Mathematics and Youth Magazine Problems 2008


Issue 367

  1. Let $S$ be the sum of $100$ terms $$S=\frac{1}{1.1 .3}+\frac{1}{2.3 .5}+\frac{1}{3.5 .7}+\frac{1}{4.7 .9}+\ldots+\frac{1}{100.199 .201}.$$ Compare $S$ with $\dfrac{2}{3}$.
  2. Let $A B C$ be an isosceles triangle $(A B=A C)$ such that $\widehat{B A C}<90^{\circ} .$ Let $B D$ and $A H$ be the altitudes. Choose a point $K$ on $B D$ such that $B K=B A$ Find the measure of angle $HAK$.
  3. Given $0<b<a \leq 4$ and $2 a b \leq 3 a+4 b$. Find the maximum value of the expression $a^{2}+b^{2}$.
  4. The equation $$5 x^{6}-16 x^{4}-33 x^{3}-40 x^{2}+8=0$$ has two roots which are reciprocal. Find these roots.
  5. Let $A B C$ be a right triangle with right angle at $A$ and $A C>A B$. Let $O$ be the midpoint of $B C,$ and $I$ be the incenter of the triangle $A B C .$ Suppose that $\widehat{O I B}=90^{\circ},$ find the ratio between three edges of the triangle $A B C$.
  6. Find all pairs of positive integers $(x ; y)$ such that $$y^{x}-1=(y-1) !$$ where $y$ is a prime number.
  7. Let $a, b, c$ be non-negative real numbers. Prove the inequality $$\left(a^{2}+b^{2}+c^{2}\right)^{2} \geq 4(a-b)(b-c)(c-a)(a+b+c).$$ When does equality occur?
  8. Let $A B C$ be an isosceles triangle at vertex $A$ and $B C \leq A C$. Choose a point $M$ on $A B$ (but not the vertices $A$ or $B$) and a point $N$ on $A C$ such that $M N$ touch the incircle of the triangle $A B C$. Find the maximum value of the ratio $\dfrac{A M}{B M \cdot C N}$ when $M$ moves along the edge $A B$.
  9. Let $O$ be the circumcenter of a triangle $A B C .$ Choose a point $P,$ different from $B$ and $C$ on the line connecting $B C$. The circumcircle of $A B C$ meets $A P$ at a point $N$ and the circle whose diameter is $A P$ at a point $E$ ($N$, $E$ are both different from $A$). $B C$ and $A E$ intersect at $M .$ Prove that $M N$ always passes through a fixed point.
  10. A sequence $\left(u_{n}\right)(n=1,2, \ldots)$ is determined by the following recursive formula $$u_{1}=1,\quad u_{n+1}=\frac{16 u_{n}^{3}+27 u_{n}}{48 u_{n}^{2}+9}.$$ Find the largest integer which is smaller than the sum $S$ of $2008$ summands $$S=\frac{1}{4 u_{1}+3}+\frac{1}{4 u_{2}+3}+\ldots+\frac{1}{4 u_{2008}+3}$$
  11. Find the smallest value of the following expression $$P=\frac{1}{\cos ^{6} a}+\frac{1}{\cos ^{6} b}+\frac{1}{\cos ^{6} c}$$ where $a$, $b$ and $c$ form an arithmetic sequence whose common difference is $\dfrac{\pi}{3}$.
  12. Let $f(x)$ be a function defined on $[0 ; 1]$ such that the following properties hold
    • $f(1)=1$
    • $f(x)=\dfrac{1}{3}\left(f\left(\dfrac{x}{3}\right)+f\left(\dfrac{x+1}{3}\right)+f\left(\dfrac{x+2}{3}\right)\right)$ for all $x \in[0 ; 1]$
    • for every $\varepsilon$ positive but can be arbitrarily small, there exists a positive number $\delta_{\varepsilon}\left(\delta_{\varepsilon}\right.$ depends on $\varepsilon$) such that: For all $x, y \in[0 ; 1]$ such that $|x-y|<\delta_{\varepsilon},$ we have $|f(x)-f(y)|<\varepsilon$.
      Prove that $f(x)=1$ for all $x \in[0 ; 1]$.

    Issue 368

    1. Consider $n$ consecutive points $A_{1}, A_{2}, A_{3}, \ldots, A_{n}$ on the same line such that $$A_{1} A_{2}=A_{2} A_{3}=A_{3} A_{4}=\ldots=A_{n-1} A_{n}.$$ Find $n,$ given that there are exactly $2025$ segments on that line whose midpoints are one of these $n$ points.
    2. Let $A B C$ be a right triangle with right angle at $A$. On the halfplane divided by $B C$ which does not contain $A,$ choose the points $D$, $E$ such that $B D$ is orthogonal with $B A$ and $B D=B A$, $B E$ is orthogonal with $B C$ and $B E=B C$. Denote by $M$ the midpoint of $C E .$ Prove that $A$, $D$ and $M$ are colinear.
    3. Find all positive integers $x$, $y$ and $z$ such that $$2 x y-1=z(x-1)(y-1).$$
    4. Solve for $x$ $$4 x-x^{2}=3 \sqrt{4-3 \sqrt{10-3 x}}.$$
    5. Let $A B C$ be a right triangle with right angle at $A$ and let $A D$ be the angle bisector at $A .$ Denote by $M$ and $N$ the bases of the altitudes from $D$ onto $A B$ and $A C$ respectively. $B N$ meets $C M$ at $K$ and $A K$ meets $D M$ at $I$. Find the measure of angle $B I D$.
    6. Let $$f(x)=2009 x^{5}-x^{4}-x^{3}-x^{2}-2006 x+1 .$$ Prove that $f(n)$, $f(f(n))$, $f(f(f(n)))$ are pairwise coprime for any positive integer $n$.
    7. Find $a$, $b$ such that $\max _{0 \leq x \leq 16}|\sqrt{x}+a x+b|$ is smallest possible. Find this minimum value.
    8. The incircle of a triangle $A B C$ touches $B C$, $C A$ and $A B$ at $A^{\prime}$, $B^{\prime},$ and $C^{\prime}$ respectively. Prove that $$A B^{\prime 2}+B C^{\prime 2}+C A^{\prime 2} \geq A B^{\prime} . B^{\prime} C^{\prime}+B C^{\prime} \cdot C^{\prime} A^{\prime}+C A^{\prime} . A^{\prime} B^{\prime} \geq B^{\prime} C^{\prime 2}+C^{\prime} A^{\prime 2}+A^{\prime} B^{\prime 2}.$$ When does equality occur?
    9. Let $k$ be a positive integer. Write $k$ as a product of prime numbers (not necessarily distinct, for instance, $k=18=3.3 .2)$ and let $T(k)$ be the sum of all factors in the factorization above. Find the largest constant $C$ such that $T(k) \geq C \ln k$ for all positive integer $k$.
    10. Let $a, b, c$ be non-negative real numbers whose sum of squares equal $3 .$ Find the maximum value of the following expression $$P=a b^{2}+b c^{2}+c a^{2}-a b c$$
    11. Let $\left(x_{n}\right)(n=1,2, \ldots)$ be a sequence, determined by the following recursive formula $$x_{1}=\frac{1}{2},\quad x_{n+1}=x_{n}-x_{n}^{2}+x_{n}^{3}-x_{n}^{4}+\ldots+x_{n}^{2007}-x_{n}^{2008},\,\forall n \in \mathbb{N}^{*}.$$ Find the limit $\displaystyle \lim_{n \rightarrow+\infty} n x_{n}$.
    12. Let $A B C D$ be a tetrahedron whose altitudes are concurrent. Denote by $R$ the radius of its circumcircle; by $h_{1}$, $h_{2}$, $h_{3}$, $h_{4}$ the lengths of the altitudes corresponding to the vertices $A$, $B$, $C$, $D$ respectively; and by $R_{1}$, $R_{2},$ $R_{3}$, $R_{4}$ the circumcircles's radius of the opposite faces of the vertices $A$, $B$, $C$, $D$ respectively. Prove the following inequality $$\frac{1}{h_1+2 \sqrt{2} R_{1}}+\frac{1}{h_{2}+2 \sqrt{2} R_{2}}+\frac{1}{h_{3}+2 \sqrt{2} R_{3}}+\frac{1}{h_{4}+2 \sqrt{2} R_{4}} \geq \frac{1}{R}.$$ When does equality occur?

    Issue 369

    1. For each integer $n$ greater than $6,$ denote by $A_{n}$ the collection of integers which are less than $n$ and not less than $\dfrac{n}{2} .$ Find $n$ such that there are no perfect square in $A_{n}$.
    2. Let $A B C$ be an acute triangle with $\widehat{B A C}=60^{\circ} . E$ and $F$ are two points on $A C$ and $A B$ respectively such that $\widehat{E B C}=\widehat{F C B}=30^{\circ}$. Prove that $$B F=F E=E C \geq \dfrac{B C}{2}.$$
    3. Find four distinct integers $a, b, c, d$ in the set $\{10 ; 21 ; 37 ; 51\}$ such that $$a b+b c-a d=637.$$
    4. Solve for $x$ $$(x+3) \sqrt{(4-x)(12+x)}=28-x.$$
    5. Let $A B C$ be a triangle with $\widehat{B A C} \neq 45^{\circ}$ and $\widehat{A I O}=90^{\circ}$ where $(O)$ and $(I)$ are its circumcircle and incircle, respectively. Choose a point $D$ on the ray $B C$ such that $B D=A B+A C$. The tangent line through $D$ touches $(O)$ at $E$. The tangent line through $B$ of $(O)$ meets $D E$ at $F$; $C F$ meets $(O)$ at another point denoted by $K$. Let $G$ be the centroid of the triangle $A B C$. Prove that $I G$ is parallel to $E K$.
    6. Let $a_{1}, a_{2}, \ldots, a_{n}$ be positive integers such that the following two properties hold
      • $a_{i}<2008$ for all $i=1,2, \ldots, n$
      • The greatest common divisor of any pair of numbers is greater than $2008$.
      Prove that $\displaystyle\sum_{i=1}^{n} \frac{1}{a_{i}}<2$.
    7. Prove the following inequality $$(3 a+2 b+c)\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right) \leq \frac{45}{2}$$ where $a, b,$ and $c$ are in the interval $[1 ; 2]$ When does equality occur?
    8. Given a triangle $A B C$ with circumcircle $(O)$ and three sides $B C=a$, $C A=b$, $A B^{\prime}=c .$ Denote by $A_{1}$, $B_{1},$ and $C_{1}$ the midpoints of $B C$, $C A,$ and $A B$ respectively; and $A_{2}$, $B_{2}$, $C_{2}$ are the midpoint of the arcs $\widehat{B C}$ (which does not contain $A$), $\widehat{C A}$ (which does not contains $B$), and $\widehat{A B}$ (which does not contain $C$). Draw the circles $\left(Q_{1}\right),\left(O_{2}\right),$ and $\left(O_{3}\right)$ whose diameters are $A_{1} A_{2}$, $B_{1} B_{2},$ and $C_{1} C_{2}$ respectively. Prove the inequality $$\mathcal{P}_{A/\left(O_{1}\right)}+\mathcal{P}_{B /\left(O_{2}\right)}+\mathcal{P}_{C /\left(O_{3}\right)} \geq \frac{(a+b+c)^{2}}{3}.$$ When does equality occur? 
    9. Find all positive integers $x, y, z, n$ such that $$x !+y !+z !=5 . n !$$ where $k !=1 \times 2 \times \ldots \times k$.
    10. Let $a$ and $b$ be two real numbers in the open interval $(0 ; 4) .$ A sequence $\left(a_{n}\right),$ $(n=0,1, \ldots)$ is constructed by the following recursive formula $$a_{0}=a,\, a_{1}=b,\quad a_{n+2}=\frac{2\left(a_{n+1}+a_{n}\right)}{\sqrt{a_{n+1}}+\sqrt{a_{n}}}.$$ Prove that the sequence $\left(a_{n}\right)$ (n=0,1, \ldots)$ converges and find its limit.
    11. Find all functions $f: \mathbb{R}^{+} \rightarrow \mathbb{R}^{+}$ such that $$f(f(x))+f(x)=\left(26^{3^{2008}}+\left(26^{32008}\right)^{2}\right) x.$$
    12. Let $S$ denote the surface area of a given tetrahedron. Prove that the sum of areas of the six angle-bisectors of this tetrahedron does not exceed $\dfrac{\sqrt{6}}{2} S$.

    Issue 370

    1. Prove the inequality $$\frac{3}{2}+\frac{7}{4}+\frac{11}{8}+\frac{15}{16}+\ldots+\frac{4 n-1}{2^{n}}<7$$ where $n$ is an arbitrary positive integer.
    2. Let $A B C$ be a right triangle with right angle at $A$. Suppose $A B=5cm$ and $I C=6cm$ where $I$ is the incenter of $A B C$. Determine the length of $B C$.
    3. Find all positive integers $x, y, z$ such that the following equality holds $$5 x y z=x+5 y+7 z+10.$$
    4. Solve for $x$ $$x^{4}-2 x^{2}-16 x+1=0.$$
    5. Let $A B C$ be an acute triangle whose altitudes $A A^{\prime}$, $B B^{\prime}$, $C C^{\prime}$ meet at $H$ Denote by $A_{1}$, $B_{1},$ and $C_{1}$ the othocenters of the triangles $A B^{\prime} C^{\prime}$, $B C^{\prime} A^{\prime}$ and $C A^{\prime} B^{\prime}$ respectively. Suppose that $H$ is the incenter of the triangle $A_{1} B_{1} C_{1}$. Prove that $A B C$ is an equilateral triangle.
    6. Let $A B C$ be an isosceles triangle with $A B=B C=a$ and $\widehat{A B C}=140^{\circ} .$ Let $A N$ and $A H$ respectively be the anglebisector and the altitude from $A$. Prove that $2 B H \cdot C N=a^{2}$
    7. Find the minimum value of the function $$f(x)=\left(32 x^{5}-40 x^{3}+10 x-1\right)^{2006}+\left(16 x^{3}-12 x+\sqrt{5}-1\right)^{2008}.$$
    8. Prove that the following equation $$A \cdot a^{x}+B \cdot b^{x}=A+B$$ where $a>1$, $0<b<1$, $A, B \in \mathbb{R}$ has at most two solutions.
    9. Let $a_{1}=\dfrac{1}{2}$ and for each $n$ greater than $1,$ let $$a_{n}=\frac{1}{d_{1}+1}+\frac{1}{d_{2}+1}+\ldots+\frac{1}{d_{k}+1}$$ where $d_{1}, d_{2}, \ldots, d_{k}$ is the collection of all distinct positive divisors of $n .$ Prove the inequality $$n-\ln n<a_{1}+a_{2}+\ldots+a_{n}<n.$$
    10. Given $n$ numbers $a_{1}, a_{2}, \ldots, a_{n}$ in $[-1 ; 2]$ such that their total sum is 0. Let $U_{k}=\dfrac{a_{k} \sqrt{4 k-1}}{(4 k-3)(4 k+1)}$ for $k=1,2, \ldots, n .$ Prove that $$\left|U_{1}+U_{2}+\ldots+U_{n}\right|<\frac{\sqrt{n}}{2}$$
    11. Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that $$f(x+y)=x^{2} f\left(\frac{1}{x}\right)+y^{2} f\left(\frac{1}{y}\right),\,\forall x, y \in \mathbb{R}^{*}.$$
    12. Let $P$ be a point on the insphere of a tetrahedron $A B C D$ and let $G_{a}$, $G_{b}$, $G_{c}$, $G_{d}$ be the centroids of the tetrahedra $P B C D$, $PCDA$, $PDAB$, $PABC$, respectively. Prove that $A G_{a}$, $B G_{b}$, $C G_{c}$ and $D G_{d}$ pass through a common point; and find the orbit of this common point when $P$ moves on the insphere of the given tetrahedron $A B C D$.

    Issue 371

    1. Which number is greater? $A=\dfrac{1}{2006}$ or $$B =\frac{1}{2008}+\left(\frac{1}{2008}+\frac{1}{2008^{2}}\right)^{2}+\ldots +\left(\frac{1}{2008}+\frac{1}{2008^{2}}+\ldots+\frac{1}{2008^{2007}}\right)^{2007}.$$
    2. In a triangle $A B C$ with altitude $A D,$ one has $A D=D C=3 B D$. Let $O$ and $H$ be the circumcenter and the orthocenter, respectively. Prove that $\dfrac{O H}{B C}=\dfrac{1}{4}$.
    3. Find all pairs of natural numbers $x$ and $y$ such that $$x^{3}=y^{3}+2\left(x^{2}+y^{2}\right)+3 x y+17.$$
    4. Let $a, b, c$ be positive real numbers. Prove the inequality $$\frac{a^{2}-b^{2}}{\sqrt{b+c}}+\frac{b^{2}-c^{2}}{\sqrt{c+a}}+\frac{c^{2}-a^{2}}{\sqrt{a+b}} \geq 0.$$ When does equality occur?
    5. A circle $\left(S_{1}\right)$ passing through the vertices $A$ and $B$ of a triangle $A B C$ meets $B C$ at another point $D .$ Another circle, $\left(S_{2}\right),$ passing through $B$ and $C$ meets $A B$ at another point $E$ and meets $\left(S_{1}\right)$ at $F$. Prove that if all four points $A$, $C$, $D$ and $E$ lie on the same circle with center at $O$, then $\widehat{B F O}=90^{\circ}$.
    6. Solve the system of equations $$\begin{cases}\dfrac{x}{a-30}+\dfrac{y}{a-4}+\dfrac{z}{a-14}+\dfrac{t}{a-10} &=1 \\ \dfrac{x}{b-30}+\dfrac{y}{b-4}+\dfrac{z}{b-14}+\dfrac{t}{b-10} &=1 \\ \dfrac{x}{c-30}+\dfrac{y}{c-4}+\dfrac{z}{c-14}+\dfrac{t}{c-10} &=1 \\ \dfrac{x}{d-30}+\dfrac{y}{d-4}+\dfrac{z}{d-14}+\dfrac{1}{d-10} &=1\end{cases}$$ where $a, b, c, d$ are distinct numbers, none of which belong to the set $\{4 ; 10 ; 14 ; 30\} .$
    7. Let $a, b, c$ be positive numbers. Prove that $$a^{b+c}+b^{c+a}+c^{a+b} \geq 1$$
    8. Choose six points $D$, $E$, $F$, $G$, $H$, $K$ in that order, on a circle with radius $R$ and center at $O$ such that $D E=F G=H K=R$. $K D$ and $E F$ meet at $A$, $E F$ and $G H$ meet at $B$ and $G H$ meets $K D$ at $C$. Prove that $$OA \cdot B C=O B \cdot C A=O C \cdot A B.$$
    9. Let $A B C D$ be a cyclic quadrilateral, inscribed in a circle centered at $I .$ Prove the following inequality $$(A I+D I)^{2}+(B I+C I)^{2} \leq(A B+C D)^{2}$$ and determine when equality occurs.
    10. Consider the quadratic equation $x^{2}-a x-1=0$ where $a$ is a positive integer. Let $\alpha$ be a positive root of this equation and construct a sequence $\left(x_{n}\right)$ by the following recursive rule $$x_{0}=a, x_{n+1}=\left[\alpha x_{n}\right], n=0,1,2, \ldots$$ Prove that there exists infinitely many integer $n$ such that $x_{n}$ is a multiple of $a$. 
    11. Given a function $f: \mathrm{N} \rightarrow \mathrm{N}$ such that for all $n \in \mathbb{N} $ $$(f(2 n)+f(2 n+1)+1)(f(2 n+1)-f(2 n)-1)=3(1+2 f(n)),\quad f(2 n) \geq f(n).$$ Denote $M=\{m \in f(\mathrm{N}): m \leq 2008\} .$ Find the number of elements of $M$.
    12. For what kind of triangle $A B C$ that the following relation among its sides and its angles holds $$\frac{b c}{b+c}(1+\cos A)+\frac{c a}{c+a}(1+\cos B)+\frac{a b}{a+b}(1+\cos C) \\ = \frac{3}{16}(a+b+c)^{2}+\cos ^{2} A+\cos ^{2} B+\cos ^{2} C.$$

    Issue 372

    1. Let $$A=\frac{1}{1^{2}}+\frac{1}{2^{3}}+\frac{1}{3^{4}}+\ldots+\frac{1}{2007^{2008}}.$$ Prove that $A$ is not an integer.
    2. Let $P(x)=x^{3}-7 x^{2}+14 x-8 .$ Prove that for every natural number $n,$ there exists a triple of distinct intergers $a_{1}$, $a_{2}$, $a_{3}$ such that the following two conditions are satisfied
      • $\left|a_{i}-a_{j}\right|<5, \forall i, j \in\{1,2,3\}$
      •  $P\left(a_{i}\right) \neq 0$ and $5^{n} \mid P\left(a_{i}\right)$ for all $i \in\{1,2,3\}$
    3. Find all pairs of intergers $x$, $y$ such that $$\sqrt[n]{x+\sqrt[n]{x+\ldots+\sqrt[n]{x}}}=y$$ ($m$ times) where $m$, $n$ are positive intergers which are greater than $2$.
    4. Given $x>2$. Prove the inequality $$\frac{x}{2}+\frac{8 x^{3}}{(x-2)(x+2)^{2}}>9.$$
    5. A pentagon $A B C D E$ is inscribed in a circle with center at $O$ and radius $R$ such that $A B=C D=E A=R$. Let $M$, $N$ be respectively the midpoints of $B C$ and $D E$ Prove that $A M N$ is an equilateral triangle.
    6. Do there exist two distinct positive integers $a$, $b$ such that $b^{n}+n$ is $a$ multiple of $a^{n}+n$ for every postive integer $n ?$.
    7. Let $S$ be the set of all pairs of real numbers $(\alpha, \beta)$ such that the equation $$x^{3}-6 x^{2}+\alpha x-\beta=0$$ has three real roots (not necessarily distinct) and they are all greater than $1$. Find the maximum value of $T=8 \alpha-3 \beta$ for $(\alpha, \beta) \in S$.
    8. Let $A B C$ be an acute triangle and denote by $Q$ the center of its Euler's circle. The circumcircle of $A B C,$ which has radius $R,$ meets $A Q, B Q,$ and $C Q$ respectively at $M$, $N$ and $P$ Prove the inequality $$\frac{1}{Q M}+\frac{1}{Q N}+\frac{1}{Q P} \geq \frac{3}{R}.$$
    9. Find the interger part of $$A=\sqrt[n]{1-\frac{x}{2008}}+\sqrt[m]{1+\frac{x}{2008}}$$ where $x$ is a real number in $[-2008 ; 2008]$, and $m$, $n$ are natural numbers, $m \geq n \geq 2$.
    10. Let $S$ be denote the set of all $n$-tuples $(n>1)$ of real numbers $\left(a_{1}, a_{2}, \ldots, a_{n}\right)$ such that $$3\sum_{i=1}^{n} a_i^{2}=502.$$ a) Prove that $\displaystyle \min _{1 \leq \leq j \leq n}\left|a_{j}-a_{i}\right| \leq \sqrt{\frac{2008}{n\left(n^{2}-1\right)}}$.
      b) Give an example of an $n$ -tuple $\left(a_{1}, a_{2}, \ldots, a_{n}\right)$ such that the above condition holds and for which there is an equality in a).
    11. A function $f(x),$ whose domain is the interval $[1 ;+\infty),$ has the following two properties $$f(1)=\frac{1}{2008},\quad f(x)+2007(f(x+1))^{2}=f(x+1),\,\forall x \in[1 ;+\infty).$$ Find the limit $$\lim_{n \rightarrow+\infty}\left(\frac{f(2)}{f(1)}+\frac{f(3)}{f(2)}+\ldots+\frac{f(n+1)}{f(n)}\right).$$
    12. The angle-bisectors $A A_{1}$, $B B_{1}$, $C C_{1}$ of a triangle $A B C$ with perimeter $p$ meet $B_{1} C_{1}$, $C_{1} A_{1},$ and $A_{1} B_{1}$ respectively at $A_{2}$, $B_{2},$ and $C_{2}$. The line through $A_{2}$ and parallel to $B C$ meets $A B$, $A C$ at $A_{3}$, $A_{4}$. Construct the points $B_{3}$, $B_{4}$ and $C_{3}$, $C_{4}$ in a similar way. Prove the inequality $$A B_{4}+B C_{4}+C A_{4}+B A_{3}+C B_{3}+A C_{3} \leq p.$$ When does equality holds?

    Issue 373

    1. Write the numbers $1,2,3, \ldots, 2007$ in an arbitrary order and let $A$ be the resulting number. Can $A+2008^{2007}+2009$ be a perfect square?.
    2. Consider the following two polynomials $$f(x)=(x-2)^{2008}+(2 x-3)^{2007}+2006 x$$ and $$g(y)=y^{2009}-2007 y^{2008}+2005 y^{2007}.$$ Let $s$ be denote the sum of all the coefficients of $f(x)$ (after expansion). Find $s$, and the value of $g(s)$.
    3. Find all positive integer solutions of the following system of two equations $$\begin{cases} x+y+z &= 15 \\ x^{3}+y^{3}+z^{3} &= 495\end{cases}.$$
    4. Let $a, b, c$ be non-negative real numbers such that $a^{2}+b^{2}+c^{2}=1 .$ Find the maximum value of the expression $$(a+b+c)^{3}+a(2 b c-1)+b(2 a c-1)+c(2 a b-1).$$
    5. Given a triangle $A B C$ where $\widehat{A B C}$ is not a right angle. Let $A H$ and $A M$ denote, the altitude and the median throught vertex $A$. Choose a point $E$ on the ray $A B$ and $F$ on the ray $A C$ such that $M E$ $=M F=M A .$ Let $K$ be reflection point of $H$ over $M .$ Prove that the four points $E$, $M$, $K$ and $F$ lie on a single circle.
    6. Solve for $x$ $$\sqrt{x+\sqrt{x^{2}-1}}=\frac{9 \sqrt{2}}{4}(x-1) \sqrt{x-1}.$$
    7. Prove that in any acute triangle $A B C$, the following inequality holds $$\frac{\tan A}{\tan B}+\frac{\tan B}{\tan C}+\frac{\tan C}{\tan A} \geq \frac{\sin 2 A}{\sin 2 B}+\frac{\sin 2 B}{\sin 2 C}+\frac{\sin 2 C}{\sin 2 A}$$
    8. The incircle of a triangle $A B C$ meets $B C$, $C A$ and $A B$ respectively at $A_{1}$, $B_{1}$, $C_{1}$. Let $p$, $S$, $R$ be respectively, half of the perimeter, the area and the circumradius of $A B C$. Let $p_{1}$ be half of the perimeter of $A_{1} B_{1} C_{1}$. Prove the inequality $$p_{1}^{2} \leq \frac{p S}{2 R}.$$ When does equality occur?
    9. Let $A(A \subset \mathbb{N})$ be a non-empty set satisfying the condition: If $a \in A$ then $4 a$ and $[\sqrt{a}]$ are also in $A([x]$ is the integer part of $x$). Prove that $A=\mathbb{N}$.
    10. Let $a$ be a natural number which is greater than $3$ and consider the sequence $\left(u_{n}\right)$ $(n=1,2, \ldots)$ defined inductively by $u_{1}=a$ and $$u_{n+1}=u_{n}-\left[\frac{u_{n}}{2}\right]+1,\,\forall n=1,2, \ldots.$$ Prove that there exists $k \in \mathbb{N}^{*}$ such that $u_{n}=u_{k}$ for all $n \geq k$.
    11. Find all polynomials with real coefficients $P(x)$, $Q(x)$ and $R(x)$ such that $$\sqrt{P(x)}-\sqrt{Q(x)}=R(x),\,\forall x.$$
    12. Let $A B C D$ be a tetrahedron with the centroid $G$ and the circumradius $R$. Prove that $$G A+G B+G C+G D+4 R \geq \frac{2}{\sqrt{6}}(A B+A C+A D+B C+C D+D B).$$

    Issue 374

    1. Find all triple of natural numbers $a, b, c$ less than $20$ such that $$a(a+1)+b(b+1)=c(c+1)$$ where $a$ is a prime number and $b$ is a multiple of $3 .$
    2. Let $f(n)=\left(n^{2}+n+1\right)^{2}+1,$ where $n$ is a positive integer and let $$P_{n}=\frac{f(1) \cdot f(3) \cdot f(5) \ldots f(2 n-1)}{f(2) \cdot f(4) \cdot f(6) \ldots f(2 n)}.$$ Prove the inequality $$P_{1}+P_{2}+\ldots+P_{n}<\frac{1}{2}.$$
    3. Find the maximum value of the expression $$T=\frac{(y+z)^{2}}{y^{2}+z^{2}}-\frac{(x+z)^{2}}{x^{2}+z^{2}},$$ where $x$, $y$, $z$ are real numbers such that $x>y$, $z>0$ and $z^{2} \geq x y$.
    4. Solve for $x$ $$\sqrt[3]{14-x^{3}}+x=2\left(1+\sqrt{x^{2}-2 x-1}\right).$$
    5. An acute triangle $A B C$ is inscribed in a fixed circle with center at $O$. Let $A I$, $B D$ and $C E$ denote the altitudes through $A$, $B$ and $C$ respectively. Prove that the perimeter of the triangle $I D E$ does not change when $A, B,$ and $C$ move on the circle $(O)$ such that the area of the triangle $A B C$ is always equal to $a^{2}$.
    6. Solve the system of equations $$\begin{cases} \sqrt{x^{2}+91} &=\sqrt{y-2}+y^{2} \\ \sqrt{y^{2}+91} &=\sqrt{x-2}+x^{2}\end{cases}.$$
    7. Let $a, b, c$ be real numbers such that $4(a+b+c)-9=0 .$ Find the maximum value of the sum $$S=\left(a+\sqrt{a^{2}+1}\right)^{b}\left(b+\sqrt{b^{2}+1}\right)^{c}\left(c+\sqrt{c^{2}+1}\right)^{a}.$$
    8. Let $I$ and $O$ denote respectively the incenter and the circumcenter of a triangle $A B C .$ Given that $\widehat{A I O}=90^{\circ},$ prove that the area of the triangle $A B C$ is less than $\dfrac{3 \sqrt{3}}{4} A I^{2}$.
    9. Let $a, b, n$ be positive integers, $b>1$ and $a$ is a multiple of $b^{n}-1 .$ Rewritten $a$ to the base $b,$ prove that the resulting contains at least $n$ non-zero digits.
    10. Let $x, y, z$ be real numbers such that $0<z \leq y \leq x \leq 8$ and $$3 x+4 y \geq \max \left\{x y ; \frac{1}{2} x y z-8 z\right\}.$$ Find the maximum value of $$A=x^{5}+y^{5}+z^{5}.$$
    11. Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that $$f\left(f(x)+y^{2}\right)=f^{2}(x)-f(x) f(y)+x y+x.$$
    12. Let $K$ denote the intersection of the two diagonals of a quadrilateral $A B C D$ where $\widehat{A B C}=\widehat{A D C}=90^{\circ} ;$ and $A C=A B+A D .$ Prove that the radii of the inscribed of the triangles $A B K$ and $A D K$ are equal.

    Issue 375

    1. Find a natural number $x$ and two decimal digits $y$, $z$ such that $$\left(5.10^{n}-2\right) x=3 . \overline{y \ldots y z}$$ for any natural number $n>1,$ where $\overline{y \ldots y z}$ (in the decimal system) contains $n-1$ digits $y$.
    2. Prove that for any $x$ and $y$  $$\frac{|x|}{2008+|x|}+\frac{|y|}{2008+|y|} \geq \frac{|x-y|}{2008+|x-y|}$$
    3. Consider an integer $n>2008$ such that both $2 n-4015$ and $3 n-6023$ are perfect squares. Find the remainder of the division of $n$ by 40
    4. Solve the quation $$\sqrt{x-\frac{1}{x}}-\sqrt{1-\frac{1}{x}}=\frac{x-1}{x}.$$
    5. Let $A B C$ be an isosceles triangle with the apex angle at $A$ and $\widehat{B A C}=150^{\circ}$. Construct the triangles $A M B$ and $A N C$ such that the rays $A M$ and $A N$ lie in the angle $B A C$ and $\widehat{A B M}=\widehat{A C N}=90^{\circ}$, $\widehat{M A B}=30^{\circ}$, $\widehat{N A C}=60^{\circ} .$ Let $D$ be a point on $M N$ such that $N D=3 M D$. $B D$ intersects with $A M$ and $A N$ at $K$ and $E,$ respectively. $B C$ and $A N$ meets at $F$. Prove that
      a) $NCE$ is an isosceles triangle;
      b) $K F$ and $C D$ are parallel.
    6. Find all pairs of integers $m$ and $n$, both greater than $1$ such that the following equality $$a^{m+n}+b^{m+n}+c^{m+n}=\frac{a^{m}+b^{m}+c^{m}}{m} \cdot \frac{a^{n}+b^{n}+c^{n}}{n}$$ is true for all real numbers $a$, $b$, $c$ satisfying $a+b+c=0$.
    7. Let $k$ be a positive integer and let $a$, $b$, $c$ be positive real numbers such that $a b c \leq 1$. Prove the equality $$\frac{a}{b^{k}}+\frac{b}{c^{k}}+\frac{c}{a^{k}} \geq a+b+c.$$
    8. Let $C$ be a point on a fixed circle whose diameter is $A B=2 R$ ($C$ is different from $A$ and $B$). The incircle of $A B C$ touches $A B$ and $A C$ at $M$ and $N$, respectively. Find the maximum value of the length of $M N$ when $C$ moves on the given fixed circle.
    9. Let $\left(x_{n}\right)$ $(n=0.1,2, \ldots)$ be a sequence such that $$x_{0}=2,\quad x_{n+1}=\frac{2 x_{n}+1}{x_{n}+2},\,\forall n=0,1,2, \ldots.$$ Determine $\displaystyle \left[\sum_{k=1}^{n} x_{k}\right]$ where $[x]$ denote the largest integer not exceeding $x$.
    10. Prove that if $a$, $b$, $c$ are positive numbers whose product $a b c=1,$ then $$\frac{a}{\sqrt{8 c^{3}+1}}+\frac{b}{\sqrt{8 a^{3}+1}}+\frac{c}{\sqrt{8 b^{3}+1}} \geq 1.$$
    11. Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a function such that $f(0)=0$ and $\dfrac{f(t)}{t}$ is a monotonic function on $\mathbb{R} \backslash\{0\}$. Prove that $$x \cdot f\left(y^{2}-z x\right)+y \cdot f\left(z^{2}-x y\right)+z \cdot f\left(x^{2}-y z\right) \geq 0$$ for all positive numbers $x$, $y$ and $z$.
    12. Let $A_{1} A_{2} A_{3} A_{4}$ be a tetrahedron. Denote by $B_{i}$ $(i=1,2,3,4)$ the feet of the altitude from a given point $M$ onto $A_{i} A_{i+1}$ (where we consider $A_{5}$ as identical to $A_{1}$). Find the smallest value of $\displaystyle\sum_{1 \leq i \leq 4} A_{i} A_{i+1} . A_{i} B_{i}$

    Issue 376

    1. Write the numbers $8^{2008}$ and $125^{2008}$ consecutively. What is the number of decimal digits of the resulting number?
    2. Find a rational number $\dfrac{a}{b}$ such that the following three conditions are satisfied
      • $-\dfrac{1}{2}<\dfrac{a}{b}<-\dfrac{2}{5}$
      • $11 a+5 b=26$
      • $200<|a|+|b|<230$
    3. Find all non-zero natural numbers $n$ such that the number $A=\dfrac{1.3 .5 .7 \ldots(2 n-1)}{n^{n}}$ is an integer, here the numerator of $A$ is the product of the first $n$ odd numbers.
    4. Prove that $$\frac{a}{b}+\frac{b}{c}+\frac{c}{a} \geq \frac{3}{2}(a+b+c-1)$$ where $a, b, c$ are positive real numbers such that $a b c=1 .$ When does equality hold? $?$
    5. In a right triangle $A B C$ with right angle at $A,$ the altitude $A H,$ the median $B M,$ and the angle-bisector $C D$ meet at a common point. Determine the ratio $\dfrac{A B}{A C}$
    6. Solve for $x$ $$\sqrt[3]{x+6}+\sqrt{x-1}=x^{2}-1$$
    7. Let $S$ denote the area of a given triangle $A B C,$ and denote $B C=a$, $C A=b$, $A B=c$. Prove the inequality $$a^{2} b^{2}+b^{2} c^{2}+c^{2} a^{2} \geq 16 S^{2}+\frac{1}{2} a^{2}(b-c)^{2}+\frac{1}{2} b^{2}(c-a)^{2}+\frac{1}{2} c^{2}(a-b)^{2}.$$ When does equality hold?
    8. Given a triangle $A B C$ with three sides $B C=a$, $A C=b$, $A B=c$ such that $a+c=2 b$ let $h_{a}$, $h_{c}$ be the altitudes from $A$ and $C$ respectively; and let $r_{a}$, $r_{c}$ denote the $A$-exradius and $C$-exradius respectively. Prove that $$\frac{1}{r_{a}}+\frac{1}{r_{c}}=\frac{1}{h_{a}}+\frac{1}{h_{c}}.$$
    9. The positive real numbers $a$, $b$, $c$, $x$, $y$ and $z$ are such that $$\begin{cases} c y+b z &=a \\ a z+c x &=b \\ b x+a y &=c\end{cases}.$$ Find the smallest possible value of the expression $$P=\frac{x^{2}}{1+x}+\frac{y^{2}}{1+y}+\frac{z^{2}}{1+z}.$$
    10. Let $f$ be a continuous function on $\mathbb{R}$ such that $f(2010)=2009$ and $f(x) \cdot f_{4}(x)=1$ for all $x \in \mathbb{R}$ (where $\left.f_{4}(x)=f(f f(f(x)))\right)$. Determine the value of $f(2008)$.
    11. Let $u_{1}, u_{2}, \ldots, u_{n}$ $(n>2)$ be a sequence of positive real numbers such that
      • $\dfrac{1004}{k}=u_{1} \geq u_{2} \geq \ldots \geq u_{n} \quad$ for some positive integer $k$;
      • $u_{1}+u_{2}+\ldots+u_{n}=2008$. Show that it is possible to select $k$ elements from the set $\left(u_{n}\right)$ such that in this collection of $k$ numbers, the smallest one is at least half of the largest.
    12. Consider the circle $(O)$ and three colinear points $X$, $Y$, $H$ that are not on this circle such that $\overline{H X} \cdot \overline{H Y} \neq \mathscr{P}_{H/(O)} .$ A straight line $d$ through $H$ meets $(O)$ at two points $M$ and $N$. $M X$ and $N Y$ intersect with $(O)$ again at $P$ and $Q$ respectively. Show that as the line $d$ through $H$ varies, the line connecting $P$ and $Q$ always passes through a fixed point.

    Issue 377

    1. Let $$A=\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\ldots+\frac{1}{2007}+\frac{1}{2008},$$ $$B=\frac{2007}{1}+\frac{2006}{2}+\frac{2005}{3}+\ldots+\frac{2}{2006}+\frac{1}{2007}.$$ Determine $\dfrac{B}{A}$.
    2. Let $A B C$ be a right isosceles triangle with right angle at $A$. $M$ is an arbitrary point on the side $B C$ (M differs from $B$, $C$ as well as the midpoint of $B C$). The altitudes from $B$ and $C$ onto $A M$ meet $A M$ at $H$ and $K$, respectively. The line through $C$ and parallel to $A M$ meets $B H$ at $N$, $A N$ meets $C K$ at $P$, $B P$ intersects with $A M$ at $I$ Prove that $I B=I P$.
    3. Let $a, b, c, d$ and $e$ be natural numbers such that $$a^{4}+b^{4}+c^{4}+d^{4}+e^{4}=2009^{2008}.$$ Prove that abcde is a multiple of $10^{4}$.
    4. Let $a, b, c$ be positive real numbers such that $a \geq b \geq c .$ Prove the inequality $$a^{2} b(a-b)+b^{2} c(b-c)+c^{2} a(c-a) \geq 0.$$
    5. Let $A M$ and $B N$ be two tangent lines from two points $A$, $B$ ($A$ differs from $B$) outside the circle $(O)$ ($M$, $N$ are on the circle, $M$ and $N$ are different). Prove that if $A M=B N$, then the line $M N$ is either parallel to $A B$ or passes through its midpoint.
    6. A number is said to be a beautiful number if it is a composite number and but it is not a multiple of either $2$, $3$ or $5$ (for example, the three smallest beautiful numbers are $49,77$ and $91$). How many beautiful numbers which are less than $1000 ?$
    7. Let $D$ and $E$ be two points on the side $B C$ of a triangle $A B C$ such that $\dfrac{B D}{C D}=2\dfrac{C E}{B E}$. The circumcircle of $A D E$ meets $A B$ and $A C$ at $M$ and $N,$ respectively. Prove that regardless of the positions of the points $D$ and $E$ on $B C,$ the centroid of the triangle $A M N$ lies on a fixed line.
    8. Find the smallest real number $k$ such that the following inequality holds for all nonnegative real numbers $a, b, c$ $$\frac{a+b+c}{3} \leq \sqrt[3]{a b c}+k \cdot \max \{|a-b|, b-c|,| c-a \mid\}.$$
    9. Determine all triple of real numbers $x, y, z$ such that $$x^{6}+y^{6}+z^{6}-6\left(x^{4}+y^{4}+z^{4}\right)+10\left(x^{2}+y^{2}+z^{2}\right) - 2\left(x^{3} y+y^{3} z+z^{3} x\right)+6(x y+y z+z x)=0.$$
    10. Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that $$f\left(x^{3}-y\right)+2 y\left(3 f^{2}(x)+y^{2}\right)=f(y+f(x)),\, \forall x, y \in \mathbb{R}.$$
    11. Consider the sequence $\left(u_{n}\right)$ $(n=1,2,\ldots)$ given by the following recursive formula $$u_{1}=u_{2}=1,\quad u_{n+1}=4 u_{n}-5 u_{n-1},\,\forall n \geq 2.$$ Prove that $\displaystyle \lim_{n \rightarrow+\infty}\left(\frac{u_{n}}{a^{n}}\right)=0$ for all real number $a>\sqrt{5}$.
    12. Choose three points $A_{1}$, $B_{1}$, $C_{1}$ on the sides of a triangle $A B C$, $A_{1} \in B C$, $B_{1} \in A C$, $C_{1} \in A B$ such that $A A_{1}$, $B B_{1}$, $C C_{1}$ meet at a common point. Again, choose three points $A_{2}$, $B_{2}$, $C_{2}$ on the sides of the triangle $A_{1} B_{1} C_{1}$, $A_{2} \in B_{1} C_{1}$, $B_{2} \in A_{1} C_{1}$, $C_{2} \in A_{1} B_{1}$. Prove that the three lines $A A_{2}$, $B B_{2}$, $C C_{2}$ meet at a common point if and only if so do $A_{1} A_{2}$, $B_{1} B_{2}$, $C_{1} C_{2}$.

    Issue 378

    1. Find all natural numbers $a$ such that both $a+593$ and $a-159$ are perfect squares.
    2. Let $A B C$ be a right triangle, with right angle at $A$ and $\widehat{A C B}=15^{\circ}$. Let $B C=a$, $A C=b$, $A B=c .$ Prove that $a^{2}=4 b c$.
    3. Find all intergers $x, y, z, t$ such that $$x^{2008}+y^{2008}+z^{2008}=2007 . t^{2008}.$$
    4. Prove the inequality $$\left(\frac{4}{a^{2}+b^{2}}+1\right)\left(\frac{4}{b^{2}+c^{2}}+1\right)\left(\frac{4}{c^{2}+a^{2}}+1\right) \geq 3(a+b+c)^{2}$$ where $a, b, c$ are positive numbers such that $a^{2}+b^{2}+c^{2}=3$.
    5. Let $A B C$ be an acute triangle. Choose a point $D,$ different from $B$ and $C,$ on the side $B C .$ Prove that the vertex $A$ and the centers of the circumcircles of the triangles $A B D$, $A C D$ and $A B C$ lie on the same circle.
    6. Find the coefficient of $x^{2}$ in the expansion of $$\left(\left(\ldots\left((x-2)^{2}-2\right)^{2}-\ldots-2\right)^{2}-2\right)^{2}$$ given that the number 2 occurs 1004 times in the expression above and there are 2008 round brackets.
    7. Find the smallest value of the following expression $$\frac{\sqrt{a_{1}+2008}+\sqrt{a_{2}+2008}+\ldots+\sqrt{a_{n}+2008}}{\sqrt{a_{1}}+\sqrt{a_{2}}+\ldots+\sqrt{a_{n}}}$$ where $n$ is a given positive natural number and $a_{1}, a_{2}, \ldots, a_{n}$ are non-negative real numbers such that $a_{1}+a_{2}+\ldots+a_{n}=n$
    8. Let $(O)$ be a circle centered at $O$ and fixed diameter $A B .$ Let $\Delta$ be a straight line which touches $(O)$ at $A .$ Choose a point $M$ on the circle $(O),$ different from $A$ and $B .$ The tangent line with $(O)$ through $M$ meets $\Delta$ at $C$. Let $(I)$ be the circle through $M$ and touches $\Delta$ at $C$. Let $C D$ be the diameter of $(I)$. Prove that
      a) $D O C$ is an isosceles triangle.
      b) The line through $D$ and perpendicular to $B C$ always passes through a fixed point when $M$ moves on the circle $(O)$.
    9. Does there exist a sequence of positive integers $a_{2003}>a_{2002}>\ldots>a_{2}>a_{1}$ with $a_{1}=2003$ such that the following two conditions are satisfied
      • All integers in the interval $\left(2003 ; a_{2 \times 13}\right)$ are either a member of this sequence or a non-prime.
      • $A=\dfrac{2004}{a_{1}}+\dfrac{2004}{a_{2}}+\ldots+\dfrac{2004}{a_{2003}}$ is an integer?.
      1. Find all continuous functions $f, g, h$ on $\mathbb{R}$ such that $$f(x+y)=g(x)+h(y)$$ for all real numbers $x$, $y$.
      2. Let $H$ and $O$ denote the orthocenter and circumcenter respectively of a triangle $A B C$. Prove the inequality $$3 R-2 O H \leq H A+H B+H C \leq 3 R+O H$$ where $R$ is its circumradius.
      3. Let $A B C D . A_{1} B_{1} C_{1} D_{1}$ be a cubic whose side is a. Let $N$, $M$ be two points on $A B_{1}$ and $B C_{1}$ respectively such that the angle between $M N$ and the plane $(A B C D)$ is $60^{\circ}$. Prove that $$M N \geq 2 a(\sqrt{3}-\sqrt{2}).$$ When does equality occur?.
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      MOlympiad.NET: Mathematics and Youth Magazine Problems 2008
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      https://www.molympiad.net/2022/04/blog-post_24.html
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