# Mathematics and Youth Magazine Problems 2013

### Issue 427

1. Let $a=123456789$. Which number is greater $2012^{9^{9^{a}}}$ or $2013^{a^{a^{9}}}$?.
2. Let $ABC$ $(AB < AC)$ be a triangle, with two altitudes $BD,CE$ and $AB=c$, $AC=b$, $BD=h_{b}$, $CE=h_{c}$. Prove that $c^{n}+h_{c^{n}} < b^{n}+h_{b}^{n},\quad\forall n\in\mathbb{N}^{*}.$
3. Find all positive integers $n$ such that $A=\left[\frac{n^{2}+n-5}{2}\right]$ is a prime number, where $[a]$ is the largest integer not exceeding $a$.
4. Find all postive integer soltuions of the equation $\sqrt{x+y}-\sqrt{x}-\sqrt{y}+2=0.$
5. Let $ABC$ be a right triangle, right angle at $A$. The bisectors $BD$ and $CE$ intersect at $O$. The area of $BOC$ is $a$. Determine the product $BD\cdot CE$ in terms of $a$.
6. Solve the system pf equations $\begin{cases} 2\sqrt{\frac{x^{4}}{3}+4} & =1+\sqrt{\frac{3}{2}}|y|\\ 2\sqrt{\frac{y^{4}}{3}+4} & =1+\sqrt{\frac{3}{2}}|x| \end{cases}.$
7. The side lengths of a traingle $ABC$ are $AB=9$, $BC=\sqrt{39}$, $CA=\sqrt{201}$. Find a point $M$ on the circle $(C;\sqrt{3})$ such that the sum $MA+MB$ is the maximum.
8. Prove that in any traingle $ABC$, \begin{align*} & \sqrt{\left(\tan\frac{A}{2}+\tan\frac{B}{2}\right)\left(\tan\frac{B}{2}+\tan\frac{C}{2}\right)}\\  + &\sqrt{\left(\tan\frac{B}{2}+\tan\frac{C}{2}\right)\left(\tan\frac{C}{2}+\tan\frac{A}{2}\right)}\\ + &\sqrt{\left(\tan\frac{C}{2}+\tan\frac{A}{2}\right)\left(\tan\frac{A}{2}+\tan\frac{B}{2}\right)}\\ \leq & 2(\cot A+\cot B+\cot C).\end{align*}
9. Let $N=1+10+10^{2}+\ldots+10^{4023}$. Find the 2013-th digit after the decimal comma of $\sqrt{N}$.
10. Find the maximum and minimum values of the expression $P=\frac{a-b}{c}+\frac{b-c}{a}+\frac{c-a}{b},$ where $a,b,c$ are positive real numbers satisfying the condition $\min\{a,b,c\}\geq\frac{1}{4}\max\{a,b,c\}.$ Let $\{S_{n}(x)\}$ be a sequence of real-valued functions defined by $S_{n}(x)=\cos^{3}x-\frac{1}{3}\cos^{3}3x+\frac{1}{3^{2}}\cos^{3}3^{2}x-\ldots+\left(\frac{-1}{3}\right)^{n}\cos^{3}3^{n}x.$
11. Find all real values of $x$ such that $\lim S_{n}(x)=\frac{3-3x}{4}.$
12. In a con-cyclic quadrilateral $ABCD$, let $A'$, $B'$, $C'$, $D'$ be the circumcenters of triangles $BCD$, $CDA$, $DAB$ and $ABC$ respectively. Let $A''$, $B''$, $C''$, $D''$ be the centers of the Euler circles of triangle $BCD$, $CDA$, $DAB$, $ABC$ respectively. Prove that the two quadrilateral $A'B'C'D'$, $A''B''C''D''$ are both convex and inversely similar.

### Issue 428

1. Determine all triple of prime numbers $a,b,c$ (not necessarily distinct) such that $abc < ab+bc+ca.$
2. Let $ABC$ be a right triangle, with right angle at $A$ and $AH$ is the altitude from $A$, $\widehat{ACB}=30^{\circ}$. Construct an equilateral triangle $ACD$ ($D$ and $B$ are in opposite side $AC$). K is the foot of the perpendicular line from $H$ onto $AC$. The line through $H$ and parallel to $AD$ meets $AB$ at $M$. Prove the points $D,K,M$ are colinear.
3. Consider a $6\times6$ board of squares with 4 corner squares being deleted. Find the smallest number of squares that can be painted black given that among the 5 squares in any figure, there is at lease one black.
4. Let $a,b,c$ be real numbers in the interval $[1,2]$. Prove the inequality $a^{2}+b^{2}+c^{2}+3\sqrt{(abc)^{2}}\geq2(ab+bc+ca).$
5. Let $ABC$ be a non-right triangle $(AB < AC)$ with altitude $AH$. $E,F$ are the orthogonal projection of point $H$ onto $AB$ and $AC$ respectively. $EF$ meets $BC$ at $D$. Draw a semicircle with diameter $CD$ on the half-plane containing $A$ with edge $CD$. The line through $B$ and perpendicular to $CD$ meets the semicircle at $K$. Prove that $DK$ is tangent to the circumcircle of triangle $KEF$.
6. Given that the equation $ax^{3}-x^{2}+ax-b=0\quad(a\ne0,\,b\ne0)$ has three positive real roots. Determine the greatest value of the following expression $P=\frac{11a^{2}-3\sqrt{3}ab-\frac{1}{3}}{9b-10(\sqrt{3}a-1)}.$
7. Solve the following system of equations $\begin{cases} \sqrt{x-\frac{1}{4}}+\sqrt{y-\frac{1}{4}} & =\sqrt{3}\\ \sqrt{y-\frac{1}{16}}+\sqrt{z-\frac{1}{16}} & =\sqrt{3}\\ \sqrt{z-\frac{9}{16}}+\sqrt{x-\frac{9}{16}} & =\sqrt{3}\end{cases}.$
8. Let $a,b$ be real constants such that $ab>0$. Let $\{u_{n}\}$ be a sequence where $n=1,2,3,\ldots$ given by $u_{1}=a,\quad u_{n+1}=u_{n}+bu_{n}^{2},\,\forall n\in\mathbb{N}^{*}.$ Determine the limit $\lim_{n\to\infty}\left(\frac{u_{1}}{u_{2}}+\frac{u_{2}}{u_{3}}+\ldots+\frac{u_{n}}{u_{n+1}}\right).$
9. Find all positive integers $k$ with the property that there exists a polynomial $f(x)$ with integer coefficients of degree greater than 1 such that for all prime numbers $p$ and natural numbers $a,b$ if $p$ divides $(ab-k)$ then it also divides $(f(a)f(b)-k)$.
10. Given $a_{i}\in[0,\alpha]$ ($i=\overline{1,n}$), ($\alpha>0$). Prove the inequality $\prod_{i=1}^{n}(\alpha-a_{i})\leq\alpha^{n}\left(1-\sum_{i=1}^{n}\frac{a_{i}}{S_{i}+\alpha}\right)$ where $S_{i}=\sum_{k=1}^{n}a_{k}-a_{i}$ for all $i=\overline{1,n}$.
11. Point $O$ is in the interior of triangle $ABC$. The ray $Ox$ parallel to $AB$ meets $BC$ at $D$, ray $Oy$ parallel to $BC$ meets $CA$ at $E$, ray $Oz$ parallel to $CA$ meets $AB$ at $F$. Prove that
a) $3S_{DEF}\leq S_{ABC}$.
b) $OD\cdot OE\cdot OF\leq27AB\cdot BC\cdot CA$.
12. The circle $(O)$ and $(O')$ meet at points $A$, $B$. Point $C$ is fixed on $(O)$ and point $D$ is fixed on $(O')$. A moving point $P$ is on the opposite ray of ray $BA$. The circumcircles of traingles $PBC$, $PDB$ intersect $BD$, $BC$ at seconde points $E$, $F$ respectively. Prove that the midpoint of line segment $EF$ is always on a fixed segment $EF$ is always on a fixed straight line.

### Issue 429

1. Find an integer size square whose area is a 4-digit number such that the last rightmost three digits are idnentical.
2. Determine the values of $a_{1},a_{2},a_{3},a_{4},a_{5},a_{6},a_{7}$. Given that $2a_{1}=3a_{2}$, $2a_{3}=4a_{4}$, $5a_{4}=2a_{5}$, $2a_{5}=5a_{6}$, $2a_{6}=3a_{7}$, $2a_{7}=3a_{1}$ and $$a_{1}+a_{2}+a_{3}+a_{4}+a_{5}+a_{6}+a_{7}=400.$$
3. Find all pairs of integers $(x,y)$ such that $x^{2}(x^{2}+y^{2})=y^{p+1}$, where $p$ is a prime number.
4. Find the maximum and minimum values of the expression $P=xy+yz+zx-xyz$ where $x,y,z$ are non-negative real numbers satisfying $x^{2}+y^{2}+z^{2}=3.$
5. Triangle $ABC$ inscribed in circle $(O)$ with $\widehat{BAC}=70^{\circ}$, $\widehat{ACB}=50^{\circ}$. The points $M,N,P,Q$ and $R$ on circle $(O)$ are such that $PA=AB=BR$, $QB=BC=CM$ and $NC=CA=AN$. Let $S$ be the intersection of arc $NQ$ and the diameter $PP'$ of $(O)$. Prove that $\Delta NRS\backsim\Delta NQR$.
6. Solve the equation $x^{3}-3x=\sqrt{x+2}.$
7. Find the measure of the angles of a triangle $ABC$ such that the expression $T=-3\tan\frac{C}{2}+4(\sin^{2}A-\sin^{2}B)$ is greatest possible.
8. Let $S.ABC$ be a triangular pyramid where the sides $SA,SB,SC$ are pairwise orthogonal, $SA=a$, $SB=b$, $Sc=c$. $H$ is the foot of the perpendicular from $S$ onto $ABC$. Prove the inequality $aS_{HBC}+bS_{HAC}+cS_{HAB}\leq\frac{abc\sqrt{3}}{2}.$
9. Let $p$ be an odd prime number, and $x,y$ are two positive integers such that $\sqrt{x}+\sqrt{y}\leq\sqrt{2p}$. Find the minimum value of the following expression $A=\sqrt{2p}-\sqrt{x}-\sqrt{y}.$
10. Does there exist a funtion $f:\mathbb{N}^{*}\to\mathbb{N}^{*}$ such that $f(mf(n))=n+f(2013m),\quad\forall m,n\in\mathbb{N}^{*}?.$
11. The non-negative real numbers $a,b,c$ are such that $$\max\{a,b,c\}\leq4\min\{a,b,c\}.$$ Prove the inequality $2(a+b+c)(ab+bc+ca)^{2}\geq9abc(a^{2}+b^{2}+c^{2}+ab+bc+ca).$
12. Let $ABC$ be a traingle inscribed in circle centered at $O$, and let $I$ be its incenter. $AI,BI,CI$ intersect $(O)$ at $A_{1},B_{1},C_{1}$; $A_{1}C_{1}$, $A_{1B_{1}}$ meet $BC$ at $M$, $N$; $B_{1}A_{1}$, $B_{1}C_{1}$ meet $CA$ at $P$, $Q$; $C_{1}B_{1}$, $C_{1}A_{1}$ meet $AB$ at $R$, $S$ respectively. Prove that $S_{MNPQRS}\leq\frac{2}{3}S_{A_{1}B_{1}C_{1}}.$

### Issue 430

1. Do there exist natural numbers $x,y,z$ such that $5x^{2}+2016^{y+1}=2017^{z}?.$
2. Point $O$ is chosen in a right triangke $ABC$, right angle at $A$, such that $\widehat{ABO}=30^{0}$ and $OA=OC$. Point $E$ on side $BC$ such that $\widehat{EOB}=60^{0}.$ Determine the three angles of traingle $ABC$ given that the line $CO$ passes through the midpoint $I$ of the line segment $AE$.
3. Find all pair of natural numbers $x,y$ such that $5^{x}=y^{4}+4y+1$.
4. Solve the system of equations $\begin{cases} x+\sqrt{y-2}+\sqrt{4-z} & =y^{2}-5z+11\\ y+\sqrt{z-2}+\sqrt{4-x} & =z^{2}-5x+11\\ z+\sqrt{x-2}+\sqrt{4-y} & =x^{2}-5y+11 \end{cases}.$
5. Let $AB=2a$ be a line segment with midpoint $O$. Two half-circles, one with center $O$ and diagonal $AB$, another with center $O'$ and diagonal $AO$ are drawn on the same half-plane divided by $AB$. Point $M$, different from $A$ and $O$, moves on the half-circle $(O')$. $OM$ meets the half-circle $(O)$ at $C$. Let $D$ be the second intersection point of $CA$ and half-circle $(O')$. The tangent line at $C$ of half-circle $(O)$ meets $OD$ at $E$. Find the position of point $M$ on $(O')$ such that $ME$ is parallel to $AB$.
6. Let $ABCD$ be a quadrilateral where the diagonals $AC,BD$ are equal and perpendicular. The triangles $AMB$, $BNC$, $CPD$, $DQA$, similar in order, are constructed outside the given quadrilateral. $O_{1}$, $O_{2}$, $O_{3}$, $O_{4}$ are the midpoints of $MN$, $NP$, $PQ$, $QM$ respectively. Prove that the quadrilateral $O_{1}O_{2}O_{3}O_{4}$ is s square.
7. Find a formula counting the number of all $2013$-digits natural numbers which are multiple of $3$ and all digits are taken from the set $X=\{3,5,7,9\}$.
8. Solve for $x$, $\log_{2}x=\log_{5-x}3.$
9. The positive integers $a_{1},a_{2},\ldots,a_{2013}$, $b_{1},b_{2},\ldots b_{2013}$ where $b_{k}>1$ for all $k$ are chosen from the set $X=\{1,2,\ldots,2013\}$. Prove that there exists a positive integer $n$ satisfying the following two conditions
• $n\leq\left(\prod_{i=1}^{2013}a_{i}\right)\left(\prod_{i=1}^{2013}b_{i}\right)+1$.
• $a_{k}b_{k}^{n}+1$ is a composite number for every $k\in X$.
10. Let $a,b,c\in\left[0,\frac{1}{2}\right]$ be such that $a+b+c=1$. Prove the inequality $a^{3}+b^{3}+c^{3}+4abc\leq\frac{9}{32}.$
11. Let $ap$ be a prime number, $p\equiv1\,(\text{mod }4)$. Determine the sum $\sum_{k=1}^{p-1}\left[\frac{2k^{2}}{p}-2\left[\frac{k^{2}}{p}\right]\right],$ where $[a]$ denotes the largest integer not exceeding $a$.
12. Let $ABC$ be a triangle inscribed inside circle $(O)$. Point $M$ not on lines $BC$, $CA$, $AB$ as weel as circle $(O)$; $AM$, $BM$, $CM$ intersect $(O)$ at $A_{1}$, $B_{1}$, $C_{1}$; $A_{2}$, $B_{2}$, $C_{2}$ are the circumcenters of triangles $MBC$, $MCA$, $MAB$ respectively. Prove that the lines $A_{1}A_{2}$, $B_{1}B_{2}$, $C_{1}C_{2}$ meet at a point on the circle $(O)$.

### Issue 431

1. Which number is greater $A=\left(1+\frac{1}{2013}\right)\left(1+\frac{1}{2013^{2}}\right)\ldots\left(1+\frac{1}{2013^{n}}\right)$ where $n$ is a positive integer, or $B=\frac{2013^{2}-1}{2012^{2}-1}$?.
2. Given four points in the plane such that no pair of points has distance less than $\sqrt{2}$ cm. Prove that there exists two of them having a distance greater than or equal to $2$ cm.
3. Find the last two digits of the number  $2003^{2004^{\mathstrut^{.^{.^{.^{2013}}}}}}.$
4. Find the maximum and minimum value of the expression $P=27\sqrt{x}+8\sqrt{y}$ where $x,y$ are non-negative real numbers satisfying $x\sqrt{1-y^{2}}+y\sqrt{2-x^{2}}=x^{2}+y^{2}.$
5. Let $ABCD$ be a cyclic quadrilateral, inscribed in circle $(O)$. $I$ and $J$ are the midpoints of $BD$ and $AC$ respectively. Prove that $BD$ is the angle bisector of angle $AIC$ if and only if $AC$ is the angle bisector of angle $BJD$.
6. Solve the following system of equations $\begin{cases} x^{3}(1-x)+y^{3}(1-y) & =12xy+18\\ |3x-2y+10|+|2x-3y| & =10 \end{cases}.$
7. Determine the greatest value of the expression $E=a^{2013}+b^{2013}+c^{2013},$ where $a,b,c$ are real numbers satisfying $a+b+c=0,\quad a^{2}+b^{2}+c^{2}=1.$
8. Let $S.ABC$ be a triangular pyramid, $G$ is the centroid of the base triangle $ABC$, $O$ is the midpoint of $SG$. A moving plane $(\alpha)$ through $O$ meets the edges $SA$, $SB$, $SC$ at $A',B'C'$ respectively. Prove that $\frac{SA'^{2}}{AA'^{2}}+\frac{SAB'}{BB'^{2}}+\frac{SC'^{2}}{CC'^{2}}\geq\frac{AA'^{2}}{SA'^{2}}+\frac{BB'^{2}}{SB'^{2}}+\frac{CC'^{2}}{SC'^{2}}.$
9. Find all natural numbers $n$ such that $A=\left[\frac{n+3}{4}\right]+\left[\frac{n+5}{4}\right]+\left[\frac{n}{2}\right]+n^{2}+3n-1$ is a prime number, where $[x]$ denotes the greatest integer not exceeding $x$.
10. Consider the real-valued function $y=a\sin(x+2013)+\cos2014x$ where $a$ is given real number. Let $M,N$ be the greatest and smallest values respectively of this function over $\mathbb{R}.$ Prove that $M^{2}+N^{2}\geq2$.
11. Let $\{a_{n}\}$ be a sequence given by $a_{1}=\frac{1}{2},\quad a_{n+1}=\frac{a_{n^{2}}}{a_{n^{2}-a_{n}+1}},\,n=1,2,\ldots$ a) Prove that the sequence $\{a_{n}\}$ converges to a finite limit and find this limit.
b) Let $b_{n}=a_{1}+a_{2}+\ldots+a_{n}$ for each positive integer $n$. Determine the integer part $[b_{n}]$ and the limit $\lim_{n\to\infty}b_{n}$.
12. Given four points $A,B,C,D$ on circle $(ABC)$ and $M$ is a point not on this circle. Let $T_{i}$ be the triangle whose three vertices are $3$ of $4$ given points, except point $i$ ($i=A,B,C,D$). Let $H_{i}$ be the triangle whose vertices are the feet of the perpendicular drawn from $M$ onto the edges (or extended edges) of triangles $T_{i}$ ($i=A,B,C,D$). Prove that
a) The circumcenter of triangles $H_{i}$ ($i=A,B,C,D$) lie on the same circle, centered at $O'$.
b) When $D$ moves on the circle $(ABC)$, $O'$ always lie on a fixed circle.

### Issue 432

1. The first $2013$ natural numbers from $1$ to $2013$ are writeen in a line in some order. Substract one from the first number, two from the second ... and 2013 from the $2013^{\text{th}}$ number. Is the product of the resulting $2013$ numbers odd or even?.
2. Let $ABC$ be an acute triangle with orthocenter $O$. $AO$ meets $BC$ at $D$. Points $E$ and $F$ are on sides $AB$ and $AC$ respectively such that $DE=DB$, $DF=DC$. Prove that $DA$ is the angle bisector of angle $EDF$.
3. Find all positive integers $a,b$ ($a\geq2$, $b\geq2$) so that $a+b$ is amultiple of $4$ and $\frac{a(a-1)+b(b-1)}{(a+b)(a+b-1)}=\frac{1}{2}.$
4. Find $x,y$ such that $\begin{cases} x\sqrt{x}+y\sqrt{y} & =2\\ x^{3}+2y^{2} & \leq y^{3}+2x\end{cases}.$
5. Given a circle centered at $O$, and diameter $AB$. Point $C$, different from $A$ and $B$, is chosen on circle $(O)$. Point $P$ on $AB$ such that $BP=AC$. The perpendicular from $P$ to $AC$ meet $AC$ at $H$. The intenal angle bisector of angle $CAB$ intersects circle $(O)$ at $E$ and intersects $PH$ at $F$. $CF$ meets circle $(O)$ at $N$. Prove that $CN$ passes through the midpoint of $AP$.
6. Let $a,b,c$ the positive real number. Prove the following inequality \begin{align*} & \left(\frac{1}{a}+\frac{2}{b+c}+\frac{3}{a+b+c}\right)^{2}+\left(\frac{1}{b}+\frac{2}{c+a}+\frac{3}{a+b+c}\right)^{2}+\left(\frac{1}{c}+\frac{2}{a+b}+\frac{3}{a+b+c}\right)^{2}\\ & \geq\frac{81}{a^{2}+b^{2}+c^{2}}.\end{align*}
7. Triangle $ABC$ is inscribed in circle $(O)$, another circle $(O')$ touches $AB$, $AC$ at $P,Q$ respectively and touches circle $(O)$ at other points $M$, $N$. Points $E$, $D$, $F$ are the perpendicular feet of point $S$ on $AM$, $MN$, $NA$ respectively. Prove that $DE=DF$.
8. The real numbers $a,b,c$ satisfying the condition that the polynomial $P(x)=x^{4}+ax^{3}+bx^{2}+cx+1$ has at least one real root. Determine all triple $(a,b,c)$ such that $s^{2}+b^{2}+c^{2}$ is smallest possible.
9. Let $a$ and $B$ be two real numbers such that $a^{p}-b^{p}$ is a positive integers for all prime number $p$. Prove that $a$ and $b$ are integers.
10. The sequence $\{u_{n}\}$ is given recursively as follows $u_{1}=\frac{1}{1+a},\quad\frac{1}{u_{n+1}}=\frac{1}{u_{n}^{2}}-\frac{1}{u_{n}}+1,\,\forall n\geq1$ where $a\in\mathbb{R},$ $a\ne-1$. Let $$S_{n}=u_{1}+u_{2}+\ldots+u_{n},\quad P_{n}=u_{1}u_{2}\ldots u_{n}.$$ Determine the value of the following expression $aS_{n}+P_{n}$.
11. Determine all funtions $f:\mathbb{R}^{+}\to\mathbb{R}^{+}$ such that
• $f$ is a decreasing function on $\mathbb{R}^{+}$.
• $f(2x)=2012^{-x}f(x)$, $\forall x\in\mathbb{R}^{+}$
where $\mathbb{R}^{+}=(0,+\infty)$.
12. Let $ABC$ be a triangle inscribed in circle $(O)$. $AD$ is a diameter of $(O)$. Point $E$ belongs to the opposite ray of ray $DA$. The perpendicular through $E$ o $AD$ meets $BC$ at $T$. $TP$ is a tangent line to $(O)$ such that $P$ and $A$ are on opposite sides of $BC$; $AP$ meets $TE$ at $Q$. $M$ is the midpoint of $AQ$; $TM$ meets $AB$, $AC$ at $X$, $Y$ respectively. Prove that $M$ is the midpoint of $XY$.

### Issue 433

1. Find all positive integers $x,y,z$ such that $x^{2}+y^{3}+z^{4}=90.$
2. Let $ABC$ be an equilateral triangle whose altitudes are $AD$, $BE$ and $CF$. Suppose $M$ is an arbitrary point inside triangle $ABC$. $I$, $K$, $L$ are the perpendicualr feet from $M$ to $AD$, $BE$, $CF$. Prove that the sum $AI+BK+CL$ does not depend on the position of $M$.
3. The rational numbers $a$, $b$ satisfy the identity $a^{2013}+b^{2013}=2a^{1006}b^{1006}.$ Prove that the equation $x^{2}+2x+ab=0$ has two rational solutions.
4. Find the minimum value of the expression $P=(x^{4}+y^{4}+z^{4})\left(\frac{1}{x^{4}}+\frac{1}{y^{4}}+\frac{1}{z^{4}}\right),$ where $x,y,z$ are positive real numbers that satisfy $x+y\leq z$.
5. Let $AH$ be the altitude from $A$ of right triangle $ABC$, right angle at $A$. Point $D$ on the oppostite ray of $HA$ such that $HA=2HD$. Point $E$ is the reflection of $B$ through $D$; $I$ is the midpoint of $AC$; $DI$ and $EI$ meet $BC$ at $M$ and $K$ respectively. Prove that $\widehat{BDK}=\widehat{MCD}$.
6. Solve the equation $\sqrt{x+\sqrt{x^{2}-1}}=\frac{27\sqrt{2}}{8}(x-1)^{2}\sqrt{x-1}.$
7. A convex quadrilateral $ABCD$ with area $S$ is inscribed in a circle whose radius is $R$ and $AB=a$, $BC=b$, $CD=c$, $DA=d$, $AC=e$. If there exists a circle touching all the opposite rays of the rays $BA$, $DA$, $CD$ and $CB$. Prove that
a) $R=\dfrac{S\cdot e}{p^{2}-e^{2}}$,
b) $a^{2}+b^{2}+c^{2}+d^{2}+\dfrac{8SR}{e}=2p^{2}$,
where $2p=a+b+c+d$.
8. Find the maximum value of the expression $\alpha(\sin^{2}A+\sin^{2}B+\sin^{2}C)-\beta(\cos^{3}A+\cos^{3}B+\cos^{3}C)$ where $A$, $B$, $C$ are three angles of an acute triangle and $\alpha$, $\beta$ are two given positive numbers.
9. Find the maximum area of a convex pectagon in the coordinate plane $Oxy$ having the following properties: all interior angles are the same, all vertices have integer coordinates, there exists a side that is parallel to the axis $Ox$, there are exactly $16$ points, including the vertices, with integer coordinates on its boundary.
10. Find all continuous functions $f$ such that $(x+y)f(x+y)=xf(x)+yf(y)+2xy,\,\forall x,y\in\mathbb{R}.$
11. Let $(a_{n})$ be a sequence where $a_{1}\in\mathbb{R}$ and $a_{n+1}=|a_{n}-2^{1-n}|$, $\forall n\in\mathbb{N}^{*}$. Find $\lim_{n\to\infty}a_{n}$.
12. A right triangle $ABC$ with right angle at $C$ is inscribed in circle $(O)$. $M$ is an arbitrary point moving on circle $(O)$, different from $A$, $B$, $C$. Point $N$ is the reflection of $M$ in $AB$, $P$ is the perpendicular foot of $N$ to $AC$, $MP$ meets $(O)$ at a second point $Q$. Prove that the circumcenter of triangle $APQ$ lies on a fixed circle.

### Issue 434

1. Find the largest possible perfect square of the form $4^{27}+4^{1020}+4^{x}$, where $x$ is a natural number.
2. Let $ABC$ be a right triangle with right angle at vertex $A$ and $\widehat{ABC}=54^{0}$. The median $AM$ meets the internal angle-bisector $CD$ at $E$. Prove that $CE=AB$.
3. Find all prime numbers $p$ such that $p-1$ and $p+1$ each has exactly $6$ divisors.
4. Solve the system of equations $\begin{cases} 3\sqrt{x}+2\sqrt{y}+\sqrt{z} & =\dfrac{1}{6}\sqrt{xyz}\\ 6\sqrt{xy}+2\sqrt{yz}+3\sqrt{zx} & =108+18\sqrt{x+4}+12\sqrt{y+9}+6\sqrt{z+36} \end{cases}.$
5. Let $AB$ and $AC$ be the tangent lines to a circle $(O)$ through an external point $A$ ($B$ and $C$ are the poins of tangency). The median $BM$ of triangle $ABC$ intersects $(O)$ at $D$, the ray $AD$ meets $(O)$ at $E$. Prove that $BE||AC$.
6. Given that $a,b,c$ are the sides of a triangle, prove the inequality \begin{align*} \sqrt{a^{2}-(b-c)^{2}}+\sqrt{b^{2}-(c-a)^{2}}+\sqrt{c^{2}-(a-b)^{2}} & \leq \sqrt{ab}+\sqrt{bc}+\sqrt{ca} \\ & \leq a+b+c \end{align*}
7. The circle $(O)$ is inscribed in a triangle $ABC$. The tangents to $(O)$ which are parallel to the sides of the triangle are drawn, they intersect these sides at points $M,N,P,Q,R$ and $S$ ($M,S\in AB$, $N,P\in AC$, $Q,R\in BC$). Let $l_{1}$, $l_{2}$, $l_{3}$ be the lengths of the internal angle-bisectors from $A,B,C$ of triangles $AMN$, $BSR$ and $CPQ$ respectively. Prove that $\frac{1}{l_{1}^{2}}+\frac{1}{l_{2}^{2}}+\frac{1}{l_{3}^{2}}\geq\frac{81}{p^{2}},$ where $p$ denotes the semiperimeter of triangle $ABC$.
8. It is given that in a triangle $ABC$, $\tan\dfrac{B}{2}\tan\dfrac{C}{2}=\dfrac{1}{3}$. Solve for $x$ $x^{2}+x-\cos A-\frac{1}{4}\cos(B-C)=0.$
9. Find all real numbers $x$ such that $\left\{ \frac{x^{2}+1}{x^{2}+x+1}\right\} =\frac{1}{2}$ where $\{a\}$ denote te fractional part of $a$, that is $\{a\}=a-[a]$.
10. Consider the sequence $(x_{n})$, $x_{n+1}=x_{n}+\frac{1}{x_{n}}+\frac{2}{x_{n}^{2}}+\ldots+\frac{2012}{x_{n}^{2012}}\quad(n\in\mathbb{N}^{*})$ where $x_{1}>0$ is given. Determine all values of $\alpha$ such that the sequence $(nx_{n}^{\alpha})$ has a non-zero limit.
11. Find all functions $f:\mathbb{R}\to\mathbb{R}$ such that $f(xf(y)+3y^{2})+f(3xy+y)=f(3y^{2}+x)+4xy-x+y$ for all $x,y\in\mathbb{R}$.
12. Let $G$ be the centroid of a tetrahedron $ABCD$. Points $X,Y<Z,T$ are chosen on the faces $(BCD)$, $(CDA)$, $(DAB)$, and $(ABC)$ respectively such that $XY$, $YZ$, $ZT$, $TX$ are parallel to $GA$, $GB$, $GC$ and $GD$. Determine the volume ratio of the two tetrahedrons $ABCD$ and $XYZT$.

### Issue 435

1. The number $s=\overline{3\ldots3}^{2}+\overline{5\ldots54\ldots4}^{2}$, written in decimal system, consist of $n+1$ digits $3$, $n-1$ digits $5$ and $n$ digits $4$. Given that $s=r^{2}$, find the value of $r$.
2. Let $AM$ be the median of triangle $ABC$. On the half-plane containing $C$ created by the side $AB$, draw line segment $AE$ perpendicular to $AB$ such that $AB=AE$. On the half-plane containing $B$ created by $AC$, draw $AF$ perpendiclar to $AC$ such that $AF=AC$. Prove that $EF=2AM$ and $EF\perp AM$.
3. Consider $n$ positive integers $a_{1},a_{2},\ldots a_{n}$ ($n>1$) satisfying $a_{1}+a_{2}+\ldots+a_{n}=a_{1}a_{2}\ldots a_{n}.$ a) Prove that for any given value of $n$, the above equation always has solution.
b) Determine all values of $n$ such that the equation $a_{1}<a_{2}<\ldots<a_{n}$.
4. The numbers $a,b,c$ satisfy $ab+bc+ca=2013abc,\quad2013(a+b+c)=1.$ Find the sum $A=a^{2013}+b^{2013}+c^{2013}$.
5. Prove that if in a trapezium $ABCD$ ($AB||CD$), $AC+CB=AD+DB$, then $ABCD$ is an isosceles trapezium.
6. Solve the inequality on $\mathbb{R}$ $(\sqrt{13}-\sqrt{2x^{2}-2x+5}-\sqrt{2x^{2}-4x+4})(x^{6}-x^{3}+x^{2}-x+1)\geq0.$
7. The three angles of an acute triangle $ABC$ are such that $A>\dfrac{\pi}{4}$, $B>\dfrac{\pi}{4}$, $C>\dfrac{\pi}{4}$. Determine the smallese value of the expression $\frac{\tan A-2}{\tan^{2}C}+\frac{\tan B-2}{\tan^{2}A}+\frac{\tan C-2}{\tan^{2}B}.$
8. Let $S.ABCD$ be a pyramid inscribed in a sphere centred at $O$, and $AB=a$, $CD=b$. Draw parallellograms $ADKB$ and $SDHC$. Determine the ratio $\dfrac{HK}{EF}$ in terms of $a$ and $b$, where $E$ is the point of intersection of $AD$ and $BC$, and $F$ is the point of intersecion of $AC$ and $BD$.
9. How many positive integers $n$ are there such that $n$ has $2013$ digits in decimal number system and $\dfrac{n}{7}$ is a positive integer with $2013$ odd digits?.
10. Find all funtions $f:\mathbb{R}\to\mathbb{R}$, $g:\mathbb{R}\to\mathbb{R}$ such that the following two conditions are satisfied
• $f(x)-2g(x)=g(y)+4y$, for all $x,y\in\mathbb{R}$;
• $f(x)g(x)\leq33x^{2}$, for all $x\in\mathbb{R}$.
11. Find all polynomials $T(x,y)$ such that $T(x,y)T(z,t)=T(xz+yt,xt+yz)$ for all $x,y,z,t\in\mathbb{R}$.
12. Let $ABCD$ be a cyclic quadrilateral. The circle whose diameter is $AB$ meets $CA$, $CB$, $DA$ and $DB$ at $E,F,I$ and $J$ respectively (all differ from $A$ and $B$). Prove that the angle-bisector of an angle between $EF$ and $IJ$ is perpendicular to the line $CD.$

### Issue 436

1. Let $n=1234567891011\ldots99100.$ Delete $100$ digits so that the remaining digits in the original order, is greatest possible.
2. Find all integers $x,y,z$ such that $(x-y)^{3}+3(y-z)^{2}+5|z-x|=35.$
3. For what values of $a$ is the numbers $a+\sqrt{15}$ and $\dfrac{1}{a}-\sqrt{15}$ are both intergers?.
4. Solve the equation $\sqrt{6-x}+\sqrt{2x+6}+\sqrt{6x-5}=x^{2}-2x-5.$
5. Let $(O)$ be a circle of diameter $AB$. Point $I$ is outside the circle, $IH$ is the perpendicular line to $AB$ through $I$ ($H$ lies between $O$ and $A$). $IA$, $IB$ meet $(O)$ at points $E$ and $F$ respectively; $EF$ meets $AB$ at $P$; $EH$ meets $(O)$ at the second point $M$; $PM$ intersects $(O)$ at the second point $N$. Let $K$ denote the midpoint of $EF$, $O'$ is the circumcenter of triangle $HMN$. Prove that $O'H||OK$.
6. Prove that in any triangle $ABC$, the following inequality holds $\left(\frac{h_{a}}{l_{a}}-\sin\frac{A}{2}\right)\left(\frac{h_{b}}{l_{b}}-\sin\frac{B}{2}\right)\left(\frac{h_{c}}{l_{c}}-\sin\frac{C}{2}\right)\leq\frac{r}{4R},$ where $h_{a},h_{b},h_{c}$ and $l_{a},l_{b},l_{c}$ are the length of the altitudes and the internal angle bisector from $A,B$ and $C$ respectively; any $R,r$ are its circumradius and inradius.
7. Let $S.ABC$ be a triangular pyramid where $\widehat{BAC}=90^{0}$, $BC=2a$, $\widehat{ACB}=\alpha$. The plane $(SAB)$ is perpendicular to $(ABC)$. Find the volume of this pyramid, given that $SAB$ is an isosceles triangle at vertex $S$ and $SBC$ is a right triangle.
8. Solve the system of equations $\begin{cases}x^{8}y^{8}+y^{4} & =2x\\ 1+x & =x(1+y)\sqrt{xy} \end{cases}.$
9. Let $A=2013^{30n^{2}+4n+2013}$ where $n\in\mathbb{N}$. Let $X$ be the set of remainders when dividing $A$ by $21$ for some natural number $n$. Determine the set $X$.
10. Let $(x_{n})$ be a sequence of natural numbers with $x_{1}=2,\quad x_{n+1}=\left[\frac{3}{2}x_{n}\right],\forall n=1,2,3,\ldots,$ where $[x]$ denotes the greates integer not exceeding $x$. Prove that in the sequence $(x_{n})$, there are infinitely many odd numbers and infinitely many even numbers.
11. Given a real number $x\geq1$. Find the limit $\lim_{n\to\infty}(2\sqrt[n]{x}-1)^{n}.$
12. Let $S$ denote the area of a given triangle, and let $P$ be an arbitrary point. $A',B',C'$ are the midpoints of $BC$, $CA$ and $AB$ respectively; $h_{a},h_{b},h_{c}$ denote the corresponding altitudes. Prove that $PA^{2}+PB^{2}+PC^{2}\geq\frac{4}{\sqrt{3}}S\max\left\{ \frac{PA+PA'}{h_{a}},\frac{PB+PB'}{h_{b}},\frac{PC+PC'}{h_{c}}\right\}.$

### Issue 437

1. Let $A$ be the sum (where $n!$ denote the product $1.2.3.\ldots n$) $A=\frac{1}{1.1!}+\frac{1}{2.2!}+\ldots+\frac{1}{n.n!}+\ldots+\frac{1}{2013.2013!}.$ Prove that $A<\dfrac{3}{2}$.
2. In a right triangle $ABC$ with right angle at $A$, let $D$ be the midpoint of $AC$, $E$ be the point on side $BC$ such that $BE=2CE$. Prove that $BD=3ED$.
3. Determine all possible triples of integers $x,y,z$ satisfying the equation $6(y^{2}-1)+3(x^{2}+y^{2}z^{2})+2(z^{2}-9x)=0.$
4. Let $a,b,c$ be three real numbers satisfying the conditions $a\ne0$ and $2a+3b+6c=0$. Find the smallest possible distance between the two roots of equation $ax^{2}+bc+c=0$.
5. Given that in a triangle $ABC$, $AB=2$, $AC=3$, $BC=4$. Prove that $\widehat{BAC}=\widehat{ABC}+2\widehat{ACB}.$
6. Let $a,b,c$ be positive real numbers satisfuing $a+b+c=\frac{1}{a}+\frac{1}{b}+\frac{1}{c}.$ Prove the inequality $3(a+b+c)\geq\sqrt{8a^{2}+1}+\sqrt{8b^{2}+1}+\sqrt{8c^{2}+1}.$
7. A circle centered at $O$ and radius $R$ circumscribes about a triangle $ABC$ whose sides are $AB=c$, $BC=a$, $CA=b$. Let $E$ be the center of the Euler circle of triangle $ABC$. Prove that if $a^{2}+b^{2}+c^{2}=5R^{2}$ then $E$ lies on the circle $(O;R)$.
8. Solve the equation $3^{x}-x-1=\log_{3}\frac{(2x+1)\log_{3}(2x+1)}{x}.$
9. For which positive integers $n$ will the following equation has positive integer solutions $\frac{1}{x_{1}^{2}}+\frac{1}{x_{n}^{2}}+\ldots+\frac{1}{x_{n}^{2}}=4.$
10. Prove that in any triangle $ABC$, the following inequality holds $\frac{\cos\left(\frac{B}{2}-\frac{C}{2}\right)}{\sin\frac{A}{2}}+\frac{\cos\left(\frac{C}{2}-\frac{A}{2}\right)}{\sin\frac{B}{2}}+\frac{\cos\left(\frac{A}{2}-\frac{B}{2}\right)}{\sin\frac{C}{2}} \leq 2\left(\frac{\tan\frac{A}{2}}{\tan\frac{B}{2}}+\frac{\tan\frac{B}{2}}{\tan\frac{C}{2}}+\frac{\tan\frac{C}{2}}{\tan\frac{A}{2}}\right).$
11. Let $(a_{n})$ be the sequence of real numbers such that $a_{1}=34,\quad a_{n+1}=4a_{n}^{3}-104a_{n}^{2}-107a_{n}$ for all positive integers $n$. Find all prime numbers $p$ satisfying the following two conditions $p\equiv3(\text{mod }4)$ and $a_{2013}+1$ is divisible by $p$.
12. Given five points $A,B,C,D,E$ on the same circle. Let $M$ be the midpoint of $DE$. The Euler circles of triangles $ADE$ and $BDE$ meet at $C'$ (different from $M$); the Euler circles of triangles $BDE$ and $CDE$ meet at $A'$ (different from $M$); the Euler circles of triangles $CDE$ and $ADE$ meet at $B'$ (different from $M$). Prove that the lines $AA'$, $BB'$ and $CC'$ are concurrent.

### Issue 438

1. Determine all triples of (not necessarily distinct) prome numbers $a,b,c$ such that $a(a+1)+b(b+1)=c(c+1).$
2. In an isosceles right triangle $ABC$, right angle at vertex $A$, let $E$ be the midpoint of side $AC$. The line perpendicular to $BE$ through $A$ meets $BC$ at $D$. Prove that $AD=2ED$.
3. Find all positive integers $x,y$ and $t$ such that $t\leq6$ and the following equation is satisfied $x^{2}+y^{2}-4x-2y-7t-2=0.$
4. Find the greatest integer not exceeding $A$, where $A=\frac{1}{\sqrt{1}}+\frac{1}{\sqrt{2}}+\ldots+\frac{1}{\sqrt{2013}}.$
5. Let $ABC$ be an acute triangle with altitudes $BE$ and $CF$. The line segments $FI$ and $EJ$ are perpendicular to $BC$ ($I,J$ belong to $BC$). Point $K,L$ on $AB$, $AC$ respectively such that $KI||AC$, $LJ||AB$. Prove that the lines $EI$, $FJ$ and $KL$ are concurrent.
6. Solve for $x$ $\sqrt{13}\sqrt{2x^{2}-x^{4}}+9\sqrt{2x^{2}+x^{4}}=32.$
7. Given that $x,y$ satisfy the conditions $x-y\geq0,\,x+y\geq0,\,\sqrt{\left(\frac{x+y}{2}\right)^{3}}+\sqrt{\left(\frac{x-y}{2}\right)^{3}}=27.$ Find the smallest possible value of $x$.
8. In a triangle $ABC$, let $m_{a},l_{b},l_{c}$ and $p$ denote the median length from $A$, the lengths of the internal angle bisectors from $B,C$ and its semiperimeter. Prove that $m_{a}+l_{b}+l_{c}\leq p\sqrt{3}.$
9. a) Let $a_{1},a_{2},\ldots,a_{n}$ be $n$ rational numbers. Prove that if $a_{1}^{m}+a_{2}^{m}+\ldots+a_{n}^{m}$ is an integer for all positive integers $m$, then $a_{1},a_{2},\ldots a_{n}$ are integer numbers.
b) Does the conclusion above remain valid if one only assumes that $a_{1},a_{2},\ldots a_{n}$ are real numbers?.
10. Find all continuous funtions $f:\mathbb{R}\to\mathbb{R}$ such that the following identity is satisfied for all $x,y$ $f(x)+f(y)+2=2f\left(\frac{x+y}{2}\right)+2f\left(\frac{x-y}{2}\right).$
11. Let $ABC$ be a triangle with side lengths $BC=a$, $AC=b$, $AB=c$ and median lengths $m_{a},m_{b},m_{c}$ drawn from vertex $A,B$ and $C$ respectively. Prove that for any point $M$ \begin{align*}\left(\frac{m_{b}}{b}+\frac{m_{c}}{c}\right)\frac{MA}{a}+\left(\frac{m_{c}}{c}+\frac{m_{a}}{a}\right)\frac{MB}{a}+\left(\frac{m_{a}}{a}+\frac{m_{b}}{b}\right)\frac{MC}{c} & \geq3,\\ \left(\frac{b}{m_{b}}+\frac{c}{m_{c}}\right)\frac{MA}{a}+\left(\frac{c}{m_{c}}+\frac{a}{m_{a}}\right)\frac{MB}{a}+\left(\frac{a}{m_{a}}+\frac{b}{m_{b}}\right)\frac{MC}{c} & \geq4.\end{align*}
12. Let $ABC$ be an isosceles triangle at vertex $A$. A circle $\omega$ which touches the sides $AB,AC$ meets $BC$ at $K,L$. $AK$ meets $\omega$ at $M$. Let $P,Q$ be the reflection points of $K$ through points $B,C$ respectively. Let $O$ be the circumcenter of triangle $MPQ$. Prove that $M$, $O$ and the center of circle $\omega$ are collinear.

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