Διασκεδαστικά Μαθηματικά : Mathematics and Youth Magazine Problems 2005 (Issue 331

Πέμπτη 30 Μαρτίου 2023

Mathematics and Youth Magazine Problems 2005 (Issue 331 - 342)

Issue 331

Can we find two positive integers $x, y$ (written in decimal system) such that $$x+y=\underbrace{99 \ldots 9}_{n \text { times }}$$ and $y$ is obtained by a permutation of the digits of $x$ in the case where $n=2004$ ? and in the case where $n=2005$?
Prove that the number $$\sqrt{(a b-c d)(b c-d a)(c a-b d)}$$ is a rational, where $a, b, c$ are rationals satisfying the condition $a+b+c+d=0$.
Find all integral solutions of the equation $$x^{2}+2003 x+2004 y^{2}+y=x y+2004 x y^{2}+2005.$$
Prove that $$\frac{1}{1+a+b}+\frac{1}{1+b+c}+\frac{1}{1+c+a} \leq \frac{1}{2+a}+\frac{1}{2+b}+\frac{1}{2+c}$$ where $a, b, c$ are positive real numbers satisfying the condition $a b c=1$.
Consider positive numbers $a, b, c$, $x, y, z$ satisfying the conditions $a+b+c=4$ and $a x+b y+c z=x y z$. Prove that $$x+y+z>4.$$
Let be given a triangle $A B C$ with its angled bisector $A M(M$ lies on the side $B C)$. The line perpendicular to $B C$ at $M$ cuts the line $A B$ at $N$. Prove that the angle $B A C$ is right when and only when $M N=M C$.
The diagonals $A C$ and $B D$ of quadrilateral $A B C D$ intersect at $K$ so that $K A=$ $K D$ and $\widehat{A K D}=120^{\circ}$. From a point $M$ on the side $B C$, draw $M N || A C$ and $M Q || B D$ ($N$ lies on $A B$, $Q$ lies on $C D$). Find the locus of the circumcenters of triangles $M N Q$ when $M$ moves on the side $B C$.
Let be given two primes $p, q$ satisfying $p>q>2$. Find all integers $k$ so that the equation $$(p x-q y)^{2}=k x y z$$ has integral solution $(x, y, z)$ satisfying $x y \neq 0$.
Consider the sequence of numbers $\left(u_{n}\right)(n=1,2,3, \ldots)$ defined by $$u_{n}=n^{2^{n}},\,\forall n=1,2, \ldots$$ Put $\displaystyle x_{n}=\frac{1}{u_{1}}+\frac{1}{u_{2}}+\ldots+\frac{1}{u_{n}}$. Prove that the sequence $\left(x_{n}\right)$ has a limit when $n$ tends to infinity and the limit is an irrational.
Find all positive integers $n \geq 3$ so that the following inequality occurs for $n$ arbitrary real numbers $a_{1}, a_{2}, \ldots, a_{n}\left(a_{n+1}=a_{1}\right):$ $$\sum_{1 \leq i<j \leq n}\left(a_{i}-a_{j .}\right)^{2} \leq\left(\sum_{i=1}^{n}\left|a_{i}-a_{i+1}\right|\right)^{2}$$
Determine the form of triangle $A B C$ knowing that its angles satisfy the condition $$\frac{\tan\frac{A}{2}}{1+\tan\frac{B}{2} \tan\frac{C}{2}}+\frac{\tan\frac{B}{2}}{1+\tan\frac{C}{2} \tan\frac{A}{2}}+\frac{\tan\frac{C}{2}}{1+\tan\frac{A}{2} \tan\frac{B}{2}}=\frac{1}{4 \tan\frac{A}{2} \tan\frac{B}{2} \tan\frac{C}{2}}.$$
Consider the rectangular parallelepipeds $A B C D A_{1} B_{1} C_{1} D_{1}$ such that the lengths of the side $A B=a$, $A D=b$, $A A_{1}=c$ and the distance between the lines $A C$ and $B C_{1}$ are natural numbers. Find the least value of the volumes of these parallelepipeds.

Issue 332

Find the remainder in the integer division of the number $S=a^{b}+b^{a}$ by 5 , where $a=\overline{22 \ldots .2}$ with 2002 digits $2, b=\overline{33 \ldots . .3}$ with $2004$ digits $3$ (written in decimal system).
Let $A B C$ be a triangle with $A B=A C$. On the perpendicular to $A C$ at $C$, take a point $D$ such that $B, D$ lie on different sides of the line $A C$. Let $K$ be the point of intersection of the line perpendicular to $A B$ at $B$ and the line passing through the midpoint $M$ of $C D$, perpendicular to $A D .$ Compare the measures of $K B$ and $K D$.
Consider the equation $x^{2}-2 k x y^{2}+k\left(y^{3}-1\right)=0$ where $k$ is a positive integral parameter. Prove that this equation has integral solution $(x, y)$ with $x>0$, $y>1$ when and only when $k$ is a perfect square.
Solve the equation $$\sqrt{x-\sqrt{x-\sqrt{x-\sqrt{x-5}}}}=5.$$
Prove that $$\sqrt{a}+\sqrt[3]{a}+\sqrt[6]{a} \leq a+2$$ where $a$ is a non negative real number.
Let $A B C$ be a triangle with $\widehat{A} \geq \widehat{B} \geq \widehat{C}$ and let $h_{\alpha}$, $h_{b}$, $h_{c}$ be its altitudes issued respectively from $A$, $B$, $C$. Prove that $$\frac{h_{a}^{2}}{h_{b}^{2}}+\frac{h_{b}^{2}}{h_{c}^{2}}+\frac{h_{c}^{2}}{h_{a}^{2}} \geq \frac{h_{a}}{h_{b}}+\frac{h_{b}}{h_{c}}+\frac{h_{c}}{h_{a}} .$$
Let $A B C D$ be a parallelogram with $A B < B C$. The bisector of angle $B A D$ cuts $B C$ at $E$. The perpendicular bisectors of $B D$, $C E$ intersect at $O$. The line passing through $C$, parallel to $B D$ cuts the circle with center $O$ and radius $O C$ at $F$. Calculate the measure of angle $A F C$.
Prove that the polynomial $$P(x)=x^{4}-2003 x^{3}+(2004+a) x^{2}-2005 x+a$$ with integral parameter $a$ has at most one integral root and has no multiple integral root (with multiplicity $>1$).
Prove that $$\left(x^{3}+3\right)\left(y^{3}+3\right)\left(z^{3}+3\right) \geq \frac{4}{27}(3 x y+3 y z+3 z x+x y z)^{2}$$ where $x, y, z$ are real numbers.
Find all functions $f(x)$, defined on the interval $(0,+\infty)$, having derivative at $x=1$, and satisfying the condition $$f(x . y)=\sqrt{x} \cdot f(y)+\sqrt{y} \cdot f(x)$$ for all positive real numbers $x, y$.
Let $A_{1} A_{2} \ldots A_{n}$ be a regular $n$-gone inscribed in a circle with radius 1 and let $M$ be a point on the minor arc $\widehat{A_{1} A_{n}}$. Prove that
a) $M A_{1}+M A_{3}+\ldots+M A_{n-2}+M A_{n}<\frac{n}{\sqrt{2}}$ when $n$ is odd,
b) $M A_{1}+M A_{3}+\ldots+M A_{n-3}+M A_{n-1} \leq \frac{n}{\sqrt{2}}$ when $n$ is even.
When does equality occur?
Let $O$ be the centroid of a regular triangle $A B C$ and $d$ be the line orthogonal to the plane $(A B C)$ at $O$. For every point $S$ (distinct from $O$) on $d$, consider the pyramid $S A B C$. Let $\alpha$ be the angle between a lateral face and the base, let $\beta$ be the angle between two adjacent lateral faces of the pyramid. Prove that the quantity $$F(\alpha, \beta)=\tan^{2} \alpha\left(3 \tan^{2} \frac{\beta}{2}-1\right)$$ does not depend on $\alpha, \beta$ when $S$ moves on $d$.

Issue 333

The fractions with positive numerators and denominators are arranged in the following order $$\frac{1}{1}, \frac{2}{1}, \frac{1}{2}, \frac{3}{1}, \frac{1}{3}, \frac{4}{1}, \frac{3}{2}, \frac{2}{3}, \frac{1}{4} \ldots, \frac{n}{1}, \frac{n-1}{2}, \ldots, \frac{n-k}{k+1}, \ldots,\frac{2}{n-1}, \frac{1}{n}, \ldots,$$ where there are no fractions of the form $\dfrac{m}{m}$, $m>1$. At which place in this sequence lies the fraction $\dfrac{2004}{2005} ?$
Let $A B C$ be a triangle with altitude $A H$. Let $M$, $N$ be the orthogonal projections of $H$ respectively on $A B$ and $A C$. Prove that the condition $B M=C N$ implies that $A B C$ is an isosceles triangle with base $B C$.
Find all couple of positive integers $x$, $y$ such that $\dfrac{x^{4}+2}{x^{2} y+1}$ is a positive integer.
Solve the equation $$16 x^{4}+5=6\sqrt[3]{4 x^{3}+x}.$$
Prove the inequality $$\frac{a+b}{a b+c^{2}}+\frac{b+c}{b c+a^{2}}+\frac{c+a}{c a+b^{2}} \leq \frac{1}{a}+\frac{1}{b}+\frac{1}{c}$$ where $a, b, c$ are positive real numbers.
Let $A B C$ be a triangle with orthocenter $H$ (distinct from $A$, $B$, $C)$, and $M$ be the midpoint of $B C$. The line passing through $H$ perpendicular to $M H$ cuts the line $A B$ at $E$ and cuts the line $A C$ at $F$. Prove that $M E F$ is an isosceles triangle with base $E F$.
From a point $M$ in the interior of a rectangle $A B C D$, draw $A M$, $B M$ then $C E \perp B M$ at $E$, $D F \perp A M$ at $F$. Let $N$ be the point of intersection of $C E$ and $D F$. Find the locus of the midpoint of $M N$ when $M$ moves in the interior of $A B C D$.
Prove that for every positive integer $n$, the difference $$s_{n}=\left(\sum_{k=1}^{n}\left[\frac{n}{k}\right]\right)-[\sqrt{n}]$$ is an even integer, where $[x]$ denotes the integer part of $x$.
Solve the system of equations $$\begin{cases}x^{2}(x+1) &=2\left(y^{3}-x\right)+1 \\ y^{2}(y+1) &=2\left(z^{3}-y\right)+1 \\ z^{2}(z+1) &=2\left(x^{3}-z\right)+1\end{cases}$$
Prove that $$\sum_{i=1}^{n} \sqrt{x_{i}^{2}+\frac{1}{x_{i}^{2}}} \geq\left(n+\frac{1}{n}\right) \cdot \sqrt{n^{2}+\frac{1}{n^{2}}} \cdot\left(\sum_{i=1}^{n} \frac{x_{i}}{1+x_{i}^{2}}\right)$$ where $x_{1}, x_{2}, \ldots, x_{n}$ are $n$ positive numbers satisfying the condition $x_{1}+x_{2}+\ldots+x_{n} \leq 1$.
Find the greatest value of the expression $$F=\sin A \cdot \sin ^{2} B \cdot \sin ^{3} C$$ where $A$, $B$, $C$ are angles of a triangle.
Let $A B C D$ be a tetrahedron inscribed in a sphere $(O)$ with center $O$, let $G$ be the centroid of $A B C D$, let $M$ be a point lying in the interior of or on the sphere with diameter $O G$. The lines $M A$, $M B$, $M C$, $M D$ cut again $(O)$ respectively at $A_{1}$, $B_{1}$, $C_{1}$, $D_{1}$. Prove that $$V\left(A_{1} B_{1} C_{1} D_{1}\right) \geq V(A B C D)$$ where $V$ denotes volume.

Issue 334

Calculate the following sum $S$ (consisting of $23$ terms) $$S=\frac{1}{1.2 .3}+\frac{1}{2.3 .4}+\ldots+\frac{1}{(n-1) n(n+1)}+\ldots+\frac{1}{23.24 .25}$$
Let $A B C$ be a triangle with $\widehat{A} \neq 90^{\circ}, \hat{B} \neq 135^{\circ}$. Let $M$ be the midpoint of $B C$. At the outside of $\triangle A B C$, construct the isosceles, right triangle $A B D$ with base $A B$. The line passing through $A$ perpendicular to $A B$ and the line passing through $C$ parallel to $M D$ intersect at $E$. The line $A B$ cuts $C E$ at $P$ and cuts $D M$ at $Q$. Prove that $Q$ is the midpoint of $B P$.
Find the least odd natural number $n$ such that $n^{2}$ is a sum of an odd number of perfect squares.
Find postitive numbers $a_{1}, a_{2}, a_{3}, a_{4}$ satisfying the following conditions $$\frac{a_{1}^{2}}{a_{2}+a_{3}}+\frac{a_{2}^{2}}{a_{3}+a_{4}}+\frac{a_{3}^{2}}{a_{4}+a_{1}}+\frac{a_{4}^{2}}{a_{1}+a_{2}}=1$$ and $$a_{1}^{2}+a_{2}^{2}+a_{3}^{2}+a_{4}^{2} \geq 1.$$
Find the least value of the expression $$T=\frac{a^{2}}{a^{2}+(b+c)^{2}}+\frac{b^{2}}{b^{2}+(c+a)^{2}}+\frac{c^{2}}{c^{2}+(a+b)^{2}},$$ where $a, b, c$ are real numbers $(a b c \neq 0)$.
The incircle with center $I$ of triangle $A B C$ touches the sides $B C$, $C A$, $A B$ respectively at $D$, $E$, $F$. The line passing through $A$ perpendicular to $IA$ cuts the lines $D E$, $D F$ respectively at $M$, $N$. The line passing through $B$ perpendicular to $IB$ cuts the lines $E F$, $E D$ respectively at $P$, $Q$. The line passing though $C$ perpendicular to $I C$ cuts the line $F D$, $F E$ respectively at $S$, $T$. Prove that $$M N+P Q+S T \geq A B+B C+C A.$$
Let be given an isosceles, right triangle $A B C$ with base $B C$. Find the locus of points $M$ satisfying the condition $$M B^{2}-M C^{2}=2 M A^{2}$$
Let be given $n$ distinct positive numbers $(n \geq 4)$. Prove that among them there are at least two numbers such that their sum and their difference do not coincide with any number of $n-2$ other given nubers.
Prove that the sum $\displaystyle S_{n}=\sum_{k=0}^{n} \frac{1}{C_{n}^{k}}$ has a finite limit when $n$ tends to infinity and find this limit ($C_{n}^{k}$ are binomial coefficients).
Find the least value of the sum $$P=\tan^{2} x \cdot \tan^{2} y+\tan^{2} y \cdot \tan^{2} z+\tan^{2} z \cdot \tan^{2} x$$ where $x, y, z$ are positive numbers satisfying the conditions $$x+y+z=\frac{\pi}{2} ;\quad \cos (x-z) \leq \frac{7}{5} \sin y ;\quad \cos (x-y) \geq 3 \sin z.$$
Let $d_{a}$, $d_{b}$, $d_{c}$ be the lengths of the inner angle bisectors of triangle $A B C$ issued respectively from the vertices $A$, $B$, $C$. Let $p$ be the semi-perimeter of $\triangle A B C$. Prove that $$d_{d} \cdot \cos \frac{A}{2}+d_{b} \cdot \cos \frac{B}{2}+d_{c} \cos \frac{C}{2} \geq p(\cos A+\cos B+\cos C).$$
Let $A B C$ be a triangle right at $A$; let $M$ be the midpoint of $B C$. On the line $d$ passing through $M$ perpendicular to the plane $(A B C)$, take a point $S$ distinct from $M$. The plane $(Q)$ containing $B C$, perpendicullar to the plane $(S A B)$, cuts the line $S A$ at $D$. Determine the position of $S$ on the line $d$ so that the volume of the tetraheron $A B C D$ attains its greatest value.

Issue 335

Find the ratio of $A$ and $B$, where $$\begin{align}A= &\frac{1}{1.1981}+\frac{1}{2.1982}+\ldots+\frac{1}{n(1980+n)}+\ldots+\frac{1}{25.2005},\\ B= & \frac{1}{1.26}+\frac{1}{2.27}+\ldots+\frac{1}{m(25+m)}+. .+\frac{1}{1980.2005}.\end{align}$$ ($A$ consists of $25$ terms, $B$ consists of $1980$ terms.)
Let $A B C$ be an isosceles, right triangle with base $B C$. Let $M$ and $N$ be respectively the midpoints of $A B$ and $A C$. Draw $N H$ perpendicular to $CM$ at $H$, draw $H E$ perpendicular to $A B$ at $E$. Prove that the triangle $A B H$ is isosceles and the line $H M$ is the bisector of angle $B H E$.
Find integral solutions of the equation $$2\left(x^{2}-y^{2}\right)^{2}=x^{2}+y^{2}+2 z^{2}.$$
Solve the system of equations $$\begin{cases} x+y+z &=1 \\ \dfrac{x}{y}+\dfrac{y}{z}+\dfrac{z}{x} &= \dfrac{x+y}{y+z}+\dfrac{y+z}{x+y}+1\end{cases}$$ where $x, y, z$ are positive numbers.
Find the least value of the expression $$P=\left(3+\frac{1}{a}+\frac{1}{b}\right)\left(3+\frac{1}{b}+\frac{1}{c}\right)\left(3+\frac{1}{c}+\frac{1}{a}\right)$$ where the positive numbers $a, b, c$ satisfy the condition $a+b+c \leq \dfrac{3}{2}$.
Let $A B C D$ be a parallelogram with obtuse angle $B A D$. In the interior of angle $B A D$, construct the isosceles right triangle $A D E$ with base $A E$ and the isosceles right triangle $A B F$ with base $A F$. Let $M$ be the midpoint of $E F$. The segment $M B$ cuts $C F$ at $K$, the segment $M D$ cuts $C E$ at $H$. Prove that $H K$ is parallel to $B D$.
Let $A B C$ be an isosceles triangle with $\widehat{A B C}=120^{\circ}$ and let $D$ be the point of intersection of the line $B C$ with the tangent at $A$ of the circumcircle of triangle $A B C$. The line passing through $D$ and the circumcenter $O$ cuts the lines $A B$ and $A C$ respectively at $E$ and $F$. Let $M$ and $N$ be respectively the midpoints of $A B$ and $A C$. Prove that the lines $A O$, $M F$, $N E$ are concurrent.
Consider the polynomial $$T(x)=x^{3}+17 x^{2}-1239 x+2001 .$$ Put $T_{1}(x)=T(x), T_{n+1}(x)=T\left(T_{n}(x)\right)$ for every $n=1,2,3, \ldots$ Prove that there exists an integer $n>1$ such that $T_{n}(x)-x$ is divisible by 2003 for every integer $x$.
Consider the sequence of numbers $\left(x_{n}\right)$ $(n=1,2,3, \ldots)$ defined by $$x_{1}=2,\quad x_{n+1}=\frac{1}{2}\left(x_{n}^{2}+1\right),\,\forall n=1,2,3, \ldots$$ Put $\displaystyle S_{n}=\frac{1}{1+x_{1}}+\frac{1}{1+x_{2}}+\ldots+\frac{1}{1+x_{n}}$. Find the integral part of $S_{2005}$ and find the limit of $S_{n}$ when $n$ tends to infinity.
Consider the sequence of numbers $\left(a_{n}\right)(n=1,2,3, \ldots)$ defined by $$a_{1}=\frac{1}{2},\quad a_{n+1}=\left(\frac{1-\left(1-a_{n}^{2}\right)^{1 / 2}}{2}\right)^{1 / 2},\,\forall n=1,2,3, \ldots$$ Prove that $a_{1}+a_{2}+\ldots+a_{2005}<1,03$.
Prove that for every triangle $A B C$, we have $$\cos A+\cos B+\cos C \leq 1+\frac{1}{6}\left(\cos ^{2} \frac{A-B}{2}+\cos ^{2} \frac{B-C}{2}+\cos ^{2} \frac{C-A}{2}\right).$$
In a triangle $A B C$, let $B C=a$, $C A=b$, $A B=c$ and let $S$ be its area. Let the points $M$, $N$, $P$ lie respectively on the sides $B C$, $C A$, $A B$. Prove that $$a b \cdot M N^{2}+b c \cdot N P^{2}+c a \cdot P M^{2} \geq 4 S^{2}$$ when does equality occur?

Issue 336

Compare the following fractions (not by direct calculations) $$\frac{222221}{222222} ; \frac{444443}{444445} ; \frac{666664}{666667} ; \frac{888885}{888889}$$
Let $A B C$ be a triangle with $\widehat{A C B}=45^{\circ}$ and obtuse angle $A$. Draw the ray $B D$ cutting the opposite ray of $C A$ at $D$ so that $\widehat{C B D}=\widehat{A B C}$. Draw $A H$ perpendicular to $B D$ at $H .$ Calculate $\widehat{C H D}$.
If the lengths of the sides of a right triangle are integers, can its area be a perfect square?
Solve the following system of equations, where $a, b, c$ are given positive numbers $$\begin{cases}\dfrac{a}{x}-\dfrac{b}{z} &=c-z x \\ \dfrac{b}{y}-\dfrac{c}{x} &=a-x y \\ \dfrac{c}{z}-\dfrac{a}{y} &=b-y z\end{cases}$$
Prove the inequality $$\frac{1}{a}+\frac{1}{b}-\left(\sqrt{\frac{a}{b}}-\sqrt{\frac{b}{a}}\right)^{2} \geq 2 \sqrt{2}$$ where $a, b$ are real positive numbers satisfying $a^{2}+b^{2}=1$.
Let $A B C$ be an acute triangle with orthocenter $H$. Prove that $$\frac{H A}{B C}+\frac{H B}{C A}+\frac{H C}{A B} \geq \sqrt{3}.$$ When does equality occur?
Let $M$ be a point in the interior of a triangle $A B C$ with $B C=a$, $C A=b$, $A B=c$. Let $h_{a}$, $h_{b}$, $h_{c}$ be respectively the distances from $M$ to the lines $B C$, $C A$, $A B$. Determine the position of $M$ so that the product $h_{a} \cdot h_{b} \cdot h_{c}$ attains its greatest value and calculate this value.
Let be given an odd prime $p$ and the polynomial $Q(x)=(p-1) x^{p}-x-1 .$ Prove that there exists an infinite number of positive integers a such that $Q(a)$ is divisible by $p^{p}$.
Solve the equation $$x^{4}+4 a x^{3}+6 b^{2} x^{2}+4 c^{3} x+1=0$$ where $a, b, c$ are positive real numbers, $a \leq 1$, knowing that it has four real roots.
Calculate the sum of $2 n$ terms $$S=\frac{1}{2} C_{2 n}^{1}-\frac{1}{3} C_{2 n}^{2}+\ldots+(-1)^{k} \frac{1}{k} \cdot C_{2 n}^{k-1}+\ldots+(-1)^{2 n+1} \frac{1}{2 n+1} C_{2 n}^{2 n}$$ where $C_{n}^{k}$ are binomial coefficients.
Prove that for every triangle $A B C$, we have
a) $\cos A+\cos B+\cos C+\cot A+\cot B+\cot C \geq \frac{3}{2}+\sqrt{3}$.
b) $\sqrt{3}(\cos A+\cos B+\cos C)+\cot\frac{A}{2}+\cot\frac{B}{2}+\cot\frac{C}{2} \geq \frac{9 \sqrt{3}}{2}$.
Let be given a tetrahedron $A B C D$. Take a point $M$ in the interior of triangle $A B C$ and the points $A^{\prime}$, $B^{\prime}$, $C^{\prime}$ lying on $D A$, $D B$, $D C$ respectively so that $M A^{\prime}$, $M B^{\prime}$, $M C^{\prime}$ are parallel respectively to the planes $(D B C)$, $(D C A)$, $(D A B)$. Prove that the circumsphere of tetrahedron $A^{\prime} B^{\prime} C^{\prime} D$ passes through a fixed point distinct from $D$ when $M$ moves in the interior of triangle $A B C$.

Issue 337

Find the first four digits (on the left) of the number $S$ which is the following sum of 1000 terms $$S=1+2^{2}+3^{3}+\ldots+n^{n}+\ldots+1000^{1000}.$$
Let $A B C$ be a triangle with $A B>A C$. Take the points $M, N$ respectively on the sides $A B$ and $A C$ such that $A M=A N$. Let $K$ be the point of intersection of $B N$ and $C M$. Compare the lengths of $K B$ and $K C$.
Consider a triangle such that the measures of its sides are three consecutive integers greater than $3$ and its area is also an integer. Prove that the triangle has an altitude which divides it into two small triangles such that the measures of the sides of both small triangles are integers.
Solve the equation $$x^{3}-3 x^{2}+2 \sqrt{(x+2)^{3}}-6 x=0$$
Find the greatest value of the expression $T=2 a c+b d+c d$, where $a, b, c, d$ are real numbers satisfying the conditions $$4 a^{2}+b^{2}=2,\quad c+d=4.$$
Let $A B C$ be a triangle. Its angle bisectors $B M$, $C N$ ($M$ on the side $A C$, $N$ on the side $A B$) intersect at $D$. Prove that $\triangle A B C$ is right at $A$ when and only when $$2 B D \cdot C D=B M \cdot C N.$$
Let be given an angle $\widehat{x P y}=30^{\circ}$. $A$ is an arbitrary point on the ray $P x$, $B$ is an arbitrary point on the ray $P y$ such that $A B=d$ ($d$ is a given constant). Find the greatest value of the perimeter and the greatest value of the area of triangle $P A B$.
Let $f: \mathbb Z \rightarrow \mathbb Z$ be a function satisfying the conditions $$f(0)=1,\quad f(f(x))=x+4 f(x),\,\forall x \in \mathbb Z.$$ Find all natural numbers $n$ $(n \geq 1)$ such that $f_{n}(0)$ is divisible by $20^{11^{2005}}$, where $f_{1}(x)=f(x)$, $f_{n}(x)=f\left(f_{n-l}(x)\right)$ for all $n \geq 2$.
Find the greatest value of the expression $$P=(x-y)(y-z)(z-x)(x+y+z)$$ where $x, y, z$ are real numbers belong to the segment $[0 ; 1]$.
Let be given a natural number $n \geq 2$ and positive real numbers $a, b$ with $a<b$. Find the greatest value of the expression $$Q=\sum_{1 \leq i<j \leq n}\left(x_{i}-x_{j}\right)^{2},$$ where $x_{1}, x_{2}, \ldots, x_{n}$ are $n$ real numbers belong to the segment $[a, b]$.
A line passing through the incenter of triangle $A B C$ cuts the sides $A B$ and $A C$ respectively at $M$ and $N$. Prove that $$\frac{B M \cdot C N}{A M \cdot A N} \leq \frac{B C^{2}}{4 A B \cdot A C}$$
Let be given a tetrahedon $A B C D$. Let $A_{1}$, $B_{1}$, $C_{1}$, $D_{1}$ be respectively the centroids of the faces opposite to the vertices $A$, $B$, $C$, $D$. The lines $A A_{1}$, $B B_{1}$, $C C_{1}$, $D D_{1}$ cut the circumsphere of $A B C D$ again at $A_{2}$, $B_{2}$, $C_{2}$, $D_{2}$ respectively. Prove that $$\frac{A A_{1}}{A A_{2}}+\frac{B B_{1}}{B B_{2}}+\frac{C C_{1}}{C C_{2}}+\frac{D D_{1}}{D D_{2}} \leq \frac{8}{3}.$$

Issue 338

Can the expression $x^{4}+y^{4}+z^{4}$ take the value 2004 for positive fractions $x, y, z$?
Let be given a triangle $A B C$. Take the point $D$ on the half-plane with boundary $A B$ not containing $C$ such that $D A \perp A B$ and $A D=A B$. Take the point $E$ on the half-plane with boundary $A C$ not containing $B$ such that $E A \perp A C$ and $A E=A C$. Compare the areas of the triangles $A D E$ and $A B C$.
Find all integral solutions of the equation $$(2 x-y-2)^{2}=7\left(x-2 y-y^{2}-1\right).$$
Solve the equation $$\sqrt{5 x-1}+\sqrt[3]{9-x}=2 x^{2}+3 x-1.$$
Prove the inequality $$\frac{a}{p b+q c}+\frac{b}{p c+q d}+\frac{c}{p d+q a}+\frac{d}{p a+q b} \geq \frac{4}{p+q}$$ for positive numbers $a, b, c, d, p, q$ satisfying $p \geq q$. Does the inequality hold for $p<q$?
In plane let be given two lines $d_{1}$, $d_{2}$ intersecting at $K$ and let $M$ be a point not lying on $d_{1}$, $d_{2}$. A line $d$ passing through $M$ cuts $d_{1}$ and $d_{2}$ respectively at $A$ and $B$ (distinct from $K$). Draw $A P \perp d_{2}$ at $P$, $B Q \perp d_{1}$ at $Q$. Prove that the line $P Q$ passes through a fixed point when the line $d$ turns around $M$.
Let $A B C$ be a triangle right at $C$, let $C D$ be its altitude and let $S$ be its area. Let $(O)$ be the circle with diameter $A B$, let $\left(O_{1}\right)$, $\left(O_{2}\right)$ be the circles touching $(O)$, touching the segment $C D$ and touching the segment $A B$ respectively at $E$ and $F$. Prove that $$S=\frac{A D \cdot B D \cdot A E \cdot B F}{2 E D \cdot F D}$$
Find the least positive integer $n$ such that there exists a polynomial of degree $n$ with integral coefficients $P(x)$ satisfying the following conditions

$P(0)=1$, $P(1)=1$,
for every positive integer $m$, the remainder of the division of $P(m)$ by $2003$ is $0$ or $1$.

Find all functions $f: R \rightarrow R$ satisfying the condition $$f\left(x^{2}+f(y)\right)=y+x f(x)$$ for all real numbers $x, y$.
Let $\left(F_{n}\right)(n=1,2, \ldots)$ be the Fibonacci sequence $$F_{1}=F_{2}=1,\quad F_{n+1}=F_{n}+F_{n-1},\,\forall n=2,3,4,\ldots$$ Prove that if $a \neq-\frac{F_{n+1}}{F_{n}}$ for every $n=1,2,3$, then the sequence of numbers $\left(x_{n}\right)$, where $$x_{1}=a,\quad x_{n+1}=\frac{1}{1+x_{n}},\,\forall n=1,2,3, \ldots$$ is defined and it has a finite limit when $n$ tends to infinity and find this limit.
Let $A B C$ be a triangle with $B C=a$, $C A=b, A C=b$ and with area $S$. Let $m_{a}$, $m_{b}$, $m_{c}$ be respectively the lengths of the medians issued from $A$, $B$, $C$. Prove that $$S \leq \frac{a^{2} m_{a}^{2}+b^{2} m_{b}^{2}+c^{2} m_{c}^{2}}{\sqrt{3}\left(a^{2}+b^{2}+c^{2}\right)}$$ When does equality occur?
Let $R$ and $r$ be respectively the radii of the circumsphere and the inscribed sphere of a tetrahedron $A B C D$ with $A B=C D$, $A C=B D$, $B C=A D$. Prove the inequality $$\frac{\sin A+\sin B+\sin C}{\sqrt{\cos A \cdot \cos B \cdot \cos C}}>\frac{3 R}{2 r}$$ where $A$, $B$, $C$ are the angles of triangle $A B C$.

Issue 339

How many digits has the number $5^{50}$ (written in decimal system)?
Find the least value of the fractions of the form $\dfrac{a b}{a c+b d}$, where $a, b, c, d$ are positive integers satisfying the condition $a+b=c+d=2006$.
Has the equation $$x^{2005}+y^{2005}=2007^{2005}$$ integral solutions?
Solve the equation $$2 \sqrt[4]{27 x^{2}+24 x+\frac{28}{3}}=1+\sqrt{\frac{27}{2} x+6}$$
Find the least value of the expression $$P=\frac{1}{1+x_{1} x_{2}}+\frac{1}{1+x_{2} x_{3}}+\frac{1}{1+x_{3} x_{4}}+\frac{1}{1+x_{4} x_{5}}+\frac{1}{1+x_{5} x_{1}}$$ where $x_{1}, x_{2}, x_{3}, x_{4}, x_{5}$ are positive real numbers satisfying the condition $$x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+x_{4}^{2}+x_{5}^{2}=2005.$$
Let be given a square $A B C D$. The line perpendicular to $A C$ at $C$ cuts the lines $A B$, $A D$ respectively at $E$, $F$. Prove that $$B E \cdot \sqrt{C F}+D F \cdot \sqrt{C E}=A C \cdot \sqrt{E F}.$$
Let $I$ be the incenter of triangle $A B C$ and let $m_{a}$, $m_{b}$, $m_{c}$ be the measures of the medians of $A B C$ issued respectively from $A$, $B$, $C$. Prove that $$\frac{I A^{2}}{m_{a}^{2}}+\frac{I B^{2}}{m_{b}^{2}}+\frac{I C^{2}}{m_{c}^{2}} \leq \frac{4}{3} .$$
Let be given two positive real numbers $u$, $v$. Consider the expression $$P=x^{2}+u y^{2}+v z^{2},$$ where $x, y, z$ are arbitrary real positive numbers satisfying the condition $x y+y z+z x=1$. Prove that the least value of $P$ equals $2 t$, where $t$ is the root lying in the interval $(0 ; \sqrt{u v})$ of the equation $$2 x^{3}+(u+v+1) x^{2}-u v=0.$$ Find prime numbers $u, v$ so that $2 t$ is a rational number.
Consider the sequence of numbers $\left(x_{n}\right)$ $(n=1,2,3, \ldots)$ defined by $x_{n}=a_{n}^{a_{n}}$, where $$a_{n}=\frac{(2 n) !}{(n !)^{2} \cdot 2^{2 n}},\,\forall n=1,2,3, \ldots$$ Prove that the sequence $\left(x_{n}\right)$ has a limit when $n$ tends to infinity and find this limit.
Let $a$ be a real number belonging to the interval $(0 ; 1)$. Find all functions $f:\mathbb R \rightarrow \mathbb R$, continuous at $x=0$, satisfying the condition $$f(x)-2 f(a x)+f\left(a^{2} x\right)=x^{2}$$ fore every $x \in \mathbb R$.
In plane, let be given a cirle with $O P=d>0$. Two arbitrary chords $A B$, $C D$ passing through $P$, form an angle with constant measure $\alpha$ $\left(0^{\circ}<\alpha \leq 90^{\circ}\right)$. Find the greatest value and the least value of the sum $A B+C D$, when the chords $A B$, $C D$ vary and determine the positions of $A B$, $C D$ in these cases.
Let $P A B C$ be a tetrahedron such that $P A$, $P B$, $P C$ are perpendicular each to the others. Let $S=S_{A B C}$, $S_{1}=S_{P A B}$, $S_{2}=S_{P B C}$, $S_{3}=S_{P A C}$. Prove that $$\frac{S_{1}^{2}}{S^{2}+S_{1}^{2}}+\frac{S_{2}^{2}}{S^{2}+S_{2}^{2}}+\frac{S_{3}^{2}}{S^{2}+S_{3}^{2}} \leq \frac{3}{4} .$$

Issue 340

Let $x$ be the sum of the digits of the number $a=3^{2004}+2005$, let $y$ be the sum of the digits of the number $x$ and let $z$ be the sum of the digits of the number $y$. Find $z$.
Find the least value of the expression $$A=|7 x-5 y|+|2 z-3 x|+|x y+y z+z x-2000|+t^{2}-t+2005,$$ where $x, y, z, t$ are rational numbers.
Find the least value of the expression $A=x^{2}+y^{2}$, where $x, y$ are positive integers and $A$ is divisible by $2004$.
Solve the equation $$13 \sqrt{x-1}+9 \sqrt{x+1}=16 x.$$
Find the least value and the greatest value of the expression $$P=\frac{x+y}{1+z}+\frac{y+z}{1+x}+\frac{z+x}{1+y}$$ where $x, y, z$ are real numbers belonging to the segment $\left[\frac{1}{2} ; 1\right]$.
Let $A B C$ be a triangle. For a point $M$ inside the triangle, let $E$ be the point of intersection of $A M$ and $B C$, let $F$ be the point of intersection of $C M$ and $A B$. Let $N$ be the reflection of $B$ in the midpoint of $E F$. Prove that the line $M N$ passes through a fixed point when $M$ move inside triangle $A B C$.
Let $S$ be the area of triangle $A B C$ with $B C=a$, $C A=b$, $A B=c$. Prove that $$S \leq \frac{\sqrt{3}}{4} \cdot \sqrt[3]{a^{2} b^{2} c^{2}}.$$ When does equality occur?
Let $f(x)$ be a polynomial of degree $3$ with integral coefficients and leading coefficient $1$. Suppose that $f(0)+f(1)+f(-1)$ is not divisible by $3$. Find $\displaystyle\lim_{n\to\infty} \sqrt[3]{f(n)}$ when the integer $n$ tends to infinity.
Does there exist a function $f:(0 ;+\infty) \rightarrow(0 ;+\infty)$ satisfying the condition $$f^{2}(x) \geq f(x+y)(f(x)+y)$$ for all positive real numbers $x, y$?
Prove that $$\sum_{i=1}^{n} \sum_{j=1}^{n} \frac{a_{i} a_{j}}{C_{k+i+j}^{k+2}} \geq 0$$ where $n, k$ are non negative integers, $n>1$, $a_{1}, a_{2}, \ldots, a_{n}$ are $n$ arbitrary real numbers, and $C_{m}^{r}=\dfrac{m !}{r !(m-r) !}$
Let $I$ be the incenter of a triangle $A B C$ with $B C=a$, $C A=b$, $A B=c$. Put $I A=d_{a}$, $I B=d_{b}$, $I C=d_{c}$. Prove that $$\sqrt{a\left(b c-d_{a}^{2}\right)}+\sqrt{b\left(c a-d_{b}^{2}\right)}+\sqrt{c\left(a b-d_{c}^{2}\right)} \leq \sqrt{6 a b c}$$
Let $P$ be a plane turning around the centroid of a regular tetrahedron $A_{1} A_{2} A_{3} A_{4}$ with side $c$. Let $B_{i}$ be the projection of $A_{i}$ $(i=$ $1,2,3,4)$ on the plane $P .$ Find the greatest value of the sum $$T=A_{1} B_{1}^{4}+A_{2} B_{2}^{4}+A_{3} B_{3}^{4}+A_{4} B_{4}^{4}$$ in term of $c$ and determine the position of $P$ when the sum attains its greatest value.

Issue 341

Find the last decimal digit of the following sum of $502$ terms $$S=2^{1}+3^{5}+4^{9}+\ldots+n^{4 n-7}+\ldots+503^{2005}.$$
Let $A B C$ be an isosceles triangle and $D$ be the point on its base $B C$ such that $C D=2 B D$. Compare the measures of the angles $\widehat{B A D}$ and $\dfrac{1}{2} \widehat{C A D}$.
Find all positive integral solutions of the equation $$x^{y}+x^{z}+x^{t}=x^{2005}.$$
Solve the equation $$\left(x^{2}-12 x-64\right)\left(x^{2}+30 x+125\right)+8000=0 .$$
Find the least value of the sum $$S=\frac{x y}{z}+\frac{y z}{x}+\frac{z x}{y}$$ where $x, y, z$ are positive real numbers satisfying the condition $x^{2}+y^{2}+z^{2}=1$.
Let $A B C$ be an equilateral triangle and $D$ be the reflection of $B$ in the line $A C$. A line passing through $B$ cuts the lines $A D$, $C D$ respectively at $M$, $N$. The lines $A N$ and $C M$ intersect at $E$. Prove that the points $A$, $C$, $D$, $E$ are concyclic.
Let $A B C$ be an equilateral triangle and $D$ be the reflection of $B$ in the line $A C$, and $M$ be the point on the ray $B C$ such that $B M=\dfrac{4}{3} B C$. The line $A M$ cuts $C D$ at $N$. Take a point $E$ on the segment $A B$ and a point $F$ on the segment $A D$ so that the lines $C E$, $N F$ are parallel. Calculate the measure of the angle $E O F$, where $O$ is the midpoint of $A C$.
Prove that for every positive integer $n>2$, there exist $n$ distinct positive integers such that the sum of these numbers is equal to their least common multiple and is equal to $n !$. T9/341. Prove that $$2 x^{2}+y^{2}+5 z^{2}+6 x y+7 x z+2 y z>0$$ for real numbers $x, y, z$ satisfying the conditions $x+y+z<0$ and $4 x z>y^{2}$.
Find the least value of the expression $$P=\sqrt{(x-a)^{2}+(y-b)^{2}}+\sqrt{(x-c)^{2}+(y-d)^{2}}$$ where $a, b, c, d, x, y$ are real numbers satisfying the following conditions $$\begin{cases}a^{2}+b^{2}+40 &=8 a+10 b,\\ c^{2}+d^{2}+12& =4 c+6 d,\\ 3 x &=2 y+13 .\end{cases}$$
Consider the convex quadrilaterals $A B C D$ having inscribed circle. Let $M$, $N$, $P$, $Q$ be the touching points of the inscribed circle with the sides $A B$, $B C$, $C D$, $D A$ respectively. Find the least value of the expression $$T=\frac{A M^{2}}{x_{1} x_{2}}+\frac{B N^{2}}{x_{2} x_{3}}+\frac{C P^{2}}{x_{3} x_{4}}+\frac{D Q^{2}}{x_{4} x_{1}}$$ where $\left\{x_{1}, x_{2}, x_{3}, x_{4}\right\}$ is a permutation of the measure $a=A B$, $b=B C$, $c=C D$, $d=D A$.
Let $O A B C$ be a tetrahedron such that the sides $O A$, $O B$, $O C$ are orthogonal each to others. Let $H$ be the orthocenter of triangle $A B C$. The line $A H$ cuts $B C$ at $K$. The line passing through the incenters of the triangles $O B K$, $O C K$ cuts $O B$, $O C$ at $M$, $N$ respectively. The plane bisecting the dihedral angle $[B,O A,H]$ cuts $B C$ at $D$, the plane bisecting the dihedral angle $[C, O A, H]$ cuts $B C$ at $E$. Prove the inequality for volumes $$V_{O A D E} \cdot V_{O A M N} \leq \frac{\sqrt{2}-1}{2} V_{O A B C}^{2}$$

Issue 342

Find all whole numbers $$A=2005^{n}+n^{2005}+2005 n$$ is divisible by $3$.
Let $A B C$ be a triangle with $\widehat{A B C}=\widehat{A C B}=36^{\circ}$. On the ray bisecting the angle $A B C$ take the point $N$ so that $\widehat{B C N}=12^{\circ}$. Compare the measures of $C N$ and $C A$.
Find all integral solutions of the equation $$\left(x^{2}+y^{2}+1\right)^{2}-5 x^{2}-4 y^{2}-5=0 .$$
Find the value of the expression $$P=a^{2005}+b^{2005}+c^{2005}$$ where $a, b, c$ are real numbers, distinct from $O$, satisfying the following conditions
Prove the following inequality for nonnegative real numbers $$\frac{2 \sqrt{2}}{\sqrt{x+1}}+\sqrt{x} \leq \sqrt{x+9}.$$ When does equality occur?
Let $A B C$ be a triangle with $A B=A C$, $\widehat{B A C}=80^{\circ}$. Take the point $M$ inside the triangle so that $\widehat{M A C}=20^{\circ}$, $\widehat{M C A}=30^{\circ}$. Find the measure of $\widehat{M B C}$.
Let be given a circle $(O)$ with center $O$, two chords $C A$, $C B$ not passing through $O$, $B A \neq B C$. The line passing through $A$, perpendicular to the line $O B$, cuts the line $C B$ at $N$. Let $M$ be the midpoint of $A N$. The line $B M$ cuts the circle $(O)$ at $B$ and $D$. Let $E$ be the point such that $O E$ is a diameter of the circle passing through $B$, $D$, $O$. Prove that the points $A$, $C$, $E$ are collinear.
Let $M$ be a set consisting of $2005$ positive numbers $a_1,\ldots,a_{2005}$. Consider all positive numbers non empty subsets $T_{i}$ of $M$ and let $s_{i}$ be the sum of the numbers belonging to $T_{i}$. Prove that the set of numbers $s_{i}$ can be partitioned into $2005$ non empty disjoint subsets so that the ratio of two arbitrary numbers belonging to a such subset does not exceed $2$.
The sequence of numbers $\left(x_{n}\right)$ $(n=1,2, \ldots)$ is defined by $$x_{1}=1, \quad x_{n+1}=\sqrt{x_{n}\left(x_{n}+1\right)\left(x_{n}+2\right)\left(x_{n}+3\right)+1},\,\forall n=1,2, \ldots.$$ Put $\displaystyle y_{n}=\sum_{i=1}^{n} \frac{1}{x_{i}+2}$ $(n=1,2, \ldots)$. Find $\displaystyle\lim_{n \rightarrow \infty} y_{n}$.
Prove the inequality $$\left(a^{2}+\frac{1}{a b}\right)^{\alpha}+\left(b^{2}+\frac{1}{b c}\right)^{\alpha}+\left(c^{2}+\frac{1}{c a}\right)^{\alpha} \geq 3.2^{\alpha}$$ where $a, b, c$ are positive numbers and $\alpha$ is a rational number greater than $1$. When does equality occur?
Let $r$ and $R$ be respectively the inradius and the circumradius of a triangle $A B C$. Prove that $$\cos A \cdot \cos B \cdot \cos C \leq \frac{r^{2}}{2 R^{2}}.$$ When does equality occur?
In space, let be given a sphere $(S)$ and a line $\Delta$ not intersecting $(S)$. For each point $M$ on $\Delta$, take three arbitrary tangent planes to $(S)$, passing through $M$ and touching $(S)$ at $A$, $B$, $C$. Prove that the planes $A B C$ contain a fixed line when $M$ moves on $\Delta$.