Mathematics and Youth Magazine Problems 2012

Issue 415

1. Let $$A=\dfrac{2011^{2011}}{2012^{2012}},\quad B=\dfrac{2011^{2011}+2011}{2012^{2012}+2012}.$$ Which number is greater, $A$ or $B$?.
2. Given \[A=\sqrt{6+\sqrt{6+\ldots+\sqrt{6}}},\:B=\sqrt[3]{6+\sqrt[3]{6+\ldots+\sqrt[3]{6}}},\] where there are exactly $n$ square roots in $A$ and $n$ cube roots in $B$. Write $[x]$ for the greatest integer not exceeding $x$. Determine the value of $\left[\dfrac{A-B}{A+B}\right]$.
3. Find all pairs of natural numbers $x,y$ such that \[x^{2}-5x+7=3^{y}.\]
4. Prove the inequality \[\left(1+\frac{1}{2}\right)\left(1+\frac{1}{2^{2}}\right)\ldots\left(1+\frac{1}{2^{n}}\right)<3.\]
5. Let $ABCD$ be aparallelogram. Points $H$ and $K$ are chosen on lines $AB$ and $BC$ such that triangles $KAB$ and $HCB$ are isosceles ($KA=AB$, $HC=CB$). Prove that
   a) Triangle $KDH$ is also isosceles.
   b) Triangle $KAB$, $BCH$ and $KDH$ are similar.
6. In a triangle $ABC$ with $a=BC$, $b=CA$, $c=AB$, $A_{1}$ is the midpoint of $BC$; $O$ and $I$ are its circumcenter and incenter respectively. Prove that if $AA_{1}$ isperpendicular to $OI$ then \[\min\{b,c\}\leq a\leq\max\{b,c\}.\]
7. The real numbers $x,y$ and $z$ are such that \[\begin{cases}\sqrt{x}\sin\alpha+\sqrt{y}\cos\alpha-\sqrt{z} & =-\sqrt{2(x+y+z)}\\ 2x+2y-13\sqrt{z} & =7 \end{cases},\quad\pi\leq\alpha\leq\frac{3\pi}{2}.\]Determine the value of $(x+y)z$.
8. Solve the following system of equations in two variables \[\begin{cases}\log_{2}x & =2^{y+2}\\ 2\sqrt{1+x}+xy\sqrt{4+y^{2}} & =0 \end{cases}.\]
9. A collection of prime numbers (each prime can be repeated) is said to be beautiful if their product is exactly ten times their sum. Find all beautiful collections. 
10. Points $A,B,C,D,E$ in clockwise order, lie on the same circle. $M,N,P,Q$ are the feet of perpendicular lines from $E$ onto $AB$, $BC$, $CD$, $DA$. Prove that $MN$, $NP$, $PQ$, $QM$ are tangent lines to a certain parabole whose focus point if $E$. 
11. The sequence $(a_{n})$ is defined recursively by the following rules \[a_{1}=1,\quad a_{n+1}=\frac{1}{a_{1}+\ldots+a_{n}}-\sqrt{2},\:n=1,2,\ldots.\] Find the limit of the sequence $(b_{n})$ where \[b_{n}=a_{1}+\ldots+a_{n}.\]
12. Let $\alpha$ and $\beta$ be two real roots of the equation \[4x^{2}-4tx-1=0\] where $t$ is a parameter. Let $f(x)=\dfrac{2x-t}{x^{2}+1}$ be a funtion defined on the interval $[\alpha;\beta]$, and let \[g(t)=\max_{x\in[\alpha;\beta]}f(x)-\min_{x\in[\alpha;\beta]}f(x).\] Prove that if a triple $a,b,c\in\left(0;\frac{\pi}{2}\right)$ are such that $\sin a+\sin b+\sin c=1$, then \[\frac{1}{g(\tan a)}+\frac{1}{g(\tan b)}+\frac{1}{g(\tan c)}<\frac{3\sqrt{6}}{4}.\]

Issue 416

1. Find all natural numbers $x,y,z$ such that \[2010^{x}+2011^{y}=2012^{z}.\]
2. The natural numbers $a_{1},a_{2},\ldots,a_{100}$ satisfy the equation \[\frac{1}{a_{1}}+\frac{1}{a_{2}}+\ldots+\frac{1}{a_{100}}=\frac{101}{2}.\]Prove that there are at least two equal numbers.
3. Let $a,b,c$ be positive real numbers. Prove the inequality \[\frac{(a+b)^{2}}{ab}+\frac{(b+c)^{2}}{bc}+\frac{(c+a)^{2}}{ca}\geq9+2\left(\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}\right).\]
4. Solve the equation \[4x^{2}+14x+11=4\sqrt{6x+10}.\]
5. In a triange $ABC$, te incircle $(I)$ meets $BC$, $CA$ at $D$, $E$ respectively. Let $K$ be the point of reglection of $D$ through the midpoint of $BC$, the line through $K$ and perpendicular to $BC$ meets $DE$ at $L$, $N$ is the midpoint of $KL$. Prove that $BN$ and $AK$ are orthogonal.
6. Determine the maximum value of the expression \[A=\frac{mn}{(m+1)(n+1)(m+n+1)}\] where $m,n$ are natural numbers.
7. Triangle $ABC$ ($AB>AC$) is inscribed in circle $(O)$. The exterior angle bisector of $BAC$ meets $(O)$ at another point $E$; $M,N$ are the midpoints of $BC$, $CA$ respectively; $F$ os the perpendicular foot of $E$ on $AB$, $K$ is the intersection of $MN$ and $AE$. Prove that $KF$ and $BC$ are parallel.
8. Solve the equation \[\sin^{2n+1}x+\sin^{n}2x+(\sin^{n}x-\cos^{n}x)^{2}-2=0\] where $n$ is a given positive integer.
9. Find all polynomials $P(x)$ such that \[P(2)=12,\quad P(x^{2})=x^{2}(x^{2}+1)P(x),\:\forall x\in\mathbb{R}.\]
10. Let $r_{1},r_{2},\ldots,r_{n}$ be $n$ rational numbers such that $0<r_{i}\leq\dfrac{1}{2}$, ${\displaystyle \sum_{i=1}^{n}r_{i}=1}$ ($n>1$), and let $f(x)=[x]+\left[x+\dfrac{1}{2}\right]$. Find the greatest value of the expression ${\displaystyle P(k)=2k-\sum_{i=1}^{n}f(kr_{i})}$ where $k$ runs over the integers $\mathbb{Z}$ (the notation $[x]$ means the greatest integer not exceeding $x$).
11. Suppose that $f:\mathbb{R}\to\mathbb{R}$ is a continuous funtion such that $f(x)+f(x+1006)$ is a rational number if and only if $x\in\mathbb{R}$, \[f(x+20)+f(x+12)+f(x+2012)\] is itrational. Prove that $f(x)=f(x+2012)$ for all $x\in\mathbb{R}$.
12. Prove the following inequality \[\frac{m_{a}}{h_{a}}+\frac{m_{b}}{h_{b}}=\frac{m_{c}}{h_{c}}\leq1+\frac{R}{r},\] where $m_{a},b_{b},m_{c}$ are medians; $h_{a},h_{b},h_{c}$ are the altitudes from $A$, $B$, $C$ and $R$, $r$ are the circumradius and inradius, respectively.

Issue 417

1. Which number is bigger, $2^{3100}$ or $3^{2100}$?.
2. Let $ABC$ be an isosceles triangle with $AB=AC$. $BM$ is the median from $B$. $N$ is a point on $BC$ such that $\widehat{CAN}=\widehat{ABM}$. Prove that $CM\geq CN$.
3. Let $a,b,c$ be positive numbers such that \[|a+b+c|\leq1,\,|a-b+c|\leq1,\,|4a+2b+c|\leq8,\,|4a-2b+c|\leq8.\] Prove the inequality \[|a|+3|b|+|c|\leq7.\]
4. Solve the equation \[(x-2)(x^{2}+6x-11)^{2}=(5x^{2}-10x+1)^{2}.\]
5. Let $ABC$ be a right triangle, with right angle at$A$, $AH$ is the altitude from $A$ and $I,J$ ae the incenters of triangles $HAB$ and $HAC$, respectively. $IJ$ cuts $AB$ at $M$ and meets $AC$ at $N$. Let $X$ and $Y$ be the intersections of $HI$ with $AB$ and $HJ$ with $AC$; $BY$, $CX$ cuts $MN$ at $P$ and $Q$ respectively. Prove that \[\frac{AI}{AJ}=\frac{HP}{HQ}.\]
6. Let $x,y,z$ be real numbers such that $x^{2}+y^{2}+z^{2}=3$. Find the minimum and maximum value of the expression \[P=(x+2)(y+2)(z+2).\]
7. In a triangle $ABC$, let $m_{a},m_{b},m_{c}$ be its median lengths, and $l_{a},l_{b},l_{c}$ be the lengths of its inner bisectors, $p$ is half of its perimeter. Prove the inequality \[m_{a}+m_{b}+m_{c}+l_{a}+l_{b}+l_{c}\leq2\sqrt{3}p.\]
8. Let $S.ABC$ be a pyramid where surface $SAB$ is a isosceles triangle at $S$ and $\widehat{BSA}=120^{0}$, the plane $(SAB)$ is perpendicular to $(ABC)$. Prove that $\dfrac{S_{ABC}}{S_{SAC}}\leq\sqrt{3}$, when does the equality occur?. (Denote by $S_{DEF}$ the area of triangle $DEF$)
9. A natural number $n$ is a good number if it is possible to partition any square into $n$ smaller squares such that at least two of them are not equal.
   a) Prove that if $n$ is a good number, then $n\geq4$.
   b) Prove that both $4$ and $5$ are not good.
   c) Find all good numbers.
10. A sequence $a_{0},a_{1},\ldots,a_{n}$ ($n\geq2$) is defined by \[a_{0}=0,\quad a_{k}=\frac{1}{n+1}+\frac{1}{n+2}+\ldots+\frac{1}{n+k},\,k=1,2,\ldots,n.\] Prove the inequality \[\sum_{k=0}^{n-1}\frac{e^{a_{k}}}{n+k+1}+(\ln2-a_{n})e^{a_{n}}<1\] where ${\displaystyle e=\lim_{n\to\infty}\left(1+\frac{1}{n}\right)^{n}}$.
11. Find all functions $f:\mathbb{R}^{+}\to\mathbb{R}^{+}$ satisfying \[f(x)f(yf(x))=f(y+f(x)),\quad x,y\in\mathbb{R}^{+}.\]
12. Given a triangle $ABC$ inscribed in a circle $(O,R)$, with center $G$ and area $S$. Prove that \[a^{2}+b^{2}+c^{2}\geq\left(4\sqrt{3}+\frac{OG^{2}}{R^{2}}\right)S+(a-b)^{2}+(b-c)^{2}+(c-a)^{2}.\]

Issue 418

1. Given \[A=1^{5}+2^{5}+3^{5}+\ldots+2011^{5}.\] Find the last digit of $A$.
2. Let $ABC$ be an isosceles right triangle with right angle at $A$. On the half-plane defined by $AB$ containing $C$ draw an isosceles right triangle $ABD$ with right angle at $B$. Let $E$ be the midpoint of segment $BD$. Draw $CM$ perpendicular to $AE$ at $M$. Let $N$ be the midpoint of segment $CM$, $K$ is the intersection of $BM$ and $DN$. Find the measure of the angle $BKD$.
3. Find all positive integer solutions of the equation \[3^{x}-32=y^{2}.\]
4. Find all minimal value of the expression \[A=\frac{1}{x^{3}+xy+y^{3}}+\frac{4x^{2}y^{2}+2}{xy}\] where $x$ and $y$ are positive real numbers satisfying $x+y=1$.
5. Let $ABC$ be an acute triangle with orthocenter $H$. Prove that $ABC$ is an equilateral triangle if and only if \[\frac{AH}{BC}=\frac{BH}{CA}=\frac{CH}{AB}.\]
6. Let $ABC$ be a triangle with circumcenter $O$, and incenter $I$. $BC$ touches the circle $(I)$ at $D$. The circle whose diameter is $AI$ meets $(O)$ at $M$ ($M\ne A$) and cuts the line passing through $A$ parallel to $BC$ at $N$. Prove that $MO$ passes through the midpoint of $DN$.
7. Solve the system of equations \[\begin{cases}\sqrt{xy+(x-y)(\sqrt{xy}+2)}+\sqrt{x} & =y+\sqrt{y}\\(x+1)(y+\sqrt{xy}+x(1-x)) & =4\end{cases}.\]
8. Let $ABC$ be an acute triangle. Prove the inequaltiy \[\cos^{3}A+\cos^{3}B+\cos^{3}C+\cos A.\cos B.\cos C\geq\frac{1}{2}.\]
9. For each natural number $n$, let $(S_{n})$ be the sum of all digits of $n$ (in the decimal system). Put $S_{k}(n)=S(S(\ldots(S(n))\ldots))$ ($k$ times). Find all natural numbers $n$ such that \[S_{1}(n)+S_{2}(n)+\ldots S_{k}(n)+\ldots+S_{223}(n)=n.\]
10. Does there exist a set $X$ satisfying the following two conditions
    • $X$ contains $2012$ natural numbers.
    • The sum of any arbitrary elements in $X$ is the $k$-th power of a positive integer ($k\geq2$).
11. Find all functions $f:\mathbb{R}\to\mathbb{R}$ satisfing \[f\left(\frac{xf(y)}{2}\right)+f\left(\frac{yf(x)}{2}\right)=4xy,\:\forall x,y\in\mathbb{R}.\]
12. Fix two circles $(K)$ and $(O)$, where $(K)$ is inside $(O)$. Two circles $(O_{1})$, $(O_{2})$ are moving so that they always externally touch each other at $M$. Both also internally touch $(O)$, and externally touch $(K)$. Prove that $M$ belongs to a fixed circle.

Issue 419

1. Let $$A=\frac{1}{1^{2}}+\frac{1}{2^{2}}+\frac{1}{3^{2}}+\ldots+\frac{1}{50^{2}}$$ and $B=\dfrac{165}{101}$. Compare $A$ and $B$.
2. Let $A B C$ be a right isosceles triangle with right angle at $A .$ If there exists a point $M$ inside the triangle with $\widehat{M B A}=$ $\widehat{M A C}=\widehat{M C B}$. Find the ratio $M A: M B: M C$.
3. Find the minimum values of the natural numbers $a, b, c$ satisfying $$\begin{align} & a+(a+1)+(a+2)+\ldots+(a+6) \\ =& b+(b+1)+(b+2)+\ldots+(b+8) \\ = &c+(c+1)+(c+2)+\ldots+(c+10).\end{align}$$
4. Solve the following equation $$6(x-1) \sqrt{x+1}+\left(x^{2}+2\right)(\sqrt{x-1}-3)=x\left(x^{2}+2\right).$$
5. Let $M$ be the midpoint of the arc $A B$ of a semicircle with center $O$ and diameter $A B$. $A C$ meets $M O$ at $D$. Prove that the circumcenter of triangle $M D C$ always lies on a fixed line when $C$ moves on the semicircle.
6. Let $a, b, c$ be positive real numbers. Prove that $$6\left(a^{3}+b^{3}+c^{3}\right) \geq 18 a b c+\left(\sqrt[3]{a(b-c)^{2}}+\sqrt[3]{b(c-a)^{2}}+\sqrt[3]{c(a-b)^{2}}\right)^{3}.$$
7. Let $A B C$ be an acute triangle which is not isosceles; and $H$, $O$ be its orthocenter and circumcenter respectively; let $D$, $E$ be respectively the foot of the altitude from $A$, $B$. The lines $O D$ and $B E$ intersect at $K$, $O E$ and $A D$ intersect at $L$. Let $M$ be the midpoint of edge $A B$. Prove that $K$, $L$, $M$ are collinear if and only if $C$, $D$, $O$, $H$ lies on the same circle.
8. Find all pairs of positive integers $(n, k)$ satisfying $C_{3 n}^{n}=3^{n} n^{k},$ where $$C_{p}^{m}=\frac{p !}{m !(p-m) !} ; 0 \leq m \leq p, p \neq 0, m, p \in \mathbb{N}.$$
9. Let $a, b, c$ be three positive real numbers satisfying $$15\left(\frac{1}{a^{2}}+\frac{1}{b^{2}}+\frac{1}{c^{2}}\right)=10\left(\frac{1}{a b}+\frac{1}{b c}+\frac{1}{c a}\right)+2012.$$ Find the largest possible value of the expression $$P=\frac{1}{\sqrt{5 a^{2}+2 a b+2 b^{2}}}+\frac{1}{\sqrt{5 b^{2}+2 b c+2 c^{2}}}+\frac{1}{\sqrt{5 c^{2}+2 c a+2 a^{2}}}.$$
10. Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that for all $x, y \in \mathbb{R}$ we have $$f(x f(y))+f(f(x)+f(y))=y f(x)+f(x+f(y)).$$
11. On the interval $[a ; b],$ pick $k$ distinct points $x_{1}, x_{2}, \ldots, x_{k}$. Let $d_{n}$ be the product of the distances from $x_{n}$ to the $k-1$ remaining points; $n=1,2,3 \ldots, k .$ Find the smallest value of $\displaystyle \sum_{n=1}^{k} \frac{1}{d_{n}}$.
12. Given a triangle $A B C$ and an arbitrary point $M$. Prove that $$\frac{M A}{B C}+\frac{M B}{C A}+\frac{M C}{A B} \geq \frac{B C+C A+A B}{M A+M B+M C}.$$

Issue 420

1. Find the integer value of the expression $f(x ; y)=\dfrac{x^{2}+x+2}{x y-1}$ where $x, y$ are positive integers.
2. Let $A B C$ be an acute triangle which is not isosceles at $A$. The perpendicular bisectors of $A B$, $A C$ cut the median $A M$ at $E$, $F$ respectively. $B E$ and $C F$ meet at $K .$ Prove that $\widehat{A K B}=\widehat{A K C}$ and $\widehat{M A B}=\widehat{K A C}$.
3. Find all triples of integers $(x ; y ; z)$ such that $$2 x y+6 y z+3 z x-|x-2 y-z|=x^{2}+4 y^{2}+9 z^{2}-1.$$
4. For each positive integer $n(n=1,2, \ldots),$ put $a_{n}=\dfrac{4 n}{n^{4}+4} .$ Prove that $$a_{1}+a_{2}+\ldots+a_{n}<\frac{3}{2}$$
5. Let $A B C$ be an acute triangle. The internal angle-bisector of angle $B A C$ cuts $B C$ at $D$. $E$, $F$ are the orthogonal projections of point $D$ on $A B$ and $A C$ respectively, $K$ is the intersection of $C E$ and $B F, H$ is the intersection of $B F$ with the circumcircle of triangle $A E K$. Prove that $D H$ is perpendicular to $B F$
6. Solve the system of equations $$\begin{cases} x+6 \sqrt{x y}-y &=6 \\ x+\dfrac{6\left(x^{3}+y^{3}\right)}{x^{2}+x y+y^{2}}-\sqrt{2\left(x^{2}+y^{2}\right)} &=3 \end{cases}.$$
7. Let $a, b, c$ be non-negative real numbers whose sum equals $1$. Prove that $$\left(1+a^{2}\right)\left(1+b^{2}\right)\left(1+c^{2}\right) \geq\left(\frac{10}{9}\right)^{3}$$
8. Point $M$ inside the triangle $A B C$ with area $S$. Let $x, y, z$ be distances of $M$ to $A$, $B$, $C$ respectively. Prove that $$(x+y+z)^{2} \geq 4 \sqrt{3} S.$$ When does the equality hold?
9. A nonempty set $S \subseteq \mathbb{Z}$ posesses the following properties
   
   • There exist $a, b \in S$ such that $(a, b)=(a-2 b-2)=1$,
   • If $x, y \in S$ then $x^{2}-y \in S$ ($x$, $y$ may be identical).
   Prove that $S=\mathbb{Z}$. ($(a, b)$ is the greatest common divisor of two integers $a$ and $b$.)
10. Find the greatest number $k$ such that the inequality $$\sqrt{a+2 b+3 c}+\sqrt{b+2 c+3 a}+\sqrt{c+2 a+3 b} \geq k(\sqrt{a}+\sqrt{b}+\sqrt{c})$$ holds for all positive numbers $a, b, c$
11. Let $\left(x_{n}\right)$ be a sequence defined by $$x_{1}=\frac{1001}{1003} ,\quad x_{n+1}=x_{n}-x_{n}^{2}+x_{n}^{3}-x_{n}^{4}+\ldots+x_{n}^{2011}-x_{n}^{2012},\, \forall n \in \mathbb{N}.$$ Find $\displaystyle \lim _{n \rightarrow+\infty}\left(n x_{n}\right)$.
12. Given four distinct points $A$, $B$, $C$, $D$ lying on a circle with center $O$. Let $I$, $J$ be the feet of the perpendicular to $A B$ and $A D$ through $C$; $K$, $L$ are the feet of the perpendicular to $B C$ and $B A$ through $D$; $N$ is the midpoint of $C D$; $M$ is the intersection of $I J$ and $K L$. $I J$ meets $O D$ at $E$ and $K L$ meets $O C$ at $F$. Prove that the five points $M$, $N$, $O$, $E$ and $F$ lie on the same circle.

Issue 421

1. Given the sum of $2012$ terms $$S=\frac{1}{5}+\frac{2}{5^{2}}+\frac{3}{5^{3}}+\frac{4}{5^{4}}+\ldots+\frac{2012}{5^{2012}}$$ Compare $S$ with $\dfrac{1}{3}$.
2. Let $A B C$ be a triangle with $\widehat{A B C}=40^{\circ}, \widehat{A C B}=30^{\circ} .$ Outside this triangle, construct triangle $A D C$ with $\widehat{A C D}=\widehat{C A D}=50^{\circ} .$ Prove that the triangle $B A D$ is isosceles.
3. Find all natural numbers $a, b, c$ such that $c < 20$ and $a^{2}+a b+b^{2}=70 c$.
4. Find the largest possible value of the expression $$P=\sqrt{1-\frac{x}{y+z}}+\sqrt{1-\frac{y}{z+x}}+\sqrt{1-\frac{z}{x+y}}$$ where $x, y, z$ are side lengths of a triangle.
5. Given a circle $(O),$ with a fixed chord $B C$. $A$ is a point moving on the line $B C$, outside the circle $(O)$. $AM$ and $AN$ are the tangent lines to circle $(O)$ $(M, N \in (O))$. The line through $B$ and parallel to $A M$ meets $M N$ at $E .$ Prove that the circumcircle of triangle $B E N$ always passes through two fixed points when point $A$ moves on the line $B C$.
6. Given that $\dfrac{1}{3} < x \leq \dfrac{1}{2}$ and $y \geq 1$. Find the minimum value of $$P=x^{2}+y^{2}+\frac{x^{2} y^{2}}{((4 x-1) y-x)^{2}}.$$
7. Let $\left(a_{n}\right)$ be a sequence of positive real numbers, given by
   
   • $a_{0}=1$,
   • $a_{m}<a_{n}$, for all $m, n \in \mathbb{N}$, $m<n$.
   • $a_{n}=\sqrt{a_{n+1} \cdot a_{n-1}}+1$ and $4 \sqrt{a_{n}}=a_{n+1}-a_{n-1}$ for all $n \in \mathbb{N}^{*}$.
   Determine the sum $T=a_{0}+a_{1}+a_{2}+\ldots+a_{2012}$.
8. The base of a triangular prism $A B C \cdot A^{\prime} B^{\prime} C^{\prime}$ is an equilateral triangle with side lengths $a$ and the lengths of its adjacent sides also equal $a$. Let $I$ be the midpoint of $A B$ and $B^{\prime} I \perp(A B C)$. Find the distance from $B^{\prime}$ to the plane $\left(A C C^{\prime} A^{\prime}\right)$ in term of $a$.
9. Find all polynomials $P(x)$ with real coefficients satisfying $$P^{2}(x)-1=4 P\left(x^{2}-4 x+1\right)$$
10. Find $\alpha, \beta$ so that the largest value of $$y=|\cos x+\alpha \cos 2 x+\beta \cos 3 x|$$ is smallest possible.
11. Let $A B C$ be a triangle with side lengths $a$, $b$ and $c$. Let $S$ and $p$ be respectively the area and the semiperimeter of this triangle. Prove the inequality $$\frac{1}{a^{2}(p-a)^{2}}+\frac{1}{b^{2}(p-b)^{2}}+\frac{1}{c^{2}(p-c)^{2}} \geq \frac{9}{4 S^{2}}$$
12. Given an acute triangle $A B C$ inscribed the circle $(O)$ with $B C>C A>A B$. On the circle $(O)$, select six distinct points $M$, $N$, $P$, $Q$, $R$ and $S$ (which are also distinct from the vertices of triangle $A B C$) so that $Q B=B C=C R$, $S C=C A=A M$ and $N A=A B=B P$. Let $I_{A}$, $I_{B}$ and $I_{C}$ be the incenters of triangles $A P S$, $B N R$ and $C M Q$ respectively. Prove that $\Delta I_{A} I_{B} I_{C} \sim \Delta A B C$.

Issue 422

1. Let $$A=1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\ldots+\frac{1}{2011}-\frac{1}{2012},\quad B=\frac{1}{1007}+\frac{1}{1008}+\ldots+\frac{1}{2012}.$$ Compute the value of $\left(\dfrac{A}{B}\right)^{2012}$.
2. Let $f(x)$ be a polynomial with integer coefficients such that $f(3) \cdot f(4)=5 .$ Prove that $f(x)-6$ does not have any integer solution.
3. Find all triple of integers $a, b, c$ such that $$2^{a}+8 b^{2}-3^{c}=283.$$
4. Given a triangle $A B C$, $B C=a$, $C A=b$, $A B=c$, $\widehat{A B C}=45^{\circ}$ and $\widehat{A C B}=120^{\circ}$. Point $I$ is taken on the opposite ray of $C B$ such that $\widehat{A I B}=75^{\circ} .$ Find the length of $A I$ in term of $a$, $b$ and $c$
5. Point $K$ lies on side $B C$ of a triangle $A B C$. Prove that $$A K^{2}=A B \cdot A C - K B \cdot K C$$ if and only if $A B=A C$ or $\widehat{B A K}=\widehat{C A K}$.
6. A non-isosceles triangle $A B C$ has $B C=a$, $C A=b$, $A B=c$. Let $\left(A A_{1}, A A_{2}\right)$, $\left(B B_{1}, B B_{2}\right)$, $\left(C C_{1}, C C_{2}\right)$ be the median and the altitude from vertices $A$, $B$ and respectively. Prove that $$\frac{a^{2}}{b^{2}-c^{2}} \overline{A_{1} A_{2}}+\frac{b^{2}}{c^{2}-a^{2}} \overrightarrow{B_{1} B_{2}}+\frac{c^{2}}{a^{2}-b^{2}} \overrightarrow{C_{1} C_{2}}=\overrightarrow{0}$$
7. Let $a, b, c \in(0 ; 1)$ and $$a b+b c+c a+a+b+c=1+a b c.$$ Prove that $$\frac{1+a}{1+a^{2}}+\frac{1+b}{1+b^{2}}+\frac{1+c}{1+c^{2}} \leq \frac{3}{4}(3+\sqrt{3})$$
8. Let $A B C$ be an acute triangle with all angles greater than $45^{\circ}$. Prove that $$\frac{2}{1+\tan A}+\frac{2}{1+\tan B}+\frac{2}{1+\tan C} \leq 3(\sqrt{3}-1).$$ When does equality occur?
9. Two sequences $\left(a_{n}\right)$ and $\left(b_{n}\right)$ are defined inductively as follows $$a_{0}=3, b_{0}=-3,\quad a_{n}=3 a_{n-1}+2 b_{n-1},\,b_{n}=4_{n-1}+3 b_{n-1},\,\forall n \geq 1.$$ Find all natural numbers $n$ such that $\displaystyle\prod_{k=0}^{n}\left(b_{k}^{2}+9\right)$ is a perfect square.
10. Let $n$ be a positive integer. How many strings of length $n: a_{1} a_{2} \ldots a_{n}$ where $a_{i}$ is chosen from $\{0,1,2, \ldots, 9\}(i=1,2, \ldots, n)$ are there such that the number of occurrences of 0 is even?
11. Let $\left(u_{n}\right)$ be a sequence defined by $u_{0}=a \in[0 ; 2), u_{n}=\dfrac{u_{n-1}^{2}-1}{n}$ for all $n=1,2,$ $3, \ldots$ Find $\displaystyle\lim _{n \rightarrow+\infty}\left(u_{n} \sqrt{n}\right)$.
12. Let $A B C$ be a triangle, inscribed in the circle $(O)$ with altitudes $A D$, $B E$ and $C F$. $A A^{\prime}$ is a diameter of $(O)$. $A^{\prime} B$, $A^{\prime} C$ intersect $A C$, $A B$ at $M$, $N$ respectively. Points $P$, $Q$ are in $E F$, such that $P B$, $Q C$ are perpendicular to $B C$. The line passing through $A$ and orthogonal to $Q N$, $P M$ cuts $(O)$ at $X$, $Y$ respectively. The tangents to circle $(O)$ at $X$ and $Y$ meet at $J$. Prove that $J A^{\prime}$ is perpendicular to $B C$.

Issue 423

1. Find all numbers abcde, where all five digits are distinct and $\overline{a b c d}=(5 e+1)^{2}$
2. Find all positive integers $x, y, z$ such that $x+3=2^{y}$ và $3 x+1=4^{z}$
3. Find the last digit of the sum $$S=1^{2}+2^{2}+3^{3}+\ldots+n^{n}+\ldots+2012^{2012}.$$
4. Given a function $f$ such that $$f\left(1+\frac{\sqrt{2}}{x}\right)=\frac{(1+2011) x^{2}+2 \sqrt{2 x}+2}{x^{2}}$$ for all nonzero $x$. Determine $f(\sqrt{2012-\sqrt{2011}})$
5. Let $A B C$ be a triangle inscribed in the circle $(O)$. The tangents of $(O)$ at $B$ and $C$ meet at $T$. The line passing through $T$ and parallel to $B C$ cuts $A B$ and $A C$ respectively at $B_{1}$ and $C_{1}$ Prove that $\widehat{B_{1} O C_{1}}$ is an acute angle.
6. On the outside of triangle $A B C$, construct equilateral triangles $A B C_{1}$, $B C A_{1}$, $CAB_{1}$ and inside of $A B C$ construct equilateral triangles $A B C_{2}$, $B C A_{2}$, $C A B_{2}$. Let $G_{1}$, $G_{2}$, $G_{3}$ be respectively the centroids of $A B C_{1}$, $B C A_{1}$, $C A B_{1}$ and let $G_{4}$, $G_{5}$, $G_{6}$ be respectively the centroids of triangles $A B C_{2}$, $BCA_{2}$ and $CAB_{2}$. Prove that the centroids of triangle $G_{1} G_{2} G_{3}$ and of triangle $G_{4} G_{5} G_{6}$ coincide.
7. Solve the equation $$3^{3 x}+3^{x}=\log _{3}\left(2^{x}+x\right)+2^{x}+3^{2^{x}+x}.$$
8. Let $A$, $B$, $C$ be the three angles of an acute triangle. Prove the inequality $$\sqrt{\frac{\cos A \cos B}{\cos C}}+\sqrt{\frac{\cos B \cos C}{\cos A}}+\sqrt{\frac{\cos C \cos A}{\cos B}}>2.$$
9. Find the largest positive integer $n$ $(n \geq 3)$ such that there exists a sequence of positive integers $a_{1}, a_{2}, \ldots, a_{n}$ satisfying the condition $$a_{k+1}+1=\frac{a_{k}^{2}+1}{a_{k-1}+1},\, k \in\{2,3, \ldots, n-1\}.$$
10. Let $p$ be an odd prime number, $n$ is a positive integer so that $p-1$, $p$, $n$ and $n+1$ are pairwise coprime. Find all positive integers $x$, $y$ satisfying $$x^{p-1}+x^{p-2}+\ldots+x+2=y^{n+1}.$$
11. Solve the system of equations $$\begin{cases}\sqrt{5 x^{2}+2 x y+2 y^{2}}+\sqrt{2 x^{2}+2 x y+5 y^{2}} &=3(x+y) \\ \sqrt{2 x+y+1}+2 \sqrt[3]{7 x+12 y+8} &=2 x y+y+5\end{cases}.$$
12. Let $A B C$ be a triangle inscribed in the circle $(O)$ and let $I$ be its incenter. $A I$, $B I$, $Cl$ cut the circle $(O)$ at $A^{\prime}$, $B^{\prime}$ and $C^{\prime}$ respectively; $A^{\prime} C^{\prime}$, $A^{\prime} B^{\prime}$ cut $B C$ at $M$, $N$; $B^{\prime} A^{\prime}$; $B^{\prime} C^{\prime}$ cut $C A$ at $P$, $Q$; $C^{\prime} B^{\prime}$, $C^{\prime} A$ cut $A B$ at $R$, $S$. Prove that $$\frac{2}{3} S_{A B C} \leq S_{M N P Q R S} \leq \frac{2}{3} S_{A^{\prime} B^{\prime} C^{\prime}}.$$

Issue 424

1. Find all $2$-digit numbers such that when multiplied by $2,3,4,$ $5,6,7,8,9,$ the sum of the digits of the resulting numbers are equal.
2. Let $$S=\frac{2}{2013+1}+\frac{2^{2}}{2012^{2}+1}+\frac{2^{3}}{2013^{2^{2}}+1}+\ldots+\frac{2^{2014}}{2013^{2^{2013}}+1}.$$ Which number is greater? $S$ or $\dfrac{1}{1006}$?.
3. Find all integer solutions of the equation $$(y-2) x^{2}+\left(y^{2}-6 y+8\right) x=y^{2}-5 y+62$$
4. Let $x$, $y$ be two rational numbers such that $$x^{2}+y^{2}+\left(\frac{x y+1}{x+y}\right)^{2}=2 .$$ Prove that $\sqrt{1+x y}$. is also a rational number.
5. Let $O$ denote the point of intersection of the two diagonals $A C$ and $B D$ of a convex quadrilateral $A B C D$. Let $E$, $F$, $H$ be the feet of the altitudes from $B$, $C$ and $O$ respectively onto $A D$. Prove that $$ A D \cdot B E \cdot C F \leq A C \cdot B D \cdot O H.$$ When does equality holds?
6. $a, b, c$ are positive real numbers satisfying $a b c=1$. Prove that $$\frac{a^{3}+5}{a^{3}(b+c)}+\frac{b^{3}+5}{b^{3}(c+a)}+\frac{c^{3}+5}{c^{3}(a+b)} \geq 9$$
7. Solve the equation $$\left(x^{3}+\frac{1}{x^{3}}+1\right)^{4}=3\left(x^{4}+\frac{1}{x^{4}}+1\right)^{3}$$
8. Let $A B C$ be a triangle with acute angle $A$. Point $P$ inside the triangle $A B C$ such that $\widehat{B A P}=\widehat{A C P}$ and $\widehat{C A P}=\widehat{A B P}$. Let $M$ and $N$ be the incenters of triangles $A B P$ and $A C P$ respectively, $R$ is the circumradius of triangle $A M N$. Prove that $$\frac{1}{R}=\frac{1}{A B}+\frac{1}{A C}+\frac{1}{A P}.$$
9. Solve the equation $$[x]^{3}+2 x^{2}=x^{3}+2[x]^{2}$$ where $[t]$ denotes the largest integer not exceeding $t$.
10. In the interior of a unit square, there are $n\left(n \in \mathbb{N}^{*}\right)$ circles whose sum of areas is greater than $n-1$. Prove that the circles has at least a common point of intersection.
11. Given that the following equation $$a_{0} x^{n}+a_{1} x^{n-1}+\ldots+a_{n-1} x+a_{n}=0 $$ has $n$ distinct roots. Prove that $$\frac{n-1}{n}>\frac{2 a_{0} a_{2}}{a_{1}^{2}}.$$
12. Let $O$, $I$ and $I_{a}$ denote the circumcenter, incenter and excenter in the angle $A$ of a triangle $A B C$. $A I$ meets $B C$ at $D$. BI meets $C A$ at $E$. The line through $I$ and perpendicular to $O I_{a}$ intersects $A C$ at $M$. Prove that $D E$ passes through the midpoint of line segment $I M$.

Issue 425

1. Find all natural numbers $N$ such that $N$ decreases by a factor of $1997$ after truncating the last several digits.
2. Let $A B C$ be a right triangle with right angle at $A$ and $\widehat{A C B}=15^{\circ}$, Point $D$ on edge $A C$ such that the line passing through $D$ and perpendicular to $B D$ cuts $B C$ at $E$ and $D E=2 D A$. Find the measure of angle $A D B$.
3. Find all positive integers $n$ such that $[A]=4951$ where $A$ is the sum of $n$ terms $$A=\left(1+\frac{1}{2}\right)+\left(2+\frac{2}{2^{2}}\right)+\left(3+\frac{3}{2^{3}}\right)+\ldots+\left(n+\frac{n}{2^{n}}\right).$$ Here $[x]$ denotes the largest integer not exceeding $x$
4. Find the minimum value of the expression $$P=\frac{1+\sqrt[3]{x}+\sqrt[3]{y}+\sqrt[3]{z}}{x y+y z+z x},$$ where $x, y, z$ are positive numbers satisfying $x+y+z=3$
5. Solve the equation $$x^{2}-2 x+7+\sqrt{x+3}=2 \sqrt{1+8 x}+\sqrt{1+\sqrt{1+8 x}}.$$
6. Let $A B C$ be a non-isosceles triangle with medians $A A^{\prime}$, $B B^{\prime}$ and $C C^{\prime}$; and altitudes $A H$, $B F$ and CK. Given that $C K=B B^{\prime}$, $B F=A A^{\prime}$. Determine the ratio $\dfrac{C C^{\prime}}{A H}$.
7. $a_{1}, a_{2}, \ldots, a_{n}$ $(n \geq 3)$ are positive numbers that $$\left(a_{1}+a_{2}+\ldots+a_{n}\right)^{2}>\frac{3 n-1}{3}\left(a_{1}^{2}+a_{2}^{2}+\ldots+a_{n}^{2}\right).$$ Prove that for any triple $a_{i}, a_{j}, a_{k}$ are three edge lengths of some triangle, where natural numbers $i, j,$ $k$ satisfying $0<i<j<k \leq n$.
8. The volume of a given parallelogrambased pyramid $S.ABCD$ is $V$. Assume that plane $(P)$ cuts$S A$, $S B$, $S C$, $S D$ at $A^{\prime}$, $B^{\prime}$, $C^{\prime}$, $D^{\prime}$ respectively such that $$\frac{S A}{S A^{\prime}}+\frac{S B}{S B^{\prime}}+\frac{S C}{S C^{\prime}}+\frac{S D}{S D^{\prime}}=8.$$ Denote the volume of the pyramid $S . A^{\prime} B^{\prime} C^{\prime}$ by $V_{1}$ and that of $S . A^{\prime} C^{\prime} D^{\prime}$ by $V_{2}$. Prove the inequality $$\frac{1}{\sqrt[3]{V_{1}}}+\frac{1}{\sqrt[3]{V_{2}}} \leq \frac{4 \sqrt[3]{2}}{\sqrt[3]{V}}.$$
9. Write $2012^{2013}$ as a sum of $2013$ positive interger $a_{1}, a_{2}, a_{3}, \ldots, a_{2013} ;$ and let $$T=a_{1}^{13}+a_{2}^{13}+a_{3}^{13}+\ldots+a_{2013}^{13}.$$ Prove that $T+2012^{2013}$ is not a perfect square.
10. The incircle $(I)$ of a triangle $A B C$ touches the edges $B C$, $C A$, $A B$ at $D$, $E$, $F$, respectively. $M$ is the intersection of $B C$ and the internal angle bisector of angle $B I C$, $N$ is the intersection of $E F$ and the internal angle bisector of angle $E D F$. Prove that $A$, $M$, $N$ are collinear.
11. If $p(x)$ and $q(x)$ are polynomials with integer coefficients, write $p(x) \equiv q(x) \pmod 2$ if the coefficients of $p(x)-q(x)$ are all even. A sequence of polynomials $p_{n}(x)$ is such that $p_{1}(x)=p_{2}(x)=1$ and $$p_{n+2}(x)=p_{n+1}(x)+x p_{n}(x),\,\forall n \geq 1.$$ Prove that $p_{2^{n}}(x) \equiv 1\pmod 2, \forall n \in \mathbb{N}$.
12. Let $A B C$ be an acute triangle. Prove the inequality $$\frac{\cos B \cos C}{\cos \frac{B-C}{2}}+\frac{\cos C \cos A}{\cos \frac{C-A}{2}}+\frac{\cos A \cos B}{\cos \frac{A-B}{2}} \leq \frac{3}{4}$$

Issue 426

1. Prove that for any natural number $n>4$ there exists a pair of natural numbers $x, y$ with $\dfrac{n}{2} \leq x<n$ and $\dfrac{n}{2} \leq y<n,$ such that $x^{2}-y$ is divisible by $n$ 
2. Let $A B C$ be an isosceles right triangle with right angle at $A$. On the ray $A C$ choose two points $E$ and $F$ such that $\widehat{A B E}=15^{\circ}$ and $C E=C F$. What is the measure of the angle $C B F$?.
3. Find all positive integer solutions of the equation $$65\left(x^{3} y^{3}+x^{2}+y^{2}\right)=81\left(x y^{3}+1\right).$$
4. Solve the system of equations $$\begin{cases} 9 x^{2}+9 x y+5 x-4 y+9 \sqrt{y} &=7 \\ \sqrt{x-y+2}+1 &=9(x-y)^{2}+\sqrt{7 x-7 y} \end{cases}.$$
5. Let $A B C$ be an isosceles triangle where the angle $B A C$ is obtuse. Suppose $D$ is a point on edge $A B$ such that $B C=C D \sqrt{2}$. The line through $D$ and perpendicular to $A B$ meets $C A$ at $E$. Prove that $C D$ passes through the midpoint of $B E$.
6. Let $$S=\sqrt{2}+\sqrt[3]{\frac{3}{2}}+\sqrt[4]{\frac{4}{3}}+\sqrt[5]{\frac{5}{4}}+\ldots+2013 \sqrt[2]{\frac{2013}{2012}}.$$ Find the integer part of $S$.
7. Given that $a, b, c$ are edge lengths of a triangle. Prove the following inequality $$\sqrt{(a+b-c)(b+c-a)(c+a-b)} \leq \frac{3 \sqrt{3} a b c}{(a+b+c) \sqrt{a+b+c}}.$$
8. Let $A B C D$ be a trirectangular tetrahedron where the edges $A B$, $A C$, $A D$ are pairwise perpendicular. $M$ is an arbitrary point in the space. Given that $A B=4$, $A C=8$, $A D=12$. Find the minimum value of the expression $$P=\sqrt{7} M A+\sqrt{11} M B+\sqrt{23} M C+\sqrt{43} M D.$$
9. Let $\left(a_{n}\right)$ be a sequence of positive integers where $$a_{1}=1,\, a_{2}=2,\quad a_{n+2}=4 a_{n+1}+a_{n},\,\forall n \geq 1 .$$ Prove that
   a) $a_{n} a_{n+2}+(-1)^{n} .5$ is a perfect square for all $n \geq 1$.
   b) The equation $x^{2}-4 x y-y^{2}=5$ has infinitely many positive integer solutions.
10. Let $a$ be a real number from $(0,1)$ and $b$ is a complex number, $|b|<1$. Prove that $$|b|+\left|\frac{a-b}{1-a b}\right| \geq a.$$
11. Let $0<\alpha<\dfrac{\pi}{2}$. Prove that $$(\cot \alpha)^{\cos 2 \alpha} \geq \frac{1}{\sin 2 \alpha}$$
12. A tetrahedron $A B C D$ is inscribed in a sphere centered at $O$. Point $G$ does not belong to planes $(B C D)$, $(C D A)$, $(D A B)$, $(A B C)$ and the sphere $(O)$. $X$, $Y$, $Z$, $T$ are the centers of the circumscribed spheres of the tetrahedron $G B C D$, $G C D A$, $G D A B$, $G A B C$ respectively. Prove that $G$ is the centroid of tetrahedron $A B C D$ if and only if $O$ is the centroid of tetrahedron $X Y Z T$.