Mathematics and Youth Magazine Problems 2011

Issue 403

  1. Given the sum $$S=\frac{5}{1.2 .3}+\frac{8}{2.3 .4}+\frac{11}{3.4 .5}+\ldots+\frac{6026}{2008.2009 .2010}.$$ Compare $S$ with $2$.
  2. Let $A B C$ be a triangle with $\widehat{B A C}=50^{\circ}$, $\widehat{A B C}=72^{\circ}$. Outside of the triangle $A B C,$ draw a triangle $B D C$ such that $\widehat{C B D}=28^{\circ}$; $\widehat{B C D}=22^{\circ}$. Find the measure of the angle $A D B$.
  3. Find all possible pair of integers $x$, $y$ satisfying the following condition $$x^{2}+y^{2}=(x-y)(x y+2)+9$$
  4. Given $a, b, c, d \in[0 ; 1]$ satisfies the following condition $$a+b+c+d=x+y+z+t=1.$$ Prove the inequality $$a x+b y+c z+d t \geq 54 a b c d.$$
  5. Let $A B C$ be a triangle with $\widehat{B A C}=45^{\circ}$. The attitudes $B D$ and $C E$ intersect at $H .$ Let $I$ be a midpoint of $D E .$ Prove that the line $H I$ goes through the centroid of the triangle $A B C$.
  6. Solve the equation $$\sqrt{x}+\sqrt[4]{x}+4 \sqrt{17-x}+8 \sqrt[4]{17-x}=34.$$
  7. A number is said to be an interesting number if it has $10$ digits, all are distinct, and is a multiple of $11111$. How many interesting numbers are there?
  8. Let $A_{1} A_{2} A_{3} \ldots A_{n}$ be a convex polygon $(n \geq 3)$ on the plane $(P)$ and let $S$ be a point outside $(P)$. Another plane $(\alpha)$ intersects the sides $S A_{1}, S A_{2}, \ldots, S A_{n}$ at $B_{1}, B_{2},\ldots, B_{n}$ respectively such that $$\frac{S A_{1}}{S B_{1}}+\frac{S A_{2}}{S B_{2}}+\ldots+\frac{S A_{n}}{S B_{n}}=a$$ where $a$ is a given positive number. Prove that such a plane $(\alpha)$ always contains a fixed point.
  9. Two circles $\omega_{1}$, $\omega_{2}$ intersect at points $A$, $B$. $C D$ is a common tangent line of $\omega_{1}$, $\omega_{2}$ $(C \in \omega_{1}, D \in \omega_{2}$) where point $B$ is closer to $C D$ than point $A$. $C B$ cuts $A D$ at $E$, $D B$ cuts $CA$ at $F$ and $E F$ cuts $A B$ at $N$. $K$ is the orthogonal projection of $N$ onto $C D$.
    a) Prove that $\widehat{C A B}=\widehat{D A K}$.
    b) Let $O$ be the circumcenter of the triangle $A C D$ and $H$ is the orthocenter of the triangle $K E F .$ Prove that $O$, $B$, $H$ are collinear
  10. Let $\left(x_{n}\right)$ be the sequence where $$x_{1}=5,\quad x_{n+1}=\frac{x_{n}^{2010}+3 x_{n}+16}{x_{n}^{2009}-x_{n}+11},\,n=1,2, \ldots$$ For each positive number $n,$ put $\displaystyle y_{n}=\sum_{i=1}^{n} \frac{1}{x_{i}^{2009}+7}$. Determine $\displaystyle \lim_{n\to\infty}y_{n}$.
  11. Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ satisfy the following equation $$f(f(x-y))=f(x) \cdot f(y)+f(x)-f(y)-x y,\,\forall x, y \in \mathbb{R}.$$
  12. For each $n \in \mathbb{N}$, let $a_{n}$ be a number of bijections $f:\{1,2, \ldots, n\} \rightarrow\{1,2, \ldots, n\}$ such that $f(f(k))=k$ for all $k \in\{1,2, \ldots, n\}$.
    a) Prove that $a_{n}$ is an even number for every $n \geq 2$.
    b) Prove that for every $n \geq 10$ and $n$ is divisible by $3$ then $a_{n}-a_{n-9}$ is divisible by $3$.

Issue 404

  1. Given seven distinct prime numbers $a$, $b$, $c$, $a+b+c$, $a+b-c$, $a-b+c$, $-a+b+c$ in which the sum of two of three numbers $a$, $b$, $c$ equals $800$. Let $d$ be the difference between the largest and the smallest number among these seven integers. What is the maximum value of $d ?$
  2. A triangle $A B C$ has sides $A B=2cm$, $A C=4cm$ and median $A M=\sqrt{3}cm$. Find the measure of the angle $B A C$, the length of side $B C$ and the area of triangle $A B C$.
  3. Find the largest natural number $k$ so that $n^{5}-2011 n$ is divisible by $k$ for all natural number $n$.
  4. Solve the equation $$\left(x^{4}-625\right)^{2}-100 x^{2}-1=0$$
  5. Let $A B C$ be an acute triangle $(A B \neq A C)$ inscribed in the circle $(O),$ and $H$ is its orthocenter. Let $d$ be an arbitrary line passing through $H$. Draw the line $d^{\prime}$ symmetric to $d$ through $B C$. Find the position of the line $d$ so that $d^{\prime}$ touches the circumcircle $(O)$.
  6. Find the largest constant $k$ such that $$\frac{a^{2}}{b}+\frac{b^{2}}{c}+\frac{c^{2}}{a} \geq k\left(a^{2}+b^{2}+c^{2}\right)$$ for all positive real numbers $a$, $b$, and $c$ whose sum equals $1 .$
  7. Let $A B C$ be a triangle inscribed in the circle $(O)$, and let $G$ be its centroid; $D$, $E$, and $F$ are the circumcenters of triangles $G B C$, $G C A$, $G A B$ respectively. Prove that $O$ is the centroid of $D E F$.
  8. Solve the equation $$\sqrt[3]{\cos 5 x+2 \cos x}-\sqrt[3]{2 \cos 5 x+\cos x}=2 \sqrt[3]{\cos x}(\cos 4 x-\cos 2 x).$$
  9. Let $x$, $y$, $z$ be real numbers such that $x \geq 1$, $y \geq 2$, $z \geq 3$ and $$\frac{1}{x+\sqrt{x-1}}+\frac{2}{y+\sqrt{y-2}}+\frac{3}{z+\sqrt{z-3}}=12$$ Find the maximum and minimum value of the function $f(x, y, z)=x+y+z$.
  10. Let $\left(x_{n}\right)$ be a sequence given by $$x_{1}=\frac{5}{2},\quad x_{n+1}=\sqrt{x_{n}^{3}-12 x_{n}+\frac{20 n+21}{n+1}},\,\forall n \in \mathbb{N}^{*}.$$ Prove that the sequence $\left(x_{n}\right)$ converges and find its limit.
  11. Find all functions $f: \mathbb{R} \rightarrow(0 ; 2011]$ such that $$f(x) \leq 2011\left(2-\frac{2011}{f(y)}\right),\,\forall x>y.$$
  12. Given four points $A_{I}$ $(i=1,2,3,4),$ no three of them are colinear and a point $M$ so that $A_{i}$ $(i=1,2,3,4)$ and $M$ do not lie on the same circle. Let $T_{i}$ be a triangle having $A_{j}$ $(j=1,2,3,4 ; j \neq i)$ as its vertices, $C_{i}$ is the circle (or the line) passing through the feet of the projections through $M$ onto three sides (or extended sides) of triangle $T_{i}$. Prove that $C_{I}$ $(i=1,2,3,4)$ have a common point.

Issue 405

  1. Which of the following two numbers is greater? $$A=\frac{326}{1955}+\frac{988}{1975}+\frac{662}{1985},\quad B=\dfrac{3951}{3950}+\dfrac{1}{5955}+\dfrac{1}{11730}.$$
  2. Let $A B C$ be an isosceles triangle with $\widehat{B A C}=96^{\circ} .$ A point $M$ is inside the triangle such that $\widehat{M B C}=12^{\circ}$, $\widehat{M C B}=24^{\circ} .$ Prove that $M A=M C$.
  3. Find the maximum value of the expression $P=\max \{a, b, c\}-\min \{a, b, c\}$ where $a, b, c$ are real numbers satisfying the condition $$a+b+c=a^{3}+b^{3}+c^{3}-3 a b c=2.$$
  4. Solve the equation $$21 x-25+2 \sqrt{x-2}=19 \sqrt{x^{2}-x+2}+\sqrt{x+1}$$
  5. $A B C$ is a triangle inscribed the circle $(O)$ with $\widehat{B A C}=60^{\circ}$, $A K$ is the angle-bisector of $\widehat{B A C}$ ($K$ is on the circle $(O)$). Let $F$ be the midpoint of $A K,$ the ray $O F$ meets the altitude $C E$ of triangle $A B C$ at $H .$ Prove that $B H$ is perpendicular to $A C$.
  6. Find the minimum value of the expression $$P=\left(5 a+\frac{2}{b+c}\right)^{3}+\left(5 b+\frac{2}{c+a}\right)^{3}+\left(5 c+\frac{2}{a+b}\right)^{3}$$ where $a, b, c$ are positive real numbers satisfying $a^{2}+b^{2}+c^{2}=3$
  7. $OABC$ is a trirectangular tetrahedron at vertex $O$. $O H$ is the altitude from $O$ of tetrahedron. Let $R$ be the circumradius of triangle $A B C .$ Prove that $O H \leq \dfrac{R \sqrt{2}}{2} .$ When does equality occur?
  8. a) Find all distinct permutations of the word $TOANHOCTUOITRE$.
    b) How many permutations are there that has three consecutive $T - TTT$?
    c) How many permutations are there without adjacent $T$s?
  9. Let $k$ be a positive integer, $\alpha$ is an arbitrary real number. Find the limit of sequence $\left(a_{n}\right)$ where $$a_{n}=\frac{\left[1^{k} \cdot \alpha\right]+\left[2^{k} \cdot \alpha\right]+\ldots+\left[n^{k} \cdot \alpha\right]}{n^{k+1}},\, n=1,2, \ldots$$ here the notation $[x]$ is the largest integer that does not exceed $x .$
  10. Find all functions $f: \mathbb{N} \rightarrow \mathbb{N}$ which satisfy the following conditions
    • $f$ is strictly increasing;
    • $f(f(n))=4 n+9$ for all $n \in \mathbb{N}^{*}$
    • $f(f(n)-n)=2 n+9$ for all $n \in \mathbb{N}^{*}$
  11. Does there exist a positive integer $n \geq 2$ so that $$f(x)=1+4 x+4 x^{2}+\ldots+4 x^{2 n}$$ is a perfect square polynomial?
  12. Let $A B C$ be a triangle inscribed the circle $(O)$ and $A^{\prime}$ is a fixed point on $(O)$. $P$ moves on $B C$, $K$ belongs to $A C$ so that $P K$ is always parallel to a fixed line $d$. The circumcircle of triangle $A P K$ cuts the circle $(O)$ at a second point $E$. $A E$ cuts $B C$ at $M$. $A^{\prime} P$ cuts the circle $(O)$ at a second point $N .$ Prove that the line $M N$ passes through a fixed point.

Issue 406

Issue 407

  1. Denote $T(a)$ the number of digits of the natural number $a$. If $T\left(5^{n}\right)-T\left(2^{n}\right)$ is even, is $n$ necessarily odd or even?
  2. A triangle $A B C$ has $\widehat{B A C}<90^{\circ}$ and $H A=B C$ where $H$ is its orthocenter. Find the measure of angle $B A C$.
  3. Find all pair of integers $x, y$ such that $$7^{x}+24^{x}=y^{2}$$
  4. Let $x, y, z$ be arbitrary positive real numbers, prove the inequality $$\frac{x^{2}-z^{2}}{y+z}+\frac{z^{2}-y^{2}}{x+y}+\frac{y^{2}-x^{2}}{z+x} \geq 0.$$ When does equality occur?
  5. From point $M$ outside circle $(O),$ draw tangent $M A$ and secant $M B C$ ($B$ is between $M$ and $C$). Let $H$ be the projection of $A$ onto $M O, K$ is the intersection of segment $M O$ with $(O)$. Prove that
    a) The quadrilateral $O H B C$ is cyclic.
    b) $B K$ is the internal angle-bisector of angle $H B M$
  6. Solve the system of equations $$\begin{cases}\sqrt{\dfrac{x^{2}+y^{2}}{2}}+\sqrt{\dfrac{x^{2}+x y+y^{2}}{3}} &=x+y \\ x \sqrt{2 x y+5 x+3} &=4 x y-5 x-3 \end{cases}$$
  7. Let $\left(x_{n}\right)$ be a sequence defined by $$3 x_{n+1}=x_{n}^{3}-2,\,n=1,2, \ldots.$$ For what values of $x_{1}$ does the sequence $\left(x_{n}\right)$ converge? Determine this limit when it converges.
  8. $S . A B C$ is a tetrahedron with isosceles perpendicular to plane $(A B C)$. $D$ is the midpoint of $B C .$ Let $\alpha$ is the angle between edge $S B$ and plane $(A B C)$; $\beta$ is the angle between edge $S B$ and plane $(S A D)$. Prove that $$\cos ^{2} \alpha+\cos ^{2} \beta>1$$
  9. Let $x, y, z$ be positive real numbers such that $x^{2}+y^{2}+z^{2}+2 x y z=1 .$ Prove the inequality $$8(x+y+z)^{3} \leq 10\left(x^{3}+y^{3}+z^{3}\right) + 11(x+y+z)(1+4 x y z)-12 x y z.$$
  10. Let $p$ be an odd prime, $n$ is a positive integer so that $(n, p)=1$. Find the number of tuples $\left(a_{1}, a_{2}, \ldots, a_{p-1}\right)$ such that the sum $\displaystyle\sum_{k=1}^{p-1} k a_{k}$ is divisible by $p,$ and $a_{1}, a_{2}, \ldots, a_{p-1}$ are natural numbers which do not exceed $n-1$.
  11. Find all the functions $f$ which is defined on $\mathbb{R},$ take value on $\mathbb{R}$ and satisfying the equation $f(x+y+f(y))=f(f(x))+2 y,$ for all real numbers $x$, $y$.
  12. Let $p, r, r_{a}, r_{b}, r_{c}$ be semiperimeter, inradius, and exradius opposite angles $A$, $B$, $C$ of triangle $A B C$ having side lengths $B C=a$, $C A=b$, $A B=c$. Prove the inequality $$\sqrt{a b}+\sqrt{b c}+\sqrt{c a} \geq p+\sqrt{r r_{a}}+\sqrt{r r_{b}}+\sqrt{r r_{c}}$$ When does equality hold?

Issue 408

  1. How many integers $n$ are there such that $-1964 \leq n \leq 2011$ and the fraction $\dfrac{n^{2}+2}{n+9}$ is reducible?
  2. Let $a_{1}, a_{2}, a_{3}, \ldots$ be a sequence satisfying the conditions $$a_{2}=3,\, a_{50}=300,\quad a_{n}+a_{n+1}=a_{n+2},\quad \forall n \geq 1 .$$ Find the sum of the first $48$ terms $S=a_{1}+a_{2}+\ldots+a_{48}$.
  3. Let $p$ be a prime number. Let $x, y$ be nonzero natural numbers such that $\dfrac{x^{2}+p y^{2}}{x y}$ is also a natural number. Prove that $$\frac{x^{2}+p y^{2}}{x y}=p+1$$
  4. Solve the system of equations $$\begin{cases}x-2 \sqrt{y+1} &=3 \\ x^{3}-4 x^{2} \sqrt{y+1}-9 x-8 y &=-52-4 x y \end{cases}$$
  5. Let $A B$ be a fixed line segment. Point $M$ is such that $M A B$ is an acute triangle. Let $H$ be the orthocenter of $M A B$, $I$ is the midpoint of $A B$, $D$ is the projection of $H$ onto $MI$. Prove that the product $M I$. $D I$ does not depend on the position of $M$.
  6. Let $a, b, c$ be real numbers such that $\sin a+\sin b+\sin c \geq \dfrac{3}{2}$. Prove the inequality $$\sin \left(a-\frac{\pi}{6}\right)+\sin \left(b-\frac{\pi}{6}\right)+\sin \left(c-\frac{\pi}{6}\right) \geq 0$$
  7. Denote by $[x]$ the largest integer not exceeding $x$. Solve the equation $$x^{2}-(1+[x]) x+2011=0.$$
  8. Let $E$ be the center of the nine-point circle (the Euler's circle) of triangle $A B C$ with edge-lengths $B C=a$, $A C=b$, $A B=c$; $E_{1}$, $E_{2}$, $E_{3}$ are respectively the projections of $E$ onto $B C$, $C A$, $A B$ and let $R$ be the circumradius of triangle $A B C$. Prove that $$\frac{S_{E_{1} E_{2} E_{3}}}{S_{A B C}}=\frac{a^{2}+b^{2}+c^{2}}{16 R^{2}}-\frac{5}{16}$$
  9. Find the minimum value of the expression $$\tan B+\tan C-\tan A \tan A+\tan C-\tan B \quad \tan A+\tan B-\tan C$$ where $A$, $B$, $C$ are three angles of an acute triangle $A B C$ and $C \geq A$
  10. a) Prove that for each positive integer $n,$ the equation $$x+x^{2}+x^{3}+\ldots+2011 x^{2 n+1}=2009$$ has a unique real root.
    b) Let $x_{n}$ be denote the real solution in part a). Prove that $0<x_{n}<\dfrac{2010}{2011}$.
  11. Let $\left(u_{n}\right)$ be a sequence given by $$u_{1}=2011,\quad u_{n+1}=\frac{\pi}{8}\left(\cos u_{n}+\frac{\cos 2 u_{n}}{2}+\frac{\cos 3 u_{n}}{3}\right),\, \forall n \geq 1.$$ Prove that the sequence $\left(u_{n}\right)$ has a finite limit.
  12. Let $A_{1} B_{1} C_{1} D_{1}$ and $A_{2} B_{2} C_{2} D_{2}$ be two squares in opposite direction (that is, if the vertices $A_{1}$, $B_{1}$, $C_{1}$, $D_{1}$ are in clockwise order, then $A_{2}$, $B_{2}$, $C_{2}$, $D_{2}$ are ordered counterclockwise) with centers $O_{1}$, $O_{2}$ suppose that $D_{2}$, $D_{1}$ are respectively in $A_{1} B_{1}$, $A_{2} B_{2}$. Prove that the lines $B_{1} B_{2}$, $C_{1} C_{2}$ and $O_{1} O_{2}$ are concurrent.

Issue 409

  1. Without taking common denominator, find the integer $x$ given that $$\left(\frac{2009}{2010}+\frac{2010}{2011}+\frac{2011}{2009}\right) \cdot(x-2011)>3 x-6033$$
  2. In a triangle $A B C$, $M$, $N$, $P$ are midpoints of sides $B C$, $C A$ and $A B$ respectively. Choose the points $A_{1}$, $A_{2}$, $B_{1}$, $B_{2}$, $C_{1}$, $C_{2}$ on the opposite rays of rays $B A$, $C A$, $C B$, $A B$, $A C$ and $B C$ respectively so that $B A_{1}=C A_{2}$, $C B_{1}=A B_{2}$, $A C_{1}=B C_{2} . A_{0}, B_{0}, C_{0}$ are midpoints of $A_{1} A_{2}, B_{1} B_{2}, C_{1} C_{2}$ respectively. Prove that the lines $A_{0} M, B_{0} N, C_{0} P$ meet at a common point.
  3. Let $b$ be a positive integer with the following properties
    • $b$ equals a sum of three squares.
    • $b$ possess a divisor of the form $a=3 k^{2}+3 k+1$ $(k \in \mathbb{N})$.
    Prove that $b^{n}$ is also a sum of three squares for any positive integer $n$.
  4. Given three non-negative numbers $a, b, c$, prove the inequality $$a+b+c \geq \frac{a-b}{b+2}+\frac{b-c}{c+2}+\frac{c-a}{a+2}.$$ When does equality hold?
  5. Let $A B C$ be a triangle with circumcircle $(O)$, $I$ is the midpoint of side $B C$. $M$ is chosen on $I C$ (differ from both $C$ and $I$). $A M$ meets $(O)$ at $D$. Point $E$ is on $B D$ such that $\widehat{B M E}=\widehat{M A I}$. $E M$ and $D C$ intersect at $F$. Prove that $$\frac{C F}{C D}=\frac{B E}{B D}$$
  6. Solve for $x$ $$\sqrt{\frac{x+2}{2}}-1=\sqrt[3]{3(x-3)^{2}}+\sqrt[3]{9(x-3)}$$
  7. Triangle $A B C$ is inscribed in a fixed circle $(O)$. The medians from $A$, $B$, $C$ meets $(O)$ at $A_{1}$, $B_{1}$, $C_{1}$ respectively. Which triangle makes the value of the expression $$p=\frac{A A_{1}^{2}+B B_{1}^{2}+C C_{1}^{2}}{A B^{2}+B C^{2}+C A^{2}}$$ minimum possible?
  8. In a triangle $A B C$, prove that $$\frac{\sin A \cdot \sin B}{\sin ^{2} \frac{C}{2}}+\frac{\sin B \cdot \sin C}{\sin ^{2} \frac{A}{2}}+\frac{\sin C \cdot \sin A}{\sin ^{2} \frac{B}{2}} \geq 9$$
  9. Let $A B C$ be a triangle with points $A'$, $B'$, $C'$ on sides $B C$, $C A$ and $A B$ respectively such that $$\frac{A^{\prime} B}{A^{\prime} C}=\frac{B^{\prime} C}{B^{\prime} A}=\frac{C^{\prime} A}{C^{\prime} B}.$$ $A A^{\prime}$ and $B B^{\prime}$ meet at $D$, $B B$ ' meets $C C$ ' at $E$ and $F$ is the intersection of $C C'$ and $A A '$. Parallel lines to $A A'$, $B B'$, $C C'$ through point $O$ in the interior of $A B C$ meet $B C$, $C A$, $A B$ at $A_{1}$, $B_{1}$, $C_{1}$ respectively. Prove that for any point $M$ $$A D\left(M A_{1}-O A_{1}\right)+B E\left(M B_{1}-O B_{1}\right)+C F\left(M C_{1}-O C_{1}\right) \geq 0$$
  10. Given $P=(n+1)^{7}-n^{7}-1$ $(n \in \mathbb{N})$. Prove that there are infinitely many natural numbers $n$ so that $P$ is a perfect square.
  11. Determine all functions $f: \mathbb{R} \rightarrow \mathbb{R}$, continuous on $\mathbb{R}$ such that $$f(x y)+f(x+y)=f(x y+x)+f(y),\, \forall x, y \in \mathbb{R}.$$
  12. $f: \mathbb{R} \rightarrow \mathbb{R}$ is a function with the following properties
    • $f(1)=2011$,
    • $f(x+1) f(x)=(f(x))^{2}+f(x)-1, \forall x \in \mathbb{R}$.
    Let $\displaystyle S_{1}=\sum_{i=1}^{n} \frac{1}{f(i)-1}$, $\displaystyle S_{2}=\sum_{i=1}^{n} \frac{1}{f(i)+1}$. Find $\displaystyle\lim_{n \rightarrow+\infty}\left(S_{1}+S_{2}\right)$.

Issue 410

  1. Find all pairs of coprime positive integers $x, y$ so that $$\frac{x+y}{x^{2}+y^{2}}=\frac{7}{25}$$
  2. Let $A B C$ be an equilateral triangle whose altitudes $A H$, $B K$ intersect at $G .$ The angle-bisector of angle $B K H$ meets $C G$, $A H$, $B C$ at $M$, $N$, $P$ respectively. Prove that $K M=N P$.
  3. Find the minimum value of the expression $S=2011 c a-a b-b c$ where $a, b, c$ satisfy $a^{2}+b^{2}+c^{2} \leq 2$.
  4. Let $A B C$ be an isosceles right triangle with right angle at $A .$ Let $M$, $N$, $O$ be respectively the midpoints of $A B$, $A C$, $B C$. The line perpendicular to $C M$ from $O$ cuts $M N$ at $G .$ Compare the lengths of the two segments $G M$ and $G N$.
  5. Solve the equation $$\sqrt{7 x^{2}+25 x+19}-\sqrt{x^{2}-2 x-35}=7 \sqrt{x+2}$$
  6. Let $A B C$ be a triangle. Let $A M$, $B N$, $C P$ be its internal angle-bisectors ($M \in B C$, $N \in C A$, $P \in A B$). Find the measure of angle $B A C$ so that $P M$ is perpendicular to $N M$
  7. Solve the equation $$(\sin x-2)\left(\sin ^{2} x-\sin x+1\right)=3 \sqrt[3]{3 \sin x-1}+1.$$
  8. Find all values of $a, b$ so that the equation $$x^{4}+a x^{3}+b x^{2}+a x+1=0$$ has at least one solution and the sum $a^{2}+b^{2}$ is smallest possible.
  9. Let $a, b, c$ be positive numbers. Prove that $$\left(a^{2012}-a^{2010}+3\right)\left(b^{2012}-b^{2010}+3\right)\left(c^{2012}-c^{2010}+3\right) \geq 9(a b+b c+c a).$$ When does equality occur?
  10. Let $A B C$ be a triangle. An arbitrary line cuts the lines $B C$, $C A$, $A B$ at $M$, $N$, $P$ respectively. Let $X$, $Y$, $Z$ be respectively the centroids of triangles $A N P$, $B P M$, $C M N$. Prove that $$S_{X Y Z}=\frac{2}{9} S_{A B C}$$
  11. Let $\left(a_{n}\right)$ $\left(n \in \mathbb{N}^{*}\right)$ be a sequence given by $$a_{1}=0 ,\, a_{2}=38,\, a_{3}=-90,\quad a_{n+1}=19 a_{n-1}-30 a_{n-2},\, \forall n \geq 3.$$ Prove that $a_{2011}$ is divisible by 2011 .
  12. For all positive integers $n$ greater than $2$. Find the number of functions $$f:\{1,2,3, \ldots, n\} \rightarrow\{1,2,3,4,5\}$$ satisfying $|f(k+1)-f(k)| \geq 3$ where $k \in\{1,2, \ldots, n-1\}$.

Issue 411

  1. The natural numbers $1,2 \ldots, 2011^{2}$ are arranged in some order in a $2011 \times 2011$ square table, each square contains one number. Prove that there exists two adjacent squares (that is two squares having a common edge or a common vertex) such that the difference between the corresponding assigned numbers is not smaller than $2012$.
  2. Find the value of the following $2009$-terms sum $$S=\left(1+\frac{1}{1.3}\right)\left(1+\frac{1}{2.4}\right)\left(1+\frac{1}{3.5}\right) \ldots\left(1+\frac{1}{2009.2011}\right)$$
  3. Find the integers $x$, $y$ satisfying the expression $$x^{3}+x^{2} y+x y^{2}+y^{3}=4\left(x^{2}+y^{2}+x y+3\right)$$
  4. $M$ is a point in the interior of a triangle $A B C .$ Let $P$, $Q$, $R$, $H$, $G$ be respectively the centroid of triangles $M B C$, $M A C$, $M A B$, $P Q R$, $A B C$. Prove that points $M$, $H$ and $G$ are colinear.
  5. $a$, $b$ and $c$ are positive real numbers whose sum is $3$. Prove the inequality $$\frac{4}{(a+b)^{3}}+\frac{4}{(b+c)^{3}}+\frac{4}{(c+a)^{3}} \geq \frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}$$
  6. The incircle $(I)$ of a triangle $A B C$ touches $B C$, $C A$, $A B$ at $D$, $E$, $F$ respectively. The line passing through $A$ and parallel to $B C$ meets $E F$ at $K$. $M$ is the midpoint of $B C$. Prove that $I M$ is perpendicular to $D K$.
  7. Solve the system of equations $$\begin{cases}\sqrt{\dfrac{x^{2}+y^{2}}{2}}+\sqrt{\dfrac{x^{2}+x y+y^{2}}{3}}&=x+y \\ x \sqrt{2 x y+5 x+3} &=4 x y-5 x-3\end{cases}$$
  8. Let $a, b, c$ be real numbers such that the equation $a x^{2}+b x+c=0$ has two real solutions, both are in the closed interval $[0 ; 1] .$ Find the maximum and mininum values of the expression $$M=\frac{(a-b)(2 a-c)}{a(a-b+c)}.$$
  9. Let $P(x)$ and $Q(x)$ be two polynomials with real coefficients, each has at least one real solution, so that $$P\left(1+x+Q(x)+(Q(x))^{2}\right)=Q\left(1+x+P(x)+(P(x))^{2}\right).$$ For any $x \in \mathbb{R}$. Prove that $P(x) \equiv Q(x)$.
  10. Let $a, b, c, d$ be positive numbers such that $a \geq b \geq c \geq d$ and $a b c d=1 .$ Find the smallest constant $k$ such that the following inequality holds $$\frac{1}{a+1}+\frac{1}{b+1}+\frac{1}{c+1}+\frac{k}{d+1} \geq \frac{3+k}{2}.$$
  11. Find all continuous functions $f: \mathbb{R} \rightarrow \mathbb{R}$ satisfying $$\{f(x+y)\}=\{f(x)+f(y)\}$$ for every $x, y \in \mathbb{R}$ ($[t]$ is the largest integer not exceed $t$ and $\{t\}=t-[t]$.)
  12. Let $A B C$ be a triangle, $P$ is an arbitrary point inside the triangle. Let $d_{a}$, $d_{b}$, $d_{c}$ be respectively the distances from $P$ to $B C$, $C A$, $A B$; $R_{a}$, $R_{b}$, $R_{c}$ are the circumradii of triangles $P B C$, $P C A$, $P A B$ respectively. Prove that $$\frac{\left(d_{a}+d_{b}+d_{c}\right)^{2}}{P A \cdot P B \cdot P C} \geq \frac{\sqrt{3}}{2}\left(\frac{\sin A}{R_{a}}+\frac{\sin B}{R_{b}}+\frac{\sin C}{R_{c}}\right)$$

Issue 412

  1. Pick $n$ numbers $(n \geq 2)$ from the first hundred natural numbers (from $1$ to $100$) so that the sum of any two distinct numbers is a multiple of $6 .$ What is the largest possible number $n$ so that this can be done?
  2. Given $A=\dfrac{5^{a}}{5^{b+c}}$ and $B=\dfrac{5^{a}+2011}{5^{b+c}+2011}$ where $a, b, c$ are the side lengths of a triangle. Compare $A$ and $B$.
  3. Do there exists three integers $x$, $y$ and $z$ such that $$|x-2005 y|+|y-2007 z|+|z-2009 x|=2011^{x}+2013^{y}+2015^{z} ?$$
  4. Determine the following sum of $2011$ terms $$S=\frac{1}{1^{4}+1^{2}+1}+\ldots+\frac{2011}{2011^{4}+2011^{2}+1}$$
  5. Given a circle $(O),$ a chord $B C$ ($B C$ is not a diameter) and point $A$ moving on the major arc $B C$. Draw a circle $\left(O_{1}\right)$ passing through $B$ and touches $A C$ at $A$, another circle $\left(O_{2}\right)$ passing through $C$ and touches $A B$ at $A .\left(O_{1}\right)$ meets $\left(O_{2}\right)$ at a second point $D$, different from $A$. Prove that line $A D$ always passes through a fixed point.
  6. A quadrilateral $A B C D$ with $A C \perp B D$ is inscribed in a fixed circle $(O ; R)$. Let $p$ be the perimeter of $A B C D$. Prove that $$\frac{A B^{2}}{p-A B}+\frac{B C^{2}}{p-B C}+\frac{C D^{2}}{p-C D}+\frac{D A^{2}}{p-D A} \geq \frac{4 R \sqrt{2}}{3}$$
  7. Solve the system of equations $$\begin{cases}(17-3 x) \sqrt{5-x}+(3 y-14) \sqrt{4-y} &=0 \\ 2 \sqrt{2 x+y+5}+3 \sqrt{3 x+2 y+11} &=x^{2}+6 x+13\end{cases}$$
  8. Prove that the following inequality holds for any triangles $A B C$ $$\cos ^{2} \frac{A-B}{2}+\cos ^{2} \frac{B-C}{2}+\cos ^{2} \frac{C-A}{2} \geq 24 \sin \frac{A}{2} \sin \frac{B}{2} \sin \frac{C}{2}.$$
  9. Two circles $\left(C_{1}\right),\left(C_{2}\right)$ are given such that the center $O$ of $\left(C_{2}\right)$ lies on $\left(C_{1}\right) .$ Let $C$, $D$ be their intersection points. Points $A$ and $B$ on $\left(C_{1}\right)$ and $\left(C_{2}\right)$ respectively such that $A C$ touches $\left(C_{2}\right)$ at $C$ and $B C$ touches $\left(C_{1}\right)$ at $C$. The line $A B$ intersects $\left(C_{2}\right)$ at $E$ and $\left(C_{1}\right)$ at $F$. $C E$ meets $\left(C_{1}\right)$ at $G, C F$ meets $G D$ at $H$. Prove that $G O$ intersects $E H$ at the circumcenter of triangle $D E F$.
  10. Let $a_{1}, a_{2} \ldots, a_{n}$ be $n$ positive real numbers such that $$a_{1}^{2}+a_{2}^{2}+\ldots+a_{k}^{2} \leq \frac{k(2 k-1)(2 k+1)}{3}$$ where any $k=\overline{1, n}$. Find the largest possible value of the expression $$P=a_{1}+2 a_{2}+\ldots+n a_{n}.$$
  11. Given a sequence $\left(x_{n}\right)$ such that $$x_{n}=2 n+a \sqrt[3]{8 n^{3}+1}, \forall n=1,2, \ldots$$ where $a$ is any real number.
    a) For what values of $a$ does the sequence has finite limit?
    b) Find $a$ such that $\left(x_{n}\right)$ is eventually increasing.
  12. Find all function $f: \mathbb{R} \rightarrow \mathbb{R}$ such that $$f\left(x^{3}+y^{3}\right)=x^{2} f(x)+y^{2} f(y)$$ where $x, y \in \mathbb{R}$

Issue 413

  1. On the cardboards, Write each five-digit numbers, from $11111$ to $99999$, on a cardboard. After mixing the cardboards, place them in a sequence in certain order. Prove that the resulting number is not a power of $2 .$
  2. Find all triple of pairwise distinct prime numbers $a, b, c$ such that $$20 a b c<30(a b+b c+c a)<21 a b c.$$
  3. Point $O$ on the median $A D$ of triangle $A B C$ is chosen such that $\dfrac{A O}{A D}=k$ $(0<k<1)$. The rays $B O$, $C O$ cut $A C$, $A B$ at $E$, $F$ respectively. Determine the value of $k$ so that $$S_{A E O F}=\frac{1}{15} S_{A B C}.$$
  4. Solve the equation $$\sqrt{x^{2}+x+19}+\sqrt{7 x^{2}+22 x+28}+\sqrt{13 x^{2}+43 x+37}=3 \sqrt{3}(x+3).$$
  5. Let $A B C$ be a right triangle with right angle at $A$. $D$ is a point within the triangle so that $C D=C A$. Choose point $M$ on the edge $A B$ so that $\widehat{B D M}=\dfrac{1}{2} \widehat{A C D}$; $N$ is the intersection of $M D$ and the altitude $A H$ of triangle $A B C$. Prove that $D M=D N$.
  6. Let $A B C$ be a triangle inscribed the circle $(O)$. $F$ is an arbitrary point on the arc $\widehat{A B}$ (not containing $C$) $(F$ differs from $A$ and $B$). $M$ is the midpoint of the arc $\widehat{B C}$ (not containing $A$); $N$ is the midpoint of the arc $\widehat{A C}$ (not containing $B$). The line passing through $C$ and parallel to $M N$ cuts the circle $(O)$ at another point $P .$ Let $I$, $I_{1}$, $I_{2}$ be the incenters of triangles $A B C$, $F A C$, $F B C$. $P I$ cuts the circle $(O)$ at $G$. Prove that the four points $I_{1}$, $F$, $G$, $I_{2}$ are concyclic.
  7. Solve the equation $$(2 \sin x-3)\left(4 \sin ^{2} x-6 \sin x+3\right)=1+3 \sqrt[3]{6 \sin x-4}.$$
  8. $x, y, z,$ and $t$ are four real numbers longing to the interval $\left[\frac{1}{2} ; \frac{2}{3}\right]$. Find the least and greatest values of the expression $$P=9\left(\frac{x+z}{x+t}\right)^{2}+16\left(\frac{x+t}{x+y}\right)^{2}$$
  9. Prove that given any prime number $p,$ there exist natural numbers $x$, $y$, $z$, $t$ so that $x^{2}+y^{2}+z^{2}-t p=0$ and $0<t<p$.
  10. Let $\left(u_{n}\right)$ be a sequence given by $$u_{1}=a ,\quad u_{n+1}=\frac{(\sqrt{2}+1) u_{n}-1}{\sqrt{2}+1+u_{n}},\,\forall n \geq 1.$$ a) Find the condition of $a$ so that all terms in the sequence are well-defined.
    b) Find the value of $a$ such that $u_{2011}=2011$.
  11. Find all functions $f: \mathbb{N}^{*} \rightarrow \mathbb{N}^{*}$ satisfying the following condition $$\frac{f(x+y)+f(x)}{2 x+f(y)}=\frac{2 y+f(x)}{f(x+y)+f(y)}, \forall x, y \in \mathbb{N}^{*}$$
  12. Let $A B C$ be a triangle, $\widehat{B A C} \neq 90^{\circ}$. $D$ is a fixed point on the edge $B C$. $P$ is a point inside the triangle $A B C$. Let $B_{1}$, $C_{1}$ be respectively the projections of $P$ onto $A C$, $A B$. $D B_{1}$ cuts $A B$ at $C_{2}$, $D C_{1}$ cut $A C$ at $B_{2}$. $Q$ is the intersection differs from $A$ of the circumcircles of triangles $A B_{1} C_{1}$ and $A B_{2} C_{2}$. Prove that the line $P Q$ always go through a fixed point when $P$ is moving.

Issue 414

  1. Do there exist two natural numbers $a, b$ such that $$(3 a+2 b)(7 a+3 b)-4=\overline{22} * 12 * 2011 ?$$
  2. Equilateral triangles $A B E$ and $B C F$ are constructed outside triangle $A B C .$ Let $G$ be the centroid of triangle $A B E$ and $I$ be the midpoint of $A C .$ Find the measure of angle $G I F$.
  3. Find the smallest positive integer $n$ such that $2^{n}-1$ is divisible by $2011$.
  4. Prove that for all integers $k$, the equation $$x^{4}-2010 x^{3}+(2009+k) x^{2}-2007 x+k=0$$ does not have two distinct integer roots.
  5. From a point $M$ outside the cycle $(O),$ draw the tangents $M A, M B$ and the secant $M C D$ to $(O), C$ lies between $M$ and $D .$ $A B$ cuts $C D$ at $N .$ Prove that $$\frac{1}{M D}+\frac{1}{N D}=\frac{2}{C D}$$
  6. Let $A B C$ be a right triangle, right angle at $A,$ satisfying $A B+\sqrt{3} A C=2 B C$. Find the position of point $M$ such that $$4 \sqrt{3} \cdot M A+3 \sqrt{7} \cdot M B+\sqrt{39} \cdot M C$$ is smallest possible.
  7. Solve the equation $$\log _{3}\left(7^{x}+2\right)=\log _{5}\left(6^{x}+19\right)$$
  8. Let $A B C$ be a triangle satisfying $$\tan \frac{A}{2} \tan \frac{B}{2}=\frac{1}{2}.$$ Prove that $A B C$ is a right triangle iff $$\sin \frac{A}{2} \sin \frac{B}{2} \sin \frac{C}{2}=\frac{1}{10}.$$
  9. Let $A B C$ be a triangle inscribed the circle $(O ; R), M$ is a point not on the circle respectively. Let $r$, $r_{1}$ be respectively the radii of the incircles of triangles $A B C$ and $A_{1} B_{1} C_{1}$. Prove that $$\left|R^{2}-O M^{2}\right| \geq 4 r \cdot r_{1}$$
  10. Find the greatest positive constant $k$ satisfying the inequality $$ \frac{k}{a^{3}+b^{3}}+\frac{1}{a^{3}}+\frac{1}{b^{3}} \geq \frac{16+4 k}{(a+b)^{3}}.$$
  11. Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that $$f(x f(y)+y)+f(x y+x)=f(x+y)+2 x y.$$
  12. For each positive integer $n$ consider a function $f_{n}$ in $\mathbb{R}$ defined by $$f_{n}(x)=\sum_{i=1}^{2 n} x^{i}+1.$$ Prove the following statements
    a) $f_{n}$ obtains its minimum value at a unique point $x_{n},$ for each positive integer $n .$ Put $S_{n}=f_{n}\left(x_{n}\right)$.
    b) $S_{n}>\dfrac{1}{2}$ for all $n \in \mathbb{N}^{*} .$ Moreover, $\dfrac{1}{2}$ is the best constant possible in the sense that there does not exist any real number $a>\dfrac{1}{2}$ such that $S_{n}>a$ for all $n \in \mathbb{N}^{*}$.
    c) The sequence $\left(S_{n}\right)$ $(n=1,2, \ldots)$ is decreasing and $\displaystyle\lim_{n\to\infty} S_{n}=\dfrac{1}{2}$.
    d) $\displaystyle\lim_{n\to\infty} x_{n}=-1$.
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MOlympiad.NET: Mathematics and Youth Magazine Problems 2011
Mathematics and Youth Magazine Problems 2011
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