Mathematics and Youth Magazine Problems 2021

Issue 523

1. Find all prime numbers $p$, $q$ and positive integers $n$ so that $$p(p+1)+q(q+1)=n(n+1).$$
2. Find all prime numbers $a, b, c, d$ satisfying $$\begin{cases}a+b^{2} &=c d \\ c+d^{2} &=17 b \end{cases}.$$
3. Find all solutions which are prime numbers of the following equation $$x^{y}+y^{x}+(x+y+1)^{3}=x^{3}+y^{3}+z+1.$$
4. Let $M$ be a point inside a square $A B C D$. The rays $A M$, $B M$, $C M$, $D M$ respectively intersect the circle circumscribing the square at $E$, $F$, $G$, $H$. The tangents of the circle at $F$ and $H$ meet at $K$ Show that three points $K$, $G$, $E$ are collinear.
5. Given positive numbers $a, b, c$ satisfying $a+b+c=3$. Show that $$\frac{a^{3}+b^{2}+c^{2}}{a^{2}+1}+\frac{b^{3}+c^{2}+a^{2}}{b^{2}+1}+\frac{c^{3}+a^{2}+b^{2}}{c^{2}+1} \geq \frac{9}{2}.$$
6. Solve the system of equations $$\begin{cases} x^{z}+y^{z}-2 &=z^{3}-z \\ y^{x}+z^{x}-2 &=x^{3}-x \\ z^{y}+x^{y}-2 &=y^{3}-y\end{cases}$$ where $x, y, z$ are integers.
7. Find real solutions of the equation $$x 2^{x^{2}}=2^{2 x+1}.$$
8. Given a triangle $A B C$ inscribed in a circle $(O)$. Let $M$ be a point on the arc $B C$ which does not contain $A$ ($M$ is different from $B$ and $C$). Draw $B E$ perpendicular to $A M$ ($E$ is on $A M$). Let $N$ be the intersection hetween $A M$ and $B C$, and $H$ the orthocenter of the triangle $C M N$. Show that the line $H E$ always passes through a fixed point when $M$ varies.
9. Given $n$ positive numbers $x_{1}, x_{2}, \ldots, x_{n}$. Find the minimum value of the expression $$S=\frac{\left(x_{1}+x_{2}+\ldots+x_{n}\right)^{1+2+\ldots+n}}{x_{1} x_{2}^{2} \ldots x_{n}^{n}}.$$
10. Find all functions $f: \mathbb{R}^{+} \rightarrow \mathbb{R}^{+}$ satisfying $$f(x f(y))+f\left(y^{2021} f(x)\right)=x y+x y^{2021},\, \forall x, y \in \mathbb{R}^{+}.$$
11. Consider the sequence $\left\{a_{n}\right\}$ with $$a_{1}=1,\, a_{2}=\frac{1}{2},\quad n a_{n}=(n-1) a_{n-1}+(n-2) a_{n-2} .$$ Find $\displaystyle\lim_{n\to\infty} \frac{a_{n}}{a_{n-1}}$.
12. Given a triangle $A B C$ and a point $M$ on the line passes through $B$, $C$ ($M$ is different from $B$, $C$). Let $K$, $L$ respectively be the second intersections between the circumcircles of the triangles $A M B$, $A M C$ and another line which passes through $M$ and different from $M A$ and $B C$. Let $P$, $Q$ respectively be the perpendicular projections of $A$ on $B K$, $C L$. $B K$ and $C L$ meet at $R$, $P O$ and $M K$ meet at $N$. Show that
    a) $\dfrac{N P}{N Q}=\dfrac{M B}{M C}$.
    b) If $M$ is the midpoint of $B C$ then $A K R L$ is a harmonic quadrilateral (a harmonic quadrilateral is a cyclic quadrilateral in which the products of the lengths of opposite sides are equal).

Issue 524

1. Find all prime numbers $x, y$ satisfying $x^{2}-2 y^{2}-1=0$.
2. Given a triangle $A B C$ $(A B < B C)$, the bisector of the angles $B A C$, $A B C$ intersect at $I$. Draw $ID$ perpendicular to $A B$ at $D$, $I E$ perpendicular to $A C$ at $E$. Let $M$, $N$ respectively be the midpoints of $B C$, $A C$. Denote $K$ the intersection between $D E$ and $M N$. Show that the points $B$, $I$, $K$ are collinear.
3. Find all pairs of integers $(x ; y)$ satisfying $$2025^{x}=y^{3}+3 y^{2}+2 y+6.$$
4. Given a triangle $A B C$ with the altitude $A H$. The incircle $I$ of the triangle is tangent to $B C$, $C A$, $A B$ respectively at $D$, $E$, $F$. Let $G$ be the intersection between $E F$ and $B I$, and $P$ the intersection between $A I$ and $F D$. Show that $G D$ is perpendicular to $H P$.
5. Suppose that $a, b, c, d$ are positive integers which satisfy $a b=cd$ and $c>a$, $d>a$. Show that $$\sqrt{b}-\sqrt{a} \geq 1.$$
6. Solve the system of equations $$\begin{cases} x^{3}+y^{3}+3 y &=x^{2}+2 y^{2}+x+8 \\ y^{3}+z^{3}+3 z &=y^{2}+2 z^{2}+y+8 \\ z^{3}+x^{3}+3 x &=z^{2}+2 x^{2}+z+8 \end{cases}$$
7. Given real numbers $a, b, c>\dfrac{1}{2}$. Show that $$\frac{1}{2 a-1}+\frac{1}{2 b-1}+\frac{1}{2 c-1}+\frac{4 a b}{1+a b}+\frac{4 b c}{1+b c}+\frac{4 c a}{1+c a} \geq 9.$$
8. Given a triangle $A B C$. Show that $$\cos ^{2} \frac{A}{2}+\cos ^{2} \frac{B}{2}+\cos ^{2} \frac{C}{2} \geq 18 \sin \frac{A}{2} \sin \frac{B}{2} \sin \frac{C}{2}.$$
9. Given positive numbers $x, y$ with $x<y$. Prove the following inequalities $$\sqrt{x^{2}-2 \sqrt{2} x+4} \cdot \sqrt{y^{2}-2 \sqrt{2} y+4} + \sqrt{x^{2}+2 \sqrt{2} x+4} \cdot \sqrt{y^{2}+2 \sqrt{2} y+4} \geq 4(x+y).$$
10. Show that $$\left(\frac{3+\sqrt{5}}{2}\right)^{3^{n}}+\left(\frac{3-\sqrt{5}}{2}\right)^{3^{n}}$$ is an integer which is greater or equal to $3^{n+1}$ and is divisible by 3 for every $n \in \mathbb{N}$.
11. Given positive numbers $\alpha$ and $\beta$. Consider the following sequences $\left(x_{n}\right)$, $\left(y_{n}\right)$ $$x_{1}=\alpha,\, y_{1}=\beta,\quad x_{n+1}=\frac{5 x_{n} y_{n}}{2 y_{n}+3 x_{n}},\, y_{n+1}=\frac{5 x_{n+1} y_{n}}{2 y_{n}+3 x_{n+1}},\, \forall n=1,2, \ldots$$ Find $\lim _{n \rightarrow \infty} x_{n}$ and $\lim _{n \rightarrow \infty} y_{n}$.
12. Given an acute triangle $A B C$ inscribed in a circle $(O)$ and suppose that $A D$ is the altitude. The tangent lines of $(O)$ at $B$ $C$ intersect at $T .$ On the line segment $A D$, choose $K$ so that $\widehat{B K C}=90^{\circ}$. Let $G$ be the centroid of $A B C . K G$ intersects $O T$ at $L .$ The points $P$, $Q$ are on the line segments $B C$ so that $L P || O B$, $L Q || O C$. The points $E$, $F$ respectively on the line segments $C A$, $A B$ so that $Q E$, $P F$ are both perpendicular to $B C$. Let $(T)$ be the circle with center $T$ which passes through $B$, $C$. Show that the circumcircle of $A E F$ is tangent to $(T)$.

Issue 525

1. Find prime numbers $a, b, c, d$ so that $a>3 b>6 c>12 d$ and $$a^{2}-b^{2}+c^{2}-d^{2}=1749.$$
2. Given an isosceles triangle $A B C$ with the vertex angle $A$ $(\hat{A}=20^{\circ})$. Let $D$, $E$ be the points on $A C$ so that $D$ is in between $A$ and $E$, and $A D=C E=B C$. Find the measurement of the angle $\widehat{D B E}$.
3. Given a polynomial $f(x)=x^{2}+a x+b$ where $a$, $b$ are integers. Prove that there always exist integers $m$ so that $f(m)=f(2021) \cdot f(2022)$.
4. Let $M$ be a point lying inside the triangle $A B C$ so that $\widehat{M B A}=\widehat{M C A}$. Draw the parallelogram $B M C D$. Show that the angle bisector of $\widehat{B A C}$ and the angle bisector of $\widehat{M C D}$ are perpendicular to each other.
5. Solve the system of equations $$\begin{cases} x y z+x+y+z &=x y+y z+z x+2 \\ \dfrac{1}{x^{2}-x+1}+\dfrac{1}{y^{2}-y+1}+\dfrac{1}{z^{2}-z+1} &=1\end{cases}$$
6. Given numbers $x, y, z$ which are greater than 1 and satisfy $x+y+z+2=x y z$. Prove that $$\sqrt{x^{2}-1}+\sqrt{y^{2}-1}+\sqrt{z^{2}-1} \geq 3 \sqrt{3}.$$ When does the equality happen?
7. Find all triangles $A B C$ whose lengths of the sides are positive integers and the length of $A C$ is equal to the length of the internal angle bisector of the angle $A$.
8. Two circles $(O)$ and $\left(O^{\prime}\right)$ intersect at two points $A$ and $B$. Through a point $C$ lying on the opposite ray of the ray $B A$ draw the tangents $C D$ and $C E$ with $(O)$. The line segment $D E$ intersects $\left(O^{\prime}\right)$ at $F$. The tangent of $\left(O^{\prime}\right)$ at $F$ intersects $C D$ and $C E$ respectively at $M$ and $N$. Show that $A B M N$ is a cyclic quadrilateral.
9. Show that, for every positive integer $n$, we have $$\frac{1}{1} C_{8 n-1}^{1}-\frac{1}{2} C_{8 n-1}^{3}+\frac{1}{3} C_{8 n-1}^{\delta}-\ldots-\frac{1}{2 n-1} C_{8 n-1}^{4 n-5}+\frac{1}{2 n-1} C_{8 n-1}^{4 n-3} \leq \frac{(8 n-1) !}{((4 n) !)^{2}}-\frac{7}{4}.$$
10. The sequence $\left(a_{n}\right)$ is determined as follows $$a_{1}=1,\, a_{2}=3,\quad \log _{2} a_{n+2}=\log _{3}\left(a_{n}+1\right),\, \forall n=1,2, \ldots.$$ Show that the sequence $\left(a_{n}\right)$ converges and find its limit.
11. Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ so that $$f\left(x+y^{4}\right)=f(x)+y f\left(y^{3}\right),\, \forall x, y \in \mathbb{R}$$
12. Given a quadrilateral $A B C D$ and a point $E$ is on $A B$. A point $F$ varies on $C D$. The points $M$ and $N$ respectively are the perpendicular projections of $C$ and $D$ on $E F$. Assume that $P$ is the intersection between the line passing through $M$ and perpendicular to $A D$ and the line passing through $N$ and perpendicular to $B C$. Show that the incenter of the triangle $M N P$ belongs to a fixed circle.

Issue 526

1. Given positive integers $m, n$ which are coprime and satisfy $m+n \neq 90$. Find the maximal value of the product $mn$.
2. Find all positive integers $n$ so that $n^{2021}+n+1$ is a prime.
3. Solve the equation $$x^{4}-6 x-1=2(x+4) \sqrt{2 x^{3}+8 x^{2}+6 x+1}.$$
4. Given acute triangle $A B C$ with $A B+A C=2 B C$ which is inscribed in the circle $(O)$ and is circumscribed the circle $(I)$. Let $M$ be the midpoint of the major arc $B C$. The line $M I$ intersects the circumcircle of $B I C$ at $N$. The line $OI$ intersects $B C$ at $P$. Show that the line $P N$ is tangent to the circumcircle of $B I C$.
5. Find all pairs of integers $(a ; b)$ so that the following system of equations $$\begin{cases}x^{2}+2 a x-3 a-1 &=0 \\ y^{2}-2 b y+x &=0\end{cases}$$ has exactly $3$ different real roots $(x ; y)$.
6. Given positive numbers $a, b, c$. Show that $$\dfrac{(a-b)^{2}}{(b+c)(c+a)}+\dfrac{(b-c)^{2}}{(c+a)(a+b)}+\dfrac{(c-a)^{2}}{(a+b)(b+c)} \geq \frac{3\left(a^{2}+b^{2}+c^{2}\right)}{(a+b+c)^{2}}-1.$$
7. Given a triangle $A B C$ with the lengths of the altitudes $h_{a}$, $h_{b}$, $h_{c}$ and its half perimeter $p$ satisfying $h_{a}^{2}+h_{b}^{2}+h_{c}^{2}=p^{2}$. Show that the triangle $A B C$ is equilateral.
8. Given a tetrahedron $S.ABC$ whose base is a right triangle with the right angle $A$ and $A C > A B$. Let $A M$ be the median and $I$ the incenter of the base. Suppose that $I M$ is perpendicular to $B I$. Compute the value of the expression $$T = \frac{V_{S . AI B}}{V_{S . A I C}}+\frac{V_{S . A I C}}{V_{S . B I C}}+\frac{V_{S . B I C}}{V_{S . A B}}.$$
9. Suppose that $a, b, c$ are the lengths of three sides of a triangle and $n \in \mathbb{N}^{*}$. Prove that $$\frac{a^{n}}{(b+c)^{n}-a^{n}}+\frac{b^{n}}{(c+a)^{n}-b^{n}}+\frac{c^{n}}{(a+b)^{n}-c^{n}} \geq \frac{3}{2^{n}-1}.$$
10. For each positive integer $n$, let $S_{n}=\sum_{k=1}^{n}\left[\sqrt{k^{2}+4 \sqrt{k^{2}+2 k+2}}\right]$, where $[x]$ is the greatest integer which does not exceed $x .$ Find all the positive integers $n$ so that $S_{n}$ is a power of a prime.
11. Find all functions $f: \mathbb{Z} \rightarrow \mathbb{Z}$ satisfying $$f(x+f(y))=f(x),\, \forall x, y \in \mathbb{Z}.$$
12. Given an acute triangle $A B C$ and $(O ; R)$ is its circumcircle. The altitude $A D=\sqrt{2} R$. Let $M, N$ respectively be the intersections between $A B$, $A C$ and the circle with the diameter $A D$. Let $P$ be the second intersection between $(O ; R)$ the circle with the diameter $A D$ and $Q$ the reflection point of $P$ in $M N$. Show that
    a) $2 S_{A M N}=S_{A B C}$.
    b) $\widehat{B Q N}=\widehat{C Q M}=\dfrac{\pi}{2}$.

Issue 527

1. Find natural numbers $a$ and $b$ given that the sum of four numbers $a+b$, $a-b$, $ab$, $a: b$ is equal to $1575$.
2. Given triangle $A B C$ with $\widehat{A}=120^{\circ}$, $\widehat{B}=40^{\circ}$. On the side $A C$ choose the point $M$ such that $A B=A M$. On the opposite ray of $A B$ choose the point $N$ such that $\widehat{A M N}=40^{\circ}$. Find the measurement of the angle $\widehat{B N C}$.
3. Find all pairs of integers $(x ; y)$ satisfying $0 \leq x+y \leq 6$ and $$x-\frac{1}{x^{3}}=y-\frac{1}{y^{3}}.$$
4. Given an acute triangle $A B C$ inscribed in a circle $(O ; R)$. The altitudes $A D$, $B E$, $C F$ meet at $H$. Let $M$ be the midpoint of $A H$. Draw $M N$ perpendicular to $B M$, $N$ is on $A C$. Show that $O N || B C$ and $E M=R \cos A$.
5. Solve the equation $$x^{2} \sqrt[4]{2-x^{4}}-x^{4}+x^{3}-1=0.$$
6. Given real numbers $x, y, z$ such that $x^{2}+y^{2}+z^{2}=3$. Prove that $$8(2-x)(2-y)(2-z) \geq(x+y z)(y+x z)(z+x y)$$
7. Find all triangles $A B C$ such that the lengths of all sides are positive integers and the length of $A C$ is equal to the length of the interior angle bisector of the angle $A$.
8. Given an acute triangle $A B C$ inscribed in a circle $(O)$. Let $I_{a}$, $I_{b}$, $I_{c}$  respectively be the centers of the excircles of the angles $A$, $B$, $C$. The line $A I_{a}$ intersects $(O)$ at $D$ which is different from $A$. On $I_{b} D$, $I_{c} D$ respectively choose the points $E$, $F$ such that $\widehat{A B C}=2 \widehat{I_{a} B E}$, $\widehat{A C B}=2 \widehat{I_{a} C F}$, $E$, $F$ are inside the trangle $I_{a} B C$. Show that $E F$ intersects $I_{b} I_{c}$ at some point on $(O)$.
9. Given a function $$f(x)=\frac{x^{2}+a x+b}{x^{2}+1}$$ with $a, b$ are integers. Suppose the range of $f(x)$ is the set of $11$ integers. Find the maximum and minimum values of the expression $M=a^{2}+b^{2}$.
10. For each positive integer $n$, let $f(n)=\left(n^{2}+n+1\right)^{2}+1$. Find the smallest positive integer $k$ such that$$f(n) \cdot f(n+1) \ldots f(n+k-1)$$ is a perfect square for some positive integer $n$.
11. Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ satisfying $$f(f(x)+y)=f(f(x))-y f(x)+f(-y)-2,\, \forall x, y \in \mathbb{R}.$$
12. Given an isosceles triangle $A B C$ with the vertex angle $A$ inscribed in a circle $(O)$. Suppose that $A D$ is a diameter of $(O)$. The points $E$, $F$ respectively on $D C$, $D B$. Let $G$ be on $E F$ such that $\dfrac{G F}{G E}=\dfrac{F B}{C E}$. Show that $C G$ and $A F$ meets each other at some point on $(O)$.

Issue 528

1. Assume that $A$ is a natural number which has two prime factors $p$ and $q$ only. Let $S$ be the sum of all positive factors of $A$. Show that $S < 2 A$.
2. Denote $a_{n}$ the integer which is closest to $\sqrt{n}$ $(n \in \mathbb{N}^{*})$, for example $$\sqrt{1}=1=a_{1} ; \quad \sqrt{2} \approx 1,4 \Rightarrow a_{2}=1 ; \quad \sqrt{3} \approx 1,7 \Rightarrow a_{3}=2 ; \ldots.$$ Compute $$\frac{1}{a_{1}}+\frac{1}{a_{2}}+\ldots+\frac{1}{2020}+\frac{1}{2021}.$$
3. Find all $4$-tuples of positive integers $(x ; y, z, t)$ satisfying $$\begin{cases}x+y+z+t=54 \\ \dfrac{x z}{(x+y)(z+t)} =\dfrac{23}{104} \\ x>z, x+y>z+t \end{cases}.$$
4. Given an acute triangle $A B C$ ($CA < CB$) with the orthocenter $H$. Let $D$ be the second intersection between $B C$ and the circumcircle of $A B H$. Denote $P$ the intersection between $D H$ and $A C$. Let $M$ be the second intersection between $A H$ and the circumcircle of $A P D$; and $N$ the second intersection between the circumcircle of $A B C$ and the circumcircle of $A P D$. Show that three points $C, M$, and $N$ are collinear.
5. Given positive numbers $a, b, c$ satisfying $4 b^{2}+c^{2}=a c$. Find the minimum value of the expression $$P=\frac{2 a}{a+2 b}+\frac{b}{b+c}+\frac{c}{c+a} .$$
6. Solve the equation $$\sin 3 x-\cos 3 x+\sin x+\cos x=\frac{1}{\sin 3 x+\cos x}-\frac{1}{\cos 3 x-\sin x} .$$
7. Given positive numbers $a, b, c$ satisfying $a^{2}+b^{2}+c^{2} \geq 2(a b+b c+c a)$. Find the minimum value of the expression $$P=a+b+c+\frac{8}{a b c} .$$
8. Given two circles $(O ; R)$ and $\left(O^{\prime} ; R^{\prime}\right)\left(R>R^{\prime}\right)$ which is internally tangent to each other at $A$. Let $M$ be a point on $\left(O^{\prime} ; R^{\prime}\right)(M \neq A) .$ The tangent of $\left(O^{\prime} ; R^{\prime}\right)$ at $M$ intersects $(O ; R)$ at $P$ and $Q .$ The circumcircle of $O^{\prime} P Q$ intersects $\left(O^{\prime} ; R^{\prime}\right)$ at $B$ and $C$. Show that $A B M C$ is a harmonic quadrilateral, i.e. a cyclic quadrilateral of which the products of opposite sides are equal.
9. Given a triangle $A B C$ and $x, y, z$ positive numbers. Show that $$\frac{x}{y+z}(1+\cos A)+\frac{y}{z+x}(1+\cos B)+\frac{z}{x+y}(1+\cos C) \geq \frac{\sqrt{3}}{2}(\sin A+\sin B+\sin C) .$$
10. Two boxes contain $25$ small white and black balls. From each box, pick randomly $1$ ball. Find the probability that we get $2$ balls with different colors assuming that the box with more balls has more black balls and the probability to get two black balls is $0,42$.
11. Find all functions $f: \mathbb{R} \rightarrow(0+\infty)$ satisfying $$\frac{1}{2015} \leq\left(\frac{f(x)}{f(r)}\right)^{\frac{1}{(x-r)^{2}}} \leq 2015, \forall x \in \mathbb{R}, \forall r \in \mathbb{Q}, x \neq r$$
12. Outside a triangle $A B C$, draw pairwise similar triangles $B C P$, $A C Q$, $A B R$, and $B A S$. Let $K$, $L$ respectively be the midpoints of $B C$ and $C A .$ Prove that two triangles $R P K$ and $S Q L$ have the same area.

Issue 529

1. Given $$A=\frac{1.2022+2.2021+3.2020+\ldots+20221}{2^{2}+6^{\prime}+12^{4}+\ldots+(k \cdot(k+1))^{1.2}+\ldots+4090506^{2033}}.$$ Compare $A$ and $\dfrac{1}{2}$.
2. Given an isosceles triangle $A B C$ with the vertex angle $A$. Choose $2$ different points $M$, $N$ inside the triangle so that $A M=C N$, $B M=A N$. The lines $B M$ and $A N$ intersect at $E$, the lines $C N$ and $A M$ intersect at $F$. Show that $A E=A F$.
3. Given real numbers $x, y, z$ satisfying $x \geq 1$, $y \geq 2$, $z \geq 3$. Prove that $$x^{2}+y^{2}+z^{2}+x y+y z+z x+25 \geq 7 x+8 y+9 z.$$
4. Given a right triangle $A B C$ with the right angle $A$ and the altitude $A H$. On the ray $H C$ we choose $D$ so that $H D=H A$. On the ray $A B$ we choose $E$ so that $A E=A C$. Prove that the distance from $E$ to $B C$ is equal to the length $D C$.
5. Solve the equation $$8 x^{4}+32 x^{3}+32 x^{2}-x-3=0.$$
6. Given a triangle $A B C$. Let $B C=a$, $C A=b$, $A B=c$. Show that $$3\left(a^{3}+b^{3}+c^{3}\right)+4 a b c \geq \frac{13}{27}(a+b+c)^{3}.$$
7. Given the numbers $x, y \in(0 ; 1)$. Find the minimum value of the expression $$P=\sqrt{x}+\sqrt{y}+\sqrt[4]{12} \sqrt{x \sqrt{1-y^{2}}+y \sqrt{1-x^{2}}} .$$
8. Given an equilateral triangle $A B C$ with the length of a side $a$ inscribed in the circle $(O)$. A point $M$ lies on the circle $(O)$. Find the minimum value of the expression $$P=\frac{1}{16} M A^{8}+M B^{8}+M C^{8} .$$
9. Given a triangle $A B C$. Assume that the measurements of the angles $A$, $B$, $C$ form a geometric sequence with the common ratio $q=2$. Let $R$, $m_{o}$, $m_{b}$, $m_{c}$ respectively be the circumradius, and the lengths of the medians $m_{a}$, $m_{b}$, $m_{c}$ of the triangle. Suppose that $x_{0}$ is a root of the equation $$x^{3}+m_{a} x^{2}+m_{b} x+m_{c}=0.$$ Show that $x_{0}^{2}<\dfrac{21 R^{2}+4}{4}$.
10. Find the positive integral solutions of the equation $$x^{2}+y^{2}=19^{5^{4}}(6-z).$$
11. Given a function $f: \mathbb{R} \rightarrow S$ where $S$ is a bounded set. Suppose furthermore that $f^{2}(0)+f(0)=0$ and $$\left|f\left(x+y^{2}+z^{3}\right)-f(x)-f^{2}(y)-f^{3}(z)\right| \leq \frac{1}{2},\,\forall x, y, z \in \mathbb{R}.$$ Show that $$|f(x)-f(-x)| \leq 2,\,\forall x \in \mathbb{R}.$$
12. Given a triangle $A B C$ inscribed in a circle $(O)$. The points $X, Y$ belong to $(O)$ so that $A X || B C$, $B Y || A C$. Let $Z$, $T$ respectively be the intersections between $X Y$ and $A C$, $B C$. Let $O_{1}$, $O_{2}$, $O_{3}$, $O_{4}$ respectively be the circumcenters of the triangles $A X Z$, $B Y T$, $CYT$, $CXZ$. Show that $O_{1} O_{2} O_{3} O_{4}$ is a parallelogram.

Issue 530

1. Find integers $x, y, z$ satisfying $$35(x y z+x+z)=52(x y+1).$$
2. We mark $n$ different real numbers on a circle $(n \geq 3)$ such that each number is the product of its two adjacent numbers. If the first two numbers are $a$ and $b$, then determine $n$ and all other numbers.
3. Given positive integers $a, b, c$ satisfying $a<b<c$ and $$\frac{1}{[a ; b]}+\frac{1}{[b ; c]}=\frac{1}{2020}.$$ Find the maximum value of $a$.
4. Given an acute triangle $A B C$ inscribed in a circle $(O)$. Let $H$ be the orthocenter of $A B C$. The perpendicular bisector of $A H$ respectively intersects the sides $A B, A C$ at $P$ and $Q$. On the opposite ray of the ray $A O$ choose a point $D$ ($D$ is different from $A$). The rays $D B$, $D C$ respectively intersect $(O)$ at $E$ and $F$ other than $B$ and $C$. Through $D$ draw perpendicular line to $D O$, this line intersects $B F$, $C E$, $O P$, $O Q$ respectively at $M$, $N$, $S$, $T$. Prove that $M T=N S$.
5. Find all pairs of integers $(x ; y)$ satisfying $$x^{2}-2 x y+2 \sqrt{2 x y-y^{2}}=1.$$
6. Consider all numbers $a, b, c$ satisfying $a \geq 0$, $b \geq 0$, $c \geq 1$ and $a+b+c=2$. Find the maximum value of the expression $$P=(a b+b c+c a)\left(a^{2} b^{2}+b^{2} c^{2}+c^{2} a^{2}+1\right) .$$
7. For an arbitrary triangle $A B C$, show that $$\left|m_{a}-m_{b}\right|+\left|m_{b}-m_{c}\right|+\left|m_{c}-m_{a}\right| \geq \sqrt{a^{2}+b^{2}+c^{2}-a b-b c-c a}$$ 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 medians from the vertices $A$, $B$, $C$.
8. Given an acute triangle $A B C$ $(A B<A C)$ inscribed in a circle $O$. Let $B E$, $C F$ be two altitudes of the triangle. The line $E F$ intersects the line $B C$ at $S$, $O A$ intersects $E F$ at $K$. Let $N$ be the midpoint of $F C$. Show that $\widehat{S A C}=\widehat{N K F}$.
9. Consider all pairs of real numbers $a, b$ such that $\sin a+\sqrt{3} \cos b=2$. Find the minimum value of the expression $$P=\sin b+\sqrt{3} \cos a.$$
10. Suppose that $x, y, z$ are positive numbers and $p$ is a prime number satisfying $0<x<y<z<p$. Given that $x^{3}$, $y^{3}$, $z^{3}$ have the same remainder when dividing by $p$. Show that $x^{2}+y^{2}+z^{2}$ is divisible by $x+y+z$
11. Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that $$(x-y) f(x+y)-(x+y) f(x-y)=4 x y\left(x^{2}-y^{2}\right)$$ for all real numbers $x, y$.
12. Given the square $A B C D$. On the side $A B$ choose an arbitrary point $K$. The angle bisectors of the angles $\widehat{A D K}$ and $\widehat{B C K}$ intersect $A B$ respectively at $E$ and $F .$ On $A C$, choose the point $M$ such that $M F$ is parallel to $B D$. From $M$ draw the line parallel to $A B$ and it intersects $D E$ at $N$. Show that $$A E^{2}+B F^{2}=2 M N^{2} .$$

Issue 531

1. For an arbitrary natural number $m$, show that $$3 \mid \left(2^{m+1}+1\right) \cdot\left(2^{m}+1\right).$$
2. Find all positive integer $n$ so that all the digits of the number $6^{n}+1$, written in the decimal system, are equal.
3. Given positive numbers $a$ and $b$ satisfying $[a]=[b]$, $a<b$. Show that $$\frac{1+b^{2}}{1+a^{2}}<2,5.$$ (The notation $[x]$ denotes the integral part of the number $x$.)
4. Given a square $A B C D$. A right angle $x A y$ rotates about $A$. The ray $A x$ intersects $B C$ and $C D$ respectively at $M$ and $N$. The ray $A y$ intersects $B C$ and $C D$ respectively at $P$ and $Q$. The line $Q M$ meets $N P$ at $R$. Let $I$ and $K$ respectively be the midpoints of $P N$ and $Q M$. Prove that $I$, $B$, $K$, $D$ are collinear.
5. Given positive numbers $a$, $b$, $c$, $d$. Prove that $$a^{4} b+b^{4} c+c^{4} a+a b c\left(a^{3}+b^{3}+c^{3}\right) \geq (a+b+c)(3 a b c-1).$$
6. Given positive numbers $a$, $b$, $c$. Show that at least one of the following equations does not have solution $$\begin{align}\sqrt{a+b} x^{2}-2 \sqrt{a} x+\sqrt{c} &=0 \\ \sqrt{b+c} x^{2}-2 \sqrt{b} x+\sqrt{a} &=0 \\ \sqrt{c+a} x^{2}-2 \sqrt{c} x+\sqrt{b} &=0 \end{align}$$
7. Given a triangle $A B C$ with $A B=c$, $B C=a$, $C A=b$. Let $G$ be the centroid of the triangle. Prove that \[ \frac{G A^{2}}{b c}+\frac{G B^{2}}{c a}+\frac{G C^{2}}{a b} \geq 1 \text {. } \]
8. Given a triangle $A B C$ $(AB <A C)$ inscribed in a circle $(O)$. The angle bisector of $A$ intersects $(O)$ at $D$, intersect $B C$ at $E$. Let $M$ be the midpoint of $B C$, and $H$ the perpendicular prejection of $M$ on $A D$. Prove that $$\sqrt{A D \cdot E H}=\frac{A C-A B}{2}.$$
9. Solve the equation $$2021^{x}=1+2019 x+\log _{2021}(1+2020 x) .$$
10. Find all pair of co-prime positive integers $(m, n)$ so that $\left(2^{m}-1\right)\left(2^{n}-1\right)$ is a perfect square.
11. Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ satisfying $$f(x f(y))+y+f(x)=f(f(x+y))+y f(x)$$ for all arbitrary real numbers $x, y$.
12. Given a convex quadrilateral $A B C D$. Let $\left(I_{1}\right)$, $\left(I_{2}\right)$ respectively be the incircles of $A D C$, $B D C$. Assume that $T_{1} T_{2}$ is the other external tangent of $\left(I_{1}\right),\left(I_{2}\right)$ (where $T$, belongs to $\left(I_{1}\right)$ and $T_{2}$ belongs to $\left.\left(I_{2}\right)\right)$. Show that $A T_{1}, B T_{2}$, $I_{1} I_{2}$ are concurrent.

Issue 532

1. Find integral solutions of the equation $$x^{2}+(x+1)^{2}=y^{4}+(y+1)^{4}.$$
2. Given a triangle $A B C$ with $\widehat{A}=120^{\circ}$, $\widehat{B}=40^{\circ}$. On the side $A C$ choose $M$ so that $A B=A M$. On the opposite ray of the ray $A B$ choose $N$ so that $\widehat{A M N}=40^{\circ}$. Find the size of the angle $\widehat{B N C}$.
3. Find integral part of the expression $$A=(\sqrt{n}+\sqrt{n+1}+\sqrt{n+2})^{2}$$ where $n$ is an positive integer.
4. Given a quadrilateral $A B C D$ inscribed in a semicircle $(O)$ with the diameter $A D$. Let $I$ be the intersection between $A C$ and $B D$. The external angle bisector of the angle $I$ of BIC intersects $A B$, $C D$ respectively at $E$ and $F$. Show that the line perpendicular to $A B$ at $E$, the line perpendicular to $C D$ at $F$, and the line $O I$ are concurrent.
5. Solve the system of equations \[\begin{cases}x^{2}(x-y)+3 y^{2}+x(1-3 y) &=y \\ 3 \sqrt{3 x^{2}-9 y+2}+\sqrt[3]{y^{3}-21}+3 &=x\end{cases}. \]
6. Given $n$ positive numbers $a_{1}, a_{2}, \ldots, a_{n}$ $(n \in \mathbb{N}, n \geq 3)$. Prove that $$\frac{a_{1}^{2}}{a_{2}\left(a_{1}+a_{2}\right)}+\frac{a_{2}^{2}}{a_{3}\left(a_{2}+a_{3}\right)}+\ldots+\frac{a_{n}^{2}}{a_{1}\left(a_{n}+a_{1}\right)} \geq \frac{n}{2} .$$
7. Given real numbers $x, y, z$ satisfying $x^{2}+y^{2}+z^{2}=1$. Find the maximum and minimum values of the expression $$P=x y(x+y)+y z(y+z)+z x(z+x).$$
8. Given a pyramid $S.ABCD$. Let $O$ be the intersection between $A C$ and $B D$. Let $h_{1}$, $h_{2}$, $h_{3}$, $h_{4}$ respectively be the distances from $O$ to the planes $(S A B)$, $(S B C)$, $(S C D)$, $(S D A)$. Show that two planes $(S A C)$ and $(S B D)$ are perpendicular if and only if $$\frac{1}{h_{1}^{2}}+\frac{1}{h_{3}^{2}}=\frac{1}{h_{2}^{2}}+\frac{1}{h_{4}^{2}}.$$
9. Given a triangle $A B C$ with the lengths of the sides $B C$, $C A$, $A B$ respectively are $a$, $b$, $c$. Let $r_{a}$, $r_{b}$, $r_{c}$ respectively be the radii of the escribed circles opposite to the angles $A$, $B$, $C$. Show that $$\frac{r_{a}}{a \sin A}+\frac{r_{b}}{b \sin B}+\frac{r_{c}}{c \sin C} \geq 3.$$
10. Find all positive integers $n$ so that $n !$ is divisible by $2^{n-1}$.
11. Given a continuous function $f: \mathbb{R} \rightarrow \mathbb{R}$ satisfying $f(x+y)=f(f(x)) \cdot f(y)$ for any pair of real numbers $x$ and $y$. Show that $f$ is a constant function.
12. Let $(O)$, $(I)$ respectively be the circumcircle and the incircle of $A B C$. Let $(D)$ be the circle which is internally tangent to $(O)$ at $X$ and is tangent to the sides $A B$, $A C$. Let $M$ be the intersection between the line perpendicular to $A X$ at $X$ and the tangent of $(D)$ parallel to $B C$ (this tangent is in the same side with $X$ determining by $B C$). Show that $A M$, $B C$ and $O F$ are concurrent.

Issue 533

1. Given $$A=\frac{1}{2021}+\frac{1}{2022}+\frac{1}{2023}+\ldots+\frac{1}{8082}+\frac{1}{8083}$$ where the denominators are consecutive integers from $2021$ to $8083$. Compare $A$ and $\dfrac{11}{6}$.
2. Find all prime numbers $p, q$ so that $p+4 q$ and $q+4 p$ are both perfect squares.
3. Find all triples $(x ; y ; p)$ satisfying $x^{5}+x^{4}+1=p^{y}$, where $x$, $y$ are positive integers and $p$ is a prime number.
4. Suppose that $A B$ is a fixed chord which is not a diameter of a circle $(O ; R)$. Let $M$ be a point moving on the major arc $A B$ ($M A < M B)$, and $N$ be the perpendicular projection of $O$ on the angle bisector of the angle $A M B$. The circumcircle of $A M N$ intersects $M B$ at $K$, intersects $A B$ at $D$. Show that the perpendicular through $K$ and parallel to $M N$ always passes through a fixed point.
5. Given positive numbers $a$, $b$, $c$, $d$ satisfying $$\frac{1}{a+3}+\frac{1}{b+3}+\frac{1}{c+3}+\frac{1}{d+3}=1.$$ Show that $$\sqrt{a b}+\sqrt{a c}+\sqrt{a d}+\sqrt{b c}+\sqrt{b d}+\sqrt{c d} \leq 6 .$$
6. Solve the system of equations $$\left\{\begin{array}{l}\sqrt{x^{2}+1} &=y+2 \sqrt{2} \\ 9 y^{2}(x+3 y) &=1-x^{3} y^{3}\end{array}\right.$$
7. Given positive numbers $a, b, c$ satisfying $a b c=1$. Show that $$\frac{3}{2} \max \{a, b, c\} \geq \frac{1}{1+a^{2}}+\frac{1}{1+b^{2}}+\frac{1}{1+c^{2}}  \geq \frac{3}{2} \min \{a, b, c\} .$$
8. Given a triangle $A B C$ inscribed in a circle $(O)$. The incircle $(I)$ of $A B C$ are tangent to $B C$, $C A$, $A B$ at $D$, $E$, $F$ respectively. Suppose that $A I$ intersects $B C$ at $S$ and intersects $(O)$ at the second point $M$. The circumcircles of the triangles $B S M$, $C S M$ intersect $M F$, $M E$ at $K$ and $L$ respectively.
   a) Show that four points $I$, $L$, $S$, $K$ both lie on a circle.
   b) Let $T$ be the second intersection between $M D$ and $(O)$. Prove that the circumcircle of the triangle $T K L$ is tangent to $(O)$.
9. Find real numbers $a$ and $b$ satisfying the following conditions
• $a<b \leq 2000\cdot 2001$;
• $(x+y+z-2000\cdot 2001)^{2} \leq \dfrac{x y z}{500 \cdot 2001}$ for all $x, y, z \in[a, b] $;
• $b-a$ obtains the maximum value.
10. Suppose that $a$, $b$, $c$ are positive numbers so that $a+10 b$, $b+10 c$, $c+10 a$ are either a power of $2$ or a power of $5$ . Show that $a b c$ is divisible by 10 but not divisible by $100$.
11. Show that we can choose some $a, b, c$ $(a, b, c \in \mathbb{R}^{+}; a, b, c \notin \mathbb{Z})$  so that the following conditions hold
• $x_{n}<1$, $\forall n \geq 1$;
• $\dfrac{x_{n+3}}{x_{n}}<1$, $\forall n \geq 1$;
• $\displaystyle\lim_{n \rightarrow+\infty}\left(x_{1}+x_{2}+\ldots+x_{n}\right)$ exists.
12. Given a scalene triangle $A B C$. Suppose that $(O)$, $(I)$ respectively are the circumcircle and the incircle of. Let $A_{0}$, $B_{0}$, $C_{0}$ respectively be the touch point between $(I)$ and $B C$, $C A$, $A B$. Let $A_{1}$, $B_{1}$, $C_{1}$ respectively be the perpendicular projections of $A_{0}$, $B_{0}$, $C_{0}$ on $B_{0} C_{0}$, $C_{0} A_{0}$, $A_{0} B_{0}$; and $A_{2}$, $B_{2}$, $C_{2}$ respectively the second intersections between $A A_{1}$, $B B_{1}$, $C C_{1}$ and $(O)$. Show that the circles $\left(A_{0} A_{1} A_{2}\right)$, $\left(B_{0} B_{1} B_{2}\right)$, $\left(C_{0} C_{1} C_{2}\right)$ have the same radical axis.

Issue 534

1. Consider the number $$A=2021^{x y}+y^{5}+3 y^{4}+4 y+12,$$ where $x$ and $y$ are non-negative integers. Find the minimum value, if exists, of the sum of all digits of $A$.
2. Given an equilateral triangle $A B C$. Let $M$ be the midpoint of $A B$. On the opposite ray of the ray $C M$, choose the point $D$ so that $C D=C A$ and on the ray $B C$, choose the point $E$ so that $E B$ is the angle bisector of $\widehat{A E D}$. Find the measurement of $\widehat{A E D}$.
3. Find the sum $S$ of all odd numbers which are less than $2022$, are not divisible by $5$ and are not divisible by $9$.
4. Given a triangle $A B C$ inscribed in a circle (O). Let $D$ be the midpoint of the $\operatorname{arc} B C$ which does not contant $A$, and $E$ the midpoint of the arc $A C$ which does not contant $B$. $E D$ intersects $A C$, $B C$ respectively at $F$ and $G$. The line which passes through $F$ and is perpendicular to $A D$ intersects the line passing through $G$ and perpendicular to $B E$ at $H$. Let $O^{\prime}$, $O^{n}$ respectively be the circumcenters of the triangles $A F E$ and $B G D$. Show that $H O || O^{\prime} O^{\prime \prime}$.
5. Given positive numbers $a, b, c$ satisfying $a^{2}+b^{2}+c^{2}=12$. Find the minimum value of the expression $$P=\frac{1}{4-a}+\frac{1}{4-b}+\frac{1}{4-c}.$$
6. Solve the equation $$\sqrt{1-2 x}+\sqrt{x^{2}-3 x+1}+\sqrt{7 x^{2}-5 x+1}=x^{2}-5 x+3$$
7. Given $a_{k}, b_{k}>0, k=\overline{1, n}, n \in \mathbb{N}, n \geq 2$. Prove that $$\frac{1}{\frac{1}{\sum_{k=1}^{n} a_{k}}+\frac{1}{\sum_{k=1}^{n} b_{k}}} \geq \sum_{k=1}^{n} \frac{1}{\frac{1}{a_{k}}+\frac{1}{b_{k}}}$$
8. Given a triangle $A B C$ with the centroid $G$. The lines $A G$, $B G$, $C G$ intersects the circumcircle of $A B C$ at $A_{1}$, $B_{1}$, $C_{1}$. Show that
   a) $A B_{1} \cdot A C_{1} \cdot B C_{1} \cdot B A_{1} \cdot C A_{1} \cdot C B_{1} \leq 4 R^{4} r^{2}$;
   b) $B A_{1} \cdot C A_{1}+C B_{1} \cdot A B_{1}+A C_{1} \cdot B C_{1} \leq 2 R(2 R-r)$ where $R$, $r$ respectively is the circumradius and inradius of the triangle $A B C$.
9. Suppose that $a, b, c$ are the lengths of 3 sides of a triangle and $p$ is the triangle's half perimeter. Show that $$\frac{\sqrt{b c}}{p-a}+\frac{\sqrt{c a}}{p-b}+\frac{\sqrt{a h}}{p-c} \geq 6 .$$
10. Find all non-negative integers $a,b,c$ satisfying $a^{2}+1$, $b^{2}+2$ are prime numbers, $c$ is not divisible by $5$, and $$\left(a^{2}+1\right)\left(b^{2}+2\right)=c^{2}+9.$$
11. Given an even positive integer $n$ and a prime number even positive integer $p>n^n$ polynomial $$Q(x)=(x-1)(x-2) \ldots(x-n)+p$$ cannot be presented as a product of two non-constant polynomials with integral coefficients.
12. Given an oblique triangle $ABC$ and $(O)$ is its circumcircle. $A D$, $B E$, $C F$ are the altitudes. $A_{b}$, $A_{c}$ respectively are the points of reflection of $B$, in $F$, $E$. $A_{a}$ is the intersection between $E F$ and $A_bA_c$. The points $B_b$, $C_c$ are determined similarly. Show that $A A_{c}$, $B B_{b}$, $C C_{c}$ are concurrent at a point on $(O)$.