Mathematics and Youth Magazine Problems 2007

Issue 355

  1. Let $a, b$ be two natural numbers satisfying $$2006 a^{2}+a=2007 b^{2}+b.$$ Prove that $a-b$ is a perfect square.
  2. Let $A B C$ be a triangle with $\widehat{B A C}=90^{\circ}, \widehat{A B C}$ $=60^{\circ} .$ Take the point $M$ on the side $B C$ such that $A B+B M=A C+C M$. Caculate the measure of $\widehat{C A M}$
  3. Find all positive integers $x, y$ greater than 1 so that $2 x y-1$ divisible by $(x-1)(y-1)$.
  4. Prove that $$\frac{a^{4} b}{2 a+b}+\frac{b^{4} c}{2 b+c}+\frac{c^{4}}{2 c+a} \geq 1$$ where $a, b, c$ are positive numbers satisfying the condition $a b+b c+c a \leq 3 a b c$. When does equality occur?
  5. Let be given two circles $\left(O_{1}\right)$, $\left(O_{2}\right)$ with centers $O_{1}$, $O_{2}$ with distinct radii, externally touching each other at a point $T$. Let $O_{1} A$ be a tangent to $\left(O_{2}\right)$ at a point $A,$ let $O_{2} B$ be a tangent to $\left(O_{1}\right)$ at a point $B$ so that the points $A, B$ are on the same side with respect to the line $O_{1} O_{2} .$ Let $H$ be the point on $O_{1} A, K$ be the point on $O_{2} B$ so that the lines $B H, A K$ are perpendicular to $O_{1} O_{2}$. The line $T H$ cuts $\left(O_{1}\right)$ again at $E,$ the line $T K$ cuts $\left(O_{2}\right)$ againt at $F$. The line $E F$ cuts $A B$ at $S$. Prove that the lines $O_{1} A$, $O_{2} B$ and $T S$ are concurrent.
  6. Let $S$ be a set consisting of 43 distinct positive integers not exceeding $100 .$ For each subset $X$ of $S$ let $t_{X}$ be the product of its elements. Prove that there exist two disjoint substs $A$ and $B$ of $S$ such that $t_{A} t_{B}^{2}$ is the cube of a natural numbers.
  7. Find the greast value of the expression $$\frac{a}{c}+\frac{b}{d}+\frac{c}{a}+\frac{d}{b}-\frac{a b c d}{(a b+c d)^{2}}$$ where $a, b, c d$ are distinct real numbers satisfying the conditions $\dfrac{a}{b}+\dfrac{b}{c}+\dfrac{c}{d}+\dfrac{d}{a}=4$ and $a c=b d$
  8. a) Let $f(x)$ be a polynomial of degree $n$ with leading coefficient $a$. Suppose that $f(x)$ has $n$ distinct roots $x_{1}, x_{2}, \ldots, x_{n}$ all not equal to zero. Prove that $$\frac{(-1)^{n-1}}{a x_{1} x_{2} \ldots x_{n}} \sum_{k=1}^{n} \frac{1}{x_{k}}=\sum_{k=1}^{n} \frac{1}{x_{k}^{2} f^{\prime}\left(x_{k}\right)}.$$ b) Does there exist a polynomial $f(x)$ of degree $n,$ with leading coefficient $a=1,$ such that $f(x)$ has $n$ distinct roots $x_{1}, x_{2}, \ldots, x_{n},$ all not equal to zero, satisfying the condition $$\frac{1}{x_{1} f^{\prime}\left(x_{1}\right)}+\frac{1}{x_{2} f^{\prime}\left(x_{2}\right)}+\ldots+\frac{1}{x_{n} f^{\prime}\left(x_{n}\right)}+\frac{1}{x_{1} x_{2} \ldots x_{n}}=0 ?$$
  9. Let $A D$, $B E$, $C F$ be the altitudes and $H$ be the orthocenter of an acute triangle $A B C .$ Let $M$, $N$ be respectively the points of intersection of $D E$ and $C F$ and of $D E$ and $B E$. Prove that the line passing through $A$ perpendicular to the line $M N$ passes through the circumcenter of triangle $B H C$.

Issue 356

  1. Caculate the following sum $S$ of 1002 terms $$S=\frac{1.3}{3.5}+\frac{2.4}{5.7}+\ldots+\frac{(n-1)(n+1)}{(2 n-1)(2 n+1)}+\ldots+\frac{1002.1004}{2005.2007}.$$
  2. Let $B E$ and $C F$ be two altitudes of a triangle $A B C .$ Prove that $A B=A C$ when and only when $A B+B E=A C+C F$.
  3. Let $A$ be a natural number greater than $9$, written in decimal system with digits $1,3,7,9$. Prove that $A$ has at least a prime divisor not less than $11 .$
  4. Find the least value of the expression $$P=\frac{b_{1}+b_{2}+b_{3}+b_{4}+b_{5}}{a_{1}+a_{2}+a_{3}+a_{4}+a_{5}}$$ where $a_{1}, a_{2}, a_{3}, a_{4}, a_{5}, b_{1}, b_{2}, b_{3}, b_{4}, b_{5}$ are non negative real numbers satisfying the conditions $a_{i}^{2}+b_{i}^{2}=1$ $(i=1,2,3,4,5)$ and $a_{1}^{2}+a_{2}^{2}+a_{3}^{2}+a_{4}^{2}+a_{5}^{2}=1$
  5. Let be given a nonright-angled triangle $A B C$ with its altitudes $A A^{\prime}$, $B B^{\prime}$, $C C^{\prime}$. Let $D$, $E$, $F$ be the centers of the escribed circles of the trangles $A B^{\prime} C^{\prime}$, $B C^{\prime} A^{\prime}$, $C A^{\prime} B^{\prime}$ opposite $A$, $A^{\prime}$, $A^{\prime}$ respectively. The escribed circle of triangle $A B C$ opposite $A$ touches the lines $B C$, $C A$, $A B$ at $M$, $N$, $P$ respectively. Prove that the circumcenter of triangle $D E F$ is the orthocenter of triangle $M N P$.
  6. Ten teams participated in a football competition where each team played against every other team exactly once. When the competition was over, it turned out that for every three teams $A$, $B$, $C$ if $A$ defeated $B$ and $B$ defeated $C$ then $A$ defeated $C$. Prove that there were four teams $A$, $B$, $C$, $D$ such that $A$ defeated $B$, $B$ defeated $C$, $C$ defeated $D$ or such that each match between them was a draw.
  7. Prove that for abitrary positive numbers $a, b, c$ such that $a b c \geq 1,$ we have $$a+b+c \geq \frac{1+a}{1+b}+\frac{1+b}{1+c}+\frac{1+c}{1+a}.$$ When does equality occur?
  8. The sequence of numbers $\left(u_{n}\right) ; n=1$, $2, \ldots,$ is defined by : $u_{0}=a$ and $$u_{n+1}=\sin ^{2}\left(u_{n}+11\right)-2007$$ for all natural number $n,$ where $a$ is a given real number. Prove that
    a) The equation $\sin ^{2}(x+11)-x=2007$ has a unique solution. Denote it by $b$.
    b) $\lim u_{n}=b$
  9. Let $A B C$ be a nonregular triangle. Take three points $A_{1}$, $B_{1}$, $C_{1}$ lying on the sides $B C$. $C A$, $A B$ respectively such that $\dfrac{B A_{1}}{B C}=\dfrac{C B_{1}}{C A}=\dfrac{A C_{1}}{A B}$. Prove that if the triangles $A B_{1} C_{1}$, $B C_{1} A_{1}$, $C A_{1} B_{1}$ have equal circumradii then $A_{1}$, $B_{1}$, $C_{1}$ are the midpoints of the sides $B C$, $C A$, $A B$ respectively.

Issue 357

  1. Let be given two bottles, the first bottle contains 1 liter of water, the second bottle is empty. One pours $\frac{1}{2}$ of the quantity of water contained in the first bottle into the second one, then one pours $\frac{1}{3}$ of the quantity of water contained in the second bottle into the first one, then one pours $\frac{1}{4}$ of the quantity of water contained in the first bottle into the second one, and so on, one pours $\frac{1}{5},$ then $\frac{1}{6},$ then $\frac{1}{7}, \ldots$ After the $2007^{\mathrm{th}}$ turn of such pouring, what are the quantities of water left in each bottle?
  2. Let $A B C$ be a right-angled triangle with right angle at $A$ and let $I$ be the point of intersection of its innner angled-bisectors. Take the orthogonal projection $E$ of $A$ on the line $B I$ then the orthogonal projection $F$ of $A$ on the line $C E .$ Prove that $2 E F^{2}=A I^{2}$
  3. Find all finite subset $A \subset \mathbb{N}^{*}$ such that there exists a finite subset $B \subset \mathbb N^{*}$ containning $A$ so that the sum of the numbers in $B$ is equal to the sum of the squares of the numbers in $A$.
  4. Prove that for every natural number $n \geq 2,$ we have $$1+\sqrt{1+\frac{4}{3 !}}+\sqrt[3]{1+\frac{9}{4 !}}+\ldots+\sqrt[n]{1+\frac{n^{2}}{(n+1) !}}<n+\frac{1}{2}$$ where $n !$ denotes $1.2 .3 \ldots n$
  5. Let $A B C D$ be a square inscribed in the circle $(O) .$ On the minor arc $\widehat{B C},$ take an arbitrary point $M$ distinct from $B, C .$ The lines $C M$ and $D B$ intersect at a point $E,$ the lines $D M$ and $A B$ intersect at a point $F$. Prove that the triangles $A B E$ and $D O F$ have equal areas.
  6. Prove that for every couple of positive integers $n, k$ the number $(\sqrt{n}-1)^{k}$ can be written in the form $\sqrt{a_{k}}-\sqrt{a_{k}-(n-1)^{k}}$ with $a_{k} \in \mathbb{N}^{*}$
  7. Prove that for every triangle $A B C$, we have $$\frac{3 \sqrt{3}}{2 \cos \frac{A}{2} \cos \frac{B}{2} \cos \frac{C}{2}}+8 \sin \frac{A}{2} \sin \frac{B}{2} \sin \frac{C}{2} \geq 5.$$ When does equality occur?
  8. Consider the sequence of numbers $\left(x_{n}\right)(n=0,1,2, \ldots)$ defined as follows $x_{0}$, $x_{1}$, $x_{2}$ are given positive numbers; $$x_{n+2}=\sqrt{x_{n+1}}+\sqrt{x_{n}}+\sqrt{x_{n-1}},\, \forall n \geq 1 .$$ Prove that the sequence $\left(x_{n}\right)(n=0,1,2, \ldots)$ has a finite limit and find its limit.
  9. Let be given a tetrahedron $A_{1} A_{2} A_{3} A_{4}$ Its inscribed sphere has center $I$, has radius $r$ and touches the faces opposite the vertices $A_{1}$, $A_{2}$, $A_{3}$, $A_{4}$ at $B_{1}$, $B_{2}$, $B_{3}$, $B_{4}$ respectively. Let $h_{1}$, $h_{2}$, $h_{3}$, $h_{4}$ be the measures of the altitudes of $A_{1} A_{2} A_{3} A_{4}$ issued from $A_{1}$, $A_{2}$, $A_{3}$, $A_{4}$ respectively. Prove that for every point $M$ in space, we have $$\frac{M B_{1}^{2}}{h_{1}}+\frac{M B_{2}^{2}}{h_{2}}+\frac{M B_{3}^{2}}{h_{3}}+\frac{M B_{4}^{2}}{h_{4}} \geq r.$$

Issue 358

  1. Let $a, b, c$ be positive numbers such that $a^{3}+b^{3}=c^{3}$. Compare $a^{2007}+b^{2007}$ and $c^{2007}$
  2. Let $A B C$ be an isosceles triangle at $A$. Let $E$ be an arbitrary point on the side $B C$. The line through $E$ perpendicular to $A B$ meets with the line through $C$ perpendicular to $A C$ at a point denoted by $D$. Let $K$ be the midpoint of $B E$ Find the measure angle $\widehat{A K D}$.
  3. Find the least positive integer $n$ $(n>1)$ such that $\dfrac{1^{2}+2^{2}+\ldots+n^{2}}{n}$ is a perfect square number.
  4. Prove the following inequality $$\left(1+\frac{2 a}{b}\right)^{2}+\left(1+\frac{2 b}{c}\right)^{2}+\left(1+\frac{2 c}{a}\right)^{2} \geq \frac{9(a+b+c)^{2}}{a b+b c+c a}$$ where $a, b, c$ are arbitrary positive real numbers. When does equality occur?
  5. Solve the equation $$x^{3}-\sqrt[3]{6+\sqrt[3]{x+6}}=6.$$
  6. Let $A B C$ be a right triangle at $A$. Draw the circle $(B)$ with center at $B$ and radius $B A$ and draw a diameter $A D .$ Choose two points $E$ and $F$ on the line $B C$ such that $B$ is their midpoint. $D E$ and $D F$ meets with $(B)$ at $M$ and $N$ respectively. Prove that $M, N$ and $C$ are colinear.
  7. Let $A B C$ be an equilateral triangle with circumcircle $(O) .$ Draw a circle $\left(O_{1}\right)$ which is tangent to both side $B C$ and arc $\widehat{B C}$. Construct the circles $\left(O_{2}\right)$, $\left(O_{3}\right)$ similarly for the remaining sides $C A$ and $A B .$ Prove that $A O_{1}=B O_{2}=C O_{3}$ iff (ie. if and only if) the circles $\left(O_{1}\right)$, $\left(O_{2}\right)$ and $\left(O_{3}\right)$ all have the same radius. (Notation: $(X)$ is a circle with center at $X$.)
  8. Let $\left(a_{n}\right)$ $(n=0,1,2, \ldots)$ be a sequence given by $$a_{0}=29,\, a_{1}=105,\,a_{2}=381,\, a_{n+3}=3 a_{n+2}+2 a_{n+1}+a_{n},\,\forall n=0,1,2, \ldots .$$ Prove that for each positive integer $m,$ there exists a number $n$ such that $a_{n}$, $a_{n+1}-1$, $a_{n+2}-2$ are all divisible by $m$
  9. Let $f$ be a continuous function on the interval [0,1] and satisfies the following properties $$f(0)=0,\,f(1)=1,\quad 6 f\left(\frac{2 x+y}{3}\right)=5 f(x)+f(y),\,\forall x \geq y \in [0,1].$$ Find $f\left(\dfrac{8}{23}\right)$.
  10. Let $a, b, c$ be non-negative real numbers with sum equal to $1 .$ Prove that $$\sqrt{a+(b-c)^{2}}+\sqrt{b+(c-a)^{2}}+\sqrt{c+(a-b)^{2}} \geq \sqrt{3}$$
  11. Let $H$ and $I$ be, respectively, the orthorcenter and the incenter of an acute triangle $A B C$. The lines $A H$, $B H$ and $C H$ meet the circumcircles of triangles $HBC$, $HCA$ and $HAB$ at $A_{1}$, $B_{1}$, $C_{1}$ respectively. $A I$, $B I$ and $C I$ meet the circumcircles of triangles $I B C$, $I C A$ and $I A B$ at $A_{2}$, $B_{2}$, $C_{2}$ respectively. Prove that $$H A_{1} \cdot H B_{1} \cdot H C_{1}+64 R^{3} \geq 9 \cdot I A_{2} \cdot I B_{2} \cdot I C_{2}.$$
  12. Let $G$ be the centroid of a tetrahedron $A B C D$. Let $A_{1}$, $B_{1}$, $C_{1}$, $D_{1}$ be arbitrary points on $G A$, $G B$, $G C$ and $G D$ respectively. GA meets the plane $\left(B_{1} C_{1} D_{1}\right)$ at $A_{2} .$ Construct the points $B_{2}$, $C_{2}$, $D_{2}$ similarly. Prove that $$3\left(\frac{G A}{G A_{1}}+\frac{G B}{G B_{1}}+\frac{G C}{G C_{1}}+\frac{G D}{G D_{1}}\right) = \frac{G A}{G A_{2}}+\frac{G B}{G B_{2}}+\frac{G C}{G C_{2}}+\frac{G D}{G D_{2}}$$

Issue 359

  1. Determine the sum of 2005 terms $$S=\frac{2^{2}}{1.3}+\frac{3^{2}}{2.4}+\frac{4^{2}}{3.5}+\ldots+\frac{2006^{2}}{2005.2007}$$
  2. Let $ABC$ be an isosceles right triangle with right angle at $A .$ Pick an arbitrary point $M$ on $A C$. The foot of the altitude with $B C$ through $M$ is $H .$ Let $I$ be the midpoint of $BM$. Find the value of the angle $\angle H A I .$
  3. Prove that there exist infinitely many positive integers $n$ such that $n !$ is a multiple of $n^{2}+1$
  4. Let $a, b, c$ be positive numbers such that $a+b+c=a b c .$ Prove that $$\frac{a}{b^{3}}+\frac{b}{c^{3}}+\frac{c}{a^{3}} \geq 1.$$
  5. Solve the equation $$\frac{4}{x}+\sqrt{x-\frac{1}{x}}=x+\sqrt{2 x-\frac{5}{x}}$$
  6. Let $A B C$ be a right triangle at $A$ with $A B<A C$, $B C=2+2 \sqrt{3}$ and its incircle has radius is $1 .$ Determine the value of the angles $\angle B$ and $\angle C$.
  7. Let $A B C$ be an acute triangle with orthocenter $H .$ Let $A G$ be the altitude through $A$ and let $D$ be the midpoint of $B C .$ The circles, whose diameters are $B C$ and $A D$ respectively, meet at $E$ and $F$. Prove that a) The circumcircles of $H, E$ and $G$ and of $H$, $F$ and $G$ are both tangent to the circle whose diameter is the side $B C$. b) $H, E$, and $F$ lie on the same line.
  8. Find the maximum value of the following expression $$x_{1}^{3} x_{2}^{2}+x_{2}^{3} x_{3}^{2}+\ldots+x_{n}^{3} x_{1}^{2}+n^{2(n-1)} x_{1}^{3} x_{2}^{3} \ldots x_{n}^{3}$$ where $x_{1}, x_{2}, \ldots, x_{n}$ are non-negative numbers whose sum is $1$ $(n \geq 2)$.
  9. Solve the system of equations $$\begin{cases}20\left(x+\dfrac{1}{x}\right) &=11\left(y+\dfrac{1}{y}\right) &=2007\left(z+\dfrac{1}{z}\right) \\ x y+y z+z x &=1 \end{cases}$$
  10. Find the limit at $x=0$ of the following function $$\frac{\sqrt{\frac{\cos 2 x+\sqrt[3]{1+3 x}}{2}}-\sqrt[3]{\frac{\cos 3 x+3 \cos x-\ln (1+x)^{4}}{4}}}{x}$$
  11. Let $\left(O_{1}\right)$ be a circle inside a triangle $A B C$ and touches both sides $A B$ and $A C$. Construct a second circle $\left(O_{2}\right)$ through $B$, $C$ and touches outside of $\left(O_{1}\right)$ at $T .$ Prove that the angle bisector of $\widehat{B T C}$ passes through the incenter of the triangle $A B C$.
  12. Let $A_{1} A_{2} A_{3} A_{4}$ be a tetrahedron with perpendicular opposite edges. Denote by $S_{i}$ the area of the opposing faces of the vertices $A_{I}$ $(i=1,2,3,4) .$ Find all possible points $M$ where the sum $$S_{1} \cdot M A_{1}+S_{2} \cdot M A_{2}+S_{3} \cdot M A_{3}+S_{4} \cdot M A_{4}$$ is smallest possible.

Issue 360

  1. Consider the following sequence $$a_{1}=3,\, a_{2}=4,\, a_{3}=6, \ldots, a_{n+1}=a_{n}+n, \ldots$$ a) Is $2006$ a number in the above sequence? b) What is the $2007^{\text {th }}$ number in this sequence? c) Calculate the sum of the first $100$ numbers in the above sequence?
  2. Let $A B C$ be a right triangle with right angle at $A,$ and $\widehat{A C B}=54^{\circ} .$ Choose a point $E$ on the open ray in opposite direction to $C A$ such that $\widehat{A B E}=54^{\circ} .$ Prove that $B C<A E$
  3. Consider the following sum of $2006$ terms $$S=\sqrt{\frac{2+1}{2}}+\sqrt[3]{\frac{3+1}{3}}+\sqrt[4]{\frac{4+1}{4}}+\ldots+200 \sqrt[2]{\frac{2007+1}{2007}}.$$ Find $[S]$. (Here $[a]$ denote the largest integer which does not exceed $a$.)
  4. Solve the following system of equations $$\begin{cases} x-\dfrac{4}{x} &=2 y-\dfrac{2}{y} \\ 2 x &=y^{3}+3\end{cases}$$
  5. Let $a, b, c$ be numbers, all greater than or equal $-\dfrac{3}{2},$ such that $$a b c+a b+b c+c a+a+b+c \geq 0.$$ Prove that $a+b+c \geq 0$.
  6. Let $M$, $N$ and $P$ be three points outside a given triangle $A B C$ such that $$\angle C A N=\angle C B M=30^{\circ},\,\angle A C N=\angle B C M=20^{\circ},\,\angle P A B=\angle P B A=40^{\circ} .$$ Show that $M N P$ is an equilateral triangle.
  7. Let $A B C$ be a right triangle with right angle at $A$ and let $A H$ be the altitute through $A$. Denote by $I$, $I_{1}$, $I_{2}$ the incenters of the triangles $A B C$, $A H B$ and $A H C$. respectively. Prove that the circumcircle of the triangle $I I_{1} I_{2}$ is exactly the incircle of the triangle $A B C$.
  8. Find all functions $f: \mathbb{R}^{+} \rightarrow \mathbb{R}^{+}$ which satisfies the following identity $$ f(x) \cdot f(y)=f(x+y f(x)), \forall x, y \in \mathbb{R}^{+}$$
  9. Find the largest real number $m$ such that there exists a number $k$ in the interval $[1 ; 2]$ such that the following inequality $$\left(\frac{a}{b}+\frac{b}{c}+\frac{c}{a}\right)^{2 k} \geq(k+1)(a+b+c)\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)+m$$ is satisfied for all positive real numbers $a, b, c$.
  10. Let $\left(u_{k}\right)$ $(k=1,2, \ldots, n,)$ be an increasing sequence and let $A_{n}$ be the set of all positive numbers of the form $u_{i}-u_{j}$ $(1 \leq j<i \leq n)$. Prove that if $A_{n}$ has fewer that $n$ elements, then $\left(u_{k}\right)$ $(k=1,2, \ldots, n)$ forms an arithmetic sequence.
  11. Let $A B C D$ be a quadrilateral with nonparallel opposite sides and let $O$ denote the intersection of the diagonals $A C$ and $B D .$ The circumcircles of the triangles $O A B$ and $O C D$ meet at $X$ and $O$. The circumcircles of the triangles $O A B$ and $O C B$ meet at $Y$ and $O .$ The circles with diameters $A C$ and $B D$ intersect at $Z$ and $T$. Prove that either $X$, $Y$, $Z$, $T$ are colinear or they lie in a circle.
  12. Consider a closed polygon $A_{1} A_{2} \ldots A_{n}$ where each of its $n$ sides is tangent to a sphere $\mathscr{C}$ at center $O$ and radius $R .$ Let $G$ be the centroid of the system of points $A_{i}$, $i=1,2, \ldots, n$ (that is, $\overline{G A_{1}}+\overline{G A_{2}}+\ldots+\overline{G A_{n}}=\overrightarrow{0}$) and denote $\angle A_{i}=\angle A_{i-1} A_{i} A_{i+1}$ (with the convention that $A_{0} \equiv A_{n}$ and $A_{n+1} \equiv A_{1}$.) Prove the inequality $$ \sum_{i=1}^{n} \frac{G A_{i}}{\sin \frac{A_{i}}{2}} \geq \frac{1}{n R} \sum_{1 \leq i<j \leq n} A_{i} A_{j}^{2}$$

Issue 361

  1. The number $(\overline{9 x})^{8}$, where $x \in\{0,1,2, \ldots, 9\}$ contains how many digits when written in decimal form?
  2. Let $H$ be the orthocenter of an acute triangle $A B C$ and denote by $M$ the midpoint of $B C$. The perpendicular line to $M H$ through $H$ meets $A B$ and $A C$ at $P$ and $Q$ respectively. Prove that $H P=H Q$.
  3. Prove that if $a, b, c$ and $d$ are distinct positive integers such that the sum $$\frac{a}{a+b}+\frac{b}{b+c}+\frac{c}{c+d}+\frac{d}{d+a}$$ is also an integer then the product $a b c d$ is a perfect square.
  4. Compare the values of the following two numbers $$A=\min _{|y| \leq 1} \max _{|x| \leq 1}\left(x^{2}+y x\right) ,\quad B=\max _{|x| \leq 1} \min _{|y| \leq 1}\left(x^{2}+y x\right).$$
  5. Solve the system of equations $$\begin{cases}\sqrt{x}+\sqrt{y}+\sqrt{z}-\dfrac{1}{\sqrt{x}}-\dfrac{1}{\sqrt{y}}-\dfrac{1}{\sqrt{z}} &=\dfrac{8}{3} \\ x+y+z+\dfrac{1}{x}+\dfrac{1}{y}+\dfrac{1}{z} &=\dfrac{118}{9} \\ x \sqrt{x}+y \sqrt{y}+z \sqrt{z}-\dfrac{1}{x \sqrt{x}}-\dfrac{1}{y \sqrt{y}}-\dfrac{1}{z \sqrt{z}} &=\dfrac{728}{27}\end{cases}.$$
  6. Let $A B C D$ be a parallelogram. Let $M$ be a point in the plane spanned by the parallelogram $A B C D$ such that $\widehat{M D A}=\widehat{M B A}$. Prove that the two triangles $M A B$ and $M C D$ share a common orthocenter.
  7. Let $AD$, $BE$, $CF$ be the three bisectors of a triangle $A B C$ ($D \in B C$,$ E \in C A$, $F \in A B)$. Denote by $O$ the circumcenter of this triangle and by $R$ its. Let $O_{1}$, $O_{2}$ and $O_{3}$ be the circumcenters of the triangles $A B D$. $B C E$ and $A C F$ respectively. Prove that $$\frac{3}{2} R \leq O O_{1}+O O_{2}+O O_{3}<2 R.$$
  8. Let $a, b, c$ be positive real numbers such that $\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c} \leq 1 .$ Find the smallest possible value of $$P=[a+b]+[b+c]+[c+a]$$ where $[x]$ is the largest integers which is smaller than $x$.
  9. Find all funtions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that the following equality holds for all $x, y \in \mathbb{R}$ $$f\left(x^{3}-y\right)+2 y\left(3 f^{2}(x)+y^{2}\right)=f(y+f(x)).$$
  10. Prove that in any triangle $A B C$, the following inequality holds $$-1<6 \cos A+3 \cos B+2 \cos C<7$$
  11. Let ABC be a triangle with circumcircle $(O ; R)$. Let $E$ be the midpoint of $A B$ and denote by $F$ the point on $A C$ such that $\dfrac{A F}{A C}=\dfrac{1}{3} .$ Construct a parallelogram $A E M F$ Prove that $$M A+M B+M C \leq \sqrt{11\left(R^{2}-O M^{2}\right)}.$$ Now suppose that $(O ; R)$ is fixed. Construct a triangle $A B C,$ inscribed in $(O, R)$ such that $$M A+M B+M C=\sqrt{11\left(R^{2}-O M^{2}\right)}.$$
  12. Let $A B C D$ be a tetrahedron whose sides $D A$, $D B$ and $D C$ are pairwise perpendicular. Denote by $x, y, z$ the angles formed from sides $A B$, $B C$ and $C A$ respectively. Prove that $$\left(2+\tan ^{2} x\right)\left(2+\tan ^{2} y\right)\left(2+\tan ^{2} z\right) \geq 64.$$

Issue 362

  1. Let $$S=\frac{1}{2^{2}}+\frac{2}{2^{3}}+\ldots+\frac{n}{2^{n+1}}+\ldots+\frac{2006}{2^{2007}}.$$ Compare $S$ with $1 .$
  2. Let $A H$ be the altitude of a right triangle $A B C,$ right angle at $A .$ Let $P$ and $Q$ be the incenters of $A B H$ and $A C H$ respectively. $P Q$ meets with $A B$ at $E$ and with $A C$ at $F .$ Prove that $A E=A F$.
  3. Find all possible pairs of positive integers $(x ; y)$ such that $4 x^{2}+6 x+3$ is a multiple of $2 x y-1$.
  4. Solve the following equation $$\sqrt{2 x^{2}+4 x+7}=x^{4}+4 x^{3}+3 x^{2}-2 x-7.$$
  5. Find the maximum value of the following expression $$A=\frac{x}{x^{2}+y z}+\frac{y}{y^{2}+2 x}+\frac{z}{z^{2}+x y},$$ where $x, y, z$ are positive real numbers such that $x^{2}+y^{2}+z^{2}=x y z$.
  6. $(O)$ is a point chosen arbitrarily inside a triangle $A B C$. $A O$, $B O$ and $C O$ meet $B C$, $C A$ and $A B$ at $M$, $N$ and $P$ respectively. Prove that the value of ratio $$\left(\frac{O A \cdot A P}{O P}\right)\left(\frac{O B \cdot B M}{O M}\right)\left(\frac{O C \cdot C N}{O N}\right)$$ does not depend on the position of the point $O$.
  7. Let $M$ be a point inside a circle $(O)$ with center at $O$ and radius $R .$ Draw two chords $C D$ and $E F$ through $M$ but not passing through $O$. The tangent lines with $(O)$ at $C$ and $D$ intersects at $A$, and the tangent lines at $E$ and $F$ meets at $B$. Prove that $O M$ and $A B$ are orthogonal.
  8. Let $p>2007$ be a prime number and $n$ is an integer exceeding $2006 p .$ Prove that $\mathrm{C}_{n}^{2006 \mathrm{p}}-\mathrm{C}_{k}^{2006}$ is a multiple of $p,$ where $k=\left[\dfrac{n}{p}\right]$ is the largest integer which is not larger than $\dfrac{n}{p}$
  9. Let $\left(u_{n}\right)$ be a sequence given by $u_{1}$ and the formula $u_{n+1}=\dfrac{k+u_{n}}{1-u_{n}},$ where $k>0$ where $n=1,2, \ldots .$ Given that $u_{13}=u_{1},$ find the value of $k$.
  10. Suppose $a, b,$ and $c$ are the three lengths of sides of a triangle. Prove that $$\frac{a}{\sqrt{a^{2}+3 b c}}+\frac{b}{\sqrt{b^{2}+3 c a}}+\frac{c}{\sqrt{c^{2}+3 a b}} \geq \frac{3}{2}$$
  11. Let $P$ be a arbitrary point in a quadrilateral $A B C D$ such that $\widehat{P A B}$, $\widehat{P B A}$, $\widehat{P B C}$, $\widehat{P C B}$, $\widehat{P C D}$, $\widehat{P D C}$, $\widehat{P A D}$ and $\widehat{P D A}$ are all acute angles. Let $M$, $N$, $K$, $L$ denote the feet of the altitude from $P$ on $A B$, $B C$, $C D$ and $D A$ respectively. Find the smallest value of the sum $$\frac{A B}{P M}+\frac{B C}{P N}+\frac{C D}{P K}+\frac{D A}{P L}$$
  12. Let $A B C . A^{\prime} B^{\prime} C^{\prime}$ be a triangular prism. $M$ and $N$ are two points, chosen on $A A^{\prime}$ and $C B$ respectively such that $$\frac{A M}{A A^{\prime}}=\frac{C N}{C B} .$$ Prove that the length of $M N$ is no less than the distance from $A^{\prime}$ to the straight line $C^{\prime} B$.

Issue 363

  1. The first $100$ positive integer numbers are written consecutively in a certain order. Call the resulting number $A$. Is $A$ a multiple of $2007$?
  2. Let $ABC$ be a nonisosceles triangle, where $AB$ is the shortest side. Choose a point $D$ in the opposite ray of $BA$ such that $BD=BC$. Prove that $\angle ACD<90^\circ$.
  3. Let $a$, $b$, $c$ be positive reals such that $a+b+c=1$. Prove that $$\left(a+\dfrac{1}{b}\right)\left(b+\dfrac{1}{c}\right)\left(c+\dfrac{1}{a}\right)\geq \left(\dfrac{10}{3}\right)^3.$$
  4. Solve the equation in $\mathbb{R}$ $$(x^4+5x^3+8x^2+7x+5)^4+(x^4+5x^3+8x^2+7x+3)^4=16$$
  5. Let $AH$ denote the altitude of a right triangle $ABC$, right angle at $A$ and suppose that $AH^2=4AM\cdot AN$ where $M$, $N$ are the feet of the altitude from $H$ to $AB$ and $AC$, respectively. Find the measures of the angles of triangle $ABC$.
  6. Find all $(x,y)\in\mathbb{Z}^2$ such that $$x^{2007}=y^{2007}-y^{1338}-y^{669}+2.$$
  7. Let $(x_n)$ be a sequence given by $$x_1=5,\quad x_{n+1}=x_n^2-2,\,\forall n\geq 1.$$ Calculate $\displaystyle\lim_{n\to\infty}\dfrac{x_{n+1}}{x_1x_2\cdots x_n}.$
  8. Let $a$, $b$, $c$ and denote the three sides of a triangle $ABC$. Its altitudes are $h_a$, $h_b$, $h_c$ and the radius of its three escribed circles are $r_a$, $r_b$, $r_c$. Prove that $$\dfrac{a}{h_a+r_a}+\dfrac{b}{h_b+r_b}+\dfrac{c}{h_c+r_c}\geq \sqrt{3}.$$
  9. In a quadrilateral $ABCD$, where $AD=BC$ meets at $O$, and the angle bisector of the angles $DAB$, $CBA$ meets at $I$. Prove that the midpoints of $AB$, $CD$, $OI$ are colinear.
  10. Prove that for all $a,b,c\in [1,+\infty)$ we have $$\left(a-\dfrac{1}{b}\right)\left(b-\dfrac{1}{c}\right)\left(c-\dfrac{1}{a}\right)\geq \left(a-\dfrac{1}{a}\right)\left(b-\dfrac{1}{b}\right)\left(c-\dfrac{1}{c}\right).$$
  11. Find the number of binary strings of length $n$ $(n>3)$ in which the substring $01$ occurs exactly twice.
  12. Let $f:\mathbb{N}\to\mathbb{R}$ be a function such that $$f(1)=\dfrac{2007}{6},\quad \dfrac{f(1)}{1}+\dfrac{f(2)}{2}+\cdots+\dfrac{f(n)}{n}=\dfrac{n+1}{2}\cdot f(n)\forall n\in\mathbb{N}.$$ Find the limit $\displaystyle\lim_{n\to\infty} (2008+n)f(n)$.

Issue 364

  1. Find the last two digit numbers of $3^{9999}-2^{9999}$.
  2. Compare the sum (consisting of $n+1$ terms) $$S_{n}=\frac{2}{101+1}+\frac{2^{2}}{101^{2}+1}+\ldots+\frac{2^{n+1}}{101^{2 \prime}+1}$$ with $0,02$.
  3. Let $a$, $b$ and $c$ be positive numbers such that $a b+b c+c a=1$. Prove the inequality $$\frac{1}{a b}+\frac{1}{b c}+\frac{1}{c a} \geq 3+\sqrt{\frac{1}{a^{2}}+1}+\sqrt{\frac{1}{b^{2}}+1}+\sqrt{\frac{1}{c^{2}}+1}.$$ When does equality occur?
  4. Solve the equation $$\sqrt{7 x^{2}-22 x+28}+\sqrt{7 x^{2}+8 x+13}+\sqrt{31 x^{2}+14 x+4}=3 \sqrt{3}(x+2).$$
  5. Let $A B C D$ be a rectangular with $A B<B C$. Let $M$ be a point, different from $A$ and $B$, on the half-circle with $A B$ as its diameter and on the same side with $CD$. $MA$ and $MB$ meet $CD$ at $P$ and $Q$ respectively. $M C$ and $M D$ meet $A B$ at $E$ and $F$ respectively. Find the position of the point $M$ on the half-circle such that the sum $P Q+E F$ is smallest possible. Calculate this smallest value.
  6. Find all pairs of positive integers $a$, $b$ such that $q^{2}-r=2007,$ where $q$ and $r$ are respectively the quotient and the remainder obtained when dividing $a^{2}+b^{2}$ by $a+b$
  7. Consider the equation $$a x^{3}-x^{2}+b x-1=0$$ where $a, b$ are real numbers, $a \neq 0$ and $a \neq b$ such that all of its roots are positive real numbers. Find the smallest value of $$P=\frac{5 a^{2}-3 a b+2}{a^{2}(b-a)}.$$
  8. Choose five points $A$, $B$, $C$, $D$ and $E$ on a sphere with radius $R$ such that $$\widehat{B A C}=\widehat{C A D} =\widehat{D A E}=\widehat{E A B}=\frac{2}{3} \widehat{B A D}=\frac{2}{3} \widehat{C A E}.$$ Prove the inequality $$A B+A C+A D+A E \leq 4 \sqrt{2} R.$$
  9. Suppose that $M$, $N$ and $P$ are three points lying respectively on the edges $A B$, $B C$. $C A$ of a triangle $A B C$ such that $$A M+B N+C P=M R+N C+P A.$$ Prove the inequality $S_{M N P} \leq \dfrac{1}{4} S_{A B C}$.
  10. Find the limit $$\lim_{n \rightarrow+\infty}\sqrt{2-\sqrt{2}} \cdot \sqrt{2-\sqrt{2+\sqrt{2}}} \cdots \underbrace{\sqrt{2-\sqrt{2+\sqrt{2+\ldots+\sqrt{2}}}}}_{n \text { signsor railiad }}$$
  11. Denote by $[x]$ the largest integer not exceeding $x$ and write $\{x\}=x-[x] .$ Find the limit $\displaystyle \lim _{n \rightarrow+\infty}(7+4 \sqrt{3})^{n}$.
  12. Let $n$ be a positive integer and $2 n+2$ real numbers $a, b, a_{1}, a_{2}, \ldots, a_{n}, b_{1}, b_{2}, \ldots, b_{n}$ such that $a_{i} \neq 0(i=1,2, \ldots, n)$ and the function $$F(x)=\sum_{i=1}^{n} \sqrt{a_{i} x+b_{i}}-(a x+b)$$ satisfies the following property: There exists distinct real numbers $\alpha, \beta$ such that $F(\alpha)=F(\beta)=0$ Prove that $\alpha$ and $\beta$ are the only real solutions of the equation $F(x)=0$

Issue 365

  1. Write $2005^{2006}$ as a sum of natural numbers, then calculate the sum of all the digits occurred in these summands. Can one obtain either $2006$ or $2007$ in this way? Why?
  2. Let $A B C$ be an isosceles right triangle, right angle at $A$. Choose a point $D$ in the half-plane on the side of $A B$ that does not contain $C$ such that $D A B$ is also an isosceles right triangle, with right angle at $D$. Let $E$ be a point (differs from $A$) on $A D$. The perpendicular line with $B E$ through $E$ intersects with $A C$ at $F .$ Prove that $E F=E B$.
  3. Find the maximum and minimum values of the following expression $$\frac{x-y}{x^{4}+y^{4}+6}.$$
  4. Solve the equation $$\sqrt{2}\left(x^{2}+8\right)=5 \sqrt{x^{3}+8}.$$
  5. The two diagonals $A C$ and $B D$ of an inscribed quadrilateral $A B C D$ intersect at $O$. The circumcircles $\left(S_{1}\right)$ of $A B O$ and $\left(S_{2}\right)$ of $CDO$ meet at $O$ and $K$. Through $O$, draw parallel lines with $A B$ and $C D$; they meet with $\left(S_{1}\right)$ and $\left(S_{2}\right)$ at $N$ and $M$, respectively. Let $P$ and $Q$ be two points on $O N$ and $O M$ respectively such that $\dfrac{O P}{P N}=\dfrac{M Q}{Q}$. Prove that $O$, $K$, $P$ and $Q$ lie on the same circle.
  6. There are nine cards in a box, labelled from $1$ to $9$. How many cards should be taken from the box so that the probability of getting a card whose label is a multiple of 4 will be greater than $\dfrac{5}{6} ?$
  7. Let $x_{1}, x_{2}, \ldots, x_{n}$ be $n$ arbitrary real numbers chosen in a given closed interval $[a ; b]$. Prove the inequality $$\sum_{i=1}^{n}\left(x_{i}-\frac{a+b}{2}\right)^{2} \geq \sum_{i=1}^{n}\left(x_{i}-\frac{1}{n} \sum_{i=1}^{n} x_{i}\right)^{2}.$$ When does equality occur?
  8. Let $H$ be the orthocenter of an acute triangle $A B C$ whose edges are $B C=a, C A=b$, and $A B=c .$ Prove that $$\frac{H A^{2}+H B^{2}+H C^{2}}{a^{2}+b^{2}+c^{2}} \leq(\cot A \cdot \cot B)^{2}+(\cot B \cdot \cot C)^{2}+(\cot C \cdot \cot A)^{2}.$$
  9. Let $p$ be a prime number, greater than $3$ and $k=\left[\dfrac{2 p}{3}\right]$. (The notation $[x]$ denote the largest integer which is not exceeding $x$.) Prove that $\displaystyle \sum_{i=1}^{k} C_{p}^{i}$ is a multiple of $p^{2}$.
  10. Find the measures of the angles of triangle $A B C$, such that $$\cos \frac{5 A}{2}+\cos \frac{5 B}{2}+\cos \frac{5 C}{2}=\frac{3 \sqrt{3}}{2}.$$
  11. Let $\left(x_{n}\right)(n=1,2, \ldots)$ be a boundedabove sequence such that $$x_{n+2} \geq \frac{1}{4} x_{n+1}+\frac{3}{4} x_{n},\,\forall n=1,2, \ldots$$ Prove that this sequence has limit.
  12. Let $\alpha$, $\alpha^{\prime}$; $\beta$, $\beta^{\prime}$ and $\gamma$, $\gamma^{\prime}$ be the dihedral angles, opposite to the edges $B C$, $DA$, $CA$, $DB$ and $A B$, $D C$ respectively, of a tetrahedron $A B C D$. Prove the inequality $$\sin \alpha+\sin \alpha^{\prime}+\sin \beta+\sin \beta^{\prime}+\sin \gamma+\sin \gamma^{\prime} \leq 4 \sqrt{2}.$$ When does equality occur?

Issue 366

  1. Let $\left(a_{n}\right)$ be $a_{1}=3$, $a_{2}=8$, $a_{3}=13$, $a_{4}=24$, $a_{5}=31$, $a_{6}=48, \ldots,$ in general, $$a_{n+2}=\begin{cases}a_{n}+4 n+8 & \text { if } n \text { is odd } \\ a_{n}+4 n+6 & \text { if } n \text { is cven }\end{cases}.$$ a) Do the numbers $2007$ and $2024$ appear in this sequence?.
    b) Find the $2007$-th number in this sequence.
  2. Let $ABC$ be a triangle with $\widehat{B A C}=40^{\circ}$ and $\widehat{A B C}=60^{\circ} .$ Denote $D$ and $E$ are two points on $A B$ and $A C$ respectively such that $\widehat{D C B}=70^{\circ}$ and $\widehat{E B C}=40^{\circ}$; $D C$ and $E B$ meets at a point $F$. Prove that $A F$ and $B C$ are orthogonal.
  3. Let $x,y,z,t$ and $u$ be positive real numbers such that $x+y+z+t+u=4$. Find the smallest possible value of the following expression $$P=\frac{(x+y+z+t)(x+y+z)(x+y)}{x y z t u}.$$
  4. Solve for $x$ $$2 \sqrt{x+1}+6 \sqrt{9-x^{2}}+6 \sqrt{(x+1)\left(9-x^{2}\right)}=38+10 x-2 x^{2}-x^{3}.$$
  5. Let $P A$ and $P B$ be two tangent lines through a point $P$ outside a circle with center $O$. $OP$ and $A B$ meet at $M$. Draw a secant $C D$ through $M$ ($C D$ does not contain $O$). The tangent lines at $C$ and $D$ meet at $Q$. Find the measure of the angle $O P Q$.
  6. Find all pairs of positive integers $(x; y)$ such that $$x^{y}+y=y^{x}+x.$$
  7. Prove that if the equation $x^{3}+a x^{2}+b x+c=0$ has three distinct real roots, then so is $$x^{3}+a x^{2}+\left(-a^{2}+4 b\right) x+a^{3}-4 a b+8 c=0.$$
  8. Let $I$ denote the incenter of an acute triangle $A B C$. The incircle $(I)$ touches $B C$, $C A$ and $A B$ at $D$, $E$ and $F$ respectively. The angle bisector of $B I C$ meets $B C$ at $M$. $A M$ meets $E F$ at $P$. Prove that $$P D \geq \frac{1}{2} \sqrt{4 D E \cdot D F-E F^{2}}.$$
  9. Let $a_{1}, a_{2}, \ldots, a_{2007}$ be pairwise distinct integers, all greater than $1$ such that $\displaystyle \sum_{i=1}^{20} a_{i}=2017035$. Could it be possible that the sum $\displaystyle \sum_{i=1}^{2007} a_{i}^{a_{i} a_{i}}$ is a perfect square?
  10. Find all polynornial with real coefficients $P(x)$ such that $$P(P(x)+x)=P(x) P(x+1),\,\forall x \in \mathbb{R}.$$
  11. Let $a_{1}, a_{2}, a_{3}, a_{4}$ and $a_{5}$ be nonnegative real numbers whose sum equal $1$. Prove the inequality $$a_{2} a_{3} a_{4} a_{5}+a_{1} a_{3} a_{4} a_{5}+a_{1} a_{2} a_{3} a_{5}+a_{1} a_{2} a_{3} a_{4} \leq \frac{1}{256}+\frac{3275}{256} a_{1} a_{2} a_{3} a_{4} a_{5}.$$ When does equality occur?
  12. Let $P$ be the intersection of the diagonals of an inscribed quadrilateral $A B C D$. Prove that all four Euler lines of the triangles $PAB$, $PBC$, $PCD$ and $PDA$ intersect in a single point. (The Euler line of a triangle is the line connecting its centroid and its orthocenter).
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MOlympiad.NET: Mathematics and Youth Magazine Problems 2007
Mathematics and Youth Magazine Problems 2007
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