Mathematics and Youth Magazine Problems 2009


Issue 379

  1. Find all pairs of integers $a$, $b$ such that $$a^{2}+a b+b^{2}=a^{2} b^{2}$$
  2. Let $ABC$ be an isosceles triangle (at vertex $A$) such that $\widehat{B A C} \geq 90^\circ$. Choose a point $M$ on $A C$, and let $A H$ and $C K$ be the altitudes from $A$ and $C$ onto $B M$ respectively $(H, K$ are the feet of these altitudes) such that $B H=H K+K C$. Find the angle $B A C$.
  3. Solve for $x$$$\sqrt{\frac{5 \sqrt{2}+7}{x+1}}+4 x=3 \sqrt{2}-1 $$
  4. Find $a$, $b$ such that the maximum value of the expression $$|||x+1|-2|-(a x+b)|$$ where $-3 \leq x \leq 4$ is smallest possible.
  5. Let $A B C D$ be a rectangular, let $I$ be the midpoint of $C D$, and let $E$ be a point on $A B$. The altitude onto $D E$ through $I$ meets $D E$ and $A D$ at $M$ and $H$ respectively. The altitude onto $C E$ through $I$ meets $C E$ and $B C$ at $N$ and $K$, respectively. $E I$ intersects with $H K$ at $G$. Prove that
    a) The points $E$, $G$, $N$, $K$, $B$ lie on the same circle.
    b) The points $E$, $G$, $M$, $H$, $A$ lie on the same circle.
  6. Let $A B C$ be a triangle. Prove the inequality $$\cos A \cos B \cos C \leq \frac{1}{8} \cos (B-C) \cos (C-A) \cos (A-B)$$
  7. Solve for $x$ $$3^{x}\left(4^{x}+6^{x}+9^{x}\right)=25^{x}+2.16^{x}$$
  8. Prove that the union of six hemispheres whose diameters are the sides of a given tetrahedron must contain the tetrahedron itself.
  9. Let $A B C$ be a triangle. Choose the pair of points $A_{1}$, $A_{2}$; $B_{1}$, $B_{2}$; $C_{1}$, $C_{2}$ on the sides $B C$, $C A$ and $A B$ respectively such that $A_{1} A_{2} B_{1} B_{2} C_{1} C_{2}$ is a convex hexagon with equal opposite sides and the triangles $A A_{1} A_{2}$, $B B_{1} B_{2}$, $CC_{1}C_{2}$ have the same areas. Prove that the lines $A_{1} B_{2}$, $B_{1} C_{2}$, $C_{1} A_{2}$ are colinear.
  10. Let $a$, $b$, $c$ be non - negative real numbers such that $$\sqrt[3]{a^{3}+b^{3}}+\sqrt[3]{b^{3}+c^{3}}+\sqrt[3]{c^{3}+a^{3}}+a b c=3.$$ Prove that the smallest value of the expression $$P=\frac{a^{3}}{b^{2}+c^{2}}+\frac{b^{3}}{c^{2}+a^{2}}+\frac{c^{3}}{a^{2}+b^{2}}$$ is $\sqrt[3]{32}m$ where $m$ is the real root of the equation $t^{3}+54 t-162=0$.
  11. Let $f: \mathbb{N} \rightarrow \mathbb{R}$ be the function with initial values $f(0)=0$, $f(1)=1$ such that $$f(n+2)-2011 f(n+1)+f(n)=0.$$ What is the probability that $f(n)$ is a prime number, where $n$ is a randomly chosen number from the set of integers $\{0,1,2, \ldots, 2008\}$?
  12. Find all functions $f(x)$ such that it is continuous on $[0 ; 1]$, differentiable on the open interval $(0 ; 1)$ and the following two conditions are satisfied $$f(0)=f(1)=\frac{2009}{11},\quad 20 f(x)+11 f(x)+2009 \leq 0,\, \forall x \in(0 ; 1).$$

Issue 380

  1. Compare $\dfrac{1}{6}$ with $$A=\frac{1}{5}-\frac{1}{7}+\frac{1}{17}-\frac{1}{31}+\frac{1}{65}-\frac{1}{127}$$
  2. Let $ABC$ be an isosceles triangle (at vertex $C$) with $\widehat{A C B}=100^{\circ}$. $M$ is a point chosen on the ray $C A$ such that $C M=A B$. Find the measure of the angle $\widehat{C M B}$. 
  3. Find all triple $(x, y, z)$ of integers such that $$\begin{cases}y^{3} &=x^{3}+2 x^{2}+1 \\ x y &=z^{2}+2\end{cases}$$
  4. Find all pair of numbers $x$ and $y$ such that the following conditions hold $x>1$, $0<y<1$, $x+y \leq \sqrt{5}$, $\dfrac{1}{x}+\dfrac{1}{y} \leq \sqrt{5}$ and $\dfrac{x}{x+1}+\dfrac{y}{1-y} \leq \sqrt{5}$.
  5. Let $A B C D$ be an isosceles trapezoid inscribed inside a circle $\left(O_{1}: R\right)$ and circumscribes the circle $\left(O_{2} ; r\right)$. Let $d=O_{1} O_{2}$. Prove that $$\frac{1}{r^{2}} \geq \frac{2}{R^{2}+d^{2}} .$$ When does equality occur? 
  6. Prove that in any triangle $A B C$, the following inequality holds $$\frac{\cot A \cdot \cot B \cdot \cot C}{\sin A \cdot \sin B \cdot \sin C} \leq\left(\frac{2}{3}\right)^{3}$$ 
  7. There are $17$ ornament betel-nut trees around a circular pond. How many ways are there to chopped off $4$ trees with the condition that no two consecutive trees be removed? 
  8. Solve the following system of equations with parameter $a$ $$\begin{cases} 2 x\left(y^{2}+a^{2}\right) &=y\left(y^{2}+9 a^{2}\right) \\ 2 y\left(z^{2}+a^{2}\right) &=z\left(z^{2}+9 a^{2}\right) \\ 2 z\left(x^{2}+a^{2}\right) &=x\left(x^{2}+9 a^{2}\right)\end{cases}$$
  9. Let $\left(x_{n}\right)$, $n=0,1,2, \ldots$ be a sequence given by the following recursive formula $$x_{0}=a,\quad x_{n+1}=x_{n}+\sin x+2 \pi,\, n=0,1,2, \ldots$$ where $a, x \in \mathbb{R}$. Prove that the limit $\displaystyle\lim_{n \rightarrow+\infty} \frac{x_{1}+\ldots+x_{n}}{n^{2}}$ exists and find its exact value. 
  10. Prove that
    a) $\displaystyle\sum_{k=0}^{n}\left(\frac{k}{n}-x\right)^{2} C_{n}^{k} x^{k}(1-x)^{n-k}=\frac{x(1-x)}{n},\, \forall x \in \mathbb{R}$
    b) $\displaystyle\sum_{k=0}^{n}\left|\frac{k}{n}-x\right|C_{n}^{k} x^{k}(1-x)^{n-k} \leq \frac{1}{2 \sqrt{n}},\, \forall x \in[0 ; 1]$. 
  11. Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that $$f\left(n^{2}\right)=f(m+n) f(n-m)+m^{2},\, \forall m, n \in \mathbb{R}$$
  12. On a given plane, choose a point $A$ outside a circle whose center is at a point $O$ and whose radius equals $R$. A chord $M N$ with constant length moves on the circle such that the two line segments $M N$ and $O A$ always intersect. Determine the positions of $M$ and $N$ such that the sum $A M+A N$ is
    a) greatest possible.
    b) smallest possible.

Issue 381

  1. Rearrange the following rational numbers in increasing order $$\frac{1005}{2002}, \frac{1007}{2006}, \frac{1009}{2010}, \frac{1011}{2014}$$
  2. In an isosceles triangle $A B C$ (at vertex $A$ ), choose a point $M$ such that $$\widehat{M A C}=\widehat{M B A}=\widehat{M C B}.$$ Compare the areas of the triangles $A B M$ and $C B M$.
  3. Determine the sum of all rational numbers of the form $\dfrac{a}{b}$ where $a$, $b$ are natural divisors of $27000$ and $\gcd(a, b)=1$.
  4. Given $a$, $b$, $c$ such that $a>0$, $b>c$, $a^{2}=b c$, $a+b+c=a b c$. Prove the inequalities $$a \geq \sqrt{3},\quad b \geq \sqrt{3},\quad 0<c \leq \sqrt{3}.$$
  5. Let $ABCD$ be a quadrilateral where $\widehat{A B C}=\widehat{A D C}=90^{\circ}$ and $\widehat{B C D}<90^{\circ}$. Choose a point $E$ on the opposite ray of $A C$ such that $D A$ is the angle-bisector of $B D E$. Let $M$ be chosen arbitrarily between $D$ and $E$, choose another point $N$ on the opposite ray of $B E$ such that $\widehat{N C B}=\widehat{M C D}$. Prove that $M C$ is the angle bisector of $D M N$.
  6. Prove that the equation $x^{3}+3 y^{3}=5$ has infinitely many rational solutions.
  7. If $a$, $b$, $c$ are the length of the sides of a triangle, prove that $$\frac{1}{\sqrt{a b+a c}}+\frac{1}{\sqrt{b c+b a}}+\frac{1}{\sqrt{c a+c b}} \geq \frac{1}{\sqrt{a^{2}+b c}}+\frac{1}{\sqrt{b^{2}+a c}}+\frac{1}{\sqrt{c^{2}+a b}}.$$
  8. Let $A B C D . A^{\prime} B^{\prime} C^{\prime} D^{\prime}$ be a parallelepiped and let $S_{1}$, $S_{2}$, $S_{3}$ denote the areas of the sides $A B C D$, $A B B^{\prime} A^{\prime}$ and $A D D^{\prime} A^{\prime}$ respectively. Given that the sum of squares of the areas of all sides of the tetrahedron $A B^{\prime} C D^{\prime}$ equals $3$, find the smallest possible value of the following expression $$T=2\left(\frac{1}{S_{1}}+\frac{1}{S_{2}}+\frac{1}{S_{3}}\right)+3\left(S_{1}+S_{2}+S_{3}\right)$$

Issue 382

  1. Compare $\dfrac{2009}{2008^{2}}$ with the sum (consisting of $2010$ terms) $$\frac{1}{2009}+\frac{2}{2009^{2}}+\frac{3}{2009^{3}}+\ldots+\frac{2009}{2009^{2009}}+\frac{2010}{2009^{2010}}$$
  2. Find a root of the polynomial $P(x)=x^{3}+a x^{2}+b x+c$ given that it has at least one root and $a+2 b+4 c=-\dfrac{1}{2}$.
  3. Let $a_{1}$, $a_{2}$, $a_{3}$, $a_{4}$, $a_{5}$, $a_{6}$, $a_{7}$, $a_{8}$, $a_{9}$ be non negative real numbers whose sum equals $1$. Put $S_{k}=a_{k}+a_{k+1}+a_{k+2}+a_{k+3}$ $(k=1,2, \ldots, 6)$. Determine the smallest possible value of $$M=\max \left\{S_{1}, S_{2}, S_{3}, S_{4}, S_{5}, S_{6}\right\}.$$
  4. Let $m$, $n$, $a$, $b$ and $c$ be real numbers such that the following conditions hold $$\begin{cases}m^{1000}+n^{1000} &=a \\ m^{2000}+n^{2000} &=\dfrac{2 b}{3}\\ m^{5000}+n^{5000} &=\dfrac{c}{36}\end{cases}.$$ Find a formula relating $a$, $b$ and $c$ which does not involve $m$, $n$.
  5. Let $A H$ be the altitude from $A$ of a triangle $A B C$. Choose a point $D$ on the half-plane created by $B C$ which contains $A$ such that $D B=D C=\dfrac{A B}{\sqrt{2}}$. Prove that the lengths of the line segments $B D$, $D H$ and $H A$ are the side lengths of a right triangle.
  6. Determine the maximum possible value of $x^{2}+y^{2}$ where $x$ and $y$ are two integers chosen arbitrarily within the interval $[-2009 ; 2009]$ such that $$\left(x^{2}-2 x y-y^{2}\right)^{2}=4.$$
  7. Consider two polynomials with real coefficients $$P(x)=x^{n}+a_{n-1} x^{n-1}+\ldots+a_{1} x+a_{0}$$ and $Q(x)=x^{2}+x+2009$. Given that $P(x)$ has $n$ distinct real roots but $P(Q(x))$ does not have any real solution. Prove that $P(2009)>\dfrac{1}{4^{n}}$.
  8. Let $ABCDEF$ be a regular hexagon and let $G$ be the midpoint of $B F$. Choose a point $I$ on $B C$ such that $B I=B G$. Let $H$ be a point on $I G$ such that $\widehat{C D H}=45^{\circ}$ and $K$ is a point on $E F$ such that $\widehat{D K E}=45^{\circ}$. Prove that $D H K$ is an equilateral triangle.

Issue 383

  1. Let $a$, $b$, $c$ be integers such that $$A=\frac{a^{2}+b^{2}+c^{2}-a b-b c-c a}{2}$$ is a perfect square. Prove that $a=b=c$.
  2. Let $A B C$ be a triangle in which $\widehat{B A C}=75^{\circ}$. Given that the altitude $A H$ has length $A H=\dfrac{B C}{2}$. Prove that $A B C$ is an isosceles triangle.
  3. Let $x$, $y$ be two nonnegative integers where $x>1$ and $2 x^{2}-1=y^{15}$. Prove that $x$ is divisible by $15$.
  4. Let $A B C D$ be a parallelogram. The ray $D x$ from $D$ is perpendicular to $D C$ and lies on the half-plane divided by $C D$ which does not contains $B$. Choose a point $E$ on $D x$ such that $D E=D C$. Draw an isosceles right triangle $B E F$ (right angle at $F$) such that $F$ and $D$ are on the same half-plane divided by $B C \cdot E H$ is the altitude onto $B C$. Prove that $F$ $D$ and $H$ are collinear.
  5. Let $a$, $b$, $c$ be three sides of a right triangle, $a$ is the hypotenuse. Prove that the equation $-a x^{2}+b x+c=0$ has two distinct roots $x_{1}$, $x_{2}$ such that $-\sqrt{2}<x_{1}<x_{2}<\sqrt{2}$.
  6. Consider a convex polygon with $2009$ vertices. A scissor is used to cut along all of its diagonals, thereby dividing the original polygon into smaller convex polygons. How many vertices are there of a resulting polygon with the greatest number of edges?
  7. Determine the smallest value of the expression $$T=\frac{a b+b c}{a^{2}-b^{2}+c^{2}}+\frac{b c+c a}{b^{2}-c^{2}+a^{2}}+\frac{c a+a b}{c^{2}-a^{2}+b^{2}}$$ where $a$, $b$, $c$ are three sides of a triangle $A B C$ and $a b c=1$.
  8. Let $A B C D . A_{1} B_{1} C_{1} D_{1}$ be a cube. On its three skew edges, choose three points $M$, $N$, $P$. Determine the positions of the points $M$, $N$, $P$ such that the triangle $M N P$ has a) The smallest perimeter possible. b) The smallest area possible.

Issue 384

  1. Replace the distinct letters by distinct numbers such that the following expression becomes a true equality $$\mathrm{VE}+\mathrm{TRUONG}+\mathrm{SA}=22 \times 12 \times 2009.$$
  2. It is well-known that the two right triangles whose side lengths are positive integers $(5,12,13)$ and $(6,8,10)$ possess additional property that the area of each triangle equals its perimeter. Are there other triangles with similar properties?
  3. Suppose given $1003$ nonzero rational numbers in which any quadruple form a proportion. Prove that at least $1000$ numbers are equal.
  4. Let $x$, $y$, $z$ be non-negative real numbers such that $x+y+z=1$. Find the maximum valuc of the following expression $$P=(x+2 y+3 z)(6 x+3 y+2 z).$$
  5. Let $A B C D$ be a cyclic quadrilateral, inscribed in a circle $(O)$. The angle bisectors of $B A D$ and $B C D$ meet at a point $K$ on the diagonal $B D$. Let $Q$ be the second intersection point (different from $A$) of $A P$ and the circle $(O)$; $M$ and $N$ be respectively the midpoints of $B D$ and $C P$. The line through $C$ and parallel to $A D$ meets $A M$ at $P$. Prove that a) $S_{A BQ}=S_{A D Q}$. b) $DN$ is perpendicular to $C P$.
  6. For each natural number $n,$ let $p(n)$ be its largest odd divisor. Determine the sum $$\sum_{n=2006}^{4012} p(n)$$
  7. Solve for $x$ $$\sqrt{3 x-2}=-4 x^{2}+21 x-22$$
  8. Let $A B C$ be a triangle whose circumcircle is $(O)$ and such that $A C<A B$. The tangent lines to $(O)$ at $B$, $C$ intersect at $T$. The line through $A$ and perpendicular to $A T$ meet $B C$ at $S$. Choose the points $B_{1}$ and $C_{1}$ on $S T$ such that $T B_{1}=T C_{1}=T B$ and that $C_{1}$ lies between $S$ and $T$. Prove that $A B C$ and $A B_{1} C_{1}$ are similar.

Issue 385

  1. Let $S(n)$ be denote the sum of the digits of $n$. Find a positive integer $n$ such that $S(n)=n^{2}-2009 n+11$.
  2. Inside a square $A B C D$ thoose two points $P$, $Q$ such that $B P$ and $D Q$ are parallel and $B P^{2}+D O^{2}=P Q^{2}$. Find the measure of the angle $P A Q$.
  3. Compare $2008$ with the sum $S$ of $2009$ terms $$S=\frac{2008+2007}{2009+2008}+\frac{2008^{2}+2007^{2}}{2009^{2}+2008^{2}}+\ldots +\frac{2008^{2009}+2007^{2009}}{2009^{2009}+2008^{2009}}.$$
  4. Let $a$, $b$, $c$ be three positive numbers such that $a+b+c=1$. Find the least value of the expression $$P=\frac{9}{1-2(a b+b c+c a)}+\frac{2}{a b c}.$$
  5. Let $B$, $C$ be two fixed points on the circle $(\omega)$ such that $B C$ does not pass through the center of $(\omega) .$ On the major arc $B C$, choose a point $A$ differs from $B$ and $C$. Another point $M$ moves on the line segment $B C$. The lines passing through $M$ and parallel to $A B$, $A C$ intersect $A C$ and $A B$ at $F$ and $E$, respectively. When $M$ moves on the line segment $B C$ and for each point $A$, let $x$ be the least possible length of $EF$. Find the positions of $A$ and $M$ such that $x$ is greatest possible.
  6. Find the greatest and the least value of the expression $$A=\frac{y^{2}}{25}+\frac{t^{2}}{144}$$ where $x$, $y$, $z$, $t$ satisfy the system of equations $$\begin{cases}x^{2}+y^{2}+2 x+4 y-20 &=0 \\ t^{2}+z^{2}-2 t-143 &= 0\\ x t+y z-x+t+2 z-61 &\geq 0\end{cases}$$
  7. Consider the sequence $\left(u_{n}\right)$ defined as follow $$u_{0}=9,\, u_{1}=161,\quad u_{n}=18 u_{n-1}-u_{n-2},\,\forall n=2,3, \ldots$$ Prove that for any $n$, $\dfrac{u_{n}^{2}-1}{5}$ is always a perfect square.
  8. Let $P$ be an arbitrary point inside a given triangle $A B C$. Let $A'$, $B'$, $C'$ be the orthogonal projection of $P$ on $B C$, $C A$, $A B$ respectively. Let $I$ be the incenter and $r$ be the inradius of the triangle $A B C$. Find the least value of the expression $$P A^{\prime}+P B^{\prime}+P C^{\prime}+\frac{P I^{2}}{2 r}$$

Issue 386

  1. Find the last two decimal digits of the sum $2008^{2009}+2009^{2008}$
  2. Let $ABC$ be an isosceles right triangle with the right angle at $A$. Let $G$ be a point on $A B$ such that $A G=\dfrac{1}{3} A B$, let $M$ be the midpoint of $B C$ and $E$ be the foot of the altitude from $M$ to $CG$. The two lines $M G$ and $A C$ meet at $D$. Prove that $D E=B C$.
  3. Find all integer solutions of the equation $$4 x^{4}+2\left(x^{2}+y^{2}\right)^{2}+x y(x+y)^{2}=132$$
  4. Let $a$, $b$ and $c$ satisfy the conditions $a \leq b \leq c$ and $$a+b+c=\frac{1}{a}+\frac{1}{b}+\frac{1}{c}.$$ Find the least value of the expression $P=a b^{2} c^{3}$.
  5. A triangle $A B C$ inscribed in a circle centered at $O$, radius $R$, $A D$ is its anglebisector. Let $E$, $F$ be the circumeenters pf the triangles $A B D$ and $A C D$. respectively. Given that $B C=a$. Determine the area of the quadrilateral $A E O F$.
  6. Let $A B C$ be a triangle with incenter $I$ such that its centroid $G$ lies inside $(I)$. Let $a$, $b$, $c$ be the lengths of the sides $B C$, $A C$, $A B$, respectively. Find the greatest and least value of the following expression $$P=\frac{a^{2}+b^{2}+c^{2}}{a b+b c+c a}$$
  7. Let $x$, $y$, $z$ be positive numbers satisfying $$x^{2}+y^{2}+z^{2}=\frac{1-16 x y z}{4}.$$ Find the least value of the expression $$S=\frac{x+y+z+4 x y z}{1+4 x y+4 y z+4 z x}.$$
  8. Solve for $x$ $$2^{x}+5^{x}=2-\frac{x}{3}+44 \log _{2}\left(2+\frac{131 x}{3}-5^{x}\right)$$
  9. Let $A B C$ be a right triangle, right angle at $A$, $M$ is the midpoint of $B C$. Construct a right angle $P M Q$ with $P \in A B$, $Q \in A C$. Prove that $$P Q^{2} \geq A P \cdot C Q+A Q \cdot B P$$
  10. Let $a_{n}$ be the last non-zero digit (counting from left to right) when expressing $n !$ in the decimal number system. Is the sequence $\left(a_{n}\right)$ for $n=1.2 .3 \ldots$ periodic? (That is, there exist the positive integers $T$ and $N$ such that $a_{i+T}=a_{i} \forall i \geq N$).
  11. Given $n$ non-negative numbers $a_{1}, a_{2}, \ldots, a_{n}$ $(n \geq 3)$ satisfying $$a_{1}^{2}+a_{2}^{2}+\ldots+a_{n}^{2}=1.$$ Prove that $$\frac{1}{\sqrt{3}}\left(a_{1}+a_{2}+\ldots+a_{n}\right) \geq a_{1} a_{2}+a_{2} a_{3}+\ldots+a_{n} a_{1}.$$ When does equality occur?
  12. Let $\left(x_{n}\right)(n=1,2, \ldots)$ be a sequence given by $$x_{1}=a \ (a>1),\, x_{2}=1,\quad x_{n+2}=x_{n}-\ln x_{n},\,\forall n \in \mathbb{N}^{*}.$$ Put $\displaystyle S_{n}=\sum_{k=5}^{n-1}(n-k) \ln \sqrt{x_{2 k-1}}$ $(n \geq 2)$. Find $\displaystyle\lim_{n \rightarrow \infty}\left(\frac{S_{n}}{n}\right)$.

Issue 387

  1. Prove that there exists a $2009$-digits multiple of $2007$ so that it is formed by four digits $0,2,7$ and $9$ and the sum of its digits is $7209$.
  2. Let $a_{1}, a_{2}, \ldots, a_{n}$ be $n$ odd integers $(n>2007)$ satisfying the following condition $$a_{1}^{2}+a_{2}^{2}+\ldots+a_{2005}^{2}=a_{2006}^{2}+a_{2007}^{2}+\ldots+a_{n}^{2}.$$ Determine the least possible value of $n$ and construct an example of such a collection $\left(a_{1}, a_{2}, \ldots, a_{n}\right)$ for the smallest value found above.
  3. Determine the value of $S$, given that $$S=\frac{2^{3}-1}{2^{3}+1} \times \frac{3^{3}-1}{3^{3}+1} \times \ldots \times \frac{2009^{3}-1}{2009^{3}+1}$$
  4. Let $A B C$ be a right triangle with right angle at $A$ and $A C>A B$. Choose a poind $D$ on $A C$ such that $A B=A D .$ Let $E$ be the foot of the altitude from $D$ onto $B C .$ The line passing through $A$ and parallel to $B C$ meets $D E$ at $H .$ Prove that $$A H<\frac{\sqrt{2}}{2} A C.$$
  5. Let $A B$ be a fixed chord of a given circle $(O)$ and $E$ is a point moving on $A B$ (but distinct from $A$ and $B$). From $E$, draw another chord $C D$. $P$ and $Q$ are two points on the rays $D A$ and $D B$ respectively such that $P$ is the reflection of $Q$ through $E$. Prove that the circle $(I)$ passing through $C$ and touches $P Q$ at $E$ always passes through a fixed point.
  6. Of all pentagons whose sum of the squares of its diagonals equals $1,$ which one has the smallest possible sum of cube of its sides.
  7. Solve the system of equations $$\begin{cases}\sqrt{2 x}+2 \sqrt[4]{6-x}-y^{2} &=2 \sqrt{2} \\ \sqrt[4]{2 x}+2 \sqrt{6-x}+2 \sqrt{2} y &=8+\sqrt{2}\end{cases}$$
  8. Prove that $$\frac{\sin A}{\tan \frac{B}{2}}+\frac{\sin B}{\tan \frac{C}{2}}+\frac{\sin C}{\tan \frac{A}{2}} \geq \frac{9}{2}$$ where $A$, $B$, $C$ are the measures of the angles of a triangle. When does equality occur?.
  9. In a triangle $A B C$, let $P$ be a point such that $P A=P B+P C$. $R$ is the midpoint of the chord $A B$ (the one that contains $P$) of the circumcircle of the triangle $A B P$ and $S$ is the midpoint of the chord $A C$ (containing $P$) of the circumcircle of the triangle $A C P$. Prove that circumcircles of the triangles $B P S$ and $CPR$ touch each other.
  10. Does there exist a function $\mathbb{R} \rightarrow \mathbb{R}$ such that
    • $f$ is continuous on $\mathbb{R}$; and
    • $f(x+2008)(f(x)+\sqrt{2009})=-2010, \forall x \in \mathbb{R} . ?$
  11. Let $\left(u_{n}\right)(n=1,2, \ldots)$ be a sequence given by $$u_{1}=2,1;\quad  u_{n+1}=\frac{-2 u_{n}^{2}+5 u_{n}}{-u_{n}^{2}+2 u_{n}+1},\,\forall n=1,2, \ldots$$ Find $\displaystyle\lim_{n\to\infty} \left(u_{1}+u_{2}+\ldots+u_{n}\right)$.
  12. $A B C$ and $A^{\prime} B^{\prime} C^{\prime}$ are two triangles in a given plane whose inradii are $r, r^{\prime}$ respectively. Let $R'$ be the radius of the circumcircle of the triangle $A^{\prime} B^{\prime} C^{\prime} .$ Prove that $$\left(\sin \frac{B^{\prime}}{2}+\sin \frac{C^{\prime}}{2}\right) P A +\left(\sin \frac{C^{\prime}}{2}+\sin \frac{A^{\prime}}{2}\right) P B +\left(\sin \frac{A^{\prime}}{2}+\sin \frac{B^{\prime}}{2}\right) P C \geq \frac{6 r r^{\prime}}{R^{\prime}}$$ for any point $P$ in the same plane. When does equalitiy occur?

Issue 388

  1. Prove that a number of the form $(\overline{33 \ldots 3})^{2}$ with $k$ $(k>0)$ digits $3$ can always be written as the difference between of the number whose digits are 1 and the one whose digits are $2$.
  2. Find all real numbers $a$ such that $|3 a-2| \leq 1$ and $A=\dfrac{3 a-1}{4 a^{4}+a^{2}}$ is an integer.
  3. Find the greatest value of the expression $$P=3 \sqrt{x}+8 \sqrt{y}$$ where $x, y$ are two non-negative numbers satisfying $17 x^{2}-72 x y+90 y^{2}-9=0$
  4. Solve the equation $$\sqrt{x-2}+\sqrt{4-x}+\sqrt{2 x-5}=2 x^{2}-5 x$$
  5. From a point $A$ outside a given circle witlı center $O$, construct two tangent lines $A B$ and $A C$ with $B, C$ are the tangency points. On $O B$, choose $N$ such that $B N=2 O N$. The perpendicular bisector of the line segment $C N$ meets $O A$ at $M .$ Determine the ratio $\dfrac{A M}{A O}$
  6. Find the least value of the expression $$A=\frac{1}{a^{4}(b+1)(c+1)}+\frac{1}{b^{4}(c+1)(a+1)}+\frac{1}{c^{4}(a+1)(b+1)}$$ where $a, b, c$ are positive real numbers satisfying $a b c=1$.
  7. Let $a, b$ be two real numbers in the interval $(0 ; 1) .$ A sequence $\left(u_{n}\right)$ $(n=0,1,2, \ldots)$ is given by $$u_{0}=a, u_{1}=b,\quad u_{n+2}=\frac{1}{2010} \cdot u_{n+1}^{4}+\frac{2009}{2010} \sqrt[4]{u_{n}},\,\forall n \in \mathbb{N}.$$ Prove that $\left(u_{n}\right)$ has a finite limit, and find that limit.
  8. Let $l_a$, $l_b$, $l_c$ be respectively the lengths of the interior angle-bisectors from the three vertices $A$, $B$, $C$ of a triangle $A B C$. Let $R$ be its circumradius. Prove that $$\frac{l_{a}+l_{b}+l_{c}}{R} \leq 2\left(\cos \frac{A}{2} \cos \frac{B}{2}+\cos \frac{B}{2} \cos \frac{C}{2}+\cos \frac{C}{2} \cos \frac{A}{2}\right).$$ When does the equality occur?
  9. Let $A B$ be a fixed chord on a given circle $(O) .$ A point $C$ moves on the circle and let $M$ be a point on the zigzag $A C B$ (consisting of two line segments $A C$ and $C B)$ such that it divides this zigzag into two parts of equal length. Find the locus of $M$ when $C$. moves around the circle $(O)$.
  10. Let $m, n, d$ be positive integers such that $m<n$ and $\gcd(d, m)=1$, $\gcd(d, n)=1$. (Here, $\gcd(a, b)$ denotes the greatest common divisor of $a$ and $b$.) Prove that in an arithmetic progression of $n$ terms with common difference $d$, there always exist two distinct terms whose product is a multiple of $m n$.
  11. A function $f: \mathbb{R} \rightarrow \mathbb{R}$ is given with the following properties
    • $f(0)=1$
    • $f(x) \leq 1,\, \forall x \in \mathbb{R}$
    • $\displaystyle f\left(x+\frac{11}{24}\right)+f(x)=f\left(x+\frac{1}{8}\right)+f\left(x+\frac{1}{3}\right)$
      Put $\displaystyle F(x)=\sum_{n=0}^{2009} f(x+n)$. Find $F(2009)$
    1. Given two positive degrees polynomials with real coefficients $$P(x)=x^{n}+a_{1} x^{n-1}+\ldots+a_{n-1} x+a_{n}$$ and $$Q(x)=x^{m}+b_{1} x^{m-1}+\ldots+b_{m-1} x+b_{m}$$ where $Q(x)$ has exactly $m$ real roots and $P(x)$ is divisible by $Q(x)$. Prove that if there exists a $k \in\{1,2, \ldots, m\}$ such that $\left|b_{k}\right|>C_{m}^{k} .2009^{k},$ then there also exists an $i \in\{1,2, \ldots, n\}$ such that $|a,|>2008$

    Issue 389

    1. Prove that the number of decimal digits in $2008^{2009}+2^{2009}$ and $2008^{2009}$ are equal.
    2. Find the least value of the expression $A=1-x y,$ where $x$ and $y$ are real numbers satisfying the following condition $$x^{2009}+y^{2009}=2 x^{1004} y^{1004}$$
    3. Does there exist a positive integer number $n$ such that $n^{6}+26^{n}=21^{2009} ?$
    4. Let $A B C$ be a right triangle with right angle at $A .$ On the sides $A B, B C$ and $C^{\prime} A$, choose $D, E$ and $F$ respectively such that $D E \perp B C$ and $D E=D F$. $M$ is the midpoint of EF. Prove that $\widehat{B C M}=\widehat{B F E}$.
    5. Let $x, y, z$ be real numbers in the interval $(0 ; 1)$. Prove that $$\frac{1}{x(1-y)}+\frac{1}{y(1-z)}+\frac{1}{z(1-x)} \geq \frac{3}{x y z+(1-x)(1-y)(1-z)}.$$ When does the equality occur?
    6. Solve the equation $$\sqrt[3]{x^{2}+4 x+3}+\sqrt[3]{4 x^{2}-9 x-3} -\sqrt[3]{3 x^{2}-2 x+2}+\sqrt[3]{2 x^{2}-3 x-2}$$
    7. Let $A_{1} A_{2} \ldots A_{n}$ be a convex polygon $(n \geq 3)$ circumscribed around the circle centered at $J$. Prove that for any point $M$, $$\sum_{i=1}^{n} \cos \frac{A_{i}}{2}\left(M A_{i}-J A_{i}\right) \geq 0$$
    8. Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ satifying the condition $$f(x y+f(z))=\frac{x f(y)+y f(x)}{2}+z$$ for all $x, y, z$ in $\mathbb{R}$.
    9. Let $a, b, c$ be positive numbers. Prove that $$\frac{a^{2}}{b}+\frac{b^{2}}{c}+\frac{c^{2}}{a} \geq \sqrt{a^{2}-a b+b^{2}}+\sqrt{b^{2}-b c+c^{2}}+\sqrt{c^{2}-c a+a^{2}}.$$
    10. Let $A B C$ be an acute triangle and the altitudes $A D, B E, C F$ meet at $H .$ On $D E$ choose a point $K$ such that $D K=D H$. On $D H,$ choose a point $I$ such that $\widehat{I K D}=90^{\circ} .$ Prove that the circle centered at $I$ with radius $I K$ touches the circle whose diameter is $B C$.
    11. Let $\left(u_{n}\right)$ be a sequence of positive numbers. Put $S_{n}=u_{1}^{3}+u_{2}^{3}+u_{3}^{3}+\ldots+u_{n}^{3}$ for $n=1,2, \ldots$. Assume that $$u_{n+1} \leq\left(\left(S_{n}-1\right) u_{n}+u_{n-1}\right) \frac{1}{S_{n+1}},$$ for any $n=2,3, \ldots$ Find $\displaystyle\lim_{n\to\infty} u_{n}$
    12. Let $p$ be a prime number and $m, n, q$ be natural numbers satisfying $2 \leq n \leq m$ and $(p, q)=1 .$ Prove that $C_{q p^{n}}^{n}$ is divisible by $p^{n+n+1}$ (The binomial coefficient $C_{n}^{k}$ is the number of ways of picking $k$ unordered elements from a set of $n$ elements).

    Issue 390

    1. Find all triples $(a, b, c)$ of positive integers such that $(a+b+c)^{2}-2 a+2 b$ is a perfect square.
    2. Given a triangle $A B C$ with $A B=A C$ and $\widehat{B A C}=80^{\circ} .$ Choose a point $I$ inside the triangle so that $\widehat{I A C}=10^{\circ}$, $\widehat{I C A}=20^{\circ} .$ Find the measure of the angle $\widehat{C B I}$.
    3. Let $G$ and $I$ be respectively the centroid and the incenter of a given triangle $A B C .$ Prove that if $A B^{2}-A C^{2}=2\left(I B^{2}-I C^{2}\right)$ then $G I$ is parallel to $B C$.
    4. Solve the equation $$\left(x^{2}+1\right)\left|x^{2}+2 x-1\right|+6 x\left(1-x^{2}\right)=\left(x^{2}+1\right)^{2}$$
    5. Let $\left(a_{n}\right)$ be a sequence given by $$a_{1}=1,\quad a_{n+1}=\sqrt{a_{n}\left(a_{n}+1\right)\left(a_{n}+2\right)\left(a_{n}+3\right)+2},\,\forall n \in \mathbb{N}^{*}.$$ Compare $\dfrac{1}{2}$ with the sum $$S=\frac{1}{a_{1}+2}+\frac{1}{a_{2}+2}+\frac{1}{a_{3}+2}+\ldots+\frac{1}{a_{2009}+2}.$$
    6. Let $a, b, c$ be positive real numbers such that $a+b+c=6 .$ Prove that $$\frac{a}{\sqrt{b^{3}+1}}+\frac{b}{\sqrt{c^{3}+1}}+\frac{c}{\sqrt{a^{3}+1}} \geq 2$$
    7. Find all triangles whose inradius equal 3 and the side lengths form the first three terms of an arithmetic progression with common difference $d$ distinct from $0 .$
    8. Let $A B C$ be a triangle with $A B=3 R$, $B C=R \sqrt{7}, C A=2 R .$ Let $M$ be an arbitrary point on the spherical surface $(C ; R) $. Find the least value of $M A+2 M B$.
    9. There are $294$ people in a meeting. Those who are acquainted shake hands with each other. Knowing that if $A$ shakes hands with $B$ then one of them shakes hands at most 6 times. What is the greatest number of possible handshakes?
    10. Given a positive integer $m$, find all functions $f: \mathbb{N} \rightarrow \mathbb{N}$ such that for every $x, y \in \mathbb{N}$ we have
      • If $f(x)=f(y)$ then $x=y$
      • $f(f(f(\ldots)))=x+y$. Here, $f$ appears $m$ times on the left hand side.
    11. Let $f(x)$ be a continuous function on the closed interval $[0 ; 1],$ and differentiable on the open interval $(0 ; 1)$ such that $f(0)=0$ $f(1)=1 .$ Prove that for two arbitrary real numbers $k_{1}, k_{2},$ there exist two distinct numbers $a, b$ in the open interval $(0 ; 1)$ such that $$\frac{k_{1}}{f(a)}+\frac{k_{2}}{f(b)}=k_{1}+k_{2}.$$
    12. Let $A B C$ be a triangle with orthocenter $H$. Prove that the common tangent, distinct from $A H,$ of the incircles of the triangles $A B H$ and $A C H$ passes through the midpoint of $B C$.
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    MOlympiad.NET: Mathematics and Youth Magazine Problems 2009
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