Διασκεδαστικά Μαθηματικά : Mathematics and Youth Magazine Problems 2017 (Issue 475

Τρίτη 28 Μαρτίου 2023

Mathematics and Youth Magazine Problems 2017 (Issue 475 - 486)

Issue 475

Given a natural number $n$. Find all prime numbers $p$ such that the following number $$A=1010n^{2}+2010(n+p)+10^{10^{1954}}$$ can be written as a difference of two perfect squares.
Given a triangle $ABC$ and let $M$ be the midpoint of $BC$. Suppose that $\widehat{ABC}+\widehat{MAC}=90^{0}$. Prove that $ABC$ is either an isosceles triangle or a right triangle.
Solve the equation $$
\frac{x}{2\sqrt{x}+1}+\frac{x^{2}}{2\sqrt{x}+3}=\frac{\sqrt{x^{3}}+x}{4}.$$
Give a isoceles trapezoid $ABCD$ inscribed in a circle $(O)$ with $AB$ is parallel to $CD$ and $AB<CD$. Let $M$ be the midpoint of $CD$and let $P$ be any point on the side $MD$. Suppose that $AP$ intersects $(O)$ at the second point $Q$, and $BP$ intersects $(O)$ at the second point $R$. Assume that QR intersects $CD$ at $E$. Let $F$ be the symmetry point of $P$ over the point $E$. Suppose that $EA$ is tangent to $(O)$, prove that $AF$ is perpendicular to $AQ$.
Let $x,y$ belong to $(0,1)$. Find the maximum value of the expression $$
P=x\sqrt{1-y^{2}}+y\sqrt{1-x^{2}}+\frac{1}{\sqrt{3}}(x+y).$$
Given $f\left(x\right)=x^{2}-6x+12$. Solve the equation $f(f(f(f(x))))=65539$.
Find all real solutions of the following system of equations:
$$\begin{cases} \sqrt{2^{2^{x}}} & =2(y+1)\\ 2^{2^{y}} & =2x \end{cases}.$$
Given a uniform triangular prism $ABC,A'B'C'$ (the base $ABC$ is an equilateral triangle). Let $\alpha$ be the angle between $A'B$ and $(ACC'A')$ and let $\beta$ be the angle between $(A'B'C')$ and $(ACC'A')$. Prove that $\alpha<60^{0}$ and $\cot^{2}\alpha-\cot^{2}\beta=\frac{1}{3}$.
Let $p$, $q$ be two coprime numbers. Let $$ T=\left[\frac{p}{q}\right]+\left[\frac{2p}{q}\right]+\ldots+\left[\frac{(q-1)p}{q}\right],$$ where $[x]$ is the gretest integral number, that isn't execeed $x$ and is called the integral part of $x$.
a) Find $T$.
b) Find $p$, $q$ such that $T$ is a prime number.
Let $S$ be the number of all the binary strings of length $n$ with the property that the sum of any $3$ consecutive numbers on any of these strings is always positive. Prove that $$2\left(\frac{3}{2}\right)^{n}\leq S\leq\frac{13}{2}\cdot\left(\frac{20}{9}\right)^{n},\quad\forall n\geq3.$$
Find all functions $f:\mathbb{R}^{+}\to\mathbb{R}^{+}$ such that $$ f\left(\frac{x}{x-y}\right)+f\left(xf(y)\right)=f\left(xf(x)\right),$$ for every $x>y>0$.
Given a triangle $ABC$. The incircle $(I)$ of $ABC$ is tangent to $BC$, $CA$ and $AB$ at $D$, $E$ and $F$ respectively. Let $B_{1}$(resp. $C_{1}$) be the intersection between the lines which go through $AB$ and $DE$ (resp. $AC$ and $DF$). Let $H$ and $K$ respectively be the orthocenter of $ABC$ and $AB_{1}C_{1}$. Prove that the line which goes through $HK$ contains the point $I$.

Issue 476

Consider the sum in the following form $$\frac{1}{n}+\frac{1}{n+1}+\frac{1}{n+2}+\ldots+\frac{1}{n+9}=\frac{p}{q}$$ where $n$ is a natural number and $\frac{p}{q}$ cannot be simplidied. Find the smallest $n$ such that $q$ is divisible by $2006$.
Given an isosceles right triangle $ABC$ with the right angle $A$. Let $M$ be the point which is inside the triangle such that $BM=BA$ and $\widehat{ABM}=30^{0}$. Prove that $MA=MC$.
Given positive numbers $a,b,c$ such that $a+b+c=3$. Find the maximum value of the expression $$P=2(ab+bc+ca)-abc.$$
Given a semicircle $O$ with the fixed diameter $AB$. Let $Ax$ be the ray such that $Ax$ is tangent to the semicircle at $A$ and $Ax$ and the semicircle are on the same half-plane determined $AB$. A point $M$, which is different from $A$, varies on the ray $Ax$. Assume that $MB$ intersects the semicircle at the second point $K$ which is different from $B$. On the ray $AB$ choose $N$ such that $AN=AM$. Prove that when $M$ varies, the line which goes through $K$ and is perpendicular to $KN$ always goes through a fixed point.
Solve the equation $$\sqrt{2x^{3}-2x^{2}+x}+2\sqrt[4]{3x-2x^{2}}=x^{4}-x^{3}+3.$$
Solve the system of equations $$\begin{cases} 3^{x}+2^{y} & =5\\ 3^{y}+2^{x} & =5 \end{cases}$$
Given $n$ positive numbers $a_{1},a_{2},\ldots a_{n}$ ($n\geq2$). Let $$S=a_{1}^{n}+a_{2}^{n}+\ldots+a_{n}^{n}, \quad P=a_{1}a_{2}\cdots a_{n}.$$ Prove that $$\frac{1}{S-a_{1}^{n}+P}+\frac{1}{S-a_{2}^{n}+P}+\ldots+\frac{1}{S-a_{n}^{n}+P}\leq\frac{1}{P}.$$
Given a triangle $ABC$. Show that $$\begin{align*} & 2(\sin A+\sin B+\sin C)\left(\cos\frac{A}{2}+\cos\frac{B}{2}+\cos\frac{C}{2}\right)\\ \leq & \frac{45}{4}+(\cos A+\cos B+\cos C)\left(\sin\frac{A}{2}+\sin\frac{B}{2}+\sin\frac{C}{2}\right).\end{align*}$$
Find $$\min_{a,b}\left\{ \max_{-1\leq x\leq1}\left|ax^{2}+bx+1\right|\right\} ,\quad\min_{a,b}\left\{ \max_{-1\leq x\leq1}\left|ax^{2}+x+b\right|\right\} .$$
Given a natural number $a\geq2$. Prove that there exists infinitely many natural numbers $n$ such that $a^{n}+1$ is divised by $n$.
Let $a$ be a real number which is different from $0$ and $-1$. Find all functions $f:\mathbb{R}\to\mathbb{R}$ such that $$f(f(x)+ay)=(a^{2}+a)x+f(f(y)-x),\forall x,y\in\mathbb{R}.$$
Given an acute triangle $ABC$ with $AB<AC$ and let $AD$, $BE$ and $CF$ be its altitude. Assume that $EF$ intersects $BC$ at $G$. Let $K$ be the perpendicular projection of $C$ on $AG$. Suppose that $AD$ intersects $CK$ at $H$ and $AC$ intersects $HF$ at $L$. Prove that $A$ is the incenter of the triangle $FKL$.

Issue 477

Let $$S=\frac{1}{1.2.3.4}+\frac{1}{2.3.4.5}+\ldots+\frac{1}{20.21.22.23}.$$ Compare $S$ and $\frac{1}{18}$.
Given an isosceles triangle $ABC$ with the base $BC$. Let $BD$ be the angle bisector. On the ray $BA$, choose $E$ such that $BE=2CD$. Show that $\widehat{EDB}=90^{0}$.
Let $x,y,z$ be positive numbers such that $x+y+z=xyz$. Find the minimum value of the expression $$ S=\frac{x}{y^{2}}+\frac{y}{z^{2}}+\frac{z}{x^{2}}.$$
Given an isosceles triangle $ABC$ with base $BC$ and let $M$ be a point inside the triangle such that $\widehat{ABM}=\widehat{BCM}$. Let $H$, $I$ and $K$ respectively be the perpendicular projections of $M$ on $AB$, $BC$ and $CA$. Suppose that $E$ is the intersection between $MB$ and $IH$, and $F$ is the intersection between $MC$ and $IK$. Assume that the circumcircles of the triangles $MEH$ and $MFK$ intersect at $N\ne M$. Prove that the line $NM$ contains the midpoint of $BC$.
Solve the equation $$ (x^{2}+x+1)(\sqrt[3]{(3x-2)^{2}}+\sqrt[3]{3x-2}+1)=9.$$
Suppose that the equation $$ x^{3}-ax^{2}+bx-c=0$$ has $3$ positive solutions. Prove that if $$2a^{3}+3a^{2}-7ab+9c-6b-3a+2=0$$ then $1\leq a\leq2$.
Let $a,b,c,d$ and $e$ be the real numbers such that $$\sin a+\sin b+\sin c+\sin d+\sin e = 0$$ $$\cos2a+\cos2b+\cos2c+\cos2d+\cos2e =-3.$$ Find the maximum and minimum values of the expression $$\cos a+2\cos2b+3\cos3c.$$
Assume that $H$ is a point inside the triangle $ABC$. Let $K$ be the orthocenter of $ABH$. The line which goes through $H$ and is perpendicular to $BC$ intersects $AK$ at $E$. The line which goes through $H$ and is perpendicular to $AC$ intersects $HK$ at $F$. Prove that $CH\bot EF$.
For $n\in\mathbb{N}$, let $S_{n}=1+\frac{1}{2}+\ldots+\frac{1}{n}$. Let $\{S_{n}\}$ denote the fractional part of $S_{n}$. Prove that for every $\epsilon\in(0,1)$, there always exists $n\in\mathbb{N}^{+}$ such that $\{S_{n}\}\leq\epsilon$.
Find the smallest $k$ such that we can use $k$ colors to color the numbers $1,2,\ldots,20$ in the following way. For each number, we use exactly one color, we can use one color for more than one number, and no $3$ numbers with the same color forms an arithmetic sequence.
Consider the sequence $\{x_{n}\}$ determined as follows $$x_{1}=-\pi,\,x_{2}=-1,$$ $$x_{n+2}=x_{n+1}+\log_{2}\frac{9+3(\cos x_{n+1}-\cos x_{n})-\cos x_{n+1}\cos x_{n}}{8+\sin^{2}x_{n}},\quad n=1,2,3,\ldots.$$ Prove that the sequence $(x_{n})$ has a finite limit and find that limit.
Given a quadrilateral $ABCD$ inscribed a circle $(O)$. The external angle bisectors of the angles $\widehat{DAB}$, $\widehat{ABC}$, $\widehat{BCD}$, $\widehat{CDA}$ respectively intersects the external angle bisector of the angle $\widehat{ABC}$, $\widehat{BCD}$, $\widehat{CDA}$, $\widehat{DAB}$ at $X,Y,Z,T$. Let $E$ and $F$ respectively be the midpoints of $XZ$ and $YT$. Prove that
a) $XYZT$ is a cyclic quadrilateral and $XZ\bot YT$.
b) $O$, $E$ and $F$ are collinear.

Issue 478

Find all the $4$-digit perfect squares such that when we reverse their digits we also get perfect squares.
Given an isosceles triangle $ABC$ with the vertex angle $A$. On the half plane determined by $BC$ which does not contain $A$ choose a point $D$ such that $\widehat{BAD}=2\widehat{ADC}$ and $\widehat{CAD}=2\widehat{ADB}$. Prove that $CBD$ is an isosceles triangle with the vertex angle $D$.
Prove thta $3^{n+2}|10^{3^{n}}-1$ for any natural number $n$.
Given a trapezoid $ABCD$ ($AB\parallel CD$) with $AB<CD$. Let $P$ and $Q$ respectively be on the diagonals $AC$ and $BD$ such that $PQ$ is not parallel to $AB$. The ray $QP$ intersects $BC$ at $M$ and the ray $PQ$ intersects $AD$ at $N$. Let $O$ be the intersect of $AC$ and $BD$. Suppose futhermore that $MP=PQ=QN$. Prove that \[ \frac{OP}{OA}+\frac{OQ}{OB}=1.\]
Let $a,b$ and $c$ be positive numbers such that $\sqrt{a}+\sqrt{b}+\sqrt{c}=1$. Find the maximum value of the expression \[ P=\sqrt{abc}\left(\frac{1}{\sqrt{(a+b)(a+c)}}+\frac{1}{\sqrt{(b+c)(b+a)}}+\frac{1}{\sqrt{(c+b)(c+a)}}\right).\]
Solve the equation $f(f(x))=x$ in terms of the parameter $m$ where \[ f(x)=x^{2}+2x+m. \]
Suppose that $a,b$ and $c$ are the lengths of three sides of a triangle. Let \[ F(a,b,c)=a^{3}+b^{3}+c^{3}+3abc-ab(a+b)-bc(b+c)-ca(c+a). \] Prove that \[F(a,b,c)\leq\min\left\{ F(a+b,b+c,c+a),4a^{2}b^{2}c^{2}F\left(\frac{1}{a},\frac{1}{b},\frac{1}{c}\right)\right\}. \]
Given an acute triangle $ABC$. Let $a,b$ and $c$ respectively be the lengths of $BC$, $CA$ and $AB$. Let $R$ and $r$ respectively be the circumradius and the inradius of $ABC$. Prove that \[\frac{r}{2R}\leq\frac{abc}{\sqrt{2(a^{2}+b^{2})(b^{2}+c^{2})(c^{2}+a^{2})}}.\]
Suppose that $a$ and $b$ are real numbers such that $0<a\ne1$ and that the equation $a^{x}-\dfrac{1}{a^{x}}=2\cos(bx)$ has exactly $2017$ real roots and they are all different real numbers. How many distinct real roots does the equation \[a^{x}+\frac{1}{a^{x}}=2\cos(bx)+4\] have?.
In a country, the length of any direct road between two cities (if any) is smaller than $100$ km and we can travel from a city to any other one by roads which have total length is smaller than $100$ km. When a road is closed under construction, we still can travel from one city to another by other roads. Prove that we can choose a route which has the total length is smaller than $300$ km.
Prove that $\gcd(1,2,\ldots,2n)$ is divisible by $C_{2n}^{n}$ for any positive interger $n$.
Given a triangle $ABC$ with $AB<AC$. Let $M$ be the midpoint of $BC$. Let $H$ be the perpendicular projection of $B$ on $AM$. Suppose that $Q$ is the point of the opposite ray of $AM$ such that $AQ=4MH$. Assume that $AC$ intersects $BQ$ at $D$. Prove that the circumcenter of $ADQ$ lies on the circumcircle of $DBC$.

Issue 479

Place the numbers from $1$ to $20$ on a circle such that the sum of any two numbers which are next to each other is a prime number.
Given a right triangle $ABC$ with the right angle $A$ and $\widehat{ABC}<45^{\circ}$. On the haft plane determined by $AB$, containing $C$, choose two points $D$ and $E$ such that $BD=BA$, $\widehat{DBA}=90^{0}$, $\widehat{EBC}=\widehat{CBA}$, and $ED$ is perpendicular to $BD$. Prove that $BE=AC+DE$.
Given two real numbers $x$ and $y$ such that $x^{2}+2xy+2y^{2}=1$. Find the maximum and minimum values of the expression \[P=x^{4}+y^{4}+(x+y)^{4}.\]
Given a triangle $ABC$ with $AB<AC$. Let $(O)$ be the incircle of $ABC$. The side $BC$ is tangent to $(O)$ at $D$. Choose $I$ on $AD$ such that $OI$ is perpendicular to $AD$. The ray $IO$ intersect the perpendicular bisector of $BC$ at $K$. Prove that $BIKC$ is an inscribed quadrilateral.
Solve the equation \[\frac{\sqrt{x^{2}+28x+4}}{x+2}+8=\frac{x+4}{\sqrt{x-1}}+2x.\]
Given non-negative numbers $a,b$ and $c$ such that $ab+bc+ca=1$. Prove that \[\sqrt{3}\leq\sqrt{1+a^{2}}+\sqrt{1+b^{2}}+\sqrt{1+c^{2}}-a-b-c\leq2.\]
Solve the system of equations \[\begin{cases} x^{8} & =21y+13\\ \dfrac{(x+y)^{25}}{2^{18}} & =(x^{3}+y^{3})^{3}(x^{4}+y^{4})^{4}\end{cases}.\]
Given a triangle $ABC$ and $P$ is a point lying inside $ABC$. Let $E$ (resp. $AC$ and $AB$). Let $K$ be the perpendicular projection of $K$ on $BC$. On the side $AF$, choose an arbitrary point $M$. Assume that $N$ is the intersection between $MK$ and $PE$. Prove that $\widehat{KHM}=\widehat{KHN}$.
Given a bijection $f:\mathbb{N}^{*}\to\mathbb{N}^{*}$. Prove that there exist positive intergers $a,b,c$ and $d$ such that $a<b<c<d$ and $f(x)+f(d)=f(b)+f(c)$.
Given a sequence $(u_{n})$ whose terms are positive integers satisfying \[0\leq u_{m+n}-u_{m}-u_{n}\leq2\quad\forall m,n\geq1.\] Prove that there exist two positive numbers $a_{1},a_{2}$ such that \[[a_{1}n]+[a_{n}n]-1\leq u_{n}\leq[a_{1}n]+[a_{2}n]+1\] for all $n\leq2017$. ($[x]$ is the greatest integer which does not exceed $x$).
Find all continuous functions $f:(0,+\infty)\to(0,\infty)$ such that \[f(x+2016)=f\left(\frac{x+2017}{x+2018}\right)+\frac{x^{2}+4033x+4066271}{x+2018},\forall x>0.\]
Given a circle $(O)$ and two fiexd points $B,C$ on $(O)$. A point $A$ is moving on $(O)$ such that $ABC$ is always an acute triangle and $AB<AC$. Choose $D$ on the side $AC$ such that $AB=AD$. The line $BD$ intersects $(O)$ at $E$ which is different from $B$. The perpendicular projection of $E$ (resp. $D$) on $AC$ (resp. $AE$) is $H$ (resp. $M$). The circumcircle of $AMH$ intersects $(O)$ at $K$. Let $N$ be perpendicular projection of $K$ on $AB$.
a) Prove that $MN$ goes through the midpoint $I$ of $BD$.
b) The circumcircles of $BCD$ and $AMH$ intersect each other at $P$ and $Q$. Prove that the midpoint of $PQ$ always lies on a fixed line when $A$ is moving on $(O)$ in the given way.

Issue 480

Find positive integers $x,y,z$ such that $$x^{y}+y^{z}+z^{x}=105.$$
Given a right triangle $ABC$ with the right angle $A$, $AB=AC$ and $BC=a\sqrt{2}$. Let $M,D$ and $E$ be arbitrary points on the sides $BC,AB$ and $AC$ respectively. Find the minimum value of $MD+ME$
Solve the system of equation \begin{align*} \sqrt{x}-\frac{2}{\sqrt{y}}+\frac{3}{\sqrt{z}} & =1\\ \sqrt{y}-\frac{2}{\sqrt{z}}+\frac{3}{\sqrt{x}} & =2\\ \sqrt{z}-\frac{2}{\sqrt{x}}+\frac{3}{\sqrt{y}} & =3 \end{align*}
Two circles $(O_{1},R_{1})$ and $(O_{2,}R_{2})$ intersect at $A$ and $B$. A line $d$ is tangent to $(O_{1},R_{1})$ and $(O_{2},R_{2})$ at $C$ and $D$ respectively. Let $R_{3},R_{4}$ respectively be the circumradii of $ACD$ and $BCD$. Prove that $R_{1}\cdot R_{2}=R_{3}\cdot R_{4}$
Suppose that $a,b,c$ are three positive numbers and given $\alpha,\beta,\lambda\geq2$. Prove that \[\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}\] \[\geq\frac{3}{2}+\frac{(a-b)^{2}}{\alpha(b+c)(c+a)}+\frac{(b-c)^{2}}{\beta(c+a)(a+b)}+\frac{(c-a)^{2}}{\lambda(a+b)(b+c)}.\]
Solve the equation \[ 4x^{3}-24x^{2}+45x-26=\sqrt{-x^{2}+4x-3}.\]
Let $x_{1},x_{2}$ be two solutions of the equation $x^{2}-2ax-1=0$ where a is an integer. Prove that for every natural number $n$, $\frac{1}{8}(x_{1}^{2n}-x_{2}^{2n})(x_{1}^{4n}-x_{2}^{4n})$ is always a product of three consecutive natural numbers.
The incircle $I$ of the triangle $ABC$ is tangent to $BC$, $CA$ and $AB$ at $D,E$ and $F$ respectivly. Suppose that $X$ is the intersection of the lines $DE$ and $AB$, and $Y$ is the intersection of the lines $CF$ and $DE$. Prove that $IY\perp CX$.
Given three positive numbers $a,b,c$ such that $24ab+44bc+33ca\leq1$. Find the minimum value of the expression $P=\frac{1}{a}+\frac{1}{b}+\frac{1}{c}$.
Find all sets of $19$ distinct positive integers such that for each set the sum of the numbers is 2017 and the sums of all the digits of the numbers are all equal.
The sequence $(a_{n})$, $n=0,1,2,\ldots$, is determined as follows: $a_{0}=610$, $a_{1}=89$ and $a_{n+2}=7a_{n+1}-a_{n}$. Find all $n$ such that $2a_{n+1}a_{n}-3$ is a fourth power of an integer.
Given a triangle $ABC$ where $AB\ne AC$ and $\widehat{A}=90^{\circ}$. Let $BE$ and $CF$ be the altitudes from $B$ and $C$. Let $(O_{1})$, $(O_{2})$ be the circles which pass through $A$ and are tangent to $BC$ at $B,C$ respectively. Assume that $D$ is the second intersection of $(O_{1})$ and $(O_{2})$. Let $M,N$ (resp. $P,Q$) respectively be the second intersections of $BA,BD$ (resp. $CA$, $CD$) and $(O_{2})$ (resp. $(O_{1})$). Let $R$ be the intersection of $AD$ and $EF$, $S$ the intersection of $MP$ and $NQ$. Show that $RS\perp BC$.

Issue 481

Find natural numbers $a,b,c$ satisfying $9^{a}+952=(b+41)^{2}$ and $a=2^{b}\cdot c$.
Prove that \[A=\frac{1}{1000}+\frac{1}{1002}+\ldots+\frac{1}{2000}<\frac{1}{2}.\]
Find positive integral solutions of the equation \[x^{5}+y^{5}+2016=(x+2017)^{5}+(y-2018)^{5}.\]
Given a quadrilateral $ABCD$ inscribed in a circle $(O)$ with $AB=BD$. The lines $AB$ and $DC$ intersect at $N$. The line $CB$ and the tangent line of the circle $(O)$ at the point $A$ intersect at $M$. Prove that $\widehat{AMN}=\widehat{ABD}$.
Solve the equation \[(1-2x)\sqrt{2-x^{2}}=x-1.\]
Solve the system of equations \[ \begin{cases} \dfrac{x+\sqrt{1+x^{2}}}{1+\sqrt{1+y^{2}}} & =x^{2}y\\ 1+y^{4}-\dfrac{4y}{x}+\dfrac{3}{x^{4}} & =0 \end{cases}.\]
Given an integer $n>1$. Prove that \[\frac{2^{2n}}{2\sqrt{n}}<C_{2n}^{n}<\frac{2^{2n}}{\sqrt{2n+1}}.\]
Given a quadrilateral $ABCD$ circumsribing a circle $(O)$. Let $E,F$ and $G$ respectively be the intersection of three pairs of line $(AB,CD)$, $(AD,CB)$ and $(AC,BD)$. Let $K$ be the intersection between $EF$ and $OG$. Prove that $\widehat{AKG}=\widehat{CKG}$ and $\widehat{BKG}=\widehat{DKG}$.
Given three nonnegative numbers $x,y,z$ such that $2^{x}+4^{y}+8^{z}=4$. Find the maximum and minimum values of the expression \[S=\frac{x}{6}+\frac{y}{3}+\frac{z}{2}.\]
Find all pairs of natural number $(m,n)$ so that $2^{m}\cdot3^{n}-1$ is a perfect square.
Let $(u_{n})$ be a sequence defined as follows \[u_{1}=0,\quad\log_{\frac{1}{4}}u_{n+1}=\left(\frac{1}{4}\right)^{u_{n}}\quad\forall n\in\mathbb{N}^{*}.\] Prove that the sequence has a finite limit and find that limit.
Given a triangle $ABC$ and its circumcircle $(O)$. Let $(O_{B})$ and $(O_{C})$ respectively be the mixtilinear incircles corresponding to $B$ and $C$ (recall that a mixtilinear incircle corresponding to $B$ is the circle which is internally tangent to $BA$, $BC$ and the circumcircle $(O)$). Let $M$ (resp. $N$) be the tangent point between $(O)$ and $(O_{B})$ (resp. $(O_{C})$). Let $(M)$ and $(N)$ be the circles with centers at $M$ and $N$ and are tangent to $AC$ and $AB$ respectively. Prove that the center of dilation of $(M)$ and $(N)$ belongs to $BC$.

Issue 482

For each natural number $n$, find the last digit of \[S_{n}=1^{n}+2^{n}+3^{n}+4^{n}.\]
Suppose that $ABC$ is an isosceles right triangle with $B$ is the right angle. Let $O$ be the midpoint of $AC$. Choose $J$ on the segment $OC$ such that $3AJ=5JC$. Let $N$ be the midpoint of $OB$. Prove that $AN\perp NJ$.
Solve the system of equations \[\begin{cases} 2(y^{2}-x^{2}) & =\dfrac{14}{x}-\dfrac{13}{y}\\ 4(x^{2}+y^{2}) & =\dfrac{14}{x}+\dfrac{13}{y} \end{cases}.\]
Given a triangle $ABC$ with $\widehat{BAC}=120^{0}$. Suppose that on the side $BC$ there exists a point $D$ such that $\widehat{BAD}=90^{0}$ and $AB=DC=1$. Find the length of $BD$.
Given three positive numbers $a,b,c$ such that $a+b+c=\sqrt{6051}$. Find the maximum calue of the expression \[P=\frac{2a}{\sqrt{a^{2}+2017}}+\frac{2b}{\sqrt{b^{2}+2017}}+\frac{2c}{\sqrt{c^{2}+2017}}.\]
Solve the equation \[\dfrac{1}{\sqrt[3]{x}}+\dfrac{1}{\sqrt[3]{3x+1}}=\dfrac{1}{\sqrt[3]{2x-1}}+\dfrac{1}{\sqrt[3]{2x+2}}\] assuming that $x>\dfrac{1}{2}$.
Suppose that $p,q,r$ are three distinct integer roots of the equation $x^{3}+ax^{2}+bx+c$ where $a,b,c$ are integers and $16a+c=0$. Prove that \[\frac{p+4}{p-4}\cdot\frac{q+4}{q-4}\cdot\frac{r+4}{r-4}\] is an integer.
Let $(O)$ be the circumcircle of the equilateral triangle $ABC$ whose sides are equal to $a$. On the arc $BC$ which does not contain $A$ choose an arbitrary point $P$ ($P\ne B$, $P\ne C$). Suppose that $AP$ intersect $BC$ at $Q$. Prove that \[PQ\leq\frac{a}{2\sqrt{3}}.\]
Let $a,b,c$ be postive numbers. Prove that \[\frac{a}{b}+\frac{b}{c}+\frac{c}{a}+\frac{9\sqrt[3]{abc}}{a+b+c}\geq6.\]
For each natural number $n>2$, let $u_{n}$ be the number of $0$'s at the end in case representation of $n!$. Find the maximum value of the expression \[P=\frac{u_{n}}{n-1}.\]
Given a convex decafon $A_{1}A_{2}\ldots A_{9}A_{10}$. We color its sides and its diagonals by 5 different colors as follows
a) Each side of each diagonal is colored by at most 1 color.
b) The sides and the diagonals which are colored have no common vertex and do not intersect (notice that the sides and the diagonals here are line segments, not he whole lines passing through them).
In how many ways can we color by the above rule?.
Given a right triangle $ABC$ with the right angle $A$. A circle $(I,r)$ is tangent to the line segments $AB$, $BC$ and $CA$ at $P,Q$ and $R$ respectively. Let $K$ be the midpoint of $AC$. The line $IK$intersects $AB$ at $M$. The line segment $PQ$ intersect the altitude $AH$ (of $ABC$) at $N$. Prove that $N$ is the orthocenter of $MQR$.

Issue 483

Find all single digit numbers $a,b,c$ such that the numbers $\overline{abc}$, $\overline{acb}$, $\overline{bca}$, $\overline{cab}$, $\overline{cba}$ are prime numbers.
Given a right triangle $XYZ$ with the right angle $X$ ($XY<XZ$). Draw $XU$ perpendicular to $YZ$. Let $P$ be the midoint of $YU$. Choose $K$ on the half plane determined by $YZ$ which does not contain $X$ such that $KZ$ is perpendicular to $XZ$ and $XY=2KZ$. The line $d$ which passes through $Y$ and is parallel to $XU$ intersects $KP$ at $V$. Let $T$ be the intersection of $XP$ and $d$. Prove that $\widehat{VTK}=\widehat{XKZ}+\widehat{VXY}$.
There are 3 people wanting to buy sheeps from Mr. An. The first one wants to buy $\dfrac{1}{a}$ of the herd, the second one wants to buy $\dfrac{1}{b}$ of the herd, and the third one wants to buy $\dfrac{1}{c}$ of the herd and it happens that
- $a,b,c\in\mathbb{N}^{*}$ and $a<b<c$,
- the numbers of sheeps each person wants to buy are positive integers,
- after shelling, Mr. An still has exactly one sheep left.
What are the possible numbers of sheeps Mr. An has?.
Given a circle $(O)$ with a diameter $AB$. On $(O)$ choose a point $C$ such that $CA<CB$. On the open line segment $OB$ choose $E$. $CE$ intersect $(O)$ at $D$. The line which goes through $A$ and is parallel to $BD$ intersects $BC$ at $I$. The lines $OI$ and $CE$ meet at $F$. Prove that $FA$ is a tangent to the circle $(O)$.
Given real numbers $a,b,c$ satisfying $a+b+c=3$ and $abc\geq-4$. Prove that \[3(abc+4)\geq5(ab+bc+ca).\]
Solve the following system of equations ($a$ is a parameter) \[\begin{cases}2x(y^{2}+a^{2}) & =y(y^{2}+9a^{2})\\ 2y(z^{2}+a^{2}) & =z(z^{2}+9a^{2})\\ 2z(x^{2}+a^{2}) & =x(x^{2}+9a^{2})\end{cases}.\]
Suppose that $x=\dfrac{2013}{2015}$ is a solution of the polynomial \[f(x)=a_{0}+a_{1}x+a_{2}x^{2}+\ldots+a_{n}x^{n}\in\mathbb{Z}[x].\] Can the sum of the coefficients of $f(x)$ be 2017?.
Given three circles $(O_{1}R_{1})$, $(O_{2}R_{2})$, $(O_{3}R_{3})$ which are pariwise externally tangent to each other at $A,B,C$. Let $r$ be the radius of the incircle of $ABC$. Prove that \[r\leq\frac{R_{1}+R_{2}+R_{3}}{6\sqrt{3}}.\]
Given positive numbers $x,y,z$ satisfying \[x^{2}+y^{2}-2z^{2}+2xy+yz+zx\leq0.\] Find the minimum value of the expression \[P=\frac{x^{4}+y^{4}}{z^{4}}+\frac{y^{4}+z^{4}}{x^{4}}+\frac{z^{4}+x^{4}}{y^{4}}.\]
Find the maximum value of the expression \[T=\frac{a+b}{c+d}\left(\frac{a+c}{a+d}+\frac{b+d}{b+c}\right)\] where $a,b,c,d$ belong to $[\frac{1}{2},\frac{2}{3}]$.
Let $R(t)$ be a polynomial of degree 2017. Prove that there exist infinitely many polynomials $P(x)$ such that \[P((R^{2017}(t)+R(t)+1)^{2}-2)=P^{2}(R^{2017}(t)+R(t)+1)-2.\] Find a relation between those polynomials $P(x)$.
Given a triangle $ABC$. The incircle $(I)$ of $ABC$ is rangent to $AB$, $BC$ and $CA$ at $K,L$ and $M$. The line $t$ which passes through $B$ and is different from $AB$ and $BC$ intersects $MK$ at $ML$ respectively at $R$ and $S$. Prove that $\widehat{RIS}$ is an acute angle.

Issue 484

Compute \[A=3+4+6+9+13+18+\ldots+4953\] where the terms are determined by the formular $a_{n+1}=a_{n}+n$, $n\in\mathbb{N}^{*}$.
Find all natural numbers $x,y$ such that \[(1+x!)(1+y!)=(x+y)!\] where $n!=1.2...n$.
In each square in a $8\times8$ chess board we place some small stones such that the sum of the stones in an row or any column is even. Prove that the sum of the stones in the black squares is even.
Let $ABCD$ be a rectangle with $AB=BC\sqrt{2}$. Choose some point $M$ on the line segment $CD$ such that $M$ is different from $D$. Draw $BI\perp AM$ ($I\in AM$). Assume that $CI$ and $DI$ intersect $AB$ at $E$ and $F$ respectively. Prove that $AE$, $BF$ and $AB$ can be the lengths of the three sides of a right triangle.
Solve the equation \[2x^{4}-8x^{3}+60x^{2}-104x-240.\]
Find the real roots of the following equation $$ 3\sqrt{x^{2}+y^{2}-2x-4y+5}+2\sqrt{5x^{2}+5y^{2}+10x+50y+130} +\sqrt{5x^{2}+5y^{2}-30x+45} \\ = \sqrt{102x^{2}+102y^{2}-204x+204y+1360}.$$
Prove that the following inequalities hold for every positive integer $n$ \[\ln\frac{n+1}{2}<\frac{1}{2}+\frac{1}{3}+\ldots+\frac{1}{n}<\log_{2}\frac{n+1}{2}.\] And hence deduce that \[\lim_{n\to+\infty}\left(\frac{1}{2}+\frac{1}{3}+\ldots+\frac{1}{n}\right)=+\infty.\]
Assume that the incircle $I$ of the triangle $ABC$ is tangent to $BC$, $CA$ and $AB$ respectively at $D$, $E$ and $F$. Draw $DG$ perpendicular to $EF$ ($G$ belongs to $EF$). Let $J$ be the midpoint of $DG$. The line $EJ$ intersects the circle $(I)$ at $H$. Let $K$ be the circumcenter of the triangle $FGH$. Prove that $IK||BH$.
Given positivenumbers $x,y,z$ satisfying \[\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=\frac{16}{x+y+z}.\] Find the minimum value of the expression \[P=\frac{x}{y}+\frac{y}{z}+\frac{z}{x}.\]
For each positive integer $n$, let $f(n)$ be the sum of the squares of the positive divisors of $n$. Find all positive integers $n$ such that \[\sum_{k=1}^{n}f(k)\geq\frac{10n^{3}+15n^{2}+2n}{24}.\]
Given the sequence $(x_{n})$ as follows \[x_{1}=1,\,x_{2}=\frac{1}{2},\quad x_{n+2}x_{n}=x_{n+1}^{2}+4^{-n},\,n\in\mathbb{N}^{*}.\] Find ${\displaystyle \lim_{n\to\infty}(3-\sqrt{5})^{n}x^{n}}$.
Given a triangle $ABC$. Let $(O)$ and $I$ respectively be the circumcircle and the incenter of $ABC$. Assume that $AI$ intersects $BC$ at $A_{1}$ and intersects $(O)$ at another point $A_{2}$. Similarly we get the points $B_{1}$, $B_{2}$ and $C_{1}$, $C_{2}$. Suppose that $B_{1}C_{1}$ intersects $B_{2}C_{2}$ at $A_{3}$, $A_{1}C_{1}$ intersects $A_{2}C_{2}$ at $B_{3}$, $A_{1}B_{1}$ intersects $A_{2}B_{2}$ at $C_{3}$. Prove that $A_{3}$, $B_{3}$, $C_{3}$ both belong to a line which is perpendicular to $OI$.

Issue 485

Let $a=n^{3}+2n$ and $b=n^{4}+3n^{2}+1$. For each $n\in\mathbb{N}$, find the greatest common divisor ($\gcd$) of $a$ and $b$.
Given an isoscesles triangle $ABC$ with $\widehat{A}=100^{0}$, $BC=a$, $AC=AB=b$. Outside $ABC$, we construct the isosceles triangle $ABD$ with $\widehat{ADB}=140^{0}$. Compute the perimeter of $ABD$ in terms of $a$ and $b$.
Find all pairs of intergers $x,y$ such that \[x^{3}y+xy^{3}+2x^{2}y^{2}-4x-4y+4=0.\]
Given an isosceles trapezoid $ABCD$ with $AB//CD$ and $DA=AB=BC$. Let $(K)$ be the circle which goes through $A$, $B$ and tangent to $AD$, $BC$. Let $P$ be a point on $(K)$ and inside $ABCD$. Assume that $PA$ and $PB$ respectively intersect $CD$ at $E$ and $F$. Assume that $BE$ and $BF$ respectively intersect $AD$ and $BC$ at $M$ and $N$. Prove that $PM=PN$.
Solve the system of equations \[\begin{cases}x^{2}+y^{2} & =4y+1\\x^{3}+(y-2)^{3} & =7\end{cases}.\]
Let $a,b,c$ be he length of three sides of a triangle. Prove that \[\frac{a^{3}(a+b)}{a^{2}+b^{2}}+\frac{b^{3}(b+c)}{b^{2}+c^{2}}+\frac{c^{3}(c+a)}{c^{2}+a^{2}}\geq a^{2}+b^{2}+c^{2}.\]
Given a function $f(x)$ which is continuous on $[a,b]$ and differentiable on $(a,b)$, where $0<a<b$. Prove that there exists $c\in(a,b)$ such that \[f'(c)=\frac{1}{a-c}+\frac{1}{b-c}+\frac{1}{a+b}.\]
Given a triangle $ABC$ inscribed the circle $(O)$. Construct the altitude $AH$. Let $M$ be the midpoint of $BC$. Assume that $AM$ intersects $OH$ at $G$. Prove that $G$ belongs to the radial axis of the circumcircle of $BOC$ and the Euler circle of $ABC$.
Given three non-negative real numbers $a,b,c$ satisfying $ab+bc+ca=1$. Find the minimum value of the expression \[P=\frac{(a^{3}+b^{3}+c^{3}+3abc)^{2}}{a^{2}b^{2}+b^{2}c^{2}+c^{2}a^{2}}.\]
Find positive integers $n\leq40$ and positive numbers $a,b,c$ satisfying \[\begin{cases} \sqrt[3]{a}+\sqrt[3]{b}+\sqrt[3]{c} & =1\\ \sqrt[3]{a+n}+\sqrt[3]{b+n}+\sqrt[3]{c+n} & \in\mathbb{Z} \end{cases}.\]
Given three positive integers $a,b,c$. Each time, we tranform the triple $(a,b,c)$ into the triple \[\left(\left[\frac{a+b}{2}\right],\left[\frac{b+c}{2}\right],\left[\frac{c+a}{2}\right]\right).\] Prove that after a finite number of such transformations, we will get a triple wit equal components. (The notation $[x]$ denotes the biggest integer which does not exceed $x$).
Given a triangle $ABC$. Let $(D)$ be he circle which is tangent to the rays $AB$, $AC$ and is internally tangent to the circumcircle of $ABC$ at $X$. Let $J$, $K$ respectively be the incenters of the triangles $XAB$, $XAC$. Let $P$ be the midpoint of the arc $BAC$. Prove that $P(AXJK)$ is a harmonic range.

Issue 486

Find all natural number of the form $\overline{abba}$ such that \[\overline{abba}=\overline{ab}^{2}+\overline{ba}^{2}+a-b.\]
Given a right isosceles triangle $ABC$ with the right angle $A$. Inside the triangle, choose a point $D$ such that $\angle ABD=15^{0}$, $\angle BAD=30^{0}$. Prove that
a) $BC=2BD$.
b) $\angle BCD>\angle ACD$.
Find all integer solutions of the equation \[\sqrt{3x+4}=\sqrt[3]{y^{3}+5y^{2}+7y+4}.\]
On a semicircle $O$ with the diameter $AB$ choose two points $E$, $F$ ($E$ is on the arc $BF$). A point O varies on the opposite ray of the ray $EB$. The circumcircle of $ABP$ intersects the line through $BF$ at the second point $Q$. Let $R$ be the midpoint of $PQ$. Prove that the circle with the diameter $AR$ always goes through a fixed point.
Solve the system of equations \[\begin{cases} x^{3}-7x+\sqrt{x-2} & =y+4\\ y^{3}-7y+\sqrt{y-2} & =z+4\\ z^{3}-7z+\sqrt{z-2} & =x+4\end{cases}\]
Given three non-negative numbers $a,b,c$ such that $a+b+c=3$, $a^{2}+b^{2}+c^{2}=5$. Prove that \[a^{3}b+b^{3}c+c^{3}a\leq8.\]
Solve the following equation with $m,n,k\in\mathbb{N}$, $n\geq m$, $k\geq2$. \[\frac{1}{4}(|\sin x|^{n}+|\cos x|^{n})=\frac{|\sin x|^{m}+|\cos x|^{m}}{|\sin2x|^{k}+|\cos2x|^{k}}\]
Given a triangle $OBC$ with $\angle AOB=120^{0}$, $OA=a$, and $OB=b$. Let $H$ be the perpendicular projection of $O$ on $AB$. Prove that \[aHA+bHB\leq\sqrt{3}ab.\]
Prove that there exists infinitely many positive integers $n$ such that $2018^{n-2017}-1$ is divisible by $n$.
Find all natural numbers $n$ satisfying $4^{n}+15^{2n+1}+19^{2n}$ is divisible by $18^{17}-1$.
Find all funtions $f:\mathbb{R}\to\mathbb{R}$ such that $f(0)$ is rational and \[f(x+f^{2}(y))=f^{2}(x+y),\,\forall x,y\in\mathbb{R}.\]
Given a triangle $ABC$. Prove that \[\cos\frac{A}{2}+\cos\frac{B}{2}+\cos\frac{C}{2}\geq\sqrt{\frac{3}{2}}\left(\sqrt{\sin\frac{A}{2}}+\sqrt{\sin\frac{B}{2}}+\sqrt{\sin\frac{C}{2}}\right).\]

Πηγή: molympiad