Issue 343
- Find the numbers $x$ and $y$ satisfying the condition $$|x-2005|+|x-2006|+|y-2007|+|x-2008|=3.$$
- Let $A B C$ be a triangle with $\widehat{B A C}=55^{\circ}$ $\widehat{A B C}=115^{\circ} .$ On the bisector of angle $A C B$ take the point $M$ so that $\widehat{M A C}=25^{\circ} .$ Calculate the measure of angle $\angle B M C$.
- Find the natural numbers $x, y, z$ satisfying the following conditions
- $x^{3}+y^{3}=2 z^{3}$.
- $x+y+z$ is a prime number.
- Solve the equation $$\sqrt[3]{x+86}-\sqrt[3]{x-5}=1.$$
- Find the least value of the expression $$A=\frac{a^{4}}{(b-1)^{3}}+\frac{b^{4}}{(a-1)^{3}}$$ where $a$, $b$ are numbers greater than $1$, satisfying the condition $a+b \leq 4$.
- Let $A B C$ be an triangle with $B C=a$, $A B=A C=b$ $(a>b)$. Suppose that the measure of the angled bisector $B D$ is equal to $b$. Prove that $$\left(1+\frac{a}{b}\right)\left(\frac{a}{b}-\frac{b}{a}\right)=1.$$
- Let $A B C$ be a triangle with angled bisectors $A A_{1}$, $B B_{1}$, $C C_{1}$. Suppose that $\widehat{A_{1} B_{1} C_{1}}=90^{\circ}$. Calculate the measure of angle $A B C$.
- For every positive number $x$, let $a(x)$ denote the number of prime numbers not exceeding $x$ and for every positive integer $m,$ let $b(m)$ denote the number of prime divisors of $m$ Prove that for every positive integer $n,$ we have $$a(n)+a\left(\frac{n}{2}\right)+\ldots+a\left(\frac{n}{n}\right)=b(1)+b(2)+\ldots+b(n).$$
- Solve the equation $$\sqrt[3]{x^{2}}-2 \sqrt[3]{x}-(x-4) \sqrt{x-7}-3 x+28=0.$$
- Not using calculators, find the exact measure of acute angle $x$ satisfying $$\cos x=\frac{1}{\sqrt{1+(\sqrt{6}+\sqrt{2}-\sqrt{3}-2)^{2}}}.$$
- Let $A B C$ be a triangle satisfying the condition $a^{2}=4 S c o \operatorname{tg} A,$ where $B C=a$ and $S$ is the area of $\triangle A B C .$ Let $O$ and $G$ be respectively the circumcenter and the centroid of triangle $A B C .$ Calculate the measure of the angle formed by the lines $A G$ and $O G .$.
- Let $A B C D$ be a tetrahedron such that its altitudes are concurrent. Let $R$ and $r$ be respectively the circumradius and the inradius of the tetrahedron $ABCD$. Let $R_A$, $R_B$, $R_C$, $R_D$ be respectively the circumradii of the tetrahedra $OBCD$, $OACD$, $OABD$, $OABC$ where $O$ is the circumcenter of the tetrahedron $A B C D$. Prove that
a) $\displaystyle \frac{1}{R_{A}^{2}}+\frac{1}{R_{B}^{2}}+\frac{1}{R_{C}^{2}}+\frac{1}{R_{D}^{2}} \geq \frac{16}{9 R^{2}}$.
b) $\displaystyle \frac{R_{A}}{\sqrt{3 R^{2}+4 R_{A}^{2}}}+\frac{R_{B}}{\sqrt{3 R^{2}+4 R_{B}^{2}}}+\frac{R_{C}}{\sqrt{3 R^{2}+4 R_{C}^{2}}}+\frac{R_{D}}{\sqrt{3 R^{2}+4 R_{D}^{2}}} \leq \frac{\sqrt 3}{3}\frac{R}{r}$.
Issue 344
- Find natural number $n$ such that the sum of $2 n$ terms $$\frac{1}{1.3}+\frac{1}{2.4}+\frac{1}{3.5}+\ldots+\frac{1}{(2 n-1)(2 n+1)}+\frac{1}{2 n(2 n+2)}$$ is equal to $\dfrac{14651}{19800}$.
- Let $A B C$ be an isosceles right angled triangle. Let $M$ be the midpoint of the hypotenuse $B C$, $E$ be the orthogonal projection of $M$ on the line $C G,$ where $G$ is the point on the side $A B$ such that $A G=\dfrac{1}{3} A B$. The lines $M G$ and $A C$ intersect at $D$. Compare the lengths of the segments $D E$ and $B C$.
- Solve the equation $$\frac{2+\sqrt{x}}{\sqrt{2}+\sqrt{2+\sqrt{x}}}+\frac{2-\sqrt{x}}{\sqrt{2}-\sqrt{2-\sqrt{x}}}=\sqrt{2}.$$
- Solve the system of equations $$\begin{cases}3 x^{3}-y^{3} &= \dfrac{1}{x+y} \\ x^{2}+y^{2} &=1\end{cases}.$$
- Pind the least value of the expression $$M=\frac{a b^{2}+b c^{2}+c a^{2}}{(a b+b c+c a)^{2}}$$ where $a$, $b$, $c$ are positive numbers satisfying the condition $a^{2}+b^{2}+c^{2}=3$.
- Let $X$ be a point on the side $A B$ of a parallelogram $A B C D$. The line passing through $X,$ parallel to $A D$ cuts $A C$ at $M$ and cuts $B D$ at $N .$ The line $X D$ cuts $A C$ at $P$ and the line $X C$ cuts $B D$ at $Q .$ Prove that $$\frac{M P}{A C}+\frac{N Q}{B D} \geq \frac{1}{3}.$$ When does equality occur?
- Let $A B C$ be a triangle with altitudes $A M$, $B N$ and with circumcircle $(O) .$ Let $D$ be a point on $(O),$ such that $D$ is distinct from $A$, $B$ and $D A$ is not parallel to $B N .$ The line $D A$ intersects the line $B N$ at $Q$. The line $D B$ intersects the line $A M$ at $P$. Prove that when $D$ moves on the circle $(O)$. the midpoint of the segment PQ lies on a fixed line.
- Let $p$ be a given odd prime number Prove that the difference $$\sum_{j=0}^{p}\left(\begin{array}{c} p \\ j \end{array}\right)\left(\begin{array}{c} p+j \\ j \end{array}\right)-\left(2^{p}+1\right)$$ is divisible by $p^{2}$, where $\left(\begin{array}{l}p \\ j\end{array}\right)$ is binomial coefficient.
- Consider the sequence $\left(f_{n}(x)\right)$ $(n=0,1,2, \ldots)$ of functions defined on $[0: 1]$ such that $$f_{0}(x)=0,\quad f_{n+1}(x)=f_{n}(x)+\frac{1}{2}\left(x-\left(f_{n}(x)\right)^{2}\right),\,\forall n=0,1,2, \ldots$$ Prove that $\dfrac{n x}{2+n \sqrt{x}} \leq f_{n}(x) \leq \sqrt{x}$ for all $n \in \mathrm{N}$, $x \in[0 ; 1]$
- Consider the polynomial $P(x)=x^{2}-1$. Find the number of distinct real roots of the equation $$P(P(\ldots, P(x)) \ldots)=0$$ where there are $2006$ notations $P$ on the left hand side of the equation.
- Suppose that $A_{1} B_{1} C_{1}$, $A_{2} B_{2} C_{2}$, $A_{3} B_{3} C_{3}$ are three triangles satisfying the conditions $$\widehat{C_{1}}=\widehat{C_{2}}=\widehat{C_{3}},\quad A_{1} B_{1}=A_{2} B_{2}=A_{3} B_{3},\\ B_{1} C_{1}+C_{2} A_{2}=B_{2} C_{2}+C_{3} A_{3}=B_{3} C_{3}+C_{1} A_{1}.$$ Prove that these three triangles are congruent.
- Consider a convex hexagon $A B C D E F$ inscribed in a circle. The diagonal $B F$ cuts $A E$, $A C$ respectively at $M$, $N$. The diagonal $B D$ cuts $C A$, $C E$ respectively at $P$, $Q$. The diagonal $D F$ cuts $E C$, $EA$ respectively at $R$, $S$. Prove that $M Q$, $N R$ and $P S$ are concurrent.
Issue 345
- Let $$A=\left(1-\frac{1}{1+2}\right)\left(1-\frac{1}{1+2+3}\right) \cdot\left(1-\frac{1}{1+2+3+\ldots+n}\right)$$ (consisting of $n-1$ factors) and $B=\dfrac{n+2}{n}$. Calculate $\dfrac{A}{B}$.
- Let $A B C$ be an isosceles triangle $(A B=A C)$ and $O$ be a point inside $A B C$ such that $\widehat{A O B}<\widehat{A O C}$. Compare the measures of $OB$ and $O C$.
- Find the numbers $x$ such that $$\frac{\sqrt{x}}{x\sqrt{x}-3 \sqrt{x}+ 3}$$ is an integer.
- Find the greatest value of the expression $$ P=\frac{x}{1+y^{2}}+\frac{y}{1+x^{2}}$$ where $x$, $y$ are non negative real numbers not exceeding $\dfrac{\sqrt{2}}{2}$.
- Prove that $$\frac{3 \sqrt{3}}{4} \leq \frac{b c}{a(1+b c)}+\frac{c a}{b(1+c a)}+\frac{a b}{c(1+a b)} \leq \frac{a+b+c}{4}$$ where $a, b, c$ are positive real numbe satisfying the condition $a+b+c=a b c$. When do equalities occur?
- Two arbitrary points $E$, lie respectively on the sides $A B$, $A C$ of a triangle $A B C$ so that $\dfrac{A E}{E B}=\dfrac{C D}{D A}$. The lines $B D$, $C E$ intersect at $M$. Determine the positions of $E$ and $D$ so that the area of triangle $B M C$ attains its greatest value and calculate this value in terms of the area of triangle $A B C$.
- Let $A B C$ be a triangle inscribed in a circle $(O)$. The bisector of angle $B A C$ cuts the circle $(O)$ at $A$ and $D .$ The circle with center $D$ and radius $D B$ cuts the line $A B$ at $B$ and $Q$, cuts the line $A C$ at $C$ and $P$. Prove that the line $A O$ is perpendicular to the line $P Q$.
- Determine non empty subsets $A$, $B$, $C$ of the set $N=\{0,1,2, \ldots\}$ satisfying the following conditions
- $A \cap B=B \cap C=C \cap A=\varnothing$;
- $A \cup B \cup C=N$;
- if $a \in A, b \in B, c \in C$ then $a+c \in A$ $b+c \in B, a+b \in C$
- Prove that $$\left|x_{1}+x_{2}+\ldots+x_{2007}\right| \leq \frac{2007}{3}$$ where $x_{1}, x_{2}, \ldots, x_{2007}$ are real numbers belonging to the segment $[-1 ; 1],$ so that the sum of their cubes is equal to $0$. When does equality occur?
- Find all functions $f: \mathbb{Z} \rightarrow \mathbb{Z}$ satisfying the following conditions $f(1)>0$ and $$f(f(m)-n)=f\left(m^{2}\right)+f(n)-2 n f(m),\,\forall m, n \in \mathbb{Z} .$$
- Let $A A_{1}$, $B B_{1}$, $C C_{1}$ be the medians of a triangle $A B C$. Prove that if the radii of the incircles of the triangles $B C B_{1}$, $C A C_{1}$, $A B A_{1}$ are all equal then $A B C$ is an equilateral triangle.
- Let be given a sphere with center $O$ and radius $R$. A pyramid $S . A B C$ moves so that the sides $S A$, $S B$, $S C$ touch the sphere $(O)$ respectively at $A$, $B$, $C$ and so that $\widehat{A S B}=90^{\circ}$, $\widehat{B S C}=60^{\circ}$, $\widehat{C S A}=120^{\circ}$. Find the locus of the apex $S$.
Issue 346
- Compare the number $\dfrac{1}{1002}$ with the following sum (consisting of $2006$ terms) $$A=\frac{2}{2005+1}+\frac{2^{2}}{2005^{2}+1}+\ldots+\frac{2^{n+1}}{2005^{2^{n}}+1}+\ldots+\frac{2^{2006}}{2005^{2^{2005}}+1}.$$
- Let $a, b, c$ be three distinct integers different from $0$ such that $a+b+c=0$. Calculate the value of the expression $$P=\left(\frac{a-b}{c}+\frac{b-c}{a}+\frac{c-a}{b}\right)\left(\frac{c}{a-b}+\frac{a}{b-c}+\frac{b}{c-a}\right).$$
- Let $a, b, c$ be positive real numbers satisfying $a b c \leq 1$. Prove that $$\frac{a}{c}+\frac{b}{a}+\frac{c}{b} \geq a+b+c.$$ When does equality occur?
- Solve the equation $$2 \sqrt{2 x+4}+4 \sqrt{2-x}=\sqrt{9 x^{2}+16}.$$
- Find the least value of the expression $$\left(x^{2}+1\right) \sqrt{x^{2}+1}-x \sqrt{x^{4}+2 x^{2}+5}+(x-1)^{2}.$$
- Let $A B C$ be a triangle with obtuse angle $\widehat{A B C}$. Prove that $$\sin (x+y)=\sin x \cdot \cos y+\sin y \cdot \cos x$$ where $x=\widehat{B A C}$ and $y=\widehat{B C A}$.
- Let $A B C D$ be a cyclic quadrilateral such that the sides $A B$, $C D$ are not parallel and let $I$ be the point of intersection of its diagonals. Let $M$, $N$ be respectively the midpoints of $B C$, $C D$. Prove that if $N I$ is perpendicular to $A B$ then $M I$ is perpendicular to $A D$.
- Let $a, b, c, d, e, f$ be six positive integers satisfying $a b c=d e f$. Prove that $$a\left(b^{2}+c^{2}\right)+d\left(e^{2}+f^{2}\right)$$ is a composite number.
- Consider all quadratic trinomials $f(x)=a x^{2}+b x+c$ ($a, b, c$ are integers, $a>0)$ having two distinct roots belonging to the interval $(0 ; 1)$. Find the trinomial such that the coefficient $a$ attains its least value.
- Prove that $$a b+b c+c a \geq 8\left(a^{2}+b^{2}+c^{2}\right)\left(a^{2} b^{2}+b^{2} c^{2}+c^{2} a^{2}\right)$$ where $a, b, c$ are nonnegative real numbers satisfying $a+b+c=1$.
- The incircle $(I)$ of a triangle $A_{1} A_{2} A_{3}$ has radius $r$ and touches the sides $A_{2} A_{3}$, $A_{3} A_{1}$, $A_{1} A_{2}$ respectively at $M_{1}$, $M_{2}$, $M_{3}$. Let $\left(I_{i}\right)$ be the circle touching the sides $A_{i} A_{j}$, $A_{j} A_{k}$ and externally touching $(I)$ ($i, j, k \in\{1,2,3\}$, $i \neq j \neq k \neq i$). Let $K_{1}$, $K_{2}$, $K_{3}$ be the touching points respectively of $\left(I_{1}\right)$ with $A_{1} A_{2}$, of $\left(I_{2}\right)$ with $A_{2} A_{3}$, of $\left(I_{3}\right)$ with $A_{3} A_{1}$. Put $A_{i} I_{1}=a_{i}$, $A_{i} K_{i}=b_{i}$ $(i=1,2,3)$. Prove that $$\frac{1}{r} \sum_{i=1}^{3}\left(a_{i}+b_{i}\right) \geq 2+\sqrt{3}.$$When does equality occur?
- Let be given a sphere with center $O$ and a chord $A B$, not passing through $O$. Let $M M^{\prime}$, $N N^{\prime}$, $P P^{\prime}$ be three chords (not coinciding with $A B$) passing through the midpoint $I$ of $A B$. Let $E$, $E^{\prime}$ be the points of intersection of the line $A B$ respectively with the planes $(MNP)$, $\left(M^{\prime} N^{\prime} P^{\prime}\right)$. Prove that $I E=I E^{\prime}$.
Issue 347
- Compare $\dfrac{5}{8}$ with $\left(\dfrac{389}{401}\right)^{10}$.
- Let $E$, $F$ be points respectively on the sides $A C$, $A B$ of a triangle $A B C$ such that $\widehat{A B E}=\dfrac{1}{3} \widehat{A B C}$, $\widehat{A C F}=\dfrac{1}{3} \widehat{A C B}$. The lines $B E$ and $C F$ intersect at $O$. Suppose that $O E=O F$. Prove that $A B=A C$ or $\widehat{B A C}=90^{\circ}$.
- Find integral solutions of the system of equations $$\begin{cases}4 x^{3}+y^{2} &=16 \\ z^{2}+y z &=3\end{cases}$$
- Consider all quadratic equations $a x^{2}+b x+c=0$ having two roots belonging to the segment $[0 ; 2]$. Find the greatest value of the expression $$P=\frac{8 a^{2}-6 a b+b^{2}}{4 a^{2}-2 a b+a c}.$$
- Consider all triangles $A B C$ such that the measures $a, b, c$ of their sides satisfy the relation $$1964 a b+15 b c+10 c a=2006 a b c.$$ Find the least value of the expression $$M=\frac{1974}{p-a}+\frac{1979}{p-b}+\frac{25}{p-c}$$ where $p$ is the semiperimeter of triangle $A B C$.
- Let be given a quadrilateral $A B C D$. Take two points $M, P$ respectively on the sides $A B$, $A C$ such that $\dfrac{A M}{A B}=\dfrac{C P}{C D}$. Find the locus of the midpoints $I$ of the segments $M P$ when $M$, $P$ moves respectively on $A B$, $A C$.
- Let $A B C$ be a triangle with $\widehat{B A C}=135^{\circ}$ and $A M$, $B N$ be two of its altitudes ($M$ on $B C$, $N$ on $C A$). The line $M N$ cuts the perpendicular bisector of $A C$ at $P$. Let $D$ and $E$ be the midpoints respectively of $N P$ and $B C$. Prove that $A B C$ is a right isosceles triangle.
- Let be given $167$ sets $A_{1}, A_{2}, \ldots, A_{167}$ satisfying the following conditions
- $\sum_{i=1}^{167}\left|A_{i}\right|=2004$;
- $\left|A_{j}\right|=\left|A_{i} \| A_{i} \cap A_{j}\right|$ for all $i, j \in\{1,2, \ldots,167\}$ and $i \neq j$.
- Find all continuous functions $f$, defined on $\mathbb{R}$, satisfying the condition $$f_{3}(x)+f(x)=2 x,\,\forall x \in \mathbb{R}$$ where $f_{3}(x)=f(f(f(x)))$.
- Find the least value of the expression $$H=\frac{x^{2}}{y+z}+\frac{y^{2}}{z+x}+\frac{z^{2}}{x+y},$$ where $x, y, z$ are positive numbers satisfying $$\sqrt{x^{2}+y^{2}}+\sqrt{y^{2}+z^{2}}+\sqrt{z^{2}+x^{2}}=2006.$$
- Let $A B C$ be an acute triangle with altitudes $A D$, $B E$, $C F$ and let $O$ be its circumcenter. Let $M$, $N$, $P$ be the midpoints respectively of the segments $B C$, $C A$, $A B$. Let $D_{1}$, $E_{1}$, $F_{1}$ be the reflexions respectively of $D$ in $M$, of $E$ in $N$, of $F$ in $P_{1}$. Prove that $O$ lies inside the triangle $D_{1} E_{1} F_{1}$.
- Let $G_{1}$, $G_{2}$, $G_{3}$, $G_{4}$ be the centroids respectively of the faces $B C D$, $CDA$, $D A B$, $A B C$ of a tetrahedron $A B C D$. Let $A_{1}$, $B_{1}$, $C_{1}$, $D_{1}$ be the points of intersection of the circumsphere of the tetrahedron respectively with $A G_{1}$, $B G_{2}$, $C G_{3}$, $D G_{4}$. Prove that $$\frac{A G_{1}}{A A_{1}}+\frac{B G_{2}}{B B_{1}}+\frac{C G_{3}}{C C_{1}}+\frac{D G_{4}}{D D_{1}} \leq \frac{8}{3} .$$
Issue 348
- Find all four-digit numbers $\overline{a b c d}$ satisfying the condition $$\overline{a b c d}=a^{2}+2 b^{2}+3 c^{2}+4 d^{2}+2006.$$
- Let $A B C$ be a right-angled triangle with right angle at $A .$ On the side $A C$ take the point $E$ so that $\widehat{E B C}=2 \widehat{A B E}$. On the segment $B E$ take the point $M$ such that $E M=B C$. Compare the measures of the angles $\widehat{M B C}$ and $\widehat{B M C}$.
- Solve the equation $$\frac{1}{4 x-2006}+\frac{1}{5 x+2004}=\frac{1}{15 x-2007}-\frac{1}{6 x-2005}.$$
- Prove that $$a(b+c)+b(c+a)+c(a+b)+2\left(\frac{1}{1+a^{2}}+\frac{1}{1+b^{2}}+\frac{1}{1+c^{2}}\right) \geq 6$$ for arbitrary numbers $a, b, c$ not less than $1$.
- Find the greatest value of the expression $$P=3 x y+3 y z+3 z x-x y z$$ where $x, y, z$ are positive numbers satisfying the condition $x^{3}+y^{3}+z^{3}=3$.
- Let be given a triangle $A B C$. $P$ is a point on the line $B C$. On the opposite ray of the ray $A P$, take the point $D$ such that $A D=\dfrac{B C}{2}$. Let $E$ and $F$ be the midpoints respectively of the segments $D B$ and $D C$. Prove that the circle with diameter $E F$ passes through a fixed point when $P$ moves on the line $B C$.
- Let $A B C$ be a triangle with $A B=A C=a$. Construct a circle with center $A$, with radius $R$ $(R<a)$. From $B$ and $C$, draw the tangents $B M$, $C N$ to this circle ($M$ and $N$ are touching points) so that they are not symmetric with respect to the altitude $A H$ of triangle $A B C$. Let $I$ be the point of intersection of $B M$ and $C N$.
a) Find the locus of $I$ when $R$ varies.
b) Prove that $I B \cdot I C=\left|a^{2}-d^{2}\right|$ where $A I=d$. - Let be given $n$ real positive numbers $a_{1}, a_{2}, \ldots, a_{n}$ satisfying the condition $$\sum_{i=1}^{k} a_{i} \leq \sum_{i=1}^{k} i(i+1),\,\forall k=1,2, \ldots, n.$$ Prove that $$\frac{1}{a_{1}}+\frac{1}{a_{2}}+\ldots+\frac{1}{a_{n}} \geq \frac{n}{n+1}$$
- Determine the number of distinct real roots belonging to the interval $(0 ; 2 \pi)$ of the equation $$e^{2 \cos ^{2} x}\left(8 \sin ^{6} x-12 \sin ^{4} x+10 \sin ^{2} x\right)=e+\frac{5}{2}.$$
- Find all polynomials $P(x)$ with real coefficients satisfying the condition $$P(x) \cdot P\left(2 x^{2}\right)=P\left(x^{3}+x\right),\,\forall x \in \mathbb{R}.$$
- Let $O$ be the point of intersection of the diagonals $A C, B D$ of a convex quadrilateral $A B C D$. Let $G_{1}$ and $G_{2}$ be the centroids respectively of the triangles $O A B$ and $O C D$. Let $H_{1}$ and $H_{2}$ be the orthocenters respectively of the triangles $O B C$ and $O D A$. Prove that $G_{1} G_{2}$ is perpendicular to $H_{1} H_{2}$
- Let $I$ and $r$ be respectively the center and the radius of the sphere inscribed in al tetrahedron $A B C D$. Let $r_{A}$, $r_{B}$, $r_{C}$, $r_{D}$ be the radii of the spheres inscribed respectivelly in the tetrahedra $I B C D$, $I A C D$, $I A B D$, $I A B C$. Prove the inequality $$\frac{1}{r_{A}}+\frac{1}{r_{B}}+\frac{1}{r_{C}}+\frac{1}{r_{D}} \leq \frac{4+\sqrt{6}}{r}.$$
Issue 349
- Let $S$ be the following sum of $2006$ terms $$S=\frac{2}{2^{1}}+\frac{3}{2^{2}}+\ldots+\frac{n+1}{2^{n}}+\ldots+\frac{2007}{2^{2006}} .$$ Compare $S$ with $3$.
- Let $A B C$ be a triangle with its two medians $A D$, $B E$ meeting at $M$. Prove that if $$\widehat{A M B} \leq 90^{\circ}$ then $A C+B C>3 A B.$$
- Prove that for every given positive integer $r$ less than $59$, there exists a unique positive integer $n$ less than $59$ such that $\left(2^{n}-r\right)$ is divisible by $59$.
- Solve the equation $$2 x^{2}-5 x+2=4 \sqrt{2\left(x^{3}-21 x-20\right)}.$$
- Prove that $$4 a b c \left[\frac{1}{(a+b)^{2} c}+\frac{1}{(b+c)^{2} a}+\frac{1}{(c+a)^{2} b}\right]+\frac{a+c}{b}+\frac{b+c}{a}+\frac{a+b}{c} \geq 9$$ for arbitrary positive real numbers $a, b, c$.
- Let $A B C$ be a right-angled triangle, right at $B$ and $A B=2 B C$. Let $D$ be the point on side $A C$ such that $B C=C D$, let $E$ be the point on side $A B$ such that $A D=A E$. Prove that $A D^{2}=A B \cdot B E$.
- In plane, let be given two lines $\Delta_{1}$, $\Delta_{2}$ intersecting at $O$. A point $M$ moves in plane so that $O M$ is equal to a constant $R$ and $M$ does not lie on $\Delta_{1}$, $\Delta_{2}$. Let $H$, $K$ be the orthogonal projections of $M$ on $\Delta_{1}$, $\Delta_{2}$ respectively. Find the locus of the incenter of triangle $M H K$.
- Let be given three prime numbers $p_{1}$, $p_{2}, p_{3}$ $\left(p_{1}<p_{2}<p_{3}\right)$. Put $$A=\left\{n \mid n \in \mathbb{N}^{*}, 1 \leq n \leq p_{1} p_{2} p_{3}, p_{1} \nmid n, p_{2} \nmid n, p_{3} \nmid n\right\}.$$ Prove that $|A| \geq 8$ ($|A|$ denotes the number of elements of the set $A$). When does equality occur?
- Let be given six real numbers $a, b, c$, $a_{1}$, $b_{1}$, $c_{1}$ $\left(a a_{1} \neq 0\right)$ satisfying the condition $$\left(\frac{c}{a}-\frac{c_{1}}{a_{1}}\right)^{2}+\left(\frac{b}{a}-\frac{b_{1}}{a_{1}}\right) \cdot \frac{b c_{1}-c b_{1}}{a a_{1}}<0.$$ Prove that each of the following equations $a x^{2}+b x+c=0$ and $a_{1} x^{2}+b_{1} x+c_{1}=0$ has two distinct roots and by representing these roots on the number line, the roots of one equation alternate with the roots of the other equation.
- Find all polynomials with real coefficients $P(x)$ satisfying the condition $$P(x) \cdot P(x+1)=P\left(x^{2}+2\right),\,\forall x \in \mathbb{R}$$
- Let $A A_{1}$, $B B_{1}$, $C C_{1}$ be the inner angled bisectors of triangle $A B C$ and $A_{2}$, $B_{2}$, $C_{2}$ be the touching points of the incircle of triangle $A B C$ with the sides $B C$, $C A$, $A B$ respectively. Let $S$, $S_{1}$, $S_{2}$ be the areas of triangles $A B C$, $A_{1} B_{1} C_{1}$, $A_{2} B_{2} C_{2}$ respectively. Prove that $$\frac{3}{S_{1}}-\frac{2}{S_{2}} \leq \frac{4}{S}.$$
- Let $Sxyz$ be a trihedral angle with $\widehat{x S y}=121^{\circ}$, $\widehat{x S z}=59^{\circ}$. $A$ is a point on $S x$, $O A=a$. On the ray bisecting the angle $\widehat{z S y}$, take the point $B$ such that $S B=a \sqrt{3}$. Calculate the measures of the angles of triangle $S A B$.
Issue 350
- Prove that $2005^{2007^{2006}}+2006^{2005^{2007}}+2007^{2006^{2005}}$ is divisible by $102$.
- Consider the sum of $n$ terms $$S_{n}=1+\frac{1}{1+2}+\frac{1}{1+2+3}+\ldots+\frac{1}{1+2+\ldots+n}$$ for $n \in \mathbb{N}^{*}$. Find the least rational number $a$ such that $S_{n}<a$ for all $\in \mathbb{N}^{*}$.
- Find all solutions $(x, y)$ of the equation $$\left(x^{2}+4 y^{2}+28\right)^{2}=17\left(x^{4}+y^{4}+14 y^{2}+49\right)$$ such that $x, y$ are natural numbers.
- Solve the following system of equations $$\begin{cases}\dfrac{1}{x}+\dfrac{1}{y+z} &=\dfrac{1}{2} \\ \dfrac{1}{y}+\dfrac{1}{x+z} &=\dfrac{1}{3} \\ \dfrac{1}{z}+\dfrac{1}{x+y} &=\dfrac{1}{4}\end{cases}$$
- Find the greatest value and the least value of the expression $$P=\sqrt{2 x+1}+\sqrt{3 y+1}+\sqrt{4 z+1}$$ where $x, y, z$ are arbitrary non negative real numbers satisfying the condition $x+y+z=4$.
- Let $M$ be a point inside an acute triangle $A B C$ satisfying the condition $\widehat{M B A}=\widehat{M C A}$. Let $K$ and $L$ be the feet of the perpendiculars respectively to $A B$ and $A C$ passing through $M$. Prove that $K$ and $L$ are in equal distances from the midpoint of $B C$ and the median issued from $M$ of triangle $M K L$ passes through a fixed point when $M$ moves inside triangle $A B C .$
- Let be given a right-angled triangle $A B C$, right at $A$ and $A H$ be its altitude issued from $A$. A circle passing through $B$ and $C$ cuts $A B$ and $A C$ at $M$ and $N$ respectively. Consider the rectangle $A M D C$. Prove that $H N$ is perpendicular to $H D$.
- Let $a$ be a natural number greater than 1. Consider a non empty subset $A$ of $N$ satysfying the condition: If $k \in A$ then $k+2 a \in A$ and $\left[\frac{k}{a}\right] \in A([x]$ denotes the integral part of $x$). Prove that $A=N$.
- Find all continuous functions $f: \mathbb{R} \rightarrow \mathbb{R}$ satisfying the condition $$9 f(8 x)-9 f(4 x)+2 f(2 x)=100 x,\,\forall x \in \mathbb{R}.$$
- Find the greatest value and the least value of the expression $$P=a(b-c)^{3}+b(c-a)^{3}+c(a-b)^{3}$$ where $a, b, c$ are arbitrary non negative real numbers satisfying the condition $a+b+c=1$.
- Let $I$ and $G$ be respectively the incenter and the centroid of a triangle $A B C$. Let $R_{1}$, $R_{2}$, $R_{3}$ be the circumradii respectively of the triangles $I B C$, $I C A$, $I A B$ and let $R_{1}^{\prime}$, $R_{2}^{\prime}$, $R_{3}^{\prime}$ be the circumradii respectively of the triangles $G B C$, $G C A$, $G A B$. Prove that $$R_{1}^{\prime}+R_{2}^{\prime}+R_{3}^{\prime} \geq R_{1}+R_{2}+R_{3} .$$
- Let $A B C D$ be a tetrahedron, the measures of its sides are: $B C=a$, $D A=a_{1}$, $C A=b$, $D B=b_{1}$, $A B=c$, $D C=c_{1}$ and let $G$ be its centroid. The sphere circumscribing $A B C D$ cuts $A G$, $B G$, $C G$, $D G$ respectively at $A_{1}$, $B_{1}$, $C_{1}$, $D_{1}$; let $R$ be its radius. Prove that $$\frac{4}{R} \leq \frac{1}{G A_{1}}+\frac{1}{G B_{1}}+\frac{1}{G C_{1}}+\frac{1}{G D_{1}} \leq \frac{2 \sqrt{3}}{3}\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{a_{1}}+\frac{1}{b_{1}}+\frac{1}{c_{1}}\right).$$
Issue 351
- Consider the product of $11$ factors $$T=(5 a+2006 b)(6 a+2005 b)(7 a+2004 b) \ldots(15 a+1996 b)$$ where $a$, $b$ are given integers. Prove that if $T$ is divisible by $2011$ then $T$ is divisible by $2011^{11}$.
- Calculate the sum of 2006 terms $$S=\frac{3^{3}+1^{3}}{2^{3}-1^{3}}+\frac{5^{3}+2^{3}}{3^{3}-2^{3}}+\frac{7^{3}+3^{3}}{4^{3}-3^{3}}+\ldots+\frac{4013^{3}+2006^{3}}{2007^{3}-2006^{3}}$$
- Find the prime number $p$ such that $2005^{2005}-p^{2006}$ is divisible by $2005+p$
- Solve the system of equations $$\begin{cases}x+y+z+t & =12 \\ x^{2}+y^{2}+z^{2}+t^{2} & =50 \\ x^{3}+y^{3}+z^{3}+t^{3} & =252 \\ x^{2} t^{2}+y^{2} z^{2} & =2 x y z t\end{cases}$$
- Find the least value of the expression $$P=\frac{a b+b c+c a}{a^{2}+b^{2}+c^{2}}+\frac{(a+b+c)^{3}}{a b c},$$ where $a, b, c$ are positive real numbers.
- Let be given a not obtuse triangle $A B C$ with its three altitudes $A A_{1}$, $B B_{1}$, $C C_{1}$ and its orthocenter $H$. Prove that $$H A^{2}+H A_{1}^{2}+H B^{2}+H B_{1}^{2}+H C^{2}+H C_{1}^{2} \geq \frac{5}{2}\left(H A \cdot H A_{1}+H B \cdot H B_{1}+H C \cdot H C_{1}\right)$$
- Let be given five concyclic points $A$, $B$, $C$, $D$, $E$ and let $M$, $N$, $P$, $Q$ be the orthogonal projections of $E$ respectively on the lines $A B$, $B C$, $C D$, $D A$. Prove that the orthogonal projections of $E$ on the lines $M N$, $N P$, $P Q$, $Q M$ are collinear.
- Prove that $(2 n+1)^{n+1} \leq(2 n+1) ! ! \pi^{n}$ for every natural number $n$, where $(2 n+1) ! !$ denotes the product of the first $n+1$ positive odd integers.
- Solve the equation $$x^{3}-3 x=\sqrt{x+2}.$$
- Let $f(x)$ be a continuous function defined on $[0 ; 1]$ satisfying the conditions $$f(0)=0,\, f(1)=1,\quad 6 f\left(\frac{2 x+y}{3}\right)=5 f(x)+f(y),\,\forall x \geq y ; x, y \in[0 ; 1].$$ Calculate $f\left(\dfrac{8}{23}\right)$.
- Calculate the measures of the angles of a triangle $A B C$ satisfying the condition $$\frac{h_{a}}{m_{b}}+\frac{h_{b}}{m_{a}}=\frac{4}{\sqrt{3}}$$ where $m_{a}, m_{b}$ are the measures of its two medians and $h_{a}$, $h_{a}$ are the measures of its two altitudes issued respectively from the vertices $A$, $B$.
- Let be given an equifaced tetrahedron $A B C D$ ($A B=C D$, $A C=B D$, $B C=A D$) and let $V$, $R$, $r$ be respectively its volume, its circumradius, its inradius. Prove that $$\frac{243 V^{2}}{512 R^{6}} \leq \cos A \cdot \cos B \cdot \cos C \leq \frac{9}{8}\left(\frac{r}{R}\right)^{2}$$ where $A$, $B$, $C$ are the angles of triangle $A B C$. When do equalities occur?
Issue 352
- Find a $5$-digit number such that by multiplying it by $2$ we obtain a $6$-digit number with six distinct nonzero digits and by multiplying it respectively by $5,6,7,8,11$ we obtain five $6$-digit numbers such that the digits of each number are the six above mentioned nonzero digits but written in another order.
- Let $a, b, c, d, m, n$ be positive integers such that $a b=c d$. Prove that the number $$A=a^{2 n+1}+b^{2 m+1}+c^{2 n+1}+d^{2 m+1}$$ is a composite number.
- Find integral solutions of the equation $$x^{5}-y^{5}-x y=32 .$$
- Let be given positive numbers $a, b, c$ satisfying the condition $a b c \geq 1$. Prove that $$\frac{a}{\sqrt{b+\sqrt{a c}}}+\frac{b}{\sqrt{c+\sqrt{a b}}}+\frac{c}{\sqrt{a+\sqrt{b c}}} \geq \frac{3}{\sqrt{2}}.$$
- Find real numbers $x, y$ satisfying the conditions $$x+y \geq 4,\quad \left(x^{3}+y^{3}\right)\left(x^{7}+y^{7}\right)=x^{11}+y^{11}.$$
- Let $A B C D$ be a convex quadrilateral and let $E$, $F$ be the midpoints respectively of $A D$, $B C$. The lines $A F$, $B E$ intersect at $M$, the lines $C E$, $D F$ intersect at $N$. Find the least value of $$P=\frac{M A}{M F}+\frac{M B}{M E}+\frac{N C}{N E}+\frac{N D}{N F} .$$
- Let $A$, $B$, $C$ be three points lying on a circle with center $O$ and radius $R$ so that $$C B-C A=R,\quad C A \cdot C B=R^{2} .$$ Calculate the measures of the angles of triangle $A B C$.
- The sequence of numbers $\left(a_{i}\right)$ $(i=1,2,3, \ldots)$ is defined by $$a_{1}=1,\, a_{2}=-1,\quad a_{n}=-a_{n-1}-2 a_{n-2},\,\forall n=3,4, \ldots$$ Calculate the value of the expression $$A=2 a_{2006}^{2}+a_{2006} \cdot a_{2007}+a_{2007}^{2}.$$
- Let $N_{m}$ be the set of all integers not less then a given integer $m$. Find all functions $f: N_{m} \rightarrow N_{m}$ satisfying the condition $$f\left(x^{2}+f(y)\right)=y+(f(x))^{2},\,\forall x, y \in N_{m}.$$
- Suppose that the system of equations $$\begin{cases}x^{2}+x y+x &=1 \\ y^{2}+x y+x+y &=1\end{cases}$$ has a unique solution $\left(x_{0}, y_{0}\right)$ with $x_{0}>0$, $y_{0}>0$. Prove that $$\frac{1}{x_{0}}+\frac{1}{y_{0}}=8 \cos ^{3} \frac{\pi}{7}.$$
- The measures of the sides of a triangle $A B C$ are $B C=a$, $C A=b$, $A B=c$ and the measures of its altitudes issued respectively from $A$, $B$, $C$ are $h_{a}$, $h_{b}$, $h_{c}$. Take $A_{1}$ on the side $B C$ so that the incircles of triangles $A B A_{1}$, $A C A_{1}$ have equal radii $r_{A}$. One defines $r_{B}$, $r_{C}$ analogously. Prove that $$2\left(r_{A}+r_{B}+r_{C}\right)+p \leq h_{a}+h_{b}+h_{c}$$ where $p$ is the semiperimeter of triangle $A B C$.
- Let be given a triangular pyramid $S.MNP$ such that $$\widehat{M S N}+\widehat{N S P}+\widehat{P S M}=180^{\circ}.$$ Prove that $\cos \alpha+\cos \beta+\cos \gamma=1$ where $\alpha$, $\beta$, $\gamma$ are the measures of the dihedral angles with sides $S M$, $S N$, $S P$ respectively.
Issue 353
- Find $2 n$-digit number of the form $\overline{a_{1} a_{2} \ldots a_{2 n-1} a_{2 n}}$ satisfying the condition $$\overline{a_{1} a_{2} \ldots a_{2 n-1} a_{2 n}}=a_{1} \cdot a_{2}+\ldots+a_{2 n-1} \cdot a_{2 n}+2006.$$
- Do there exist three numbers $a, b, c$ satisfying $$\frac{a}{b^{2}-c a}=\frac{b}{c^{2}-a b}=\frac{c}{a^{2}-b c}$$
- Find all positive integers $x, y, z$ satisfying simultaneously the two conditions
- $\dfrac{x-y \sqrt{2006}}{y-z \sqrt{2006}}$ is a rational number,
- $x^{2}+y^{2}+z^{2}$ is a prime number.
- Find the greatest value and the least value of the expression $P=x y z$ where $x, y, z$ are real numbers satisfying $$\frac{8-x^{4}}{16+x^{4}}+\frac{8-y^{4}}{16+y^{4}}+\frac{8-z^{4}}{16+z^{4}} \geq 0.$$
- Prove that $$\frac{2}{9} \leq a^{3}+b^{3}+c^{3}+3 a b c < \frac{1}{4}$$ where $a, b, c$ are the measures of three sides of a triangle with perimeter $a+b+c=1$.
- Consider convex quadrilateral $A A^{\prime} C^{\prime} C$ such that the lines $A C$, $A^{\prime} C^{\prime}$ intersect at a point $I$. Take a point $B$ on the side $A C$ and a point $B^{\prime}$ on the side $A^{\prime} C^{\prime}$. Let $O$ be the point of intersection of the lines $A C^{\prime}$, $A^{\prime} C$; let $P$ be that of $A B^{\prime}$, $A^{\prime} B$; let $Q$ be that of $B C^{\prime} \cdot B^{\prime} C$. Prove that the points $P$, $O$, $Q$ are collinear.
- Let be given an isosceles triangle $A B C$ with $A B=A C$. Take a point $D$ on the side $A B$ and a point $E$ on the side $A C$ so that $D E=B D+C E$. The bisector of angle $B D E$ cuts the side $B C$ at $I$. a) Find the measure of angle $\angle D I E$. b) Prove that the line $D I$ passes through a fixed point when $D$ moves on $A B$ and $E$ moves on $A C$.
- Find all positive integers $n$ greater than 1 such that every integer $k$, $1<k<n$ satisfying $\gcd(k, n)=1$, is a prime.
- Find all polynomials $P(x)$ satisfying the condition $$P\left(x^{2006}+y^{2006}\right)=(P(x))^{2006}+(P(y))^{2006}$$ for all real numbers $x, y$.
- Solve the equation $$2 \sqrt{x^{2}-\frac{1}{4}+\sqrt{x^{2}-\frac{1}{4}+\sqrt{\ldots+\sqrt{x^{2}-\frac{1}{4}+\sqrt{x^{2}+x+\frac{1}{4}}}}}}=2 x^{3}+3 x^{2}+3 x+1$$ where on the left side there are $2006$ signs of radical.
- Let be given a quadrilateral $A B C D$ inscribed in a circle with center $O$, radius $R$. The lines $A B$, $C D$ intersect at $P$, the lines $A D$, $B C$ intesect at $Q$. Prove that $$\overrightarrow{O P} \cdot \overrightarrow{O Q}=R^{2}.$$
- Let $M$ be a point lying inside the tetrahedron $A B C D$. The lines $M A$, $M B$, $M C$, $M D$ cut the faces $B C D$, $C D A$, $D A B$, $A B C$ respectively at $A^{\prime}$, $B^{\prime}$, $C^{\prime}$, $D^{\prime}$. Prove that the volume of the tetrahedron $A^{\prime} B^{\prime} C^{\prime} D^{\prime}$ does not exceed $\dfrac{1}{27}$ that of the tetrahedron $A B C D$.
Issue 354
- a) Find all natural number, each of which can be written as the sum of two relatively prime integers greater than $1$.
b) Find all natural numbers, each of which can be written as the sum of three pairwise relatively prime integers greater than $1$. - Let $A B C$ be a triangle with acute angle $\widehat{A B C}$. Let $K$ be a point on the side $A B$, and $H$ be its orthogonal projection on the line $B C$. A ray $B x$ cuts the segment $KH$ at $E$ and cuts the line passing through $K$ parallel to $B C$ at $F$. Prove that $\widehat{A B C}=3 \widehat{C B F}$ when and only when $E F=$ $2 B K$.
- Find all natural numbers $n$ such that the product of the digits of $n$ is equal to $$(n-86)^{2}\left(n^{2}-85 n+40\right).$$
- Prove that $a b+b c+c a<\sqrt{3} d^{2}$, where $a, b, c, d$ are real numbers satisfying the following conditions $$0<a, b, c<d,\quad \frac{1}{a}+\frac{1}{b}+\frac{1}{c}-\sqrt{\frac{1}{a^{2}}+\frac{1}{b^{2}}+\frac{1}{c^{2}}}=\frac{2}{d}.$$
- Solve the equation $$x^{4}+2 x^{3}+2 x^{2}-2 x+1=\left(x^{3}+x\right) \sqrt{\frac{1-x^{2}}{x}}.$$
- Let $A B C D$ be a square with sides equal to $a$. On the side $A D$, take the point $M$ such that $A M=3 M D$. Draw the ray $B x$ cutting the side $C D$ at $I$ such that $\widehat{A B M}=\widehat{M B I}$. The angle bisector of $\widehat{C B I}$ cuts the side $C D$ at $N$. Calculate the area of triangle $B M N$.
- Let $B C$ be a fixed chord (which is not a diameter) of a circle. On the major arc $B C$ of the circle, take a point $A$ not coinciding with $B$, $C$. Let $H$ be the orthocenter of triangle $A B C$. The second points of intersection of the line $B C$ with the circumcircles of triangles $A B H$ and $A C H$ are $E$ and $F$ respectively. The line $E H$ cuts the side $A C$ at $M$ and the line $F H$ cuts the side $A B$ at $N$. Determine the position of $A$ so that the measure of the segment $M N$ attains its least value.
- How many are there natural $9$-digit numbers with $3$ distinct odd digits, $3$ distinct even digits and every even digit in each number appears exactly two times (in this number).
- For every positive integer $n$, consider the function $f_{n}$ defined on $\mathbb{R}$ by $$f_{n}(x)=x^{2 n}+x^{2 n-1}+\ldots+x^{2}+x+1$$ a) Prove that the function $f_{n}$ attains its least value at a unique value $x_{n}$ of $x$.
b) Let $S_{n}$ be the least value of $f_{n}$. Prove that - $S_{n}>\dfrac{1}{2}$ for all $n$ and there does not exist a real number $a>\dfrac{1}{2}$ such that $S_{n}>a$ for all $n$.
- $\left(S_{n}\right)$ $(n=1,2, \ldots)$ is a decreasing sequence and $\lim S_{n}=\dfrac{1}{2}$.
- $\displaystyle\lim_{n\to\infty} x_{n}=-1$.
- Let $$A=\sqrt{x^{2}+\sqrt{4 x^{2}+\sqrt{16 x^{2}+\sqrt{100 x^{2}+39 x+\sqrt{3}}}}}.$$ Find the greatest integer not exceeding $A$ when $x=20062007$.
- Let $A B C$ be a triangle with $B C=d$ $C A=b$, $A B=c$, with inradius $r$ and with incenter $I$. Let $A_{1}$, $B_{1}$, $C_{1}$ be respectively the touching points of the sides $B C$, $C A$, $A B$ with the incircle. The rays $I A$, $I B$, $I C$ cut the incircle respectively at $A_{2}$, $B_{2}$, $C_{2}$. Let $B_{i} C_{i}=a_{1}$, $C_{1} A_{i}=b_{i}$, $A_{i} B_{i}=c_{i}$ $(\mathrm{i}=1,2)$. Prove that $$\frac{a_{2}^{3} b_{2}^{3} c_{2}^{3}}{a_{1}^{2} b_{1}^{2} c_{1}^{2}} \geq \frac{216 r^{6}}{a b c}.$$ When does equality occur?
- Let $O A B C$ be a tetrahedron with $$\widehat{A O B}+\widehat{B O C}+\widehat{C O A}=180^{\circ}.$$ $O A_{1}$, $O B_{1}$, $O C_{1}$ are internal angle bisectors respectively of the triangles $O B C$, $O C A$, $O A B$; $O A_{2}$, $O B_{2}$, $O C_{2}$ are internal angle bisectors respectively of the triangles $O A A_{1}$, $O B B_{1}$, $O C C_{1}$. Prove that $$\left(\frac{A A_{1}}{A_{2} A_{1}}\right)^{2}+\left(\frac{B B_{1}}{B_{2} B_{1}}\right)^{2}+\left(\frac{C C_{1}}{C_{2} C_{1}}\right)^{2} \geq(2+\sqrt{3})^{2}.$$ When does equality occur?