Mathematics and Youth Magazine Problems 2019

Issue 499

1. Find positive integers $x, y$ such that $x^{y}+y^{x}=100$
2. Given an acute triangle $A B C$. Outside the triangle, we construct two isosceles right triangles with the right angles $A A B D$ and $A C E .$ Let $J$ and $K$ respectively be the perpendicular projections of $D$ and $E$ on $B C$. Prove that $A J K$ is an isosceles right triangle. 
3. Solve the equation $$\frac{1}{\sqrt{3 x^{2}+x^{3}}}+2 \sqrt{\frac{x}{3 x+1}}=\frac{3}{2}.$$
4. Given a right triangle $A B C$ with the right angle $A$ and $A C=2 A B$. Let $A H$ be an altitude of $A B C$. On the opposite ray of $A H$ choose the point $K$ such that $A H=2 A K$. Find the sum $\widehat{A K B}+\widehat{H A B}$.
5. Given real numbers $x, y$ satisfying $y-2 x+4<0$. Find the minimum value of the expression $$P=x^{2}-4 y+\frac{4\left(x^{2}-4 y\right)}{(y-2 x+4)^{2}}.$$
6. Find all positive numbers $x, y, z$ satisfying $$\begin{cases}x^{2}+y^{2}+z^{2}+x y z &=4 \\ \left(\dfrac{1}{x^{9}}+\dfrac{1}{y^{9}}+\dfrac{1}{z^{9}}\right)(1+2 x y z) &=9\end{cases}.$$
7. Given positive numbers $a, b, c$ such that $a^{3}+b^{3}+c^{3}=3$. Find the minimum value of the expression $$M=\frac{a^{2}}{b}+\frac{b^{2}}{c}+\frac{c^{2}}{a}.$$
8. Let $A B C$ be inscribed in a circle $(O)$ and assume that $I$ is the incenter of $A B C$. Assume that $M$ is the midpoint of $A I$. Choose $K$ on $B C$ such that $I K \perp I O$. Choose $Q$ on $(O)$ such that $A Q \parallel I K$. Suppose that $Q M$ intersects $B C$ at $H$. Choose $G$ on $A K$ such that $H G \parallel Q I$. $Q K$ intersects $(O)$ at $L$ other than $Q$. Show that $\widehat{G I K}=\widehat{I A L}$.
9. For each natural number $n$, let $$A_{n}=\left[\frac{n+3}{4}\right]+\left[\frac{n+5}{4}\right]+\left[\frac{n}{2}\right]$$ (the notation $[x]$ denotes the maximal integer which does not exceed $x$). Find all natural numbers $n$ such that $\dfrac{n^{6}-1}{A_{n}}$ is a perfect square.
10. We call the trace of a triangle the ratio between the length of its shortest side and the length of its longest side. Given a square with the side equals 2019 length units. Does there exist a positive integer $n$ and a decomposition of this square into $n$ triangles satisfying both following conditions
• Any side of any of these triangles is less than and equal the side of the square,
• The sum of all the traces of these $n$ triangles does not exceed $\dfrac{n^{2}+6 n-9}{n^{2}}$.
11. The sequence $\left\{u_{n}\right\}_{n \in \mathbb{N}^{*}}$ is determined as follows $$u_{1}=1,\quad u_{n+1}=\sqrt{n(n+1)}\frac{u_{n}}{u_{n}^{2}+n},\, \forall n=1,2 \ldots.$$ a) Find $\displaystyle \lim _{n \rightarrow+\infty} u_{n}$.
    b) Find $\left[\dfrac{\sqrt{2018}}{u_{2018}}\right]$ where $[x]$ denotes the maximal integer which does not exceed $x$. 
12. Given an acute triangle $A B C$. The points $M$, $N$ are on the line segment $B C$ and the points $P$, $Q$ respectively are on the line segments $CA$, $A B$ such that $MNPQ$ is a square. The incircles of $A P Q$, $B Q M$, $C P N$ are respectively tangent to $P Q$, $Q M$, $P N$ at $X$, $Y$, $Z$. Prove that $A X \perp Y Z$.

Issue 500

1. Find all integer $x$ such that $$(x-2018)^{3}+(x-2019)^{2}=2020-x.$$
2. Let $x_{n}=2^{2^{n}}+1$, $n=1,2, \ldots, 2019$. Show that $$\frac{1}{x_{1}}+\frac{2}{x_{2}}+\frac{2^{2}}{x_{3}}+\ldots+\frac{2^{2018}}{x_{2019}}<\frac{1}{3}.$$
3. Given $0<x<y \leq z \leq 2$ and $3 x+2 y+z=9$. Find the maximum value of the expression $$A=3 x^{2}+2 y^{2}+z^{2}.$$
4. Given triangle $A B C$ with $A B<B C$. On the side $B C$ choose $D$ so that $C D=A B$. Through $D$ draw the line which is parallel to $A C$. This line intersects the angle bisector of $\widehat{A B C}$ at $I$. From $I$ draw $IH$ perpendicular to $B C$ ($H$ is on $B C$). From $H$ draw $H E$ perpendicular to $A B$ ($E$ is on $A B$) and draw $H F$ perpendicular to $A D$ ($F$ is on $A D$). Prove that $\widehat{A E I}=\widehat{A F I}$.
5. Solve the system of equations $$\begin{cases} x y(x+y)-x(y-1)-\dfrac{3}{8}\left(y^{2}+1\right) &=0 \\ x y^{2}+x-y &=0 \end{cases}.$$
6. Find $x \in[-1 ; 0]$ such that $$\sqrt{x+1}+\sqrt[3]{x^{2}+1}=2.$$
7. Solve the equation $$(2+\sqrt{3})^{|\sin x|}+\left(\frac{\sqrt{2}+\sqrt{6}}{2}\right)^{|\cos x|}=\frac{2+\sqrt{2}+\sqrt{6}}{2}.$$
8. Given a triangle $A B C$ with $h_{\alpha}$, $h_{b}$, $h_{c}$ and $a$, $b$, $c$ are the lengths of altitudes and the corresponding sides. Prove that $$\frac{a}{h_{c}}+\frac{b}{h_{b}}+\frac{c}{h_{c}} \geq 2\left(\tan \frac{A}{2}+\tan \frac{B}{2}+\tan \frac{C}{2}\right).$$
9. Find all positive integers $n$ such that $$3 n=2 S^{3}(n)+7 S^{2}(n)+16$$ where $S(n)$ is the sum of all the digits of $n$
10. Given $0<\lambda<1$ and $2019$ positive numbers $x_{1}, x_{2}, \ldots, x_{2019}$ such that $\displaystyle \sum_{i=1}^{200} x=2019$. Prove that $$673 \leq \sum_{i=1}^{2019} f\left(\frac{2019-\lambda x_{i}}{2019-\lambda}\right) \leq \sum_{i=1}^{2019} f\left(x_{i}\right)$$ where $f(x)=\dfrac{1}{x^{2}+x+1}$.
11. Two sequences $\left(x_{n}\right)$, $\left(y_{n}\right)$ are determined by $x_{1} \in\left[\dfrac{1}{2},\dfrac{\sqrt{3}}{2}\right]$, $y_{1} \in\left[\dfrac{1}{2}, \dfrac{\sqrt{3}}{2}\right]$ and $$x_{n+1}=\sqrt{1-y_{2}^{2}},\,y_{m+}=\frac{1}{\sqrt{21}} \sqrt{9-5 x_{n}^{2}},\,\forall n \in \mathbb{N}.$$ Show that the sequences $\left(x_{n}\right)$, $\left(y_{n}\right)$ converse and then find $\displaystyle \lim_{n\to\infty} x_{2}$ and $\displaystyle \lim_{n\to\infty} y_{n}$.
12. Given a triangular pyramid $S . A B C$ with $\widehat{A S B}=60^{\circ}$, $\widehat{B S C}=90^{\circ}$, $\widehat{C S A}=120^{\circ}$; $S A=2$, $S B=9$, $S C=4$. Let $X$ be the set of all values $d$ such that there exists a point $M$ from which the distances to the planes $(S B C)$, $(S C A)$, $(S A B)$, $(A B C)$ are respectively $\sqrt{3} d$, $\dfrac{d}{2}$, $d$, $d$. Find the cardinality of $X$ and also find the largest possible value for $d$.

Issue 501

1. Find natural numbers $x$ such that $$\frac{x-1}{2018}+\frac{x-7}{503}=\frac{x-3}{1008}+\frac{x-9}{670}.$$
2. Find all pairs non-negative numbers $(x, y)$ such that $$1+3^{x+1}+2.3^{3 x}=y^{3}.$$
3. Given $0 \leq a \leq 2$, $0 \leq b \leq 2$, $0 \leq c \leq 2$ and $a+b+c=3$. Show that $$3 \leq a^{3}+b^{3}+c^{3} \leq 9.$$
4. From a point $M$ which is outside a circle $(O)$ we draw a tangent $M A$ and a secant $M B C$ to $(O)$ ($B$ is in between $M$ and $C$). Let $D$, $E$, $K$ respectively be the midpoints of $M A$, $M B$, $M E$. Suppose that $H$ is the point of reflection of $D$ through $K$. The line $H E$ intersects $C D$ at $N$. Prove that $C N K H$ is a cyclic quadrilateral.
5. Solve the system of equations $$\begin{cases} 2 x \sqrt{x}+3 x \sqrt{y}+\sqrt{y} &=138 \\ y \sqrt{y}+6 \sqrt{x}y+8 \sqrt{x} &=213\end{cases}.$$
6. Solve the equation $$12 \sqrt[3]{x^{2}+4} \cdot \sqrt{2 x-3}=\left(x^{2}+16 x-12\right) \sqrt[3]{x-1}.$$
7. Find the coefficient of $x^{3}$ in the expansion of $$(1+x)(1+2 x)(1+3 x) \ldots(1+n x).$$
8. Given a triangle $A B C$. Let $B C=a$, $C A=b$, $B A=c$ and $r_{a}$, $r_{b}$, $r_{c}$ respectively the radii of the excircles relative to $A$, $B$, $C$. Prove that $$\frac{r_{a}}{r_{b}}+\frac{r_{b}}{r_{e}}+\frac{r_{c}}{r_{o}} \geq \frac{a}{b}+\frac{b}{c}+\frac{c}{a}.$$
9. Given positive numbers $a$, $b$, $c$, $d$ such that $a \geq b \geq c \geq d$ and $a b c d=1$. Find the smallest constant $k$ so that the following inequality holds $$\frac{1}{a+1}+\frac{1}{b+1}+\frac{1}{c+1}+\frac{k}{d+1} \geq \frac{3+k}{2}.$$
10. Let $n$ be an integer which is greater or equal to $6 .$ Find the largest positive integer $m$ so that among $n$ arbitrary distinct positive integers which are less than or equal to $m$ there always exist $4$ numbers so that one of them is equal to the sum of the remains.
11. Find the smallest integer $k$ so that there exist two sequences $\left\{a_{i}\right\},\left\{b_{i}\right\}$ satisfying
    a) $a_{i}, b_{i} \in\left\{1,2018,2018^{2}, 2018^{3}, \ldots\right\}$, $ i=1,2, \ldots, k$
    b) $a_{i} \neq b_{i}, i=1,2, \ldots, k$
    c) $a_{i} \leq a_{i+1}, b_{i} \leq b_{i+1}$
    d) $\displaystyle\sum_{i=1}^{k} a_{i}=\sum_{i=1}^{k} b_{i}$
12. Given a triangle $A B C$ with $A B+A C=2 B C$. Let $O$ be its circumcenter, and $H$ its orthocenter. Let $M_{a}$ be the midpoint of $B C$. Prove that the circles with diameters $HM_{a}$ and $A O$ are tangent to each other.

Issue 502

1. Show that $$A=10^{10}+10^{10^1}+10^{10^2}+\ldots+10^{10^{10}}-5$$ is divisible by $7$.
2. Given integers $a$, $b$ and $c$. Find the natural pumber $d$ so that $$|a-b|+|b-c|+|c-a|=2018^{d}+2019.$$
3. Given positive numbers $a, b, c$ such that $a^2+b^2+c^{2}=\dfrac{3}{4}$. Find the maximum value of the expression $$P=\left(1-\frac{1}{a}\right)\left(1-\frac{1}{b}\right)\left(1-\frac{1}{c}\right).$$
4. Given a triangle $A B C$ and $M$ is inside the triangle so that $\widehat{A B M}=\widehat{A C M}$. Let $D$ be the symmetrical point of $M$ about the line $A C$. Draw the parallelogram $B M C E$. The ray $B M$ intersects $A C$ at $l$. The ray $D l$ intersects $B E$ at $F$. Show that the points $A$, $D$, $C$, $E$, $F$ are on the same circle.
5. Solve the equation $$x^{6}-7 x^{2}+\sqrt{6}=0.$$
6. Solve the system of equations $$\begin{cases}\log_{\frac{1}{4}} x &=\left(\dfrac{1}{4}\right)^{2}+1 \\ 128 x^{3} \sqrt{4 x^{2}+y^{2}} &=y^{3} \sqrt{64 x^{2}+y^{2}}\end{cases}.$$
7. Suppose that $$(1+x+x^2+\ldots+x^{2018})^{2019} = a_0 +a_1 x +a_2 x^2 + \ldots +a_{4074342}x^{4074342}.$$ Prove that $$C_{2019}^0 a_{2019}-C_{2019}^1 a_{2018}+C_{2019}^{2} a_{2017}-C_{2019}^{3} a_{2016}+\ldots+C_{2019}^{2018} a_{1}-C_{2019}^{2019} a_{0}=-2019.$$
8. Given an aculc triangle $A B C$ inscribed in a circle $(O)$. Let $H$ be the orthocenter of the triangle, $P$ the midpoint of the minor are $\widehat{B C}$. Assume that $E$ is the intersection between $B H$ and $A C$. $F$ is the intersection between $\mathrm{CH}$ and $A B$. Let $Q$, $R$ respectively be the second interscctions between $P E$, $P F$ and $(O)$. Let $K$ be the interscetion between $B Q$ and $C R$. Show that $K H \parallel A P$.
9. Given non-ncgative numbers $a, b, c$ with at most one of them is equal to $0$. Prove that $$\frac{a^{3}}{b^{2}-b c+c^{2}}+\frac{b^{3}}{c^{2}-c a+a^{2}}+\frac{c^{3}}{a^{2}-a b+b^{2}} \geq a+b+c.$$
10. Let $\displaystyle S_n = \sum_{k=2}^n k\cos\dfrac{\pi}{k}$. Find $\displaystyle \lim_{n \rightarrow+\infty} \dfrac{S_{n}}{n^{2}}$.
11. There are $100$ marbles distributed into $k$ groups. A distribution is called "special" if any two groups have different numbers of marbles and if we divide any group into two smaller ones then among $k+1$ new groups there are groups with equal numbers of marbles. Find the maximum and minimum values for $k$ so that there exist corresponding special distributions. 
12. Outside a cyclic quadrilateral $A B C D$, draw the squares $A B M N$, $B C P Q$, $C D R S$, $D A U V$. Let $B^{\prime}$ be the intersection between $P Q$ and $M N$, $D^{\prime}$ the intersection between $U V$ and $RS$. Show that the midpoint of $B^{\circ} D^{\circ}$ belongs to $B D$.

Issue 503

1. Given integers $a, b, c$ satisfying $a=b-c=\dfrac{b}{c}$. Prove that $a+b+c$ is a cube of some integer.
2. Given an isosceles $A B C$ with the acute vertex angle $A$. Draw the altitude $B H$ ($H$ is on $A C)$. The line through $H$ and parallel to $B C$ and the line through $C$ and parallel to $B H$ intersect at $E .$ Let $M$ be the midpoint of $H E .$ Find the value of the angle $\widehat{A M C}$.
3. Solve the equation $$x^{2}-3 x+1=-\frac{\sqrt{3}}{3} \sqrt{x^{4}+x^{2}+1}.$$
4. Given a half circle with the center $O$ and the diameter $A B=2 R$. Let $M$ be a point on the opposite ray of the ray $A B$. Draw a secant $M C D$ of the circle $(C$ is between $M$ and $D$). Assume that $A D$ intersects $B C$ at $I$. Determine the position of $M$ given that $M C I O$ is a cyclic quadrilateral.
5. Given real numbers $a$, $b$ and $c$ satisfying $\left(1+a^{2}\right)\left(4+b^{2}\right)\left(9+c^{2}\right) \leq 100$. Show that $$-4 \leq 3 a b+2 a c+b c \leq 16.$$
6. Solve the system of equations $$\begin{cases}\sqrt{x^{2}+x y+y^{2}}+\sqrt{y^{2}+y z+z^{2}}+\sqrt{z^{2}+z x+x^{2}} &=\sqrt{3}(x+y+z) \\ \sqrt{x y z}-(\sqrt{x}+\sqrt{y}+\sqrt{z})+2 &=0 \end{cases}.$$
7. For any triangle $A B C$, show that $$ \tan \frac{A}{4}+\tan \frac{B}{4}+\tan \frac{C}{4}+\tan \frac{A}{4} \tan \frac{B}{4} + \tan \frac{B}{4} \tan \frac{C}{4}+\tan \frac{C}{4} \tan \frac{A}{4} \leq 3(9-5 \sqrt{3}).$$ When does the equality happen?
8. Given a triangle $A B C$ inscribed in a circle $(O)$. The medians $A A_{1}$, $B B_{1}$, $C C_{1}$ respectively intersect $(O)$ at $A_{2}$, $B_{2}$, $C_{2}$. Let $A B=c$, $B C=a$, $C A=b$. Show that $$\frac{A_{1} A_{2}}{a}+\frac{B_{1} B_{2}}{b}+\frac{C_{1} C_{2}}{c} \geq \frac{\sqrt{3}}{2}.$$
9. Given non-negative numbers $a$, $b$, $c$ satisfying $a+b+c=\dfrac{4}{3}$. Prove that $$3\left[a(a-1)^{2}+b(b-1)^{2}+c(c-1)^{2}\right] \geq a b+b c+c a.$$ When does the equality happen?
10. Find all pairs of positive integers $(n ; k)$ so that $$(n+1)(n+2) \ldots(n+k)-k$$ is a complete square. 
11. Find all functions $f(x)$ which are continuous on $[a ; b]$, are differentiable on $(a ; b)$, and satisfy $$f^{\prime}(x) \leq \frac{f(b)-f(a)}{b-a},\, \forall x \in(a ; b),$$ where $a$, $b$ are given real numbers with $a < b$.
12. Given a circle $(O)$ and a point $K$ lying outside the circle. Draw the tangents $K I$, $K J$ to the circle at $I$, $J$. On the opposite ray of $I O$ take an arbitrary point $O$. The circle with center $O'$ radius $O^{\prime} J$ intersects $(O)$ at the other point $A$. $A I$ intersects $\left(O^{\prime}\right)$ at the other point $D$. The line through $K$ perpendicular to $O^{\prime}D$ meets $\left(O^{\prime}\right)$ at $B$ and $C$. Show that $I$ is the center of the incircle of $A B C$.

Issue 504

1. Find the natural number $n$ given that $n^{5}+n+1$ has only one prime factor.
2. Given a right isosceles triangle $A B C$ with the vertex angle $A$. Let $M$, $N$, $I$ respectively be the midpoints of $A B$, $A C$ and $N C$. Assume that $K$ is the perpendicular projection of $N$ on $B C$. Show that $A K$, $B I$ and $C M$ are concurrent.
3. Given positive numbers $a$, $b$, $c$, $d$ such that $a+b=c+d=2019$ and $a b \geq c d$. Find the minimum value of the expression $$P=\frac{a+3 \sqrt[3]{b}+2}{\sqrt[3]{c}+\sqrt[3]{d}}.$$
4. Given a triangle $A B C$ with the angle $\hat{A} > 90^{\circ}$. Choose a point $I$ inside the line segment $B C$ so that $B A$ intersects the circumcircle $\Delta A C I$ at $D$ $(D \neq A)$ and $C A$ intersects the circumcircle $\Delta A B I$ at $E$ $(E \neq A)$, $B E$ intersects $CD$ at $N$. Let $M$ be the midpoint of $B C$, $M A$ intersects the circumcircle $\triangle B N C$ at $F$. Show that $A$, $D$, $E$, $F$, $N$ both belong to a circle.
5. Solve the system of equations $$\begin{cases} x^{4}+3 x &=y^{4}+y \\ x^{2}-y^{2} &=2\end{cases}.$$
6. Find the sum of the squares of all real roots of the equation $$x^{5}+2018 x^{2}+2019=x^{4}+2019 x^{3}+2020 x.$$
7. Given $3$ positive numbers $x, y, z$ such that $x y z=1$. Prove that $$\frac{1}{x^{k+1}(y+z)}+\frac{1}{y^{k+1}(z+x)}+\frac{1}{z^{k+1}(x+y)} \geq \frac{3}{2} \quad \left(k \in \mathbb{N}^{*}\right)$$
8. Given two equilateral triangles $A B C$ and $A B^{\prime} C^{\prime}$ with the same orientation. Let $K$ be the second intersection between the circumcircles of $A B C$ and $A B^{\prime} C^{\prime}$. Let $M$ be the intersection between $B C^{\prime}$ and $C B^{\prime}$. Show that $M A=M K$.
9. Determine the coefficient of $x$ in the polynomial expansion of $$(1+x)(1+2 x)^{2} \ldots(1+n x)^{n}.$$
10. Given the real sequence $\left\{x_{n}\right\}$ determined as follows $$x_{1}=1,\quad x_{n+1}=x_{n}+\frac{1}{2 x_{n}},\, \forall n \geq 1.$$ Show that $\left[9 x_{81}\right]=81$ (where $[x]$ denotes the integral part of $x$ ).
11. Given an integer $k \geq 2$ and aninteger $n \geq \dfrac{k(k+1)}{2}$. Find the maximal positive integer $m$ so that among $n$ arbitrary distinct positive integers which do not exceed $m$ there always exist $k+1$ numbers of which some number is equal to the sum of the remaining ones.
12. Given a non-right triangle $A B C$. The altitudes $B B^{\prime}$ and $C C^{\prime}$ intersect at $H$. Let $M$ be the midpoint of $A H$. Let $K$ be an arbitrary point on $B^{\prime} C^{\prime}$ ($K$ is different from $B^{\prime}$, $C^{\prime}$). The line $A K$ intersects $M B^{\prime}$, $M C^{\prime}$ respectively at $E$, $F$. Let $N$ be the intersection between $B E$ and $C F$. Show that $K$ is the orthocenter of the triangle $N B C$.

Issue 505

1. Find the smallest positive interger $a$ so that $2 a$ is a square and $3 a$ is a cube.
2. Find different non-zero digits $a$, $b$, $c$, $d$ so that $\overline{a b c d a 1}-4 n=n^{2}$ for some postitive integer $n$ (the last digit of $\overline{abcdal}$ is $1$).
3. Find all polynomials $P(x)$ whose the coefficients are integers between $0$ and $8$ and $P(9)=32078$
4. Let $ABCD$ be a convex quadrilateral. Denote the midpoints of $A B$, $A C$, $C D$, $D B$ respectively $M$, $N$, $P$, $Q$. Let the lengths of the sides $A B$, $B C$, $C D$, $D A$ respectively be $a$, $b$, $c$, $d$. Let the area of $M N P Q$ be $S$. Assume that $A D$ and $B C$ are perpendicular. Show that $$\frac{(c-a)^{2}-(b-d)^{2}}{8} \leq S \leq \frac{(b+d)^{2}-(c-a)^{2}}{8}$$
5. Let $x$, $y$, $z$ be positive numbers such that $x+y+z=3$. Find the minimum value of the expression $$P=x^{5}+y^{5}+z^{5}+\frac{2}{x+y}+\frac{2}{y+z}+\frac{2}{z+x}+\frac{10}{x y z}.$$
6. Find all real solutions of the equation $$\sqrt[3]{\frac{x^{3}-3 x+\left(x^{2}-1\right) \sqrt{x^{2}-4}}{2}}+\sqrt[3]{\frac{x^{3}-3 x-\left(x^{2}-1\right) \sqrt{x^{2}-4}}{2}}=x^{2}-2.$$
7. Given the equation $$\frac{1}{3} x^{5}+2 x^{4}-5 x^{3}-7 x^{2}+12 x-1=0.$$ a) Show that the equation has $5$ distinct roots.
   b) Let $x_{I}$ $(i=1,5)$ be the roots of the equation. Find the sum $$S=\sum_{i=1}^{5} \frac{x_{i}-1}{x_{i}^{5}+6 x_{i}^{4}-3}.$$
8. Given any triangle $A B C$ show that $$ \left(1+\sin ^{2} \frac{A}{2}\right)\left(1+\sin ^{2} \frac{B}{2}\right)\left(1+\sin ^{2} \frac{C}{2}\right) \geq \frac{125}{64}.$$
9. Given positive numbers $a$, $b$, $c$ and a number $-2<k<2$. Prove that $$27\left(a^{2}+k a b+b^{2}\right)\left(b^{2}+k b c+c^{2}\right)\left(c^{2}+k c a+a^{2}\right) \geq (k+2)^{3}(a b+b c+c a)^{3}.$$
10. A man using a map on his phone walked from the point $A$ to the point $B$. He arrived $B$ after a few straight walks and correspondingly a few rotations of the phone (to find the right directions). Assume that each time he needed to rotate his phone clockwisely an acute angle from the previous direction. Given that the sum of all the angles is $\alpha$ which is less than $180^{\circ}$. Show that the total distance that he walked is less than or equal to $\dfrac{A B}{\cos \frac{\alpha}{2}}.$
11. Given the real sequence $\left(a_{n}\right)$ determined as follows $$a_{1}=2020, \quad a_{n+1}=1+\frac{2}{a_{n}},\, \forall n \geq 1.$$ a) Prove that $2 n<a_{1}+a_{2}+\ldots+a_{n}<2 n+2018$ for any arbitrary $n=1,2, \ldots$.
    b) Find the maximal real number $a$ such that the inequality $$\sqrt{x^{2}+a_{1}^{2}}+\sqrt{x^{2}+a_{2}^{2}}+\ldots+\sqrt{x^{2}+a_{n}^{2}} \geq n \sqrt{x^{2}+a^{2}}$$ holds for any given $x \in \mathbb{R}$, $n=1,2, \ldots$.
12. Given a triangle $A B C$ which is not an isosceles triangle with the vertex angle $A$. Let $M$ be on the side $B C$. Let $I_{1}$, $I_{2}$ respectively be the incenters of the triangles $A B M$, $A C M$. Assume that $N$, $P$, $Q$ respectively be the second intersections between $A M$, $A B$, $A C$ and the circumcircle of $A I_{1} I_{2}$. Let $J_{1}$, $J_{2}$ respectively be the incenters of the triangles $A P N$, $A Q N$. Prove that the radical center of the circumcircles of $A I_{1} I_{2}$, $A J_{1} J_{2}$, $M I_{1} I_{2}$ belongs to $B C$.

Issue 506

1. Find all integers $x$, $y$, $z$ which satisfy $$3 x^{2}+6 y^{2}+2 z^{2}+3 y^{2} z^{2}-18 x=6.$$
2. Given an isosceles triangle $A B C$ with the vertex angle $A$. Let $H$ be the point in the interior domain determined by the angle $A$ such that $H B \perp B A$, $H C \perp C A$. On the line segment $B C$ we choose $M$ such that $B M=\dfrac{1}{4} B C$. Let $N$ be the midpoint of $A C$ Calculate the angle $\widehat{H M N}$.
3. Find all pairs of integers $(x ; y)$ satisfying $$y^{3}-2(x-4) y^{2}+\left(x^{2}-9 x-1\right) y+3 x^{2}+x=0.$$
4. Given an acute triangle $A B C$ and suppose that $B E$ and $C F$ are the two altitudes. Draw $F H$ and $E K$ perpendicular to $B C$ $(H, K \in B C)$. Draw $H M$ parallel to $A C$ and $K N$ parallel to $A B$ $(M \in A B, N \in A C)$. Show that $E F \parallel M N$.
5. Solve the equation $$\sqrt{\frac{x-1}{x+1}}+\frac{2 x+6}{(\sqrt{x-1}+\sqrt{x+3})^{2}}=2.$$
6. Given two positive numbers $a$ and $b$ such that $a<b$ and $a^{b}=b^{a}$. Show that there exists a positve number $c$ such that $$a=\left(1+\frac{1}{c}\right)^{c},\quad b=\left(1+\frac{1}{c}\right)^{c+1}.$$
7. Solve the system of equations $$\begin{cases}\tan x-\tan y &=(1+\sqrt{x+y})^{y}-(1+\sqrt{x+y})^{x} \\ 3^{\sqrt{1-x}}+5^{\sqrt{1-y}} &=2(1+\sqrt{9-10 x+y})\end{cases}.$$
8. Show that for any triangle $A B C$ we always have $$\frac{(b+c) a}{m_{a}^{2}}+\frac{(c+a) b}{m_{b}^{2}}+\frac{(a+b) c}{m_{c}^{2}} \geq 8$$ where $a$, $b$, $c$, $m_{a}$, $m_{b}$, $m_{c}$ respectively are the lengths of the sides $B C$, $C A$, $A B$ and the corresponding medians.
9. Let $a$, $b$, $c$ be positive numbers such that $a+b+c=3$. Find the minimum value of the expression $$M=\sqrt[3]{\frac{a^{5}}{b^{4}}}+\sqrt[3]{\frac{b^{5}}{c^{4}}}+\sqrt[3]{\frac{c^{5}}{a^{4}}}.$$
10. Find all natural numbers $n$ so that $2^{n}+n^{2}+1$ is a perfect square.
11. Given a strictly increasing sequence of positive integers $\left(a_{n}\right)$. Let $$S_{n}=\frac{\sqrt{a_{1}}}{\left[a_{1}, a_{2}\right]}+\frac{\sqrt{a_{2}}}{\left[a_{2}, a_{3}\right]}+\ldots+\frac{\sqrt{a_{n}}}{\left[a_{n}, a_{n+1}\right]},\, \forall n=1,2, \ldots$$ (for positive integers $x, y$ we denote $[x, y]$ the least common multiple (l.c.m.) of $x$ and $y$.) Show that the sequence $\left(S_{n}\right)$ has the finite limit when $n \rightarrow+\infty$.
12. Given an acute triangle $A B C$ $(A B < A C)$. Two altitudes $B E$ and $C F$ intersect at $H$. Let $I$ be the center of the circle which passes through $A$, $B$ and is tangent to $B C$ and $J$ the center of the circle which passes through $B$, $H$ and is tangent to $B C$. Let $M$ be the midpoint of $A H$, and $S=E F \cap B C$ Show that $S M$ bisects $I J$.

Issue 507

1. Does it exist a natural number $n$ so that the last digit of the sum $1+2+3+\ldots+n$ is $2$, $4$, $7$ or $9$?.
2. Given a right triangle $A B C$ with the right angle $A$ and $A B<A C$. Let $E$ and $F$ be the points on the sides $A C$ and $B C$ respectively such that $E F \perp B C$ and $E F=F B$. Let $D$ be the point on the side $A C$ such that $A D=A B$. Prove that $E F D$ is an isosceles triangle.
3. Find positive integral solutions of the equation $$1+5^{x}=2^{y}+5.2^{2}.$$
4. Given an acute triangle $A B C$. Outside the triangle, draw two equilateral triangles $A B D$ and $A C E$. On the line segments $A D$, $CE$, $CB$ choose the points $M$, $N$, $F$ respectively so that $$\frac{A M}{A D}=\frac{C N}{C E}=\frac{C F}{C B}=\frac{1}{3} .$$ Compare the lengths of two length segments $M N$ and $E F$.
5. Given real numbers $x,y, z \geq 0$ such that $\max \{x ; y ; z\} \geq 1$. Show that $$x^{3}+y^{3}+z^{3}+(x+y+z-1)^{2} \geq 1+3 x y z.$$
6. Solve the system of equations $$\begin{cases}x^{3}+x+2 &=8 y^{3}-6 x y+2 y \\ \sqrt{x^{2}-2 y+2}+2 \sqrt[4]{x^{3}(5-4 y)} &=2 y^{2}-x+2\end{cases}.$$
7. Suppose that $$P(x)=x^{n}+x^{n-1}+a_{n-2} x^{n-2}+\ldots+a_{1} x+a_{0}$$ has $n$ distinct real roots $x_{1}, x_{2}, \ldots, x_{n}$. Show that $$\frac{x_{1}^{n}}{P^{\prime}\left(x_{1}\right)}+\frac{x_{2}^{n}}{P^{\prime}\left(x_{2}\right)}+\ldots+\frac{x_{n}^{n}}{P^{\prime}\left(x_{n}\right)}=-1$$ where $P^{\prime}(x)$ is the derivative of $P(x)$.
8. Suppose that the inscribed sphere of the tetrahedron $A_{1} A_{2} A_{3} A_{4}$ is tangent to the face which is opposite to $A_{i}$ at $B_{I}$ $(i=1,2,3,4)$. Prove that if $B_{1} B_{2} B_{3} B_{4}$ is almost-regular (opposite sides have the same length) if and only if $A_{1} A_{2} A_{3} A_{4}$ is almost-regular.
9. Find the minimum and maximum values of the expression $$P=\frac{\left(2 x^{2}+5 x+5\right)^{2}}{(x+1)^{4}+1}$$
10. Find all prime numbers $p$ and positive integers $a, b$ so that $p^{a}+p^{b}$ is a perfect square.
11. Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ satisfying $$f((x+z)(y+z))=(f(x)+f(z))(f(y)+f(z)),\,\forall x, y, z \in \mathbb{R}.$$
12. Given a triangle $A B C$ and a point $M$ on the side $B C$. The symmedians through $M$ of the triangles $M A B$, $M A C$ intersect the circles $(M A B)$, $(M A C)$ respectively at $Q$, $R$ which are different from $M$. Let $P$ be the point on $B C$ so that $AP \perp AM$. Denote by $l$ the external common tangent, which closer to $A$, of two circles $(M A B)$, $(MAC)$. Suppose that $l$ is parallel to $B C$. Show that $l$ is tangent to $(P Q R)$. (The notion $(X Y Z)$ is for the circumcircle of the triangle $X Y Z$).

Issue 508

1. Let $$A=11.13 .15+13.15 .17+\ldots+91.93 .95+93.95 .97.$$ Is $A$ divisible by $5 ?$
2. Find $2019$ numbers so that the absolute values of these numbers do not exceed 0,5 and the sum of any $3$ arbitrary numbers among these is an integer.
3. Let $x, y, z$ be positive numbers. Find the minimum value of the expression $$P=x^{2}+y^{2}+z^{2}+\frac{x^{3}}{x^{2}+y^{2}}+\frac{y^{3}}{y^{2}+z^{2}}+\frac{z^{3}}{z^{2}+x^{2}}-\frac{7}{6}(x+y+z).$$
4. Given a triangle $A B C$ with $\widehat{A B C}$ and $\widehat{A C B}$ are acute. Let $M$ be the midpoint of $A B .$ On the opposite ray of the ray $B C$ choose the point $D$ such that $\widehat{D A B}=\widehat{B C M}$. Through $B$ draw a line perpendicular to $C D$. This line intersects the perpendicular bisector of $A B$ at $E$. Show that $D E$ is perpendicular to $A C$.
5. Solve the equation $$x^{2010}-2011 x^{670}+\sqrt{2010}=0.$$
6. Given non-negative numbers $a$, $b$, $c$ with at most one of them is equal to $0$. Show that for every positive integer $n$ we have $$\frac{a^{2^{2}}+b^{2^{n}}}{a^{2^{2}}+c^{2^{n}}}+\frac{b^{2^{n}}+c^{2^{n}}}{b^{2^{n}}+a^{2^{n}}}+\frac{c^{2^{n}}+a^{2^{n}}}{c^{2^{n}}+b^{2^{n}}} \geq \frac{a+b}{a+c}+\frac{b+c}{b+a}+\frac{c+a}{c+b}.$$
7. Find the value of the expression $$f(A, B, C)=\sin A+\sin B+\sin C-\sin A \sin B \sin C$$ where $A$, $B$, $C$ are the angles of a triangle.
8. Given a tetrahedron $O A B C$ with $O A$, $O B$, $O C$ are pairwise perpendicular and $O A=a$, $O B=b$, $O C=c$. Let $r$ be the radius of the inscribed sphere of $O A B C$. Show that $$\frac{1}{r} \geq \frac{\sqrt{3}+1}{\sqrt{3}}\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right).$$
9. Suppose that the positive numbers $a, b, c, d$ form a pregression (in that order) with the common difference $m$. Show that $$e^{a^2}\left(4 m^{2}+2 m a+1\right)+e^{b^{2}} \cdot 2 m a+e^{c^{2}}\left(2 m^{2}+2 m a\right)<e^{d^{2}}$$
10. Show that, for any integer $n \geq 1$, the equation $x^{2 n+1}=x+1$ has exactly one real solution which is denoted by $x_{n}$. Then find $\displaystyle \lim _{n \rightarrow+\infty} x_{n}$
11. Find all the functions $f: \mathbb{R} \rightarrow \mathbb{R}$ satisfying $$f(x) f(y)+f(x+y)=x f(y)+y f(x)+f(x y)+x+y+1,\,\forall x, y \in \mathbb{R}.$$
12. Given a harmonic quadrilateral $A B C D$ (a cyclic quadrilateral in which the products of two opposite sides are equal) inscribed the circle $(O)$. Let $M$ be the midpoint of $A C$. Let $X$, $Y$, $Z$, $T$ respectively be the perpendicular projection of $M$ on $A B$, $B C$, $C D$, $D A$. Let $E=A B \cap C D$, $F=A D \cap C B$, $P=A C \cap B D$, $Q=X Z \cap Y T$. Show that $P Q$ passes through the midpoint of $E F$.

Issue 509

1. Compare the following numbers $$A=\frac{1}{5}+\frac{2}{5^{2}}+\frac{3}{5^{3}}+\cdots+\frac{2018}{5^{2018}} ; \quad B=\frac{2018}{2019}$$
2. Suppose that $P$ is a point inside a triangle $A B C$ so that $\widehat{P B C}=30^{\circ}$, $\widehat{P B A}=8^{\circ}$ and $\widehat{P A B}=\widehat{P A C}=22^{\circ}$. Find the value of the angle $\widehat{A P C}$.
3. Find positive solutions of the equation $$\frac{1}{5 x^{2}-x+3}+\frac{1}{5 x^{2}+x+7} +\frac{1}{5 x^{2}+3 x+13}+\frac{1}{5 x^{2}+5 x+21}=\frac{4}{x^{2}+6 x+5}.$$
4. Given a square $A B C D$ with the length of a side $a$. On the sides $A D$ and $C D$ respectively choose two points $M$, $N$ so that $M D+D N=a$. Let $E$ be the intersection of two lines $B N$ and $A D$. Let $F$ be the intersection of two lines $B M$ and $C D$. Show that $$M E^{2}-N E^{2}+N F^{2}-M F^{2}=2 a^{2}.$$
5. Given positive numbers $a, b, c$ Find the minimum value of the expression $$P=\frac{a}{\sqrt[3]{a}+\sqrt[3]{b c}}+\frac{b}{\sqrt[3]{b}+\sqrt[3]{c a}}+\frac{c}{\sqrt[3]{c}+\sqrt[3]{a b}} + \frac{9 \sqrt[3]{(a+1)(b+1)(c+1)}}{4(a+b+c)}.$$
6. Let $a, b, c, d$ be positive numbers such that $$\frac{1}{a b}+\frac{1}{b c}+\frac{1}{c d}+\frac{1}{a d}=1 .$$ Show that $$\frac{a b c d}{8}+2 \geq \sqrt{(a+c)\left(\frac{1}{a}+\frac{1}{c}\right)}+\sqrt{(b+d)\left(\frac{1}{b}+\frac{1}{d}\right)}.$$
7. Find real solutions of the following system of equations $$\begin{cases} x^{3}+2 y^{3} &=2 x^{2}+z^{2} \\ 2 x^{3}+3 x^{2} &=3 y^{3}+2 z^{2}+7 \\ x^{3}+x^{2}+y^{2}+2 x y &=2 x z+2 y z+2\end{cases}$$
8. Given a triangle $A B C$ inscribed in a circle $(O)$. Assume that $B O$ and $C O$ intersect the altitude $A D$ of the triangle respectively at $E$ and $F$. Let $I$ and $J$ respectively be the centers of the circles $(A C F)$ and $(A B E)$. Two points $K$, $H$ are on $A B$, $A C$ respectively so that $J K \parallel A O \parallel  I H$. Suppose that $I J$ intersects $A B$ and $A C$ at $M$ and $N$. Show that the intersection between $M H$ and $N K$ is on the midsegment, which is opposite to the vertex $A$, of the triangle $A B C$.
9. Solve the equation $$8^{x}+27^{\frac{1}{x}}+2^{x+1} \cdot 3^{\frac{x+1}{x}}+2^{x} \cdot 3^{\frac{2 x+1}{x}}=125.$$
10. Let $[x]$ be the maximal integer which does not exceed $x$ and let $\{x\}=x-[x]$. Consider the sequence $\left(u_{n}\right)$ with $$u_{n}=\left\{\frac{2^{2 n+1}+n^{2}+n+2}{2^{n+1}+2}\right\}.$$ Find the number of terms of the sequence $\left(u_{n}\right)$ satisfying $$\frac{2526.2^{n-99}}{2^{n}+1} \leq u_{n} \leq \frac{23}{65}.$$
11. Find all functions $f: \mathbb{N}^{*} \rightarrow \mathbb{R} \backslash\{0\}$ so that $$f(1)+f(2)+\cdots+f(n)=\frac{f(n) f(n+1)}{2}, \forall n \in \mathbb{N}^{*}.$$
12. Given a triangle $A B C$ with $A B+A C=2 B C$. Let $I_{a}$ be the center of the excircle corresponding to the angle $A$. The circle $\left(A, A I_{a}\right)$ intersects $B C$ at $E$ and $F$ with $E$ is on the ray $C B$ and $F$ is on the ray $B C$. The circle $\left(E B I_{\alpha}\right)$ meets $A B$ at $M$ and the circle $\left(F C I_{\alpha}\right)$ meets $A C$ at $N$. Show that $B C N M$ is both a cyclic and tangential quadrilateral.

Issue 510

1. Find natural numbers $x, y, z$ satisfying $$3^{x}+5^{y}-2^{z}=(2 z+3)^{3}.$$
2. Given a right triangle $A B C$ with the right angle $A$ and $\hat{B}=75^{\circ}$. Let $H$ be the point on the opposite ray of $A B$ such that $B H=2 A C$. Find the angle $\widehat{B H C}$.
3. Given that $x y(x+y)+y z(y+z)+z x(z+x)+2 x y z=0$. Show that $$x^{2019}+y^{2019}+z^{2019}=(x+y+z)^{2019}.$$
4. Given a triangle $A B C$ with $\widehat{A B C}=30^{\circ}$. Outside the triangle $A B C$, construct the isosceles triangle $A C D$ with the right angle $D$. Show that $$2 B D^{2}=B A^{2}+B C^{2}+B A \cdot B C.$$
5. Find the minimum value of the expression $$T=\frac{5-3 x}{\sqrt{1-y^{2}}}+\frac{5-3 y}{\sqrt{1-z^{2}}}+\frac{5-3 z}{\sqrt{1-x^{2}}}.$$
6. Find all possible values for the parameter $m$ so that the equation $$4^{x}+2=m \cdot 2^{x}(1-x) x$$ has a unique solution.
7. Let $a, b, c$ be non-negative numbers such that $(a+b)(b+c)(c+a)>0$. Find the minimum value of the expression $$P=\sqrt{\frac{a}{b+c}}+\sqrt{\frac{b}{c+a}}+\sqrt{\frac{c}{a+b}}+\frac{4 \sqrt{a b+b c+c a}}{a+b+c}.$$
8. Given a triangle $A B C$ with $\widehat{C}=45^{\circ}$. Let $G$ be the centroid of $A B C .$ Let $\widehat{A G B}=\alpha$. Prove that $$\frac{\sqrt{2}}{\sin A \sin B}+3 \cot \alpha=1.$$
9. Let $a, b, c$ be positive number such that $a^{2}+b^{2}+c^{2}=3$. Show that $$\frac{1}{a+b}+\frac{1}{b+c}+\frac{1}{c+a} \geq \frac{1}{2 \sqrt{2}}\left(\sqrt{a^{2}+b^{2}}+\sqrt{b^{2}+c^{2}}+\sqrt{c^{2}+a^{2}}\right).$$
10. For any real numbers $a, b, c$ we let $$T(a, b, c)=|a-b|+|b-c|+|c-a|.$$ Consider a sequence $(*)$ of integers $x_{1}, x_{2}, \ldots, x_{12}$ satisfying the conditions: there exists a polynomial $f(x)$ with integral coefficients so that $f\left(x_{1}\right), f\left(x_{2}\right), \ldots, f\left(x_{12}\right)$ are different and $$880<\sum_{k<j=k \leq 12} T\left(f\left(x_{i}\right), f\left(x_{j}\right), f\left(x_{k}\right)\right) \leq 3758.$$ Show that from the sequence $(*)$ we can always extract an arithmetic progression with at least four terms.
11. Find all functions $h(x): \mathbb{R} \rightarrow \mathbb{R}$ which satisfy all of the following conditions
• $h(2019)=0$
• $h(x+1)=h(x)$, $\forall x \in \mathbb{R}$.
• $3^{x+y}[h(x) h(y)+h(x+y)]=3^{x}(y+1) h(x)+3^{y}(x+1) h(y)+3^{x y} h(x y)$, $\forall x, y \in \mathbb{R}$.
12. Given an acute triangle $A B C$ inscribed in a circle $(\Omega)$. The points $E$, $F$ are on the sides $CA$, $A B$ respectively so that the quadrilateral $B C E F$ is cyclic. The perpendicular bisector of $C E$ intersects $B C$, $E F$ at $N$, $R$ respectively. The perpendicular bisector of $B F$ intersects $B C$, $E F$ at $M$, $Q$ respectively. Let $K$ be the reflection point of $E$ over the line $R M$. Let $L$ be the reflection point of $F$ over the line $Q N$. Suppose that the intersection between $R K$ and $Q^{B}$ is $S$; the intersection between $Q L$ and $R C$ is $T$.
    a) Show that four points $Q, R, S, T$ both belong to a circle, say $(\omega)$.
    b) Show that $(\omega)$ and $(\Omega)$ are tangent to each.