\]
6. Solve the equation \[(x+1)(x+2)(x+3)=\frac{720}{(x+4)(x+5)(x+6)}.\]
7. Determine the number of solutions of each following equation
   a) ${\displaystyle \sin x=\frac{x}{1964}}$,
   b) $\sin x=\log_{100}x$,
8. Given a triangle $ABC$ inscribed in a circle $(O)$. The bisectors of the angles $A,B,C$ respectively intersects the circle at $D,E,F$. Denote respectively by $h_{a},h_{b},h_{c},S$ the heights from $A,B,C$ and the area of $ABC$. Prove that \[AD.h_{1}+BE.h_{b}+CF.h_{c}\geq4\sqrt{3}S.\]
9. Given real numbers $a,b,c,d$ satisfying \[a^{2}+b^{2}+c^{2}+d^{2}=1.\] Find the maximum and minimum values of the expression \[P=ab+ac+ad+bc+cd+3cd.\]
10. Given $k\geq1$ and positive numbers $x,y$. For any positive integer $n\geq2$, show the following inequalities \[\sqrt{xy}\leq\sqrt[n]{\frac{x^{n}+y^{n}+k[(x+y)^{n}-x^{n}-y^{n}]}{2+k(2^{n}-2)}}\leq\frac{x+y}{2}.\]
11. A pair of positive integers is called a good pair if their quotient if either $2$ or $3$. What is the most number of good pairs we can get among $2015^{2016}$ arbitrary different positive integers?.
12. Given a triangle $ABC$. Let $E,F$ respectively be the perpendicular projections of $B,C$ on $AC,AB$; and then let $T$ be the perpendicular projection of $A$ on $EF$. Denote the midpoints of $BE$ and $CF$ by $M$ and $N$ respectively. Suppose that $TM,TN$ intersects $AB$, $AC$ respectively at $P$, $Q$. Prove that $EF$ goes through the midpoint of $PQ$,

Issue 462

1. Find the last digit of the following number \[A=1^{2015}+2^{2015}+3^{2015}+\ldots+2014^{2015}+2015^{2015}.\]
2. Given a right trapezoid $ABCD$ ($A$ and $B$ are right angles) with $AD<BC$ and $AC\perp BD$. Prove that \[AC^{2}+BD^{2}=3AB^{2}+CD^{2}.\]
3. Does there exist a pair of positive integers $(a,b)$ such that both the equation $x^{2}+2ax-b-2=0$ and the equation $x^{2}+bx-a=0$ have integer solutions?.
4. Given a right triangle $ABC$ with the right angle $A$ such that \[BC^{2}=2BC\cdot AC+4AC^{2}.\] Find the angle $\widehat{ABC}$.
5. Solve the equation \[\sqrt[5]{3x-2}-\sqrt[5]{2x+1}=\sqrt[5]{x-3}.\]
6. Solve the system of equations \[\begin{cases} x+\sqrt{x^{2}+9} & =\sqrt[4]{3^{y+4}}\\ y+\sqrt{y^{2}+9} & =\sqrt[4]{3^{z+4}}\\ z+\sqrt{z^{2}+9} & =\sqrt[4]{3^{x+4}} \end{cases}.\]
7. Given an acute triangle $ABC$ with angles measured in radian. Prove that \[\sin A+\sin B+\sin C>\frac{5}{2}-\frac{A^{2}+B^{2}+C^{2}}{\pi^{2}}.\]
8. Given a tetrahedron $ABCD$ inscribed in a sphere of radius $1$. Assume that the product of the lengths of its sides is equal to $\frac{512}{27}$. Compute the lengths of its sides.
9. Let $a,b,c$ be the lengths of three sides of a triangle. Prove that \[\frac{3(a^{2}+b^{2}+c^{2})}{(a+b+c)^{2}}+\frac{ab+bc+ca}{a^{2}+b^{2}+c^{2}}\leq2.\]
10. Find the number of the triples $(a,b,c)$ of integers in the interval $[1,2015]$ such that $a^{3}+b^{3}+c^{3}$ is divisible by $9$?. 
11. Let $a,b,c$ be real numbers such that $a^{2}+b^{2}+c^{2}=1$. Find the minimum value of the expression \[P=|6a^{3}+bc|+|6b^{3}+ca|+|6c^{3}+ab|.\]
12. Given a triangle $ABC$ with $(O)$ and $H$ respectively are the circumcircle and the orthocenter of the triangle. Let $P$ be an arbitrary point on $OH$. Let $A_{0}$, $B_{0}$, $C_{0}$ respectively be the intersections between $AH$, $BC$, $CH$ and $BC$, $CA$, $AB$. Suppose that $A_{1}$, $B_{1}$, $C_{1}$ respectively be the second intersections between $AP$, $BP$, $CP$ and $(O)$. Let $A_{2}$, $B_{2}$, $C_{2}$ respectively be the reflection points of $A_{1}$, $B_{1}$, $C_{1}$ through $A_{0}$, $B_{0}$, $C_{0}$. Prove that $H$, $A_{2}$, $B_{2}$, $C_{2}$ belong to a circle with center is on $OH$.