Mathematics and Youth Magazine Problems 2018 (Issue 487 - 498)

Issue 487

1. Find all pairs of prime numbers $(p,q)$ satisfying

$p^{q}-q^{p}=79.$

2. Find integers $a,b,c,d$ satisfying

$\sqrt[3]{a^{2}+b^{2}+c^{2}}=\sqrt{a+b+c}=d.$

3. Let $a,b,c$ be positive numbers such that $a+b+c=1$. Prove that $\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}\geq\dfrac{21}{1+36abc}.$

 

4. Given a triangle $ABC$ circumscribing a circle $(O)$. The sides $AB$, $BC$ and $CA$ are tangent to $(O)$ at $D$, $E$ and $F$ respectively and assume furthermore that $EC=2EB$. Suppose that $EI$ is a diameter of $(O)$. Through $D$ draw a line which is parallel to $BC$. This line intersects the line segment $EF$ at $K$. Prove that $A$, $I$,$K$ are collinear.

5. Find the maximal number $M$ such that the inequality $x^{2}\geq M[x]\{x\}$ holds for every $x$ (where $[x]$, $\{x\}$ respectively are the integral part and the fractional part of $x$).

6. Solve the equation \[x^{3}+x+6=2(x+1)\sqrt{3+2x-x^{2}}.\]

7. Solve the system of equations \[\begin{cases}\dfrac{1}{\sqrt{x}}+\dfrac{1}{\sqrt{y}}+\dfrac{1}{\sqrt{z}} & =3\\ \dfrac{x^{2}}{y}+\dfrac{y^{2}}{z}+\dfrac{z^{2}}{x} & =\sqrt[4]{\dfrac{x^{4}+y^{4}}{2}}+\sqrt[4]{\dfrac{y^{4}+z^{4}}{2}}+\sqrt[4]{\dfrac{z^{4}+x^{4}}{2}}\end{cases}.\]

8. Given a triangle $ABC$ with the exradii $r_{a}$, $r_{b}$, $r_{c}$, the medians $m_{a}$, $m_{b}$, $m_{c}$ and the area $S$. Prove that \[r_{a}^{2}+r_{b}^{2}+r_{c}^{2}\geq3\sqrt{3}S+(m_{a}-m_{b})^{2}+(m_{b}-m_{c})^{2}+(m_{c}-m_{a})^{2}.\]

9. Given a positive integer $n$ and positive numbers $a_{1},a_{2},\ldots a_{n}$. Find a real number $\lambda$ such that \[a_{1}^{x}+a_{2}^{x}+\ldots+a_{2}^{x}\geq n+\lambda x,\quad\forall x\in\mathbb{R}.\]

10. a) Prove that for every positive integer $n$ the equation $2012^{x}(x^{2}-n^{2})=1$ has a unique solution (denoted by $x_{n}$).

b) Find ${\displaystyle \lim_{n\to\infty}(x_{n+1}-x_{n})}$.

11. Let $T$ be the set of all positive factors of $n=2004^{2010}$. Suppose that $S$ be an arbitrary nonempty subset of $T$ satisfying the fact that for all $a$, $b$ belong to $S$ and $a>b$ then $a$ is not divisible by $b$. Find the maximal number of elements of such subset $S$.

12. Given a triangle $ABC$ whose the incircle $(I)$ is tangent to $BC$ at $D$. Let $H$ be the perpendicular projection of $A$ on $BC$. Let $N$ be the midpoint pf $AH$. The line through $D$ and $N$ intersects $CA$, $AB$ respectively at $J$ and $S$. Assume that $BJ$ intersects $CS$ at $P$. Suppose that $DA$, $DP$ intersect $(I)$ respectively at $G$, $L$. Prove that $B$, $C$, $G$, $L$ lie on some circle.

 Issue 488

1. Give a pentagon $ABCDE$. Assume that $BC$ is parallel to $AD$, $CD$ is parallel to $BE$, $DE$ is parallel to $AC$, and $AE$ is parallel to $BD$. Show that $AB$ is parallel to $CE$. 
2. Prove that \[\cfrac{1+\frac{1}{3}+\frac{1}{5}+\ldots+\frac{1}{4035}}{1+\frac{1}{2}+\frac{1}{3}+\ldots+\frac{1}{2018}}>\frac{2019}{4036}.\]
   
3. Given two triples $(a,b,c)$; $(x,y,z)$ none of them contains all $0$'s, such that \[a+b+c=x+y+z=ax+by+cz=0.\] Prove that the expression $P=\dfrac{(b+c)^{2}}{ab+bc+ca}+\dfrac{(y+z)}{xy+yz+zx}^{2}$ is a constant.
4. Outside a triangle $ABC$, draw triangles $ABD$, $BCE$, $CAF$ such that $\widehat{ADB}=\widehat{BEC}=\widehat{CFA}=90^{0}$, $\widehat{ABD}=\widehat{CBE}=\widehat{CAF}=\alpha$. Prove that $DF=AE$.
5. Show that the following sum is a positive integer \[\begin{align} S=&1+\frac{1}{2}+\frac{1}{3}+\ldots+\frac{1}{2017}+\left(1+\frac{1}{2}+\frac{1}{3}+\ldots+\frac{1}{2017}\right)^{2}+\\&+\left(\frac{1}{2}+\frac{1}{3}+\ldots+\frac{1}{2017}\right)^{2}+\ldots+\left(\frac{1}{2017}\right)^{2}.\end{align}\]
6. Solve the system of equations \[\begin{cases}2x^{5}-2x^{3}y-x^{2}y+10x^{3}+y^{2}-5y & =0\\ (x+1)\sqrt{y-5}-y+3x^{2}-x+2 & =0\end{cases}.\]
7. Prove that $x_{0}=\cos\dfrac{\pi}{21}+\cos\dfrac{8\pi}{21}+\cos\dfrac{10\pi}{21}$ is a solution of the equation \[4x^{3}+2x^{2}-7x-5=0.\]
8. In any triangle $ABC$, show that \[\cos(A-B)+\cos(B-C)+\cos(C-A)\leq\frac{1}{2}\left(\dfrac{a+b}{c}+\frac{b+c}{a}+\frac{c+a}{b}\right).\]
9. Given $6$ positive numbers $a,b,c$, $x,y,z$ and assume that $x+y+z=1$. Show that \[ax+by+cz\geq a^{x}b^{y}c^{z}.\]
10. Given an infinite sequence of positive integers $a_{1}<a_{2}<\ldots a_{n}<\ldots$ such that $a_{i+1}-a_{i}\geq8$ for all $i=1,2,3,\ldots$. For each $n$, let $s_{n}=a_{1}+a_{2}+\ldots a_{n}$. Show that for each $n$, there are at lease two square numbers inside the half-open interval $[s_{n},s_{n+1})$.
11. Given two positive sequences $(a_{n})_{n\geq0}$ and $(b_{n})_{n\geq0}$ which are determined as follows \[a_{0}=\sqrt{3},\,b_{0}=2,\quad a_{n}^{2}+1=b_{n}^{2},\,a_{n}+b_{n}=\frac{1+a_{n+1}}{1-a_{n+1}},\,\forall n\in\mathbb{N}.\] Show that they converges and find the limits.
12. Given a triangle $ABC$ with $AB\ne AC$. A circle $(O)$ passing through $B$, $C$ intersects the line segments $AB$ and $AC$ at $M$ and $N$ respectively. Let $P$ be the intersection of $BN$ and $CM$. Let $Q$ be the midpoint of the arc $BC$ which does not contian $M$, $N$. Let $K$ be the incenter of $PBC$. Show that $KQ$ always goes through a fixed point when $(O)$ varies.

 Issue 489

1. Find all pairs of integers $(x,y)$ satisfying \[x^{2}+x=3^{2018y}+1.\]
2. Given a triangle $ABC$ with $\angle B=45^{0}$, $\angle C=30^{0}$. Let $BM$ be one of the medians of $ABC$. Find the angle $\widehat{AMB}$.
3. Given real numbers $x,y$ satisfying $0<x,y<1$. Find the minimum value of the expression \[F=x^{2}+y^{2}+\frac{2xy-x-y+1}{4xy}.\]
4. Given a circle $(O)$ with a diameter $AB$. On $(O)$ pick a point $C$ ($C$ is different from $A$ and $B$). Draw $CH$ perpendicular to $AB$ at $H$. Choose $M$ and $N$ on the line segments $CH$ and $BC$ respectively such that $MN$ is parallel to $AB$. Through $N$ draw a line perpendicular to $BC$. This line intersects the ray $AM$ at $D$. On the line $DO$ choose two points $F$ and $K$ such that $O$ is the midpoint of $FK$. The lines $AF$ and $AK$ respectively intersect $(O)$ at $P$ and $Q$. Prove that $D$, $P$, $Q$ are colinear.
5. Suppose that the polynomial \[f(x)=x^{3}+ax^{2}+bx+c\] has $3$ non-negative real solutions. Find the maximal real number $\alpha$ so that \[f(x)\geq\alpha(x-a)^{3},\,\forall x\geq0.\]
6. Solve the equation \[(1-\sqrt{2}\sin x)(\cos2x+\sin2x)=\frac{1}{2}.\]
7. Given the following system of equations \[\begin{cases}\dfrac{yz(y+z-x)}{x+y+z} & =a\\ \dfrac{zx(z+x-y)}{x+y+z} & =b\\ \dfrac{xy(x+y-z)}{x+y+z} & =c\end{cases}\] where $a,b,c$ are positive parameters.
   a) Show that the system always has a positive solution.
   b) Solve the system when $a=2$, $b=5$, $c=10$.
8. Suppose that $a,b,c$ are the lengths of three sides of a triangle. Prove that \[ \frac{ab}{a^{2}+b^{2}}+\frac{bc}{b^{2}+c^{2}}+\frac{ca}{c^{2}+a^{2}}\geq\frac{1}{2}+\frac{2r}{R}\] with $R$, $r$ respectively are the inradius and the circumradius of the triangle.
9. For any integer $n$ which is greater than $3$, let \[P=\sqrt[60]{3}\cdot\sqrt[120]{4}\ldots\sqrt[n^{3}-n]{n-1}.\] Show that \[\sqrt[24n^{2}+24n]{3^{n^{2}+n-12}}\leq P<\sqrt[8]{3}.\]
10. Find natural numbers $n$ so that $4^{m}+2^{n}+29$ cannot be a perfect square for any natural number $m$.
11. The sequence $(a_{n})$ is given as follows \[a_{1}=\frac{1}{2},\quad a_{n+1}=\frac{(a_{n}-1)^{2}}{2-a_{n}},\, n\in\mathbb{N^{*}.}\] a) Find ${\displaystyle \lim_{n\to\infty}a_{n}}$.
    b) Show that ${\displaystyle \frac{a_{1}+a_{2}+\ldots+a_{n}}{n}\geq1-\frac{\sqrt{2}}{2}}$ for all $n\in\mathbb{N^{*}}$.
12. Given a triangle $ABC$ inscribed in a circle $(O)$. A point $P$ varies on $(O)$ but is different from $A$, $B$ and $C$. Choose $M$, $N$ respectively on $PB$, $PC$ so that $AMPN$ is a parallelogram.
    a) Prove that there exists a fixed point which is equidistant from $M$ and $N$.
    b) Prove that the Euler line of $AMN$ always goes through a fixed point.

 Issue 490

1. The natural number $a$ is coprime with $210$. Dividing $a$ by $210$ we get the remainder $r$ satisfying $1 < r < 120$. Prove that $r$ is prime.
2. Given non-zero numbers $a,b,c,d$ satisfying $b^{2}=ac$, $c^{2}=bd$, $b^{3}+27c^{3}+8d^{3}\ne0$. Show that \[\frac{a}{d}=\frac{a^{3}+27b^{3}+8c^{3}}{b^{3}+27c^{3}+8d^{3}}.\]
3. Find all natural solutions of the equation \[3xyz-5yz+3x+3z=5.\]
4. Given a half circle with the center $O$, the diameter $AB$, and the radius $OD$ perpendicular to $AB$. A point $C$ is moving on the arc $BD$. The line $AC$ intersects $OD$ at $M$. Prove that the circumcenter $I$ of the triangle $DMC$ always belongs to a fixed line.
5. Let $x,y$ be real numbers such that $x^{3}+y^{3}=2$. Find the minimum value of the expression \[P=x^{2}+y^{2}+\frac{9}{x+y}.\]
6. Given positive numbers $a,b,c$ satisfying $abc=1$. Prove that \[\frac{1}{a^{5}+b^{5}+c^{2}}+\frac{1}{b^{5}+c^{5}+a^{2}}+\frac{1}{c^{5}+a^{5}+b^{2}}\leq1.\]
7. Find all positive integers $a$, $b$ such that \[\sqrt{8+\sqrt{32+\sqrt{768}}}=a\cos\frac{\pi}{b}.\]
8. Given a triangle $ABC$. Let $(K)$ be the circle passing through $A$, $C$ and is tangent to $AB$ and let $(L)$ be the circle passing through $A$, $B$ and is tangent to $AC$. Assume that $(K)$ intersects $(L)$ at another point $D$ which is different from $A$. Assume that $AK$, $AL$ respectively intersect $DB$, $DC$ at $E$ and $F$. Let $M$, $N$ respectively be the midpoints of $BE$, $CF$. Prove that $A$, $M$, $N$ are colinear.
9. Given real numbers $a,b,c$ such that \[2(a^{2}b^{2}+b^{2}c^{2}+c^{2}a^{2})\geq a^{4}+b^{4}+c^{4}.\] Prove that \[|b+c-a|+|c+a-b|+|a+b-c|+|a+b+c|=2(|a|+|b|+|c|).\]
10. Find all triples of positive integers $(a,b,c)$ such that \[2^{a}+5^{b}=7^{c}.\]
11. The sequence $(u_{n})$ is determined as follows \[u_{1}=14,\,u_{2}=20,\,u_{3}=32,\quad u_{n+2}=4u_{n+1}-8u_{n}+8u_{n-1},\,\forall n\geq2. \] Show that $u_{2018}=5\cdot2^{2018}$.
12. Given a triangle $ABC$ with $(O)$ is the circumcircle and $I$ is the incenter. Let $D$ be the second intersection of $AI$ and $(O)$. Let $E$ be the intersection between $BC$ and the line pasing through $I$ and perpecdicular to $AI$. Assume that $K$, $L$ respectively are the intersections between $BC$, $DE$ and the line passing through $I$ and perpendicular to $OI$. Prove that $KI=KL$.

 Issue 491

1. Consider all pairs of integers $x$, $y$ which are greater than $1$ and satisfy $x^{2017}=y^{2018}$. Find the pair with the smallest possible value for $y$.
2. Given an isosceles triangle $ABC$ with the apex $A$. Suppose that $\hat{A}=180^{0}$, $BC=a$, $AC=b$. Outside $ABC$, construct the isosceles triangle $ABD$ with the apex $A$ and $\widehat{BAD}=36^{0}$. Find the perimeter of the triangle $ABD$ in terms of $a$ and $b$.
3. Find positive integers $n$ such that if the positive integer $a$ is a divisor of $n$ then $a+2$ is also a divisor of $n+2$.
4. Given a triangle $ABC$ inscribed the circle $(O)$ with diameter $AC$. Draw a line which is perpendicular to $AC$ at $A$ and intersects $BC$ at $K$. Choose a point $T$ on the minor $AB$ ($T$ is different from $A$ and $B$). The line $KT$ intersects $(O)$ at the second point $P$. On the tangent line to the circle $(O)$ at the point $T$ choose two points $I$ and $J$ such that $KIA$ and $KAJ$ are isosceles triangles with the apex $K$. Show that
   a) $\widehat{TIP}=\widehat{TKJ}$.
   b) The circle $(O)$ and the circumcircle of the triangle $KPJ$ are tangent to each other.
5. Find integral solutions of the equation \[y^{2}+2y=4x^{2}y+8x+7.\]
6. Solve the system of equations \[\begin{cases}\log_{2}x+\log_{2}y+\log_{2}z & =3\\ \sqrt{x^{2}+4}+\sqrt{y^{2}+4}+\sqrt{z^{2}+4} & =\sqrt{2}(x+y+z)\end{cases}\]
7. Show that \[\lim_{n\to\infty}\sqrt{1+2\sqrt{1+3\sqrt{1+\ldots\sqrt{1+(n-1)\sqrt{1+n}}}}}=3.\]
8. Let $m_{a}$, $m_{b}$, $m_{c}$ be the lengths of the medians of the triangle with the perimeter $2$. Show that \[\max\{1;3\sqrt[3]{r}\}\leq m_{a}+m_{b}+m_{c}<\frac{3}{\sqrt{2}}\] where $r$ is the inradius of the triangle.
9. Assume that $a,b,c$ are three non-negative numbers such that $a+b+c=3$. Find the maximum value of the expression \[P=a\sqrt{b^{3}+1}+b\sqrt{c^{3}+1}+c\sqrt{a^{3}+1}.\]
10. For every $n\in\mathbb N$, let $F_{n}=2^{2^{n}}+1$. For each $n\in\mathbb N$, let $q$ be a prime divisor of $F_{n}$. Show that $2^{n+1}\mid q-1$. Futhermore, if $n\geq 2$, show even more that $2^{n+2}\mid q-1$.
11. Find all funtions $f:\mathbb{R\to\mathbb{R}}$ satisfying \[f(x)f(y)+f(xy)+f(x)+f(y)=f(x+y)+2xy\] for all $x,y\in\mathbb{R}.$
12. Given a convex hexagon $ABCDEF$ circumscribing a circle $(O)$. Assume that $O$ is the circumcenter of the triangle $ACE$. Prove that the circumcenter of the triangles $OAD$, $OBE$ and $OCF$ has another common point besides $O$.

 Issue 492

1. Let $$M=\frac{1}{10}+\frac{1}{20}+\frac{1}{35}+\frac{1}{56}+\frac{1}{84}+\frac{1}{120}+\ldots$$ a) Is the fraction $\dfrac{1}{15400}$ a term of $M$ ? Why?.
   b) Compute the sum of the $8$ first terms of $M$. 
2. Given a triangle $A B C$ with $A B < A C$. The angle bisector of $\widehat{B A C}$ intersects the perpendicular bisector of $B C$ at $M$. Let $H$, $K$, and $I$ respectively be the perpendicular projections of $M$ on $A B$, $A C$ and $B C$. Prove that $H$, $I$, $K$ is collinear. 
3. Find integer solutions of the equation $$x^{3}=4 y^{3}+x^{2} y+y+13$$
4. Given a quadrilateral $A B C D$ inscribed in the circle with the diameter $A C$. Let $M$ be the point on $A B$ such that $A M=A D$. The lines $D M$ and $B C$ intersects at $N,$ and the $A N$ intersects the circle at $K$. Let $H$ be the perpendicular projection of $D$ on $A C$. Assume that $A B$ intersects $N H$ and $C K$ at $P$ and $Q .$ Show that $$\frac{1}{M P}=\frac{1}{M A}+\frac{1}{M Q}.$$
5. Solve the system of equations $$\begin{cases} 3 x^{3}+6 x+2 &=2 y^{3} \\ 3 y^{3}+6 y+2 &=2 z^{3} \\ 3 z^{3}+6 z+2 &=2 x^{3}\end{cases}$$
6. Solve the inequality $$x^{3}+6 x^{2}+9 x \leq \sqrt{x+4}-2$$
7. Let $a, b, c$ be positive numbers with the product is equal to $1$. Find the minimum value of the expression $$P=\frac{1}{a^{2017}+a^{2015}+1}+\frac{1}{b^{2017}+b^{2015}+1}+\frac{1}{c^{2017}+c^{2015}+1}$$
8. Given a triangle $A B C$ inscribed in the circle $(O)$. The tangent lines of $(O)$ at $B$ and $C$ intersect at $P .$ The line which goes through $A$ and is parallel to $B P$ intersects $B C$ at $M$. The line which goes through $A$ and is parallel to $B C$ intersects $B P$ at $N$. Suppose that $I$ is the intersections between $A P$ and $M N$. Prove that four points $B$, $I$, $O$, $C$ lie on a circle.
9. Given the equation $x^{3}+m x^{2}+n=0$. Find $m$, $n$ so the the equation has three distinct non-zero real roots $u$, $v$, $t$ satisfying $$\frac{u^{4}}{u^{3}-2 n}+\frac{v^{4}}{v^{3}-2 n}+\frac{t^{4}}{t^{3}-2 n}=3$$
10. Let $p$ be an odd prime and $a_{1}, a_{2}, \ldots, a_{p}$ is an arithmetic progression with the common difference $d$ which is not divisible by $p$. Prove that $\prod_{i=1}^{p}\left(a_{i}+a_{1} a_{2} \ldots a_{p}\right): p^{2}$
11. Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that $$f(x+f(y))=f\left(x+y^{2018}\right)+f\left(y^{2018}-f(y)\right),\, \forall x, y \in \mathbb{R}.$$
12. Given an acute triangle $A B C$ with the altitudes $B E$, $C F$. Let $S T$ be a chord of the circumcircle of $A E F$. Two circles which go through $S$, $T$ is tangent to $B C$ respectively at $P$ and $Q$. Prove that the intersection between $P E$, $Q F$ lies on the circumcircle of $A E F$.

 Issue 493

1. Find $2018$ numbers so that each of them is the square of the sum of all remaining numbers.
2. Find the following sum $$S=(1+2.3+3.5+\ldots+101.201) +\left(1^{2}+2^{2}+3^{2}+\ldots+100^{2}\right).$$
3. Find all pairs of positive integers $(m, n)$ such that $$n^{3}-5 n+10=2^{m}.$$
4. Given a triangle $A B C$ with $B C=a$, $A C=b$, $A B=c$ and $3 \hat{B}+2 \hat{C}=180^{\circ}$. Prove that $$b+c \leq \dfrac{5}{4} a.$$
5. Solve the system of equations $$\begin{cases}x^{2}-y^{2}+\sqrt{x}-y+2&=0 \\ x+8 y+4 \sqrt{x}-8 \sqrt{y}-4 \sqrt{x y} &=0\end{cases}$$
6. Given three positive numbers $a, b, c$ satisfying $a+b+c=3 .$ Show that $$\frac{1}{(a+b)^{2}+c^{2}}+\frac{1}{(b+c)^{2}+a^{2}}+\frac{1}{(c+a)^{2}+b^{2}} \leq \frac{3}{5}$$
7. Solve the system of equations $$\begin{cases}x-1&=\sqrt[4]{9+12 y-6 y^{2}} \\ y-1&=\sqrt[4]{9+12 x-6 x^{2}}\end{cases}.$$
8. Given a right prism with equilateral bases $A B C . A^{\prime} B^{\prime} C$. Let $\alpha$ be the angle between the line $B C$ and the plane $\left(A^{\prime} B C\right)$. Prove that $\sin \alpha \leq 2 \sqrt{3}-3$
9. Given positive numbers $a$, $b$. Show that $$\frac{1}{2}\left[1-\frac{\min (a, b)}{\max (a, b)}\right]^{2} \leq \frac{b-a}{a}-\ln b+\ln a \leq \frac{1}{2}\left[\frac{\max (a, b)}{\min (a, b)}-1\right]^{2}.$$
10. Find the maximal positive number $k$ so that the following inequality $$a^{2}+b^{2}+c^{2}+k(a+b+c) \geq 3+k(a b+b c+c a)$$ holds true for all positive numbers $a, b, c$.
11. Given the sequence $\left(x_{n}\right)$ $$x_{2}=x_{3}=1,\quad (n+1)(n-2) x_{n+1}=\left(n^{3}-n^{2}-n\right) x_{n}-(n-1)^{3} x_{n-1},\, \forall n \geq 3.$$ Find all indices $n$ so that $x_{n}$ is an integer.
12. Given a cyclic quadrilateral $A B C D .$ Let $K$ be the intersection between $A C$ and $B D .$ Let $M$, $N$, $P$ and $Q$ respectively be the perpendicular projection of $K$ on $A B$, $B C$, $C D$ and $D A$. And then, let $X$, $Y$, $Z$ and $T$ respectively be the perpendicular projection of $K$ on $M N$, $N P$, $P Q$, $Q M$. Prove that $A X C Z$ and $B Y D T$ have equal areas.

 Issue 494

1. Find all natural numbers $m, n$ and primes $p$ satisfying each of the following equalities
   a) $p^{m}+p^{n}=p^{m+n}$.
   b) $p^{m}+p^{n}=p^{m n}$
2. Given a triangle $A B C$. Let $M$, $N$ and $P$ respectively be the midpoints of $A B$, $A C$ and $B C$. Let $O$ be the intersection between $C M$ and $P N$, $I$ be the intersection between $A O$ and $B C$ and $D$ be the intersection between $M I$ and $A C$. Show that $A I$, $B D$, $M P$ are concurrent.
3. Solve the equation $$\frac{1-4 \sqrt{x}}{2 x+1}=\frac{2 x}{x^{2}+1}-2$$
4. Given a right triangle $A B C$ with the right angle $A$. Let $A H$ be the altitude. On the opposite ray of the ray $H A$ pick an arbitrary point $D(D \neq H)$. Through $D$ draw the line perpendicular to $B D .$ That line intersects $A C$ at $E$. Let $K$ be the perpendicular projection of $E$ on $A H .$ Show that $D K$ has a fixed length when $D$ varies.
5. Given real numbers $x$, $y$ such that $x^{2}+y^{2}=1$. Find the minimum and maximum values of the expression $$T=\sqrt{4+5 x}+\sqrt{4+5 y}$$
6. Find conditions on $a$, $b$ besides $a>b \geq-1$ so that the system $$\begin{cases} x^{2} &=(a-y)(a+y+2) \\ y^{2} &=(b-x)(b+x+2)\end{cases}$$ has unique solution.
7. Given positive real numbers $a, b, c$. Prove that $$\frac{b(2 a-b)}{a(b+c)}+\frac{c(2 b-c)}{b(c+a)}+\frac{a(2 c-a)}{c(a+b)} \leq \frac{3}{2}$$
8. Let $A$, $B$ and $C$ denote the angles of a triangle $A B C$ $\left(A, B, C \neq \frac{\pi}{2}\right)$. Show that $$\frac{\sin A}{\tan B}+\frac{\sin B}{\tan C}+\frac{\sin C}{\tan A} \geq \frac{3}{2}$$
9. Find all pairs of positive integers $(x ; y)$ satisfying $$x^{3}+y^{3}=x^{2}+72 x y+y^{2}$$
10. For any integer $n$, show that $$a_{n}=\frac{3+\sqrt{5}}{10}\left(\frac{7+3 \sqrt{5}}{2}\right)^{n}+\frac{3-\sqrt{5}}{10}\left(\frac{7-3 \sqrt{5}}{2}\right)^{n}+\frac{2}{5}$$ is a perfect square.
11. A class has $n$ students attending $n-1$ clubs. Show that we can choose a group of at least two students so that, for each club, there are an even number of students in that group attend it.
12. Given a square $A B C D$ and $P$ is an arbitrary point on the side $A B$. Let $\left(I_{1}\right)$, $\left(I_{2}\right)$ respectively be the incircles of $A D P$, $C B P$. Assume that $D I_{1}$, $C I_{2}$ intersect $A B$ respectively at $E$, $F$. The line through $E$ which is parallel to $A C$ intersects $B D$ at $M$ and the line through $F$ which is parallel to $B D$ intersects $A C$ at $N$. Show that $M N$ is a common tangent to $\left(I_{1}\right)$ and $\left(I_{2}\right)$.

 Issue 495

1. Show that it is impossible to write $2^{9^{2018}}$ as a sum of $n$ consecutive positive integers for any $n \in \mathbb{N}$, $n \geq 2$.
2. Find the last twelve digits of the number $5^{1040}$. 
3. Find integral solutions of the following systems of equations
   a) $\begin{cases}3 a^{2}+2 a b+3 b^{2}&=12 \\ a^{2}+b^{2}&=c^{2}\end{cases}.$
   b) $\begin{cases}(z-3)\left(x^{2}+y^{2}\right)-2 x y &=0 \\ x+y &=z\end{cases}.$
4. Given a right triangle $A B C$ with the right angle $A$. In the angle $\widehat{B A C}$ draw the rays $A x$, $A y$ such that $\widehat{C A x}=\dfrac{1}{2} \widehat{A B C}$, $\widehat{B A y}=\dfrac{1}{2} \widehat{A C B}$. The ray $A x$ intersects the angle bisector of $\widehat{A C B}$ at $Q$, the ray $A y$ intersects $B C$ at $K$. Compute the ratio $\dfrac{A K}{A Q}$.
5. Solve the equation $$3 \sqrt[3]{\frac{x^{2}-2 x+2}{2 x-1}}+2 x=5$$
6. Given real numbers $x$, $y$, $z$ such that $x y z=1$. Show that $$\left(\frac{x}{1+x y}\right)^{2}+\left(\frac{y}{1+z y}\right)^{2}+\left(\frac{z}{1+x z}\right)^{2} \geq \frac{3}{4}$$
7. Suppose that the polynomial $P(x)=x^{2018}-a x^{2016}+a$ ($a$ is a real parameter) has $2018$ real solutions. Show that there exists $\left|x_{0}\right| \leq \sqrt{2}$
8. Given a pyramid $S . A B C D$ with the base $A B C D$ is a parallelogram, and $S A=S B=S C=a$, $A B=2 a$, $B C=3 a$. Let $S D=x$ $(0<x<a \sqrt{14})$. Find $x$ in terms of $a$ so that the product $A C \cdot S D$ obtains its maximum value.
9. Suppose that $x, y, z$ are nonnegative numbers satisfying $x+y+z=1$. Find the maximum value of the expression $$P=x y+y z+z x+\frac{2}{9}(M-m)^{3}$$ where $M=\max \{x, y, z\}$ and $m=\min \{x, y, z\}$
10. Consider the sequence $a_{n}=n+[\sqrt[3]{n}]$, $n$ positive integers, $[\sqrt[3]{n}]$ is the integral part of $\sqrt[3]{n}$. Suppose there exists a positive integer $k$ such that the terms $a_{k} ; a_{k+1} ; \ldots ; a_{k+p}$ are $p+1$ consecutive natural numbers with $p=6015 \times 2006+1 .$ Show that $$k>8.10^{9}+6.10^{7}.$$
11. Find the least number $k$ so that in any subset of $k$ elements of $\{1,2, \ldots, 25\}$ we can always find at least a Pythagorean triple.
12. Let $A B C$ be a triangle inscribed in a circle $(O)$. Suppose that $A H$ is the altitude and the line $A O$ intersects $B C$ at $D .$ Let $K$ be the second intersection of the circumcircle of $A D C$ and the circumcircle of $A H B$. Suppose that the circumcircle of $K H D$ intersects $(O)$ at $M$ and $N$. Let $X$ be the intersection of $M N$ and $B C$. Show that $X A=X K$.

 Issue 496

1. Find a three-digit number such that it is $9$ times the sum of the squares of its digits.
2. Given triangle $A B C$ with $\widehat{B}=45^{\circ}$, $\widehat{C}=30^{\circ}$. Outside the triangle $A B C$ take the point $D$ such that $\widehat{D B C}=\widehat{D C B}=15^{\circ}$. Prove that triangle $A B D$ is equilateral.
3. Let $x, y, z$ be three non-negative numbers. Prove that $$[(x+y)(y+z)(z+x)]^{2} \geq xyz(2 x+y+z)(2 y+z+x)(2 z+x) +y)$$
4. Given a semicircle $(O ; R)$ with diameter $A B$ and tangent $A x$. On $A x$ take $A K=R$. A circle with center $K$ and radius $R$ cuts $KB$ at $I$, a circle with center $B$ and radius $B I$ cuts $(O)$ at $E$. Ray $B E$ intersects $A x$ at $C$. Prove that the median of $B C$ is a tangent to $(O)$.
5. Solve the system of equations $$\begin{cases}x^{2}\left(y-2 x^{2} y\right)+y \sqrt{(x y+1)(3-xy)} &= y^{5} \\ x^{3}\left(1-x^{2}\right)+x\left(\dfrac{1}{2}+\sqrt{\dfrac{1}{4} +\left(x^{2}-y^{2}\right)^{2}}\right) &=x^{2} y^{2}\end{cases}$$
6. Solve the following system of equations on the set of real numbers $$\begin{cases}\dfrac{1}{x}+\dfrac{1}{y}+\dfrac{1}{z} &=\dfrac{1}{ xyz} \\ x+y+z &=xy z+\dfrac{8}{27}(x+y+z)^{3}\end{cases}$$
7. Solve the equation $$\log _{2} x=\log _{5-x} 3.$$
8. Prove that the sum of the squares of the distances from the midpoint of a side of a triangle to the orthocenter and circumcenter of the triangle does not depend on which side's midpoint is chosen in the triangle.
9. With $k$ being the given positive integer, we denote the square of the sum of its digits as $f_{1}(k)$. Put $f_{n+1}(k)=f_{1}\left(f_{n}(k)\right)$ $(n=1,2, \ldots)$. Let's calculate $f_{2018}\left(2^{1990}\right)$.
10. Given the sequence $\left(x_{n}\right)$ $\left(n \in \mathbb{N}^{*}\right)$ defined by $$x_{1}=0, \quad x_{n+1}=\left(\frac{1}{4}\right)^{x_{n}},\, \forall n \in \mathbb{N}^{*}.$$ Proof that the sequence $\left(x_{n}\right)$ has a finite limit when $n \rightarrow+\infty$. Find $\displaystyle\lim_{n \rightarrow+\infty} x_{n}$.
11. Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ that satisfy $$f(x) f(y)-\frac{4}{9} xy=f(x+y ),\, \forall x, y \in \mathbb{R}.$$
12. Let quadrilateral $A B C D$ inscribed in circle $(O)$. $A C$ delivers $B D$ at $E$, $A D$ delivers $BC$ at $F$. Let $\left(O_{1}\right)$ be the circle tangent to rays $E A$, $E B$ and tangent to $(O)$; $\left(O_{2}\right)$ is a circle tangent to rays $FA$, $FB$ and externally tangent to $(O)$ at a point on arc $AB$ that does not contain $C $, $D$. Prove that the intersection of two common external tangents to $(O_{1})$ and $(O_{2})$ lies on $(O)$.

 Issue 497

1. Find all positive integers $n$ such that there exist prime numbers $p, q$ satisfying $$1 !+2 !+3 !+\ldots+n !=p^{2}+q^{2}+5895$$ Notice that $n !=1.2 .3 \ldots n$. 
2. Given an acute angle $A B C$ with the altitude $A H,$ the median $B M,$ the angle bisector $C K$. Show that if $HMK$ is an equilateral triangle then so is $A B C$. 
3. Suppose that $n$ is a positive integer so that $3^{n}+7^{n}$ is divisible by $11$. Find the remainder in the divison of $2^{n}+17^{n}+2018^{n^{2}}$ by $11$ 
4. Given a circle $(O)$ and suppose that $B C$ is a fixed chord of $(O) .$ Let $A$ be a point moving on the major arc $B C ;$ and $M,$ $N$ respectively the midpoints of $A B$ and $A C$. Show that each of the altitudes $M M^{\prime}$ and $N N^{\prime}$ of $\triangle A M N$ contains some fixed point. 
5. Solve the equation $$13 x^{2}-5 x-13=(16 x-11) \sqrt{2 x^{2}-3}$$
6. Solve the system of equations $$\begin{cases}\sqrt{4 x^{2}+5}+\sqrt{4 y^{2}+5}&=6|x y| \\ 2^{\dfrac{1}{x^{8}}+\dfrac{2}{y^{4}}}+2^{\dfrac{1}{y^{8}}+\dfrac{2}{x^{4}}}&=16 \end{cases}.$$
7. Show that the following inequality holds for all positive numbers $a, b, c$ $$a^{2}+b^{2}+c^{2} \geq \frac{1}{2}(a b+b c+c a) + \sqrt{\frac{2(a+b+c)\left(a^{3} b^{3}+b^{3} c^{3}+c^{3} a^{3}\right)}{(a+b)(b+c)(c+a)}}$$
8. Given triangle $A B C$ with the incenter $I$, the centroid $G$. There lines $A G$, $B G$ and $CG$ respectively intersect the circumcircle of $A B C$ at $A_{2}$, $B_{2}$, $C_{2}$. Show that $$G A_{2}+G B_{2}+G C_{2} \geq I A+I B+I C.$$
9. Given continuous functions $f,g:[a, b] \rightarrow[a, b]$ such that $$f(g(x))=g(f(x)) \quad \forall x \in[a, b]$$ where $a, b$ are real numbers. Show that the equation $f(x)=g(x)$ has at least one solution.
10. Find all pairs of integers $(m, n)$ so that both $m^{5} n+2019$ and $m n^{5}+2019$ are cubes of integers.
11. Let $\quad S=\{1,2,3, \ldots, 2019\}$ and suppose that $S_{1}, S_{2}, \ldots, S_{410}$ are subsets of $S$ such that for every $i \in S$, there exist exactly $40$ of them containing i. Prove that there exist different indices $j, k, l \in\{1,2, \ldots, 410\}$ such that $\left|S_{j} \cap S_{k} \cap S_{l}\right| \leq 1$
12. Given a cyclic quadrilateral $A B C D$ with $E$ is the intersection between $A C$ and $B D .$ The circumcircles of $E A D$ and $E B C$ intersect at another point $F$ (besides $E$). The perpendicular bisectors of $B D$, $A C$ respectively intersect $F B$, $F A$ at $K$, $L$. Show that $K L$ goes through the midpoint of $A B$.

 Issue 498

1. Let $$A=1^{2016}+2^{2016}+3^{2016}+\ldots+2015^{2016}+2016^{2016}.$$ Show that $A$ is not a perfect square.
2. Find all natural numbers $k$, $m$, $n$ so that $2 . k !=m !-2 . n !$ where $n !=1.2 .3 \ldots n$, $0 !=1$.
3. Find integral solutions $(x ; y)$ of the equation $$\left(y-\sqrt{y^{2}+2}\right)\sqrt{x}+\sqrt{4+2 x}=2$$
4. Given a triangle $A B C$ with the angles satisfying $\hat{A}: \widehat{B}: \widehat{C}=8: 3: 1 .$ Let $A D$ be the angle bisector of the angle $A(D$ is on $B C)$. Assume furthermore that $$\frac{1}{B D^{2}}+\frac{1}{B C^{2}}=\frac{4}{3}.$$ Compute the length of $A D$. 
5. Solve the equation $$2 x^{3}+20=9 x \sqrt[3]{x^{3}-7}$$
6. Solve the system of equations $$\begin{cases}\sqrt{2\left(x^{2}+4 y^{2}-8\right)} &=\left(\sqrt{y^{2}+x-3}+1-y\right) \left(\sqrt{y^{2}+x-3}+1+y\right) \\ 2 \sqrt[4]{y^{4}+5} &=x  \end{cases}$$
7. Solve the equation $$\sqrt{1+2 \log _{16} x^{2}}+\sqrt{4-\frac{3}{4} \log _{8} x^{4}} +\frac{\log _{2} x^{2}-3}{\log _{2}^{2} x-\frac{2}{3} \log _{2} x^{3}+2}=0$$
8. Given a triangle $A B C$ $(A B<A C)$, inscribed in a given circle $(O)$ with the point $A$ can be varied and the points $B$, $C$ are fixed, and two points $A$ and $O$ are always on the same side determined by $B C$. A circle $\left(O^{\prime}\right)$ is internally tangent to $(O)$ at $T$ ($T$ is outside the triangle $A B C$) and is tangent to the sides $A B$, $A C$ respectively at $P$, $Q$. The line $P Q$ intersects $B C$ at $R$. The lines $T B$, $T C$ meet again $\left(O^{\prime}\right)$ respectively at $E$, $F$ $(E \neq T, F \neq T)$. Prove that
   a) The line $E F$ is parallel to the line $B C$.
   b) The line $R T$ always passes through a fixed point when $A$ varies. 
9. Let $a, b, c$ be positive numbers such that $a+b+c+2=2 a b c$. Prove that $$\frac{a+2}{\sqrt{6\left(a^{2}+2\right)}}+\frac{b+2}{\sqrt{6\left(b^{2}+2\right)}}+\frac{c+2}{\sqrt{6\left(c^{2}+2\right)}} \leq 2$$
10. a) Given rational numbers $a_{1}, a_{2}, \ldots, a_{n}$ $\left(n \in \mathbb{Z}^{+}\right) $. Show that if, for any positive integer $m$, $a_{1}^{m}+a_{2}^{m}+\ldots+a_{n}^{m}$ is interger then $a_{1}, a_{2}, \ldots, a_{n}$ are integers.
    b) Is the above conclusion still correct if we only assume that $a_{1}, a_{2}, \ldots, a_{n}$ are real?
11. People come and sit on the bench, one by one. The first person can choose any seat. The next people firstly try to avoid to sit next to the previous ones. If they cannot avoid, then they can choose any empty seat. How many such arrangements are there?
12. Given a triangle $A B C$. Let $m_{a}$, $m_{b}$, $m_{c}$ respectively be the lengths of the medians corresponding to the sides $B C=a$, $A C=b$, $A B=c$. Show that $$(a b+b c+c a)\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right) \geq 2 \sqrt{3}\left(m_{a}+m_{b}+m_{c}\right).$$ When does the equality hold?