# 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$$$\sqrt{14-x^{3}}+x=2\left(1+\sqrt{x^{2}-2 x-1}\right).$$ 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$$$\sqrt{x+6}+\sqrt{x-1}=x^{2}-1$$ 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$$$\frac{a+b+c}{3} \leq \sqrt{a b c}+k \cdot \max \{|a-b|, b-c|,| c-a \mid\}.$$ 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?.  MOlympiad.NET là dự án thu thập và phát hành các đề thi tuyển sinh và học sinh giỏi toán. Quý bạn đọc muốn giúp chúng tôi chỉnh sửa đề thi này, xin hãy để lại bình luận facebook (có thể đính kèm hình ảnh) hoặc google (có thể sử dụng$\LaTeX$) bên dưới. BBT rất mong bạn đọc ủng hộ UPLOAD đề thi và đáp án mới hoặc liên hệbbt.molympiad@gmail.comChúng tôi nhận tất cả các định dạng của tài liệu:$\TeX\$, PDF, WORD, IMG,... 