Mathematics and Youth Magazine Problems 2020

Issue 511

1. Find all $6$-digit natural numbers which are both a perfect square and a cube.
2. Given a triangle $A B C$ with $\widehat{A}=30^{\circ}$, $\widehat{B}=20^{\circ}$. On the side $A B$ choose the point $D$ such that $A D=B C$. Find the value of the angle $\widehat{B C D}$.
3. Assume that $a, b \in \mathbb{R}$ and $a^{2}+b^{2}+16=8 a+6 b$. Show that
   a) $10 \leq 4 a+3 b \leq 40$.
   b) $7 b \leq 24 a$.
4. Given a half circle with the center $O,$ the diameter $B C .$ Choose a point $G$ inside the half circle so that $\widehat{B G O}=135^{\circ}$. The line which is perpendicular to $G B$ at $G$ intersects the half circle at $A$. The incircle $I$ of $A B C$ is tangent to $B C$, $C A$ respectively at $D$ and $E .$ Show that $G$ lies on $E D$.
5. Suppose that $x, y, z$ are positive numbers satisfying $x+y \leq 2 z$. Find the minimum value of the expression $$P=\frac{x}{y+z}+\frac{y}{x+z}-\frac{x+y}{2 z}.$$
6. Show the inequality $$\left(\frac{x+y}{x-y}\right)^{2020}+\left(\frac{y+z}{y-z}\right)^{2020}+\left(\frac{z+x}{z-x}\right)^{2020}>\frac{2^{1010}}{3^{1009}}$$ where $x, y, z$ are different numbers.
7. Solve the system of equations $$\begin{cases}x_{2} &=x_{1}^{3}-3 x_{1} \\ x_{3} &=x_{2}^{3}-3 x_{2} \\ \ldots  & \ldots \\ x_{2020} &=x_{2019}^{3}-3 x_{2019} \\ x_{1} &=x_{2020}^{3}-3 x_{2020}\end{cases}$$
8. Given a right triangle $A B C$ with the right angle $A$ and the altitude $A H$. On the line segment $A H$ choose a point $I$, the line $C I$ intersects $A B$ at $E .$ On the side $A C$ choose the point $F$ such that $E F$ is parallel to $B C$. The line which passes through $F$ and is perpendicular to $C E$ at $N$ intersects $B I$ at $M$. Let $D$ be the intersection between $A N$ and $B C$. Prove that four points $M$, $N$, $D$, $C$ both lies on a circle.
9. Let $x$, $y$ be real numbers. Find the minimum value of the expression $$P=\sin ^{4} x\left(\sin ^{4} y+\cos ^{4} y+\frac{9}{8} \cos ^{2} x \cdot \sin ^{2} 2 y\right)+\cos ^{4} x.$$
10. Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ satisfying $$f(2 f(x)+2 y)=x+f(2 f(y)+x),\, \forall x, y \in \mathbb{R}.$$
11. There are $n$ $(n \geq 2)$ soccer teams attending a tournament. Each team will play with all other teams once. The winning team get 3 points, the losing team gets 0 point; and if the match ties, both teams get 1 point. After the tournament, we recognize that all teams got different total points. What is the possible minimal value for the difference between the team with the most points and the team with the least points?
12. Given a triangle $A B C$ with $I$ is the center of the excircle relative to the vertex $A$. This circle is tangent to $B C$, $C A$, $A B$ respectively at $M, N, P$. Let $E$ be the intersection between $M N$ and $B I$, and $F$ be the intersection between $M P$ and $CI$. The line $B C$ intersects $A E$, $A F$ respectively at $G$, $D$. Show that $A I$ is parallel to the line passing through $M$ and the center of the Euler circle of $A G D$.

Issue 512

1. Find all natural numbers $N$ so that the sum of its factors is equal to $2 N$ and the product of its factors is equal to $N^{2}$.
2. Given natural numbers $a, b, c$ such that $\dfrac{a}{3}=\dfrac{b}{5}=\dfrac{c}{7}$. Prove that $$\frac{2019 b-2020 a}{2019 c-2020 b}>1.$$
3. Solve the system of equations $$\begin{cases} x^{2} &=2 z-1 \\ y^{2} &=x z \\ z^{2} &=2 y-1\end{cases}.$$
4. Given an acute triangle $A B C$. Draw the altitudes $C H$, $B K$ ($H$, $K$ is respectively on $A B$ and $A C$). Choose two points $P$ and $Q$ on the ray $C H$ and the ray $B K$ respectively such that $\widehat{P A Q}=90^{\circ}$. Draw $A M$ perpendicular to $P Q$ ($M$ is on $P Q$). Show that $M B$ is perpendicular to $M C$.
5. Let $a, b, c$ be positive numbers satisfying $a b+b c+c a=8$. Find the minimum value of the expression $$P=3\left(a^{2}+b^{2}+c^{2}\right)+\frac{27(a+b)(b+c)(c+a)}{(a+b+c)^{3}}.$$
6. Let $x, y, z$ be positive numbers so that $x \geq z$. Find the minimum value of the expression $$P=\frac{x z}{y^{2}+y z}+\frac{y^{2}}{x z+y z}+\frac{x+2 z}{x+z}.$$
7. Find the integral solutions of the equation $$\tan \frac{3 \pi}{x}+4 \sin \frac{2 \pi}{x}=\sqrt{x}.$$
8. Given a triangle $A B C$ inscribed in a circle $(O)$ and $(I)$ is the incircle of the triangle. Let $M$ be the midpoint of $B C$ and $X$ the midpoint of the arc $\widehat{B A C}$ of $(O)$. Let $P$, $Q$ respectively be the perpendicular projections of $M$ on $C I$, $B I$. Show that $X I \perp P Q$.
9. Given a triangle $A B C$ with area $S$ and $B C=a$, $C A=b$, $A B=c$. Solve the system of equations (variables $x, y, z$) $$\begin{cases}a^{2} x+b^{2} y+c^{2} z &=4 S \\ x y+y z+z x &=1\end{cases}.$$
10. For any positive integer $n$ show that $n$ and $2^{2^{n}}+1$ are coprime.
11. The sequences $\left(x_{n}\right)$, $\left(y_{n}\right)$ are determined as follows $$x_{1}=3,\, x_{2}=17,\quad x_{n+2}=6 x_{n+1}-x_{n},\,\forall n \in \mathbb{N}^*,$$ $$y_{1}=4,\, y_{2}=24,\quad y_{n+2}=6 y_{n+1}-y_{n},\,\forall n \in \mathbb{N}^*.$$ Show that no term in these sequences $\left(x_{n}\right)$, $\left(y_{n}\right)$ is a cube number.
12. Given a right triangle $A B C$, with the right angle $A$, inscribed in a circle $(O)$. The point $A^{\prime}$ is the reflection point of $A$ over $0$. The point $P$ is the perpendicular projection of $A^{\prime}$ on the perpendicular bisector of $B C$. Let $H_{a}$, $H_{b}$, $H_{c}$ respectively be the orthocenter of $A P A^{\prime}$, $B P A^{\prime}$, $C P A^{\prime}$. Show that the circle $(H_aH_bH_c)$ is tangent to the circle $(O)$.

Issue 513

1. Find prime numbers $p$, $q$, $r$ satisfying $$p+q^{2}+r^{3}=200.$$
2. Consider the number $$P=\frac{2^{2}+1}{2^{2}+3.2+4}+\frac{3^{2}+1}{3^{2}+3.3+4}+\ldots+\frac{98^{2}+1}{98^{2}+3.98+4}$$ (including 97 terms). Show that $$\frac{6}{10^{6}}<P<\frac{1}{83325}.$$
3. Find all positive integers $a$, $b$, $c$, $d$ satisfying $$\begin{cases}a+b+c+d-3 &=a b \\ a+b+c+d-3 &=c d\end{cases}.$$
4. Given a quadrilateral $A B C D$ with $A B=A D$, $C B=C D$ and $\widehat{A B C}=90^{\circ}$. Let $R$ and $r$ be the circumradius and inradius of $A B C D$ respectively. Prove that $R \geq r \sqrt{2}$.
5. Given positive numbers $x$, $y$, $z$ satisfying $$\sqrt{x^{2}+y^{2}}+\sqrt{y^{2}+z^{2}}+\sqrt{z^{2}+x^{2}}=2020.$$ Find the minimum value of the expression $$T=\frac{x^{2}}{y+z}+\frac{y^{2}}{z+x}+\frac{z^{2}}{x+y}.$$
6. Suppose that $a$, $b$, $c$ are positive numbers and $a b c=1$. Show that $$\sqrt{\frac{a b}{b c^{2}+1}}+\sqrt{\frac{b c}{c a^{2}+1}}+\sqrt{\frac{c a}{a b^{2}+1}} \leq \frac{a+b+c}{\sqrt{2}}.$$
7. Find the real solutions of the system of equations $$\begin{cases} x &\notin(-\pi ; \pi) \\ \sin y-\sin x &=\dfrac{2 x y(\pi+x)}{\pi^{2}+x^{2}} \\ y^{3}+\pi^{3} & =x^{3}-3 \pi x y \end{cases}.$$
8. Given a circle $(O)$ and a point $P$ inside the circle and is different from $O$. A moving line $\Delta$ passing through $P$ but not $O$ intersects $(O)$ at $E$ and $F .$ The tangents at $E$, $F$ to the circle $(O)$ meet at $T$. Let $S$ be the intersection between the line segment $T P$ and $(O)$. Let $\omega$ be the circle which passes through $S$, $T$ and is tangent to $(O)$. Show that the circle $\omega$ always passing through a fixed point when $\Delta$ varies.
9. Given real numbers $x$, $y$, $z$ satisfying $x+y+z=0$. Find the minimum value of the expression $$S=\frac{1}{4 e^{2 x}-2 e^{x}+1}+\frac{1}{4 e^{2 y}-2 e^{y}+1}+\frac{1}{4 e^{2 z}-2 e^{z}+1}.$$
10. Three sequences $\left(a_{n}\right)$, $\left(b_{n}\right)$, $\left(c_{n}\right)$ are determined as follows $a_{0}=2$, $b_{0}=9$, $c_{0}=2020$ and $$\begin{cases}a_{n} & =-\dfrac{1}{4} a_{n-1}+\dfrac{1}{2} b_{n-1}+\dfrac{1}{2} c_{n-1} \\ b_{n} &=\dfrac{1}{2} a_{n-1}-\dfrac{1}{4} b_{n-1}+\dfrac{1}{2} c_{n-1} \\ c_{n} &=\dfrac{1}{2} a_{n-1}+\dfrac{1}{2} b_{n-1}-\dfrac{1}{4} c_{n-1}\end{cases}$$ for all $n=1,2, \ldots$. Find the limits $\displaystyle \lim_{n\to\infty} a_{n}$, $\displaystyle \lim_{n\to\infty}b_{n}$, $\displaystyle \lim_{n\to\infty}c_{n}$. Can you generalize this problem?
11. Let $h$ be a positive integer so that $p:=2^{h}+1$ is a prime number. Find the smallest positive integer $k$ so that $2^{k}-1$ is divisible by $p$.
12. Suppose that $(D)$ and $(O)$ are two circles which tangent to each other at $X$. In the case of internally tangent then $(D)$ is inside $(O)$. Let $A$ be a point on $(D)$ which is different from $X$ so that the tangent to $(D)$ at $A$ intersects $(O)$. Let $B$ be any point of that intersection. Show that the radical line of $(O)$ and the circle $(B, B A)$ is a tangent line to $(D)$.

Issue 514

1. Draw on a board $2019$ plus signs $(+)$ and $2020$ minus signs $(-)$. We perform a procedure as follows. We delete two arbitrary signs. If they are both plus or both minus, we will add back a plus sign. If not, we add back a minus sign. We do it for $4038$ times. What is the remain sign on the board?
2. Given a triangle $A B C$ with $\hat{A}=60^{\circ}$ and $A B+A C=2 B C$. Show that the median $A M,$ the altitude $B H$ and the angle bisector $C I$ of the triangle are concurrent.
3. Suppose that $a$, $b$, $c$ are positive numbers and $a+b+c=a b c$. Prove that $$\frac{a}{a^{2}+1}+\frac{b}{b^{2}+1} \leq \frac{c}{\sqrt{c^{2}+1}}.$$
4. Given an acute triangle $A B C$ inscribed in a circle $O$. Draw the altitude $A D$. Let goi $E$ and $F$ respectively be the perpendicular projections of $D$ on the sides $A B$ and $A C$. Suppose that two line segments $O A$ and $E F$ meet at $I$. Show that $$A B \cdot A C \cdot A I=A D^{3}.$$
5. Consider the polynomial $f(x)=(x+a)(x+b)$ where $a$ and $b$ are integers. Show that there always exists at least an integer $m$ so that $f(m)=f(2020) \cdot f(2021)$.
6. Let $a$, $b$, $c$, $d$ be non-negative numbers whose sum is equal to $1$. Find the maximum and minimum values of the expression $$P=\frac{a}{b+1}+\frac{b}{c+1}+\frac{c}{d+1}+\frac{d}{a+1}.$$
7. Given the system of equation $$\begin{cases}a_{11} x_{1}+a_{12} x_{2}+\ldots+a_{1 n} x_{n} &=0 \\ a_{21} x_{1}+a_{22} x_{2}+\ldots+a_{2 n} x_{n} &=0 \\ \ldots & \ldots \\ a_{n 1} x_{1}+a_{n 2} x_{2}+\ldots+a_{n n} x_{n} &=0\end{cases}$$ with the coefficients satisfying
   
   • $a_{i i}>0, \forall i=\overline{1, n}$
   • $a_{i j}<0, \forall i \neq j, \forall i, j=\overline{1, n}$
   • $\displaystyle \sum_{k=1}^{n} a_{i k}>0, \forall i=\overline{1, n}$.
   Prove the system of equation has unique solution $x_{1}=x_{2}=\ldots=x_{n}=0$.
8. Given a circle $O$ and a point $M$ outside the circle. Draw a secant $M A B$ ($A$ is in between $M$ and $B$). The tangents at $A$ and $B$ intersect at $C$. Draw $C D$ perpendicular to $M O$ $D E$ perpendicular to $C A$ and $D F$ perpendicular to CB. Show that the line $E F$ always passes through a fixed point when the secant $M A B$ varies. 
9. Given positive numbers $x$, $y$ satisfying $x<\sqrt{2} y$, $x \sqrt{y^{2}-\dfrac{x^{2}}{2}}=2 \sqrt{y^{2}-\dfrac{x^{2}}{4}}+x$. Find the minimum value of the expression $$P=x^{2} \sqrt{y^{2}-\frac{x^{2}}{2}}.$$
10. Let $S=1 ! 2 ! \ldots 100 !$. Show that there exists an interger $k$, $1 \leq k \leq 100$, so that $\dfrac{S}{k !}$ is a perfect square. Is such $k$ unique? (Notice that $n !=1.2 .3 \ldots n$ with $n \in \mathbb{N}$ and $0 !:=1$.)
11. Given a non-constant function $f(x)$ which is determined on $\mathbb{R}$. Show that there always exist a real number $a$, a non-empty proper subset $A$ and two functions $g(x), h(x)$ satisfying $g(x) \geq a$, $\forall x \in A$ and $h(x)<a$, $\forall x \in \bar{A}$ $(\bar{A}=\mathbb{R} \backslash A)$ so that $f(x)=g(x)+h(x)$, $\forall x \in \mathbb{R}$.
12. Given an acute triangle $A B C$. Two points $M$, $N$ are inside the side $B C$ such that $B M=C N$. The line which passes through $M$ and is perpendicular to $\mathrm{CA}$ intersects $A B$ at $F .$ The line which passes through $N$ and is perpendicular to $A B$ intersects $A C$ at $E$. Two lines $M F$ and $N E$ intersect at $P$. On the circumcircle of $P E F$ choose $Q$ so that $P Q \parallel B C$. Let $R$ be the reflection point of $Q$ in the midpoint of $B C .$ Show that $A R \perp B C$.

Issue 515

1. Two stations $A$ and $B$ are $999km$ away. The milestones along the railway from $A$ to $B$ show the distances from that point to $A$ and $B$ as follows $$0 / 999 ; 1 / 998 ; 2 / 997 ; \ldots ; 999 / 0.$$ Among these milestones, how many of them contains only two different digits?
2. Given a triangle $A B C$ with the side $B C$ is fixed and the vertex $A$ can vary. Draw the perpendicular bisector $A D .$ Through $C$ draw a perpendicular line to $A D$ at $N$. Let $M$ be the midpoint of $A C$. Show that when $A$ is moving, $M N$ always passes through a fixed point.
3. Suppose that $a$ and $b$ are positive integers so that $(a, 6)=1$ and $3 \mid a+b$. Assume that $p, q$ are prime numbers so that both $p q+a$ and $b p+q$ are also prime numbers. Prove that $a+6$ is a prime number.
4. Given a circle $(O, R)$. From a point $A$ outside the circle we draw two tangents $A B$, $A C$ ($B$, $C$ are touch points) and a secant $A D E$ $(D$ is in between $A$ and $E)$. The line $B C$ intersects $OA$ at $H$. From $H$ draw the line parallel to $B E$. That line intersects $A B$ at $K$. Show that $B D$ passes through the midpoint of $H K$.
5. Solve the system of equations $$\begin{cases}x^{3}+x(y+z)^{2} &=26 \\ y^{3}+y(z+x)^{2} &=40 \\ z^{3}+z(x+y)^{2} &=54\end{cases}.$$
6. Solve the equation $$\sqrt{x^{3}+x+2}=x^{4}-x^{3}-7 x^{2}-x+10.$$
7. Suppose that $x$, $y$ are positive numbers so that $x+y \leq 6$. Find the minimum value of the expression $$P=x^{2}(6-x)+y^{2}(6-y)+(x+y)\left(\frac{1}{x y}-x y\right).$$
8. Given a triangle $A B C$. Let $O$ and $I$ be the circumcenter and the incenter of the triangle respectively. Let $D$ be the second intersection between $(O)$ and $AI$. Let $P$ be the intersection between $B C$ and the line which passes though $I$ and perpendicular to $AI$. Let $Q$ be the reflection point of $I$ in $O$. Show that $$\widehat{P A Q}=\widehat{P D Q}=90^{\circ}.$$
9. Find the limit $$\lim_{n \rightarrow+\infty} \frac{\mid \sqrt[3]{1}]+[\sqrt[3]{2}]+\ldots+\left[\sqrt[3]{n^{3}+3 n^{2}+3 n}\right]}{n^{4}}$$ where $n$ is a positive integer and the notion $[x]$ denote the integer which does not exceed $x$.
10. Given a positive integer $n$ so that both $6 n+1$ and $20 n+1$ are perfect squares. Show that $58 n+11$ is a composite number.
11. Suppose that $g:[a, b] \rightarrow R$ is $a$ continuous functions with $g(a) \leq g(b)$ and $f:[a, b] \rightarrow[g(a), g(b)]$ is an increasing function. Show that the equation $f(x)=g(x)$ has at least one solution.
12. Given a triangle $A B C$. Let $O$ and $H$ be the circumcenter and the orthocenter of the triangle respectively. Let $S$ be the circumcenter of the triangle $O B C$. Denote $K$ and $L$ the reflection points of $S$ in $A B$ and $A C$ respectively. Show that $K L$ passes through the the midpoint of $O H$.

Issue 516

1. Let $$A=\frac{1}{1 ! 3}+\frac{1}{2 ! 3}+\frac{1}{3 ! 5}+\ldots+\frac{1}{(n-2) ! n}$$ for $n \in \mathbb{N}, n \geq 3,$ where $n !=1.2 .3 . . n .$ Show that $A<\dfrac{1}{2}$.
2. For any natural number $n$ which is divisible by $4,$ show that the number $a=27.7^{n}+2021$ cannot be a product of $m$ consecutive natural numbers $(m \in \mathbb{N}, m \geq 2)$
3. Given positive numbers $a, b, c, d$ satisfying $\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}+\dfrac{1}{d}=4$. Show that $$2(a+b+c+d)-4 \geq \sqrt[3]{\frac{a^{3}+b^{3}}{2}}+\sqrt[3]{\frac{b^{3}+c^{3}}{2}}+\sqrt[3]{\frac{c^{3}+d^{3}}{2}}+\sqrt[3]{\frac{d^{3}+a^{3}}{2}}$$
4. Given an acute triangle $\triangle A B C$. Outside the triangle, draw the equilateral triangle $\Delta A C E$ and the isosceles triangle $\Delta A B D$ with $\widehat{A B D}=120^{\circ}$. Let $I$ be the midpoint of $D E$ and $F(F \neq E)$ the other intersection between $D E$ and the circumcircle of $\Delta A C E .$ Show that $B$, $C$, $I$, $F$ lie on a same circle.
5. Let $f(x)=x^{2}+b x+c .$ Show that if the equation $f(x)=x$ has two distinct roots and $b^{2}-2 b-3 \geq 4 c$ then the equation $f[f(x)]=x$ has four distinct roots.
6. Let $f(x)=3 x^{2}+8 x+4 .$ Find the coefficient of $x^{4}$ in the polynomial $g(x)=f(f(f(x))))$
7. Given positive numbers $a, b, c .$ Prove that $$ \frac{5 a+c}{b+c}+\frac{6 b}{c+a}+\frac{5 c+a}{a+b} \geq 9.$$
8. A triangle $ABC$ inscribed in a circle O. Let $G$ be the centroid of the triangle. The medians $A A_{1}, B B_{1}, C C_{1}$ of the triangle respectively intersect (O) at $A_{2}, B_{2}, C_{2}$. Show that $$\frac{A_{1} A_{2}}{G A_{1}}+\frac{B_{1} B_{2}}{G B_{1}}+\frac{C_{1} C_{2}}{G C_{1}} \geq 3.$$
9. Solve the system of equations $$\begin{cases}\sqrt{x^{2}+x+1}-\sqrt{y^{2}-y+1} &=\sqrt{x^{2}+y^{2}-\dfrac{1}{2}} \\ 2 x^{3} y-x^{2} &=\sqrt{x^{4}+x^{2}}-2 x^{3} y \sqrt{4 y^{2}+1}\end{cases}.$$
10. Find all the prime numbers $p$ and $q$ so that $p^{2}+1 \mid 3^{4}+1$ and $q^{2}+1 \mid 3^{\prime}+1$
11. Find all continuous functions $f: \mathbb{R} \rightarrow \mathbb{R}$ satisfying $$f(4 x y)=f\left(2 x^{2}+2 y^{2}\right)+4(x-y)^{2},\, \forall x, y \in \mathbb R$$
12. Suppose that $A B C D$ is a parallelogram with the angle $\widehat{B A D}$ is acute. $A$ moving line $d$, which always passes through $B$, intersects $C D$ at $M .$ The line $A M$ intersects $B C$ at $N$. Two lines $B M, D N$ respectively intersect the circumcircle of the triangle $C M N$ at $K$, $L$ (besides $M$, $N$). Choose $P$ and $Q$ so that $P N=P K$,  $P N \perp A D$, $Q M=Q L$ and $Q M \perp A B$. Show that $P Q$ always passes through a fixed point when $d$ varies.

Issue 517

1. Find all the prime numbers $p$ so that $2^{p}+p^{2}$ is a prime.
2. Given an isosceles right triangle $A B C$ with the vertex angle $A$. The point $D$ inside $A B C$ such that $\widehat{A B D}=\widehat{B C D}=30^{\circ} .$ Compute the vertex angle $\widehat{C A D}$.
3. Given positive numbers $a, b, c$ such that $a+b+c=1 .$ Find the maximum value of the expression $$P=\frac{1}{a+3}+\frac{1}{b+3}+\frac{1}{c+3}-\frac{1}{3(a b+b c+c a)}.$$
4. Suppose that $A B C D$ is a thombus with $\widehat{B A C}=60^{\circ}$. Let $M$ be a point on the line segment $B C$ ($M$ is different from $B$ and $C$). The line $A M$ meets the line $C D$ at $N$ and then let $E$ be the intersection between the lines $D M$ and $B N .$ Show that the line $B C$ is tangent to the circumcircle of the triangle $CEN$.
5. Solve the cquation $$8 x^{2}-11 x+1=(1-x) \sqrt{4 x^{2}-6 x+5}.$$
6. Given real numbers $x, y, z, t$ satisfying $x^{2}+y^{2}+z^{2}+t^{2} \leq 2 .$ Find the maximum value of the expression $$P(x, y, z, t)=(x+3 y)^{2}+(z+3 t)^{2} + (x+y+z)^{2}+(x+z+t)^{2}.$$
7. Find all the real roots of the equation $$\left\{\frac{2 x^{2}-5 x+2}{x^{2}-x+1}\right\}=\frac{1}{2}$$ where $\{a\}$ denotes the fractional part of the number $a$. 
8. Given a convex quadrilateral $A B C D .$ Two diagonals $A C$ and $B D$ intersect at $O .$ Let $M$, $N$, $P$, $Q$ respectively be the perpendicular projections of $O$ on the lines $A B$, $B C$, $C D$, $D A$. Show that $A C$ and $B D$ are perpendicular if and only if $$\frac{1}{O M^{2}}+\frac{1}{O P^{2}}=\frac{1}{O N^{2}}+\frac{1}{O Q^{2}}$$
9. Given positive numbers $a$, $b$, $c$ such that $a+b+c=3$. Find the minimum value of the expression $$P=\frac{a}{b}+\frac{b}{c}+\frac{c}{a}+\frac{6 a b c}{a b+b c+c a}.$$
10. Find all the pairs $(p, k)$, where $p$ is a prime number and $k$ is a positive integer. such that $$k !=\left(p^{3}-1\right)\left(p^{3}-p\right)\left(p^{3}-p^{2}\right).$$
11. Find all the functions $f: \mathbb{Z} \rightarrow \mathbb{Z}$ satisfying $$f(2 f(a)+f(b))=2 a+b-4,\, \forall a, b \in \mathbb{Z}.$$
12. Given a triangle $A B C$ inscribed in a circle $(O)$ where $B$ and $C$ are fixed and $A$ can vary. There altitudes $A D$, $B E$, $C F$ meet at $H$. Let $\left(C_{1}\right)$, $\left(C_{2}\right)$, $\left(C_{3}\right)$ be the circles with diameters $A B$, $B C$, $CA$ respectively. The line $A H$ intersects $\left(C_{2}\right)$ at $M$. ($M$ belongs to the line scgment $A H$, $B H$ intersects $\left(C_{3}\right)$ at $N$ belongs to the line segment $B H$), $CH$ intersects $\left(C_{1}\right)$ at $P$ ($P$ belongs to the line segment $C H$), $A N$ intersects $\left(C_{1}\right)$ at $S$ which is different from $A$, $A P$ intersects $\left(C_{3}\right)$ at $T$ which is different from $A .$ The lines $N P$ intersects $\left(C_{1}\right)$ at $X$ which is different from $P$, intersects $\left(C_{3}\right)$ at $Y$ which is different from $N$. $X S$ meets $Y T$ at $Z$. The line passing through $A$ and perpendicular to $N P$ intersects $\left(C_{1}\right)$ at $J$ and intersect $\left(C_{3}\right)$ at $L$.
    a) The circle with center $A$ and radius $A N$ meets $\left(C_{1}\right)$ at $V,$ meets $\left(C_{3}\right)$ at $U$. Show that $V$, $N$, $I$ are collinear and $U$, $P$, $L$ collinear.
    b) Show that the incenter of the triangle $X Y Z$ is a fixed point.

Issue 518

1. Find prime numbers $x$, $y$, $z$ which satisfy the equality $$x^{5}+y^{3}-(x+y)^{2}=3 z^{3}$$
2. Given a triangle $A B C$. Let $M$ be the point on the side $A B$ so that $M B=\dfrac{1}{4} A B,$ and $I$ the point on the side $B C$ so that $I C=\dfrac{3}{8} B C$. The point $N$ is the reflection of $A$ in the point $I$. Show that $M N \parallel A C$.
3. Solve the equation $$\sqrt[3]{x+2020}+\sqrt[3]{x+2021}+\sqrt[3]{x+2022}=0$$
4. Given an acute triangle $A B C$ $(A B>B C)$ inscribed in a circle $(O)$. Let $A D$ and $C E$ be two altitudes of the triangle $A B C$. Let $I$ be the midpoint of $D E$. The ray $A I$ intersects $(O)$ at $K$ $(K \neq A)$. Show that the circumcenter of the triangle $I D K$ lies on $B D$.
5. Find the minimum value of the expression $$\frac{a^{3}}{1-a^{2}}+\frac{b^{3}}{1-b^{2}}+\frac{c^{3}}{1-c^{2}}$$ where $a, b, c$ are positive numbers satisfying $a+b+c=1$.
6. Solve the system of equations $$\begin{cases}\dfrac{1}{\sqrt{4 x^{2}+8 x+5}}+\dfrac{1}{\sqrt{4 y^{2}-8 y+5}} &= \dfrac{2}{\sqrt{(x+y)^{2}+1}} \\ \dfrac{1}{\sqrt{x-1}} +\dfrac{1}{\sqrt{y-3}} &= \dfrac{2 \sqrt{5}}{5}\end{cases}.$$
7. Given real numbers $a$, $b$, $c$ satisfying $a+b+c=1$. Show that $$8 a b c-8 \leq(a b+b c+c a+1)^{2}$$
8. Given a triangle $ABC$ inscribed in a circle $(O)$. Let $H$ be the orthocenter of the triangle. Let $X$, $Y$, $Z$ respectively be the second intersection between the circles $(A O H)$, $(B O H)$, $(C O H)$ with $(O)$. Show that $H$ is the incenter of the triangle $X Y Z$. (The notation $(U V W)$ denotes the circumcircle of the triangle $U V W$).
9. Find all real numbers $x$, $y$, $z$ such that $$2^{x^{2}-3 y+z}+2^{y^{2}-3 z+x}+2^{z^{2}-3 x+y}=\frac{3}{2}$$
10. The integers from $51$ to $150$ are aranged in a chess board of the size $10 \times 10$. Does there exist an arrangment so that for any pair of numbers $(a ; b)$ which are adjacent in a row or in a column at least one of the following equations $$x^{2}-a x+b=0 \quad \text{and} \quad x^{2}-b x+a=0$$ has an integral solution?
11. A sequence $\left(a_{n}\right)$ $\left(n \in \mathbb{N}^{*}\right)$ is determined as follows $$a_{1}=1,\, a_{2}=2,\quad a_{n+2}=2 a_{n+1}-p a_{n},\, n \in \mathbb{N}^{*}$$ where $p$ is some prime number. Find all possible values for $p$ so that there exists a positive integer $m$ satisfying $a_{m}=-3$.
12. Given a non-isosceles triangle $A B C$ and let $I$, $J$ respectively be the center of the incircle of $A B C$ and the excircle corresponding to the angle $A$. Let $D$ be the second intersection between the circle with the diameter $A I$ and the circumcircle of $A B C$. Assume that $E$ is the intersection between $A I$ and $B C$ and $P$ is the midpoint of the arc $B A C$. Show that the intersect between $P J$ and $D E$ lies on the circumcircle of $A B C$.

Issue 519

1. Find positive integers $x$, $y$ such that $$x^{y}+y^{x}=23-x y$$
2. Given a triangle $A B C$ with right angle at $A .$ Let $A H$ be the altitude. If $\dfrac{A H}{B C}=\dfrac{12}{25},$ find $\dfrac{A B}{A C}$.
3. Suppose that $a$, $b$ are positive and $9 a^{2}+9 b^{2}+82 a b+10 a+10 b \geq 1 .$ Show that $$41 a^{2}+41 b^{2}+18 a b \geq 3-2 \sqrt{2}.$$
4. Given a rhombus $A B C D$ with $\widehat{A B C}=60^{\circ} .$ The diagonals $A C$ and $B D$ intersect at $O$. A line $d$ through $D$ intersects the opposite rays of the rays $A B$, $C B$ respectively at $E$, $F$ $(E \neq A, F \neq C) .$ Let $M$ be the intersection between $C E$ and $A F,$ and then $H$ the intersection between $O M$ and $E F .$ Show that $A$, $C$, $D$, $H$ lie on some circle.
5. Solve the equation $$\frac{2020 x^{4}+x^{4} \sqrt{x^{2}+2020}+x^{2}}{2019}=2020.$$
6. Solve the system of equations $$\begin{cases}\sqrt{x^{2}+1}+x-8 y^{2}+8 \sqrt{2 y}-8 & = \dfrac{1}{x-\sqrt{x^{2}+1}}-8 y^{2}+8 \sqrt{2 y}-4 \\ 2 y^{2}-2 \sqrt{2 y}-\sqrt{x^{2}+1}+x+2 &=0\end{cases}$$
7. Given the function $f(x)=x^{2}-x+1$. Let $$f_{1}(x)=f(x),\quad f_{n}(x)=f\left(f_{n-1}(x)\right),\,n=2,3,4, \ldots$$ Find the coefficient of $x^{2}$ in the polynomial expansion of $f_{2021}(x)$.
8. Find the point $M$ inside a given tetrahedron $A B C D$ such that the product of the distances from $M$ to the faces of $A B C D$ is maximal.
9. Does there exist a triangle $A B C$ with $\sin A=\cos B=\tan C ?$
10. The sequence determined as follows $$a_{0}=\frac{1}{2},\quad a_{n+1}=\frac{4 n+5}{4 n+6} a_{n},\, \forall n \geq 0.$$ Let $\displaystyle b_{n}=\sum_{i=0}^{n} a_{i}$, $n \in \mathbb{N}$. Find $\displaystyle \lim _{n \rightarrow+\infty} \dfrac{b_{n}}{n}$.
11. Given real numbers $a, b, c \in[1 ; 5]$ so that $a+b+c=9 .$ Find the minimum and maximum values of the expressions
    a) $P=a b c$.
    b) $F=a b+b c+c a$.
    c) $S_{\lambda}=a^{\lambda}+b^{\lambda}+c^{\lambda}$, where $\lambda$ is a constant $\lambda \in\{0\} \cup[1 ;+\infty)$.
12. Given a triangle $A B C$ which is not equilateral and $G$ is its centroid. The lines $A G$, $B G$, $C G$ respectively intersect the circumcircles of $G B C$, $G C A$, $G A B$ at $D$, $E$, $F$. Show that the Euler lines of the triangles $D B C$, $E C A$, $F A B$ are concurrent.

Issue 520

1. Find all prime numbers $p$ so that $2^{p}+p^{2}$ is a prime.
2. Given an acute triangle $A B C$. Let $D$ be the point which is equidistant to $3$ vertices of the triangle. Suppose that $E$, $F$ respectively are the midpoints of $B C$, $A C$. Let $G$ be the perpendicular projection of $E$ on $A B,$ then call $J$ the midpoint of $D E .$ Show that the triangle $B G J$ is isosceles.
3. Solve the following system of equations $$\begin{cases}x^{3}+y &=2 \\ y^{3}+z &=2 \\ z^{3}+t&=2 \\ t^{3}+x&=2\end{cases}.$$
4. Given a quadrilateral $A B C D$ with $\widehat{A B D}=\widehat{A C D}=90^{\circ} .$ Draw $B H \perp A D$ at $H$. On the diangonal $A C$ we choose the point $I$ so that $A I=A B .$ Let $O$ be the midpoint of $A D .$ The line perpendicular to $O I$ at $I$ intersects $B H$ and $C D$ at $E$ and $F$ respectively. Show that $I F=2 I E$.
5. Solve the equation $$x+\frac{2 x \sqrt{6}}{\sqrt{x^{2}+1}}=1$$
6. Solve the following system of equations $$\begin{cases}\sqrt{x}+\sqrt{y}+\sqrt{z}+1&=4 \sqrt{x y z} \\ \dfrac{1}{2 \sqrt{x}+1}+\dfrac{1}{2 \sqrt{y}+1}+\dfrac{1}{2 \sqrt{z}+1}&=\dfrac{3 \sqrt{x y z}}{x+y+z}\end{cases}.$$
7. Given non-zero numbers $a, b, c$ $d$ whose sum is 4 and each of them is greater or equal to $-2 .$ Show that $$\frac{1}{a^{2}}+\frac{1}{b^{2}}+\frac{1}{c^{2}}+\frac{1}{d^{2}} \geq \frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{d}$$
8. Given a triangle $A B C$ with $B C=a$, $C A=b$, $A B=c .$ Prove that $$\frac{b}{a^{2} c^{2}}\left[a^{2}(b+c)+b^{2}(c+a)+c^{2}(a+b)\right] \geq \frac{6 \sin ^{3} B}{\cos \frac{A}{2} \cos \frac{B}{2} \cos \frac{C}{2}}.$$ When does the equality happen?
9. Solve the equation $$\sum_{r=1}^{\infty}(-1)^{-1} \frac{x(x-1) \ldots(x-r+1)}{(x+1) \ldots(x+r)}=\frac{1}{2}$$ where $n$ is a given positive integer.
10. The sequence $\left(u_{e}\right)$ is determined as follows $$ u_1=3,\quad u_n = 4u_{n-1}+3 n^{2}-12 n^{3}+12 n-4 ,\,\forall n=2,3, \ldots$$ Show that for any odd prime number $p$, $\displaystyle 2019 \sum_{i=1}^{n-1} u_{i}$ is always divisble by $p$.
11. Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ satisfying $$f(f(x)-2 y)=6 x+f(f(y)+x),\, \forall x, y \in \mathbb{R}$$
12. Given a triangle $A B C$ and let $(O)$ be it circumcircle. Let $A_{0}$, $B_{0}$, $C_{0}$ respectively be the midpoints of $B C$, $C A$, $A B$. Assume that $A_{1}$, $B_{1}$, $C_{1}$ respectively are the perpendicular projections of $A$, $B$, $C$. Let $(O_a)$ be the circle passing through $B_0$, $C_{0}$ and is internally tangent to $(O)$ at $A_{2}$ which is different from $A$; $\left(O_{b}\right)$ the circle passing through $C_{0}$, $A_{0}$ and is internally tangent to $(O)$ at $B,$ which is different from $B$; and $\left(O_{c}\right)$ the circle passing through $A_{0}$, $B_{0}$ and is internally tangent to $(O)$ at $C_{2}$ which is different from $C$. Let $A_{3}$ be the intersection between $B_{1} C_{1}$ and $B_{2} C_{2}$, and similarly we get the points $B_{3}$, $C_{3}$. Show that $A_{3}, B_{3}$, $C_{3}$ belong to a line which is perpendicular to the Euler line of the triangle $A B C$.

Issue 521

1. Find the smallest positive integer $n$ so that for cach set of $n$ numbers chosen from $1 ; 2 ; 3 ; \ldots ; 100,$ there always exist two numbers $a, b$ so that $a+b$ is a prime number.
2. Given an odd prime number $p$. Find all pairs of positive integers $(x ; y)$ so that both $x+y$ and $x y+1$ are powers of $p$
3. Find all integers $x, y$ satisfying $$x^{3}(3 y+1)+y^{2}(3 x+1)+(x+y)\left(x^{2}-x y+y^{2}+1\right)+2 x y=343$$
4. Two circles $(O)$ and $\left(O^{\prime}\right)$ intersect at $A$ and $B$. An exterior common tangent touchs $(O)$ and $\left(O^{'}\right)$ at $C$ and $D .$ Show that $$\frac{A C}{A D}+\frac{B D}{B C} \geq 2$$
5. Solve the system of equations $$\begin{cases} 3 x^{2} y-x y-y&=1 \\-x y^{2}-y+y^{2} &=3\end{cases}.$$
6. Given real numbers $x, y \in(0 ; 1)$. Find the maximum value of the expression $$P=\sqrt{x}+\sqrt{y}+\sqrt[4]{12} \sqrt{x \sqrt{1-y^{2}}+y \sqrt{1-x^{2}}}$$
7. Find the real roots of the equation $$(x+1)\left(x^{2}+1\right)\left(x^{3}+1\right)=30 x^{3}$$
8. Given a triangle $A B C$ with the centroid $G$, the nine-point center $E$ and the circumradius $R$. Sbow that
   a) $E A+E B+E C \leq 3 R ;$
   b) $E A^{2}+E B^{2}+E C^{4} \geq G A^{2}+G B^{2}+G C^{2}$.
9. Solve the following trigonometric equation $$\sqrt{4^{n} \cos ^{4 n} x+3}+\sqrt{4^{n} \sin ^{4n} x+3}=4$$ where $n$ is an arbitrary natural number.
10. Let $a, b, c$ be positive integers. Show that there exists an natural number $k$ so that the three integers $a^{t}+b c$, $b^{4}+c a$, $c^{2}+a b$ have at least one common prime divisor.
11. Find all functions $f: Z \rightarrow Z$ satisfying $$f\left(f(x)+y f\left(x^{2}\right)\right)=x+x^{2} f(y)$$ for all $x, y \in \mathbb{Z}$
12. Given a triangle $A B C$ and $M$ is an arbitrary point on the side $B C$. The incircle $(I)$ of the triangle $A B M$ is tangent to the sides $B M$, $M A$, $A B$ respectively at $D$, $E$, $F,$ The incircle $(J)$ of the triangle $A C M$ is tangent to the sides $C M$, $M A$, $A C$ respectively at $X$, $Y$, $Z$ Let $H$ be the intersection between $D F$ and $X Z$. Show that the lines $A H$, $D E$, $X Y$ are concurrent.

Issue 522

1. Show that for every $n \in \mathbb{N}$, $4^{2^{n}}+2^{2^{n}}+1$ is divisible by $7$.
2. Given an acute triangle $A B C$ with $A B < A C$. Let $A D$ be the altitude from $A$ of the triangle. Let $M$ and $N$ so that $A B$ is the perpendicular bisector of $M D$ and $A C$ is the perpendicular bisector of $N D$. The line $M N$ intersects $A B$ at $E$ and intersects $A C$ at $F .$ The lines $B F$ and $C E$ meet at $H$. Let $I$, $K$, $O$ respectively be the midpoints of $B H$, $C H$ and $B C$. Prove that $$\widehat{E I F}=\widehat{E K F}=\widehat{E D F}=\widehat{E O F}.$$
3. Consider the following sets $$\begin{array}{l}A=\{x \in \mathbb{N}: x=3 k+2 \text {with } k \in \mathbb{N} \text { and } k \leq 668\} \\ B=\{x \in \mathbb{N}: x=5 k+1 \text { with } k \in \mathbb{N} \text { and } k \leq 668\} \\ C=\{x \in \mathbb{N}: x \in A \text { and } x \in B\}\end{array}.$$ How many elements are there in the set $C$?
4. Given a right triangle $A B C$ with the right angle $A$ and $\hat{B}=60^{\circ}$. Let $A H$ be the altitude from $A$ of the triangle. Let $I$ be the midpoint of $A B$. On the ray $I H$ choose $K$ so that $B K=B A$. Find the value of the angle $\widehat{B K C}$.
5. Given real numbers $a, b, c$ satisfying $a b \neq 0$, $2 a\left(a^{2}+b^{2}\right)=b c$, and $b\left(a^{2}+15 b^{2}\right)=6 a c$. Compute the value of the expression $$P=\frac{a^{4}+6 a^{2} b^{2}+15 b^{4}}{15 a^{4}+b^{4}}.$$
6. Solve the system of equations $$\begin{cases}x^{2}+y^{2} &=9 \\ \sqrt{5-x}+\sqrt{23+x-6 y} &=2 \sqrt{5}\end{cases}$$
7. Given non-negative numbers $a,b,c$ satisfying $25 a+45 b+52 c \leq 95$. Find the maximum value of the expression $$\frac{a+b}{a+b+1}+\frac{b+c}{b+c+1}+\frac{c+a}{c+a+1}.$$
8. Given a triangle $A B C$. Its incircle $(I)$ is tangent to $B C$, $C A$, $A B$ respectively at $D$, $E$, $F$ Let $M$ be the reflection of $F$ over $B$ and $N$ the reflection of $E$ over $C$. The altitude $D H$ of $D E F$ intersects $M N$ at $G$. Show that $D H=D G$.
9. Given non-negative numbers $x$, $y$, $z$ satisfying $x+y+z=3$. Prove that $$\left(x^{3}+y^{3}+z^{3}\right)\left(x^{3} y^{3}+y^{3} z^{3}+z^{3} x^{3}\right) \leq 36(x y+y z+z x).$$
10. The sequence $\left(u_{n}\right)$ is determined as follows $$u_{1}=16,\, u_{2}=288,\quad u_{n+2}=18 u_{n+1}-17 u_{n},\, \forall n \geq 1.$$ Find the smallest possible value for $n$ so that $u_{n}$ is divisible by $2^{2020}$.
11. Find all functions $f: \mathbb{Z}^{+} \rightarrow \mathbb{Z}^{+}$ satisfying $$f\left(\frac{f(n)}{n^{2020}}\right)=n^{2021},\,\forall n \in \mathbb{Z}.$$
12. Given a triangle $A B C$. Let $A^{\prime}$ be the reflection of $A$ over the midpoint $M$ of $B C$. The line $A A^{\prime}$ intersects $\left(A^{\prime} B C\right)$ at the second point $K$. Let $I$, $J$ respectively be the centers of $(K A B)$ and $(K A C)$. $I J$ intersects $B C$ at $S$. Show that $S A$ is a tangent line of the circumcircle $(O)$ of $A B C$.