Mathematics and Youth Magazine Problems 2022

Issue 535

1. Given odd coprime natural numbers $a, b$. Find all possible values of $$\gcd\left(2^{a}+2^{\frac{a+1}{2}}+1 ; 2^{b}+2^{\frac{b+1}{2}}+1\right)$$ where $\gcd(x, y)$ denote the greatest common divisor of two numbers $x$ and $y$.
2. Suppose that $N$ is an even number which is not a multiple of $10$. Find the tens place of $N^{20}$ and the hundreds place of $N^{200}$.
3. Given a prime number $p$. Find all pairs of integers $(x ; y)$ satisfying $$x^{2}-(p+2) x+2 p+1=p^{y} .$$
4. Given a triangle $A B C$ inscribed in a circle $(O)$. The tangents to the circle at $B$ and $C$ intersect at $M$. $O M$ intersects $B C$ at $I, A M$ intersects $(O)$ at $N$, and draw the diameter $N P$. The lines $P A$ and $B C$ intersect at $E$ and $Q$ is the intersection between $P C$ and $B N$. Show that $M E$ is tangent to the circumcircle of $O I Q$.
5. Given positive numbers $x_{1}, x_{2}, \ldots, x_{n}$. Show that $$\frac{x_{2}}{x_{1}\left(x_{1}+x_{2}\right)}+\frac{x_{3}}{x_{2}\left(x_{2}+x_{3}\right)}+\ldots+\frac{x_{1}}{x_{n}\left(x_{n}+x_{1}\right)} \geq \frac{1}{x_{1}+x_{2}}+\frac{1}{x_{2}+x_{3}}+\ldots+\frac{1}{x_{n}+x_{1}} .$$
6. Suppose that the equation $$(x-15)(x-10)(x-1964)=2021$$ has $3$ solutions $x_{1}$, $x_{2}$, $x_{3}$. Find the value of the expression $$S=\sum_{i=1}^{3}\left[\left(x_{i}-15\right)^{3}+\left(x_{i}-10\right)^{3}+\left(x_{i}-1964\right)^{3}\right] .$$
7. Given a function $f: \mathbb{N}^{*} \rightarrow \mathbb{R}$ satisfying the condition: For each positive integer $n>1$, there exists a prime divisor $p$ of $n$ so that $f(n)=f\left(\frac{n}{p}\right)-f(p)$. Suppose furthermore that $f(2021)=1$. Compute $f(2022)$.
8. Given a triangle ABC incribed circle $( O )$. A circle $( J)$ passing through intersects the sides $A B$, $A C$ respectively at $D$ a $E$; intersects $(O)$ at the second point $P$. Let $I$, $K$ respectively be the circumcenters of $B D P$ and $CEP$. Prove that $I D$, $K E$, $A O$ are $O$ of $B D P$ and point on $(J)$.
9. Given an arbitrary prime number $p$, and $q$ is an integer $0 \leq q \leq p-1$. Show that $$q !(p-q-1) !+(-1)^{q} \equiv 0 \pmod p$$ where $n !=1.2.3....n$.
10. Find all increasing continuous function $f: \mathbb{R} \rightarrow \mathbb{R}$ satisfying $$f(f(x))=f(x)+1,\,\forall x \in \mathbb{R} .$$
11. Find the smallest positive number $a>1$ satisfying the condition: For any point $P$ lies inside and on the boundary of the square $ABCD$, there always two among four triangles $PAB$, $PCD$, $PDA$, $PBC$ (some triangle can be reduced to a line segment) so that the ratio of two areas belongs to $\left[\frac{1}{a} ; a\right]$.
12. Given a triangle $A B C$ inscribed in a circle $(O)$ with orthocenter $H$. The circle with diameter $A H$ intersects $(O)$ at the second point $K$. $O H$ intersects the tangent at $A$ to the circle $(O)$ at $P$. Let $S$ be the point on $(O)$ so that $A S || O H$. The line through $S$ perpendicular to $B C$ intersects $(O)$ at $T$. Show that $K T$ passes through the orthocenter of $A P H$.

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Issue 536

1. Let $m$ be the sum of the digits of the number $9^{2022}$, $n$ the sum of the digits of $m$ and $k$ the sum of the digits of $n$. Find $k$.
2. Given a triangle $A B C$ with $\widehat{A B C}=40^{\circ}, \widehat{A C B}=10^{\circ}$. Let $d$ be the perpendicular bisector of the line segment $A B$, and $d^{\prime}$ the line through $C$ and perpendicular to $d$. Let $D$ be the intersection between $d$ and $d'$. Find the measurement of the angle $\widehat{B D A}$.
3. Solve the equation $$\sqrt[4]{2 x-1}+\sqrt{1-x}=-x^{2}+3 x-1$$
4. Given a quadrilateral $A B C D$ with $\widehat{C A D}=40^{\circ}$, $\widehat{B A C}=50^{\circ}$, $\widehat{C B D}=20^{\circ}$ and $\widehat{C D B}=25^{\circ}$. Find the measurement of the acute angle formed by the diagonals of the quadrilateral.
5. Given real numbers $a, b$ so that $a+b=2$. Find the maximum value of the expression $$P=a b+\frac{1}{a^{2}+b^{2}+1}-\frac{1}{a^{3}+b^{3}+1}+\frac{1}{a^{4}+b^{4}+1} .$$
6. Given $n$ positive numbers $a_{1}, a_{2}$, $\ldots, a_{n}\left(n \in \mathbb{N}^{*}\right)$. Let $S=a_{1}+a_{2}+\ldots+a_{n}$. Prove that $$\left(S-a_{1}\right)^{a_{1}}\left(S-a_{2}\right)^{a_{2}} \ldots\left(S-a_{n}\right)^{a_{n}} \leq\left(\frac{n-1}{n} \cdot S\right)^{S}.$$
7. Given a non-obtuse triangle $A B C$. Show that $$\left(\sin \frac{A}{2}\right)^{\sin \frac{A}{2}}+\left(\sin \frac{B}{2}\right)^{\sin \frac{B}{2}}+\left(\sin \frac{C}{2}\right)^{\sin \frac{C}{2}} \geq \frac{3 \sqrt{2}}{2} .$$
8. Given an equilateral triangle $A B C$ with the side $a$, inscribed in a circle $(O)$. Let $M$ be a point moving on $(O)$. Let $I, J, K$ respectively be the perpendicular projections of $M$ on $B C, C A, A B$. Find the minimum value of the expression $$P=4 M I^{4}+M J^{4}+M K^{4}.$$
9. Given an acute triangle $A B C$ inscribed in a circle $(O)$ with radius $R$. The medians $A A_{1}$, $B B_{1}$, $C C_{1}$ respectively intersects $(O)$ at $A_{2}, B_{2}, C_{2}$. Show that $$A_{1} A_{2}+B_{1} B_{2}+C_{1} C_{2} \geq \frac{3 R}{2}.$$
10. Given a prime $p$ and an integer $n$ which is greater than 4 . Show that the polynomial $f(x)=x^{n}-p x^{2}+p x+p^{2}$ cannot be factorized as a product of two integral coefficients polynomials whose degrees are greater or equal to $1$.
11. Find all functions $f: \mathbb{Q} \rightarrow \mathbb{Q}$ satisfying $f(1)=0$ and $$f(x y-y)=f(x) f(y)+f(x)-1,\, \forall x, y \in \mathbb{Q}.$$
12. Given a triangle $A B C$ inscribed in a circle $(O)$. The angle bisector of $\widehat{B A C}$ intersects $B C$ at $E$ and intersects $(O)$ at the second point $P$. Let $P^{\prime}$ be the reflection point of $P$ in $B C$. $A P^{\prime}$ intersects $B C$ at $K$. Let $L$ be the orthocenter of $A E I$. Show that the circle with diameter $A L$ is tangent to the Euler circle of $A B C$.

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Issue 537

1. Find all natural numbers $a, b, c$ satisfying $a^{3}+a+2=2^{b}$ and $a^{2}=2^{c}+1$.
2. Given a triangle $A B C$ with $\widehat{C A B}=30^{\circ}$, $\widehat{C B A}=100^{\circ}$. On the line segment $A B$ choose the point $D$ so that $C D=A B$. Find the measurement of the angle $\widehat{C D B}$.
3. Suppose that $a, b, n$ are postitive integers satisfying $\dfrac{a^{n}-1}{b^{n}-1}$ and $\dfrac{a^{n+1}-1}{b^{n+1}-1}$ are prime numbers. Prove that $n=1$.
4. Given an isosceles triangle $A B C$ with the vertex angle $A$ inscribed in the circle $(O ; R)$. $M$ is a point moving on the arc $B C$ which does not contain $A$. The points $D$ and $E$ respectively are the perpendicular projections of $M$ on $A B$ and $A C$. Determine the position of $M$ so that the sum $$M A+M B+M C+M D+M E$$ obtains its maximum value.
5. Given a positive number $k$ and three numbers $a, b, c$ in the interval $[0 ; k]$ so that $a+b+c=2 k$. Find the minimum value of the expression $$A=a b(a+k)+b c(b+k)+c a(c+k).$$
6. How many sets $A$ are there so that $A$ has $2020$ distinct positive numbers including $1$, $2$ and satisfies: For any two different elements $x$ and $y$ belong to $A$ then either $|x-y|$ or $x+y$ also belongs to $A$?
7. For any triangle $A B C$, show that $$\frac{r_{i}}{a}+\frac{r_{b}}{b}+\frac{r_{c}}{c} \geq \sqrt{\frac{3}{2} \cdot \frac{4 R+r}{R}}$$ where $a=B C$, $b=A C$, $c=A B$; $R$, $r$, $r_{a}$, $r_{b}$, $r_{c}$ respectively are the circumradius, inradius, exradii corresponding to $A$, $B$, $C$ of the triangle.
8. Given a triangle $A B C$ and let $(I)$ be its incircle. Let $E$, $F$ respectively be the points of tangency between $(I)$ and $A B$, $A C$. Let $M$, $N$ respectively be the points of the line segments $A B$, $A C$ so that $M N$ is tangent to $(I)$. Prove that $$\frac{E B}{E A} \cdot \frac{M A}{M B}+\frac{F C}{F A} \cdot \frac{N A}{N C}=1.$$
9. Given $x, y \geq 1$. Show that $$\frac{x^{3}+y}{y^{3}+x}-1 \geq \ln \frac{\left(x^{2}+1\right)^{2}}{x}-\ln \frac{\left(y^{2}+1\right)^{2}}{y}$$
10. Given an integer $n>1$. Show that if $4^{n}-2^{n}+1$ is a prime number then all the prime factors of $n$ must be less than $5$.
11. Consider the function $f(x)=\dfrac{2033\cdot 25^{x}}{25^{x}+5}$, $x \in \mathbb{R}$. Compute the sum $$S=\sum_{k=1}^{2020}\left[\frac{\left[f\left(\frac{k}{2021}\right)\right]}{19}\right]$$ where $[x]$ is the largest integer which does not exceed $x$.
12. Given an acute, scalene triangle $A B C$ inscribed in the circle $(O)$. The circle with center $C$ and radius $C B$ intersects $B A$ at $D$ (which is different from $B$) and intersects $(O)$ at $E$ (which is different from $B$). $D E$ intersects $(O)$ at $F$ (which is different from $E$). $C O$ intersects $D E$, $A B$ respectively at $G$, $L$. Let $Q$ be the reflection point of $E$ in the midpoint of $G F$. The point $P$ belongs to $E F$ so that $P L$ perpendicular to $B C$. The circumcircle of $L P Q$ intersects $A B$ at $R$ (which is different from $L$). Let $K$ be the reflection point of $R$ in $C F$. Show that $F K || B C$.

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Issue 538

1. Find all pairs of integers $(x ; y)$ so that $$x^{2}+y=y^{2023}.$$
2. Given a triangle $A B C$ with $\widehat{B A C}=75^{\circ}, \widehat{A B C}=45^{\circ}$. Let $D$ be the point on the side $A B$ so that $A D=2 B D$. Find the measurement of the angle $\widehat{B C D}$.
3. Find all positive integers $n$ so that $\dfrac{n+3}{2}$, $n-5$, $2 n+1$ are cube numbers.
4. Let $I$ be the intersection of the interior angle bisectors of the right triangle $A B C$ with the right angle at $A$. Let $M$ be the midpoint of $B C$; and $N$ be the intersection between $M I$ and $A B$. Given that $\widehat{B N M}=75^{\circ}$, find the measurement of the angle $\widehat{A B C}$.
5. Suppose that $x, y$ are positive numbers satisfying $x^{3} y^{3}+12 x y^{2}+8 \leq 8 y^{3}$. Find the minimum value of the expression $$Q=\frac{x^{4}+3 x^{2} y^{2}+2 y^{4}}{x y^{3}}.$$
6. Given a prime $p$, $p>3$ and let $n=\dfrac{2^{2 p}-1}{3}$. Find the greatest common divisor of $2^{2^{n}-1}-2$ and $2^{n}-1$.
7. Given a triangle $A B C$ with the lengths of the sides $B C=a$, $C A=b$, $A B=c$. Let $S$, $R$ respectively be the area and the radius of the circumcircle of $A B C$. Prove that $$a \cos ^{3} A+b \cos ^{3} B+c \cos ^{3} C \geq \frac{S}{2 R}.$$
8. Given a circle $(O ; R)$ with two perpendicular diameters $A B$ and $C D$. Let $E$ be a moving point on the minor arc $A C$. $B E$ intersects $D C$ at $N$, $D E$ intersects $A B$ at $M$. Show that $$\frac{1}{M B}+\frac{1}{N D} \geq \frac{\sqrt{2}}{R}.$$
9. Given positive numbers $a$, $b$, $c$, $d$ and a positive integer $n$. Show that $$\frac{a^{n}}{\sqrt[n]{b^{n}+c^{n}+d^{n}}}+\frac{b^{n}}{\sqrt[n]{c^{n}+d^{n}+a^{n}}}+\frac{c^{n}}{\sqrt[n]{d^{n}+a^{n}+b^{n}}} +\frac{d^{n}}{\sqrt[n]{a^{n}+b^{n}+c^{n}}} \geq \frac{a^{n-1}+b^{n-1}+c^{n-1}+d^{n-1}}{\sqrt[n]{3}}.$$
10. Find all positive integers $n$ so that $$\frac{n^{2021 n-2020}-2021 n+2019}{2021 n-2020}$$ is an integer.
11. Suppose that $x, y$ are real numbers. Find the minimum value of the expression $$P=\sin ^{4} x \cdot\left(\sin ^{4} y+\cos ^{4} y+\frac{9}{8} \cos ^{2} x \cdot \sin ^{2} 2 y\right)+\cos ^{4} x.$$
12. Given a triangle $A B C$ and its circumcircle $(O)$. A point $P$ is moving on $(O)$. Let $Q$ be the reflection point of $P$ in $B C$. Let $E$, $F$ respectively be the intersections between $B Q$, $C Q$ and $A C$, $A B$. Show that the center of the Euler circle of $A E F$ belongs to a fixed line.

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Issue 539

1. Given $A=\overline{100 \ldots 02008 a b}$ (there are $2008$ digits $0$'s between the digits 1 and $2$). Find the digits $a$ and $b$ so that $A$ is divisible by $41$.
2. Find the least possible number $n$ so that we can write $2108$ as a sum of $n$ perfect squares.
3. Find integral solutions of the equation $$11 x^{2}+2 x y+16 y^{2}+4=7 z^{2}.$$
4. Given a scalene triangle $A B C$ with $A B+A C=2 B C$. Let $(O)$ and $(I)$ respectively be the circumcircle and the incircle of $A B C$. $(I)$ is tangent to $B C$ at $D$. The ray $A I$ intersects $(O)$ at the second point $M$. The ray $M D$ intersects $(O)$ at the second point $E$. The ray $A E$ intersects $B C$ at $K$. Show that $K$, $I$, $O$ are collinear.
5. Given positive numbers $x, y, z$ satisfying $x+y+z=3$. Find the minimum value of the expression $$P=\sqrt{x+3}+\sqrt{y+3}+\sqrt{z+3}+\frac{x^{2}+y^{2}+z^{2}}{2(x y+y z+z x)}$$
6. Solve the system of equations $$\begin{cases} x^{2}+y^{2}+y &=5 \\ x^{3}+y^{3}+6 x y &=8\end{cases}$$
7. Given two positive integers $m$, $n$. Let $$\begin{align}A&=(1+n)^{n}(1+m)^{m}-(1+n)^{n}-(1+m)^{m},\\ B&=(1+n)^{m}(1+m)^{n}-(1+n)^{m}-(1+m)^{n}.\end{align}$$ Find $A-m n B \pmod{m n^{2}}$.
8. Given $5$ different points $A$, $B$, $C$, $D$, $E$ on the space. Let $M$, $N$, $P$ respectively be the midpoints of the line segments connecting the midpoints of the pairs $B C$ and $D E$, $C D$ and $E A$, $D E$ and $A B$. Show that the lines $A M$, $B N$, $C P$ are concurrent.
9. Given a triangle $A B C$ with $A B=c$, $B C=a$, $C A=b$. The medians $A A_{1}$, $B B_{1}$, $C C_{1}$ respectively intersect the circumcircle $(O)$ at $A_{2}$, $B_{2}$, $C_{2}$. Let $G$ be the centroid of $A B C$. Show that $$\frac{1}{G A_{2}}+\frac{1}{G B_{2}}+\frac{1}{G C_{2}} \leq \frac{9}{\sqrt{a^{2}+b^{2}+c^{2}}}.$$
10. Find the largest possible number $m$ so that for each positive integer $n$, the sum of the prime divisors (not necessarily different) of $n$ is not less than $m \ln n$.
11. Find all functions $\mathrm{f}: \mathbb{R}^{+} \rightarrow \mathbb{R}^{+}$ satisfying $$f(x+y f(x))=f(x)+x f(y),\, \forall x, y \in \mathbb{R}^{+}.$$
12. Given an acute triangle $A B C$ $(A B<A C)$. The altitudes $B E$, $C F$ meet at $H$. Let $M$ and $N$ respectively be the midpoints of $A H$ and $B C$. The circle with the diameter $A H$ intersects $A N$ at the second point $U$. $E F$ intersects $A H$ at $T$. Suppose that $K$ is the reflection point of $U$ in $B C$. Show that $K H$, $N T$ meet together at some point on the circumcircle of $M B C$.

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Issue 540

1. Prove that $$1+\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}+\ldots+\frac{1}{\sqrt{2022}}>2(\sqrt{2023}-1)$$
2. Let $$A=1+\frac{1}{2}(1+2)+\frac{1}{3^{2}}(1+2+3)^{2}+\ldots+\frac{1}{2021^{2020}}(1+2+\ldots+2021)^{2020}.$$ Compare $A$ with $\left(2^{2020}\right)^{20} .$
3.  Compute the sum $$S=\frac{a}{1-a^{2}}+\frac{a^{2}}{1-a^{4}}+\frac{a^{4}}{1-a^{8}}+\ldots+\frac{a^{2^{n}}}{1-a^{2^{n+1}}}$$ with $a=\sqrt[2^n]{2}$, $n=2022$.
4. Given a right triangle $A B C$ with the right angle $A$. Let $D$ be a point on the side $B C$ satisfying $$\frac{2}{A D^{2}}=\frac{1}{D B^{2}}+\frac{1}{D C^{2}}.$$ Show that either $D$ is the midpoint of $B C$ or $D$ is the intersection between the angle bisector of $A$ and $B C$.
5. Find the integral solutions of the equation $$9 x^{2}-6 x-3+32 \sqrt{3 x+1}=64.$$
6. Let $(x ; y ; z)$ be a permutation of $3$ positive numbers $a, b, c$. Show that $$\frac{x^{2}}{\sqrt{a b(a+2 c)(b+2 c)}}+\frac{y^{2}}{\sqrt{b c(b+2 a)(c+2 a)}} +\frac{z^{2}}{\sqrt{c a(c+2 b)(a+2 b)}} \geq 1.$$
7. Find all pairs of prime numbers $(p ; q)$ satisfying $$p^{3}-127=q^{6}-121 q^{3}.$$
8. Given a triangle $A B C$ and a point $M$ on the side $A B$. The line through $M$ and parallel to $B C$ intersects $A C$ at $N$. The line through $N$ and parallel to $A B$ intersects $C M$ at $I$. Show that $$S_{I M N} \leq \frac{4}{27} S_{A B C}.$$
9. Given a triangle $A B C$ with $A B=c$, $B C=a$, $C A=b$. Let $I$ be the incenter of $A B C$. Show that $$I A^{2}+I B^{2}+I C^{2} \leq \frac{a^{3}+b^{3}+c^{3}}{a+b+c}.$$
10. Let $p$ be a prime number in the form $2^{2^{n}}+1$ $(n \in \mathbb{N}, n \geq 1)$. Let $S=3+3^{2}+\ldots+3^{p-2}$. Show that
• For any positive integer $h$ with $h<p$ and $h$ is not a divisor of $p-1$ then $3^{h}+1$ is not divisible by $p$.
• $S+1 \vdots p$
11. Given $n$ real numbers $a_{1}, a_{2}, \ldots, a_{n}$ satisfying $0 \leq a_{i}<1$, $\forall i=\overline{1, n}$, $n \geq 2$ Find the least positive constant $k$ so that the following inequality holds $$\frac{1}{\prod_{i=1}^{n}\left(1-a_{i}\right)}-\frac{\sum_{i=1}^{n} a_{i}}{k}+\sum_{1 \leq i<j \leq n} a_{i} a_{j}+\sum_{i=1}^{n} a_{i}^{2} \geq 1.$$
12. Given a triangle $A B C$ and its incenter $I$, and $D$ is a point on $B C$. Let $K$ be the intersection between the line through $D$ and perpendicular to $I B$ with the external angle bisector of $\widehat{B A C}$. Let $L$ be the intersection between the line through $D$ and perpendicular to $I C$ with the external angle bisector of $\widehat{B A C}$. The points $P, Q$ respectively are on $I B$, $I C$ so that $\widehat{P D Q}=90^{\circ}$. Prove that $K P \perp L Q$.

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Issue 541

1. Find all natural numbers $x, y$ satisfying $$(x y+3)(x y+2)=2007^{n 7}+1930 .$$
2. Given a triangle $A B C$ $(A B<A C)$ with $\widehat{B A C}=\alpha$. Let $M$ be the midpoint of $B C$. Let $D$ be the point on the side $A C$ such that $A D=\dfrac{A B+A C}{2}$. Find the measurement of $\widehat{A D M}$ in terms of $\alpha$.
3. Find all positive numbers $x$ satisfying $$\frac{1945}{6 x+3}+\frac{1954}{4 x+5}+\frac{1975}{3 x+9}+\frac{2023}{2 x+14}=15 .$$
4. Outside a given triangle $A B C$ draw triangles $A B D$, $B C E$, $C A F$ such that $\widehat{A D B}=\widehat{B E C}=\widehat{C F A}=90^{\circ}$, $\widehat{A B D}=\widehat{C B E}=\widehat{C A F}=\alpha$. Prove that $D F=A E$.
5. Let $x$, $y$ be positive numbers such that $x y \geq 1$. Find the minimum value of the expression $$A=\frac{x}{y+1}+\frac{y}{x+1}+\frac{1}{x y+1}.$$
6. Find all natural numbers $x$, $y$ which are greater than $1$ and satisfy $$1+x+x^{2}+\ldots+x^{n-1}=5^{y}.$$
7. Given positive numbers $a, b, c$ such that $a+b+c=1$. Show that $$\frac{2 a-3 b c}{2 a+3 b c}+\frac{2 b-3 c a}{2 b+3 c a}+\frac{2 c-3 a b}{2 c+3 a b} \leq 1 .$$
8. Given an equilateral triangle $A B C$ inscribed in a circle $(O)$ with radius $R$. Show that, for an arbitrary point $M$ on the circle $(O)$, we always have $$6 \sqrt{2}<\frac{M A^{3}+M B^{3}+M C^{3}}{R^{3}}<3 \sqrt[4]{216}.$$
9. Let $a, b, c$ be the lengths of three sides of a triangle which has the circumference $5$ and $m_{a}$, $m_{b}$, $m_{c}$ be the lengths of the corresponding medians. Find the minimum value of the expression $$T=m_{a}^{2}+m_{b}^{2}+2 m_{c}^{2}+4 c .$$
10. Given a positive integer $n$. Show that there does not exist a $(2 n+1)$-polygon with equal sides and integral vertices i.e. the components of the coordinates of all vertices are integers.
11. For each positive integer $n$, show that there exists a unique positive integer $x_{n}$ satisfying $$(2+\sqrt{7})^{n}=\sqrt{x_{n}-3^{n}}+\sqrt{x_{n}}.$$
12. In a plane there are 4 fixed points $A$, $B$, $C$, $D$ which are collinear in that order. Find the locus of the points $M$ such that two triangles $M A C$ and $M B D$ share the same orthocenter.

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Issue 542

1. Find all natural numbers $n$ satisfying $$\frac{n+2}{n+1}+\frac{n+3}{n+2}+\ldots+\frac{n+2024}{n+2023} = \frac{n-2024}{n-2023}+\frac{n-2025}{n-2024}+\ldots+\frac{n-4046}{n-4045} .$$
2. Given an isosceles triangle $A B C$ with $\hat{A}=100^{\circ}$, the angle bisector $B D$. On the side $B C$ choose the point $N$ so that $B N=B D$. Show that $N A+N C=B D$.
3. Let $a, b, c$ be integers so that $a b-b c-c a$ is divisible by $3$. Show that if $a^3+b^3$ $+c^3$ is divisible by $3$ then $a^3+b^3+c^3$ is divisible by $27$.
4. Given a triangle $A B C$ with $\widehat{A B C}=30^{\circ}$. Outside $A B C$ draw the right isosceles triangle $A C D$ with right angle $D$. Show that $$2 B D^2=B A^2+B C^2+B A \cdot B C .$$
5. Consider pairs of real numbers $(x ; y)$ satisfying $$2 x+y-x y=1\quad (*).$$ The pairs $(a ; b)$ and $(m ; n)$, with $a>1>m$, satisfy $(*)$. Find the minimal value of the expression $$P=(a-m)^2+(b-n)^2.$$
6. Given $100$ positive integers $a_1, a_2, \ldots, a_{100}$ satisfying the conditions
   • $1 \leq a_1<a_2<\ldots<a_{99}<a_{100}$.
   • $a_{i+1}=a_i^3+2006$ with $i=1,2, \ldots, 99$.
   Show that among $100$ given numbers there is at most one perfect square.
7. Given an acute triangle $A B C$. Find the minimal value of the expression $$P=\sin A+\sin B+\sin C+\tan A+\tan B+\tan C.$$
8. Given an acute triangle $A B C$ with orthocenter $H$ and incenter $I$. The rays $A H$, ${BH}$, ${CH}$ respectively intersect the circumcircle $A B C$ at $A_2$, $B_2$, $C_2$ respectively. Show that $$\frac{1}{H A_2}+\frac{1}{H B_2}+\frac{1}{H C_2} \geq \frac{1}{I A}+\frac{1}{I B}+\frac{1}{I C}.$$
9. Show that the following inequality always holds for arbitrary real numbers $x, y, z$ we have $$\left(x^2-x+1\right)\left(y^2-y+1\right)\left(z^2-z+1\right) \geq(x-y)(y-z)(z-x) .$$ When does the equality happen?
10. Show that for every positive integer $m$, the number $2 m^2+4 m+4$ has no divisor of the form $2^n-1$, where $n$ is an integer which is bigger than $1$.
11. Given real numbers $a, b, c$ satisfying $a+b+c=3$. Show that $$\frac{1}{\sqrt{a^2+3}}+\frac{1}{\sqrt{b^2+3}}+\frac{1}{\sqrt{c^2+3}} \leq \frac{3}{2}.$$
12. Given a triangle $A B C$ and let $\omega$, $\gamma$ be its circumcircle and the incircle respectively. The circle $\gamma$ is tangent to $B C$ at $D$. $M$ is a moving point on the minor arc $B C$ of $\omega$. From $M$ draw the tangents $M E$, $M F$ with $\gamma$ (at $E$, $F$ respectively). $P$ is the reflection point of $D$ in the center of $\gamma$. Let $H$ be the orthocenter of $P E F$.
    a) Show that $H$ always belongs to a fixed circle.
    b) Choose $B_1$, $C_1$ on the opposite rays of the rays $C A$, $C B$ respectively $\left(B_1 \neq C, C_1 \neq B\right)$. Through $B_1$ draw the other tangent which is different from $A C$ with $\gamma$, and that tangent intersects $B C$, $B A$ at $A_1$, $C_2$ respectively. Through $C_1$ draw the other tangent which is different from $A B$ with $\gamma$, and that tangent intersects $C B$, $C A$ at $A_2$, $B_2$ respectively. Assume that $B_1 C_2$, $C_1 B_2$, $A M$ are concurrent and the circumcircles of $M A_1 A_2$, $M B_1 B_2$, $M C_1 C_2$ intersect $\omega$ again at $X$, $Y$, $Z$ respectively. Show that $A X$, $B Y$, $C Z$ are concurrent.

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Issue 543

1. Compute $$S=\frac{1}{2 !}+\frac{2}{3 !}+\frac{3}{4 !}+\ldots+\frac{2020}{2021 !}+\frac{2022}{2022 !}$$ with $n !=1.2 .3 \ldots n$.
2. Given two polynomials $f(x)=a x^2+b x+c$ and $g(x)=(c-b) x^2+(c-a) x+a+b$, where $a$, $b$, $c$ are integers and $b \neq c$. Assume that $f(x)$ and $g(x)$ have a common solution. Show that $a+b+2009 c$ is divisible by $3$.
3. Solve the system of equations $$\begin{cases}8 x^4 &=y\left(16+3 x^4\right) \\ 8 y^4 &=z\left(16+3 y^4\right) \\ 8 z^4 &=x\left(16+3 z^4\right)\end{cases}$$
4. Given a circle $(O ; R)$ with a diameter $B C$. A point $A$ is moving on $(O)$ but is different from $B$ and $C$. Let $H$ be the perpendicular projection of $A$ on $B C$, and $E$ the midpoint of $A H$. The line through $H$ and perpendicular to $C E$ intersects the line $B A$ at $D$. Show that the length of $C D$ is a constant.
5. Consider the real numbers $x$, $y$ satisfying $27 x^3+216 y^3=16$. Find the maximum value of the expression $$A=\frac{(x+2 y+1)^3}{3\left(x^2+y^2\right)-2(2 x+y)+3} .$$
6. Find all the primes $p$ so that the sum $A=7^p+9 p^6$ is a complete square.
7. Given positive numbers $x$, $y$, $z$. Consider all rectangles $A B C D$ so that there exists a point $P$ which is totally inside the rectangle and satisfies $P A=x$, $P B=y$, $P C=z$. Find the maximum value of the area of the rectangle $A B C D$.
8. On a given plane $(\alpha)$ fix three points $A$, $B$, $C$ so that the triangle $A B C$ is equilateral with side $a$. On the space a point $S$ is moving, but not on the plane $(\alpha)$, so that the distance from $A$ to the plane $(S B C)$ is equal to the distance between two lines $S A$ and $B C$; and $S A=a$. Show that $S$ lies on a fixed circle.
9. Consider positive numbers $x$, $y$, $z$ satisfying $x y z=1$. Find the maximum value of the expression $$P=\frac{1}{2 x^2+y^2+z\left(x^2+y^2+z\right)} + \frac{1}{2 y^2+z^2+x\left(y^2+z^2+x\right)}+\frac{1}{2 z^2+x^2+y\left(z^2+x^2+y\right)}.$$
10. Find all polynomials $f$ with integral coefficients so that: For each prime $p$ and arbitrary $u, v \in \mathbb{N}$ satisfying $p \mid u v-1$ then $p \mid f(u) f(v)-1$
11. Consider the sequence $a_n=\sin n q$, where $n \in \mathbb{N}$, $q \in \mathbb{R}$. Find the values of $q$ for which the limit $\lim _{n \rightarrow-\infty} a_n$ exists.
12. Given a triangle $A B C$ with the centroid $G$. $D$, $E$, $F$ are respectively the perpendicular projections of $G$ on $B C$, $C A$, $A B$. Show that the Lemoine point of the triangle $D E F$ lies on the Euler line of the triangle $A B C$.

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Issue 544

1. Find the sum of all largest odd divisors of the numbers $2022, 2023, ..., 4044$.
2. Given a triangle $A B C$ with $\widehat{B}=80^{\circ}$, $\widehat{C}=60^{\circ}$. On the opposite ray of the ray $C B$ choose the point $M$ so that $\widehat{M A B}=60^{\circ}$. Compare the lengths of the line segments $B M$ and $A C$.
3. Find all natural numbers $a$ so that $3 a^{2 n}+6 a^n+27$ is a perfect square for any positive integer $n$.
4. Give a trapezium $A B C D$ ($A B || C D$, $A B<C D$). Let $M$, $N$ respectively be the midpoints of $A B$, $C D$. Through $A$ draw the line $\left(d_1\right)$ perpendicular to $A D$. Through $B$ draw the line $\left(d_2\right)$ perpendicular to $B C$. Show that if $C D-A B=$ $2 M N$ then the lines $\left(d_1\right)$, $\left(d_2\right)$ and $M N$ are concurrent.
5. Solve the system of equations $$\begin{cases} 20 x+11 y &=2024 \\ x^3+4 x &=y^3+3\left(x^2+y^2\right)+4 y+4\end{cases}$$
6. Given real numbers $x, y, z$ which are greater than $1$ and satisfy $$\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=2.$$ Prove that $$x+y+z \leq x y z+\frac{9}{8}.$$
7. Find pairs of non-zero polynomials with integral coefficients $P(x)$, $Q(x)$ so that $$\frac{P(\sqrt{2}+\sqrt{17}+\sqrt{19})}{Q(\sqrt{2}+\sqrt{17}+\sqrt{19})}=\sqrt{2}+\sqrt{7}.$$
8. Given an acute triangle $A B C$ inscribed in a circle $(O)$. Two points $D$, $E$ are on the line segment $A B$, $A C$ respectively so that $A D=A E$. The perpendicular bisectors of $B D$ and $C E$ intersects the minor arcs $\overparen{A B}$, $\overparen{A C}$ of $(O)$ at the points $G$, $H$ respectively. Let $P$ be the intersection between $G D$ and $E H$; let $K$ be the circumcenter of the triangle $P G H$. Show that $O A$ is parallel to $K P$.
9. Given positive numbers $x$, $y$. Find the minimum value of the expression $$F=\frac{x y}{\left(x+\sqrt{x^2+y^2}\right)^2} .$$
10. Given a real sequence $\left(a_n\right)$ determined as follows $a_1=1$ and $$a_{n+1}=\sqrt{\frac{n}{n+1}} \cdot a_n+\frac{1}{n+1},\, n=1,2, \ldots$$ Show that the sequence $\left(a_n\right)$ has a finite limit and find that limit.
11. Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ satisfying $$f(x y)-f(f(x)-f(y))=x f(y)+x+y,\,\forall \forall x, y \in \mathbb{R}.$$
12. Given a triangle $A B C$ inscribed in a circle $(O)$ and $H$ is the orthocenter of the triangle. Let $P$ be the reflection point of $H$ in $O$. Draw $P K \perp B C$ $(K \in B C)$. Let $M$ be the midpoint of $B C$. $B P$, $C P$ intersect $A C$, $A B$ at $E$, $F$ respectively. Denote by $H_1$, $H_2$ the orthocenters of triangles $B E C$, $B F C$ respectively. Show that the points $H_1$, $H_2$, $K$, $M$ belong to the same circle.

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Issue 545

1. Find positive integers $m$, $n$ so that both $\dfrac{m^{12}}{1023 m+n}$ and $\dfrac{n^{12}}{1023 n+m}$ are prime numbers.
2. Given a square $A B C D$. A point $K$ is moving on the side $A B$ and $K$ is different from $A$. Draw the square $D K I E$ where $I$ is in the same half-plane defined by $A D$ with the point $B$. Find the measurement of the angle $\widehat{E B I}$.
3. Find all positive integers $n$ so that $\sqrt{n+1}+\sqrt{2^n+1}$ is also a positive integer.
4. Given an acute triangle $A B C$ with the orthocenter $H$. Show that $$H A+H B+H C<\sqrt{\frac{2}{3}}(A B+B C+C A) .$$
5. Given positive numbers $a, b, c,d$. Show that $$a^4+b^4+c^4+d^4-4 a b c d \geq 4(a-b)^2 \sqrt{a b c d} .$$
6. Solve the equation $$\left(x^2+3 x+3\right) \sqrt{2 x^2+x+1}-x^3-4 x^2-12 x=9 .$$
7. Denote $[x]$ the largest integer which does not exceed $x$. Let $\{x\}=x-[x]$. Find all triples of real numbers $(x ; y ; z)$ satisfying the system of equations $$\begin{cases}x+[y]+\{z\} &=1,1 \\ \{x\}+y+[z] &=2,2 \\ [x]+\{y\}+z &=3,3\end{cases}$$
8. Suppose that the line segments $A B$ and $C D$ intersect at $M$ $(M \notin\{A, B, C, D\})$. Draw the rays $B x || A C$ and $C y || B D$, and assume that they intersect at $N$. Let $E$, $F$ respectively be the midpoints of $B C$ and $A D$. Show that $E F$ and $M N$ are either parallel or coincident.
9. Find the maximum value of the expression $$\frac{3 a+2 b+c}{(a+b)(a+c)(b+c)}$$ where $a, b, c$ are positive numbers satisfying $$3 b c+4 a c+5 b a \leq 6 a b c .$$
10. For $n \in \mathbb{N}^*$, we write on a board the numbers $1,2,3, \ldots, 3 n+1$. Then in each step, we delete $4$ numbers, e.g. $a, b, c, d$, and we write again on the board the number whose value is $64(a+b+c+d)$. After $n$ steps, there is remain number is always greater than $\frac{64}{625} n^5$.
11. Given the sequence $\left(x_n\right)$ of positive numbers which satisfy only one number on the board. Show that the $$\frac{2}{x_n+2}+\frac{1}{x_{n+1}+2}+\frac{1}{x_{n+1}}=1,\, n=0,1,2, \ldots.$$ Show that $\left(x_n\right)$ has a limit which is a finite number and find that limit.
12. Given a triangle $A B C$ and its incircle $(I)$. Let $D$, $E$, $F$ repectively be the points where $(I)$ is tangent to $B C$, $C A$, $A B$. Let $A^{\prime}$, $B^{\prime}$, $C^{\prime}$ respectively be the reflection points of $A$, $B$, $C$ in $E F$, $F D$, $D E$. Let $X=E B^{\prime} \cap F C^{\prime}$, $Y=F C^{\prime} \cap D A^{\prime}$, $Z=D A^{\prime} \cap E B^{\prime}$. Show that the orthocenter of $X Y Z$ is the center of the Euler circle of $D E F$.