# Mathematics and Youth Magazine Problems 2009

### Issue 379

1. Find all pairs of integers $a$, $b$ such that $$a^{2}+a b+b^{2}=a^{2} b^{2}$$
2. Let $ABC$ be an isosceles triangle (at vertex $A$) such that $\widehat{B A C} \geq 90^\circ$. Choose a point $M$ on $A C$, and let $A H$ and $C K$ be the altitudes from $A$ and $C$ onto $B M$ respectively $(H, K$ are the feet of these altitudes) such that $B H=H K+K C$. Find the angle $B A C$.
3. Solve for $x$$\sqrt{\frac{5 \sqrt{2}+7}{x+1}}+4 x=3 \sqrt{2}-1$$ 4. Find$a$,$b$such that the maximum value of the expression $$|||x+1|-2|-(a x+b)|$$ where$-3 \leq x \leq 4$is smallest possible. 5. Let$A B C D$be a rectangular, let$I$be the midpoint of$C D$, and let$E$be a point on$A B$. The altitude onto$D E$through$I$meets$D E$and$A D$at$M$and$H$respectively. The altitude onto$C E$through$I$meets$C E$and$B C$at$N$and$K$, respectively.$E I$intersects with$H K$at$G$. Prove that a) The points$E$,$G$,$N$,$K$,$B$lie on the same circle. b) The points$E$,$G$,$M$,$H$,$A$lie on the same circle. 6. Let$A B C$be a triangle. Prove the inequality $$\cos A \cos B \cos C \leq \frac{1}{8} \cos (B-C) \cos (C-A) \cos (A-B)$$ 7. Solve for$x$$$3^{x}\left(4^{x}+6^{x}+9^{x}\right)=25^{x}+2.16^{x}$$ 8. Prove that the union of six hemispheres whose diameters are the sides of a given tetrahedron must contain the tetrahedron itself. 9. Let$A B C$be a triangle. Choose the pair of points$A_{1}$,$A_{2}$;$B_{1}$,$B_{2}$;$C_{1}$,$C_{2}$on the sides$B C$,$C A$and$A B$respectively such that$A_{1} A_{2} B_{1} B_{2} C_{1} C_{2}$is a convex hexagon with equal opposite sides and the triangles$A A_{1} A_{2}$,$B B_{1} B_{2}$,$CC_{1}C_{2}$have the same areas. Prove that the lines$A_{1} B_{2}$,$B_{1} C_{2}$,$C_{1} A_{2}$are colinear. 10. Let$a$,$b$,$c$be non - negative real numbers such that $$\sqrt{a^{3}+b^{3}}+\sqrt{b^{3}+c^{3}}+\sqrt{c^{3}+a^{3}}+a b c=3.$$ Prove that the smallest value of the expression $$P=\frac{a^{3}}{b^{2}+c^{2}}+\frac{b^{3}}{c^{2}+a^{2}}+\frac{c^{3}}{a^{2}+b^{2}}$$ is$\sqrt{32}m$where$m$is the real root of the equation$t^{3}+54 t-162=0$. 11. Let$f: \mathbb{N} \rightarrow \mathbb{R}$be the function with initial values$f(0)=0$,$f(1)=1$such that $$f(n+2)-2011 f(n+1)+f(n)=0.$$ What is the probability that$f(n)$is a prime number, where$n$is a randomly chosen number from the set of integers$\{0,1,2, \ldots, 2008\}$? 12. Find all functions$f(x)$such that it is continuous on$[0 ; 1]$, differentiable on the open interval$(0 ; 1)$and the following two conditions are satisfied $$f(0)=f(1)=\frac{2009}{11},\quad 20 f(x)+11 f(x)+2009 \leq 0,\, \forall x \in(0 ; 1).$$ ### Issue 380 1. Compare$\dfrac{1}{6}$with $$A=\frac{1}{5}-\frac{1}{7}+\frac{1}{17}-\frac{1}{31}+\frac{1}{65}-\frac{1}{127}$$ 2. Let$ABC$be an isosceles triangle (at vertex$C$) with$\widehat{A C B}=100^{\circ}$.$M$is a point chosen on the ray$C A$such that$C M=A B$. Find the measure of the angle$\widehat{C M B}$. 3. Find all triple$(x, y, z)$of integers such that $$\begin{cases}y^{3} &=x^{3}+2 x^{2}+1 \\ x y &=z^{2}+2\end{cases}$$ 4. Find all pair of numbers$x$and$y$such that the following conditions hold$x>1$,$0<y<1$,$x+y \leq \sqrt{5}$,$\dfrac{1}{x}+\dfrac{1}{y} \leq \sqrt{5}$and$\dfrac{x}{x+1}+\dfrac{y}{1-y} \leq \sqrt{5}$. 5. Let$A B C D$be an isosceles trapezoid inscribed inside a circle$\left(O_{1}: R\right)$and circumscribes the circle$\left(O_{2} ; r\right)$. Let$d=O_{1} O_{2}$. Prove that $$\frac{1}{r^{2}} \geq \frac{2}{R^{2}+d^{2}} .$$ When does equality occur? 6. Prove that in any triangle$A B C$, the following inequality holds $$\frac{\cot A \cdot \cot B \cdot \cot C}{\sin A \cdot \sin B \cdot \sin C} \leq\left(\frac{2}{3}\right)^{3}$$ 7. There are$17$ornament betel-nut trees around a circular pond. How many ways are there to chopped off$4$trees with the condition that no two consecutive trees be removed? 8. Solve the following system of equations with parameter$a$$$\begin{cases} 2 x\left(y^{2}+a^{2}\right) &=y\left(y^{2}+9 a^{2}\right) \\ 2 y\left(z^{2}+a^{2}\right) &=z\left(z^{2}+9 a^{2}\right) \\ 2 z\left(x^{2}+a^{2}\right) &=x\left(x^{2}+9 a^{2}\right)\end{cases}$$ 9. Let$\left(x_{n}\right)$,$n=0,1,2, \ldots$be a sequence given by the following recursive formula $$x_{0}=a,\quad x_{n+1}=x_{n}+\sin x+2 \pi,\, n=0,1,2, \ldots$$ where$a, x \in \mathbb{R}$. Prove that the limit$\displaystyle\lim_{n \rightarrow+\infty} \frac{x_{1}+\ldots+x_{n}}{n^{2}}$exists and find its exact value. 10. Prove that a)$\displaystyle\sum_{k=0}^{n}\left(\frac{k}{n}-x\right)^{2} C_{n}^{k} x^{k}(1-x)^{n-k}=\frac{x(1-x)}{n},\, \forall x \in \mathbb{R}$b)$\displaystyle\sum_{k=0}^{n}\left|\frac{k}{n}-x\right|C_{n}^{k} x^{k}(1-x)^{n-k} \leq \frac{1}{2 \sqrt{n}},\, \forall x \in[0 ; 1]$. 11. Find all functions$f: \mathbb{R} \rightarrow \mathbb{R}$such that $$f\left(n^{2}\right)=f(m+n) f(n-m)+m^{2},\, \forall m, n \in \mathbb{R}$$ 12. On a given plane, choose a point$A$outside a circle whose center is at a point$O$and whose radius equals$R$. A chord$M N$with constant length moves on the circle such that the two line segments$M N$and$O A$always intersect. Determine the positions of$M$and$N$such that the sum$A M+A N$is a) greatest possible. b) smallest possible. ### Issue 381 1. Rearrange the following rational numbers in increasing order $$\frac{1005}{2002}, \frac{1007}{2006}, \frac{1009}{2010}, \frac{1011}{2014}$$ 2. In an isosceles triangle$A B C$(at vertex$A$), choose a point$M$such that $$\widehat{M A C}=\widehat{M B A}=\widehat{M C B}.$$ Compare the areas of the triangles$A B M$and$C B M$. 3. Determine the sum of all rational numbers of the form$\dfrac{a}{b}$where$a$,$b$are natural divisors of$27000$and$\gcd(a, b)=1$. 4. Given$a$,$b$,$c$such that$a>0$,$b>c$,$a^{2}=b c$,$a+b+c=a b c$. Prove the inequalities $$a \geq \sqrt{3},\quad b \geq \sqrt{3},\quad 0<c \leq \sqrt{3}.$$ 5. Let$ABCD$be a quadrilateral where$\widehat{A B C}=\widehat{A D C}=90^{\circ}$and$\widehat{B C D}<90^{\circ}$. Choose a point$E$on the opposite ray of$A C$such that$D A$is the angle-bisector of$B D E$. Let$M$be chosen arbitrarily between$D$and$E$, choose another point$N$on the opposite ray of$B E$such that$\widehat{N C B}=\widehat{M C D}$. Prove that$M C$is the angle bisector of$D M N$. 6. Prove that the equation$x^{3}+3 y^{3}=5$has infinitely many rational solutions. 7. If$a$,$b$,$c$are the length of the sides of a triangle, prove that $$\frac{1}{\sqrt{a b+a c}}+\frac{1}{\sqrt{b c+b a}}+\frac{1}{\sqrt{c a+c b}} \geq \frac{1}{\sqrt{a^{2}+b c}}+\frac{1}{\sqrt{b^{2}+a c}}+\frac{1}{\sqrt{c^{2}+a b}}.$$ 8. Let$A B C D . A^{\prime} B^{\prime} C^{\prime} D^{\prime}$be a parallelepiped and let$S_{1}$,$S_{2}$,$S_{3}$denote the areas of the sides$A B C D$,$A B B^{\prime} A^{\prime}$and$A D D^{\prime} A^{\prime}$respectively. Given that the sum of squares of the areas of all sides of the tetrahedron$A B^{\prime} C D^{\prime}$equals$3$, find the smallest possible value of the following expression $$T=2\left(\frac{1}{S_{1}}+\frac{1}{S_{2}}+\frac{1}{S_{3}}\right)+3\left(S_{1}+S_{2}+S_{3}\right)$$ ### Issue 382 1. Compare$\dfrac{2009}{2008^{2}}$with the sum (consisting of$2010$terms) $$\frac{1}{2009}+\frac{2}{2009^{2}}+\frac{3}{2009^{3}}+\ldots+\frac{2009}{2009^{2009}}+\frac{2010}{2009^{2010}}$$ 2. Find a root of the polynomial$P(x)=x^{3}+a x^{2}+b x+c$given that it has at least one root and$a+2 b+4 c=-\dfrac{1}{2}$. 3. Let$a_{1}$,$a_{2}$,$a_{3}$,$a_{4}$,$a_{5}$,$a_{6}$,$a_{7}$,$a_{8}$,$a_{9}$be non negative real numbers whose sum equals$1$. Put$S_{k}=a_{k}+a_{k+1}+a_{k+2}+a_{k+3}(k=1,2, \ldots, 6)$. Determine the smallest possible value of $$M=\max \left\{S_{1}, S_{2}, S_{3}, S_{4}, S_{5}, S_{6}\right\}.$$ 4. Let$m$,$n$,$a$,$b$and$c$be real numbers such that the following conditions hold $$\begin{cases}m^{1000}+n^{1000} &=a \\ m^{2000}+n^{2000} &=\dfrac{2 b}{3}\\ m^{5000}+n^{5000} &=\dfrac{c}{36}\end{cases}.$$ Find a formula relating$a$,$b$and$c$which does not involve$m$,$n$. 5. Let$A H$be the altitude from$A$of a triangle$A B C$. Choose a point$D$on the half-plane created by$B C$which contains$A$such that$D B=D C=\dfrac{A B}{\sqrt{2}}$. Prove that the lengths of the line segments$B D$,$D H$and$H A$are the side lengths of a right triangle. 6. Determine the maximum possible value of$x^{2}+y^{2}$where$x$and$y$are two integers chosen arbitrarily within the interval$[-2009 ; 2009]$such that $$\left(x^{2}-2 x y-y^{2}\right)^{2}=4.$$ 7. Consider two polynomials with real coefficients $$P(x)=x^{n}+a_{n-1} x^{n-1}+\ldots+a_{1} x+a_{0}$$ and$Q(x)=x^{2}+x+2009$. Given that$P(x)$has$n$distinct real roots but$P(Q(x))$does not have any real solution. Prove that$P(2009)>\dfrac{1}{4^{n}}$. 8. Let$ABCDEF$be a regular hexagon and let$G$be the midpoint of$B F$. Choose a point$I$on$B C$such that$B I=B G$. Let$H$be a point on$I G$such that$\widehat{C D H}=45^{\circ}$and$K$is a point on$E F$such that$\widehat{D K E}=45^{\circ}$. Prove that$D H K$is an equilateral triangle. ### Issue 383 1. Let$a$,$b$,$c$be integers such that $$A=\frac{a^{2}+b^{2}+c^{2}-a b-b c-c a}{2}$$ is a perfect square. Prove that$a=b=c$. 2. Let$A B C$be a triangle in which$\widehat{B A C}=75^{\circ}$. Given that the altitude$A H$has length$A H=\dfrac{B C}{2}$. Prove that$A B C$is an isosceles triangle. 3. Let$x$,$y$be two nonnegative integers where$x>1$and$2 x^{2}-1=y^{15}$. Prove that$x$is divisible by$15$. 4. Let$A B C D$be a parallelogram. The ray$D x$from$D$is perpendicular to$D C$and lies on the half-plane divided by$C D$which does not contains$B$. Choose a point$E$on$D x$such that$D E=D C$. Draw an isosceles right triangle$B E F$(right angle at$F$) such that$F$and$D$are on the same half-plane divided by$B C \cdot E H$is the altitude onto$B C$. Prove that$FD$and$H$are collinear. 5. Let$a$,$b$,$c$be three sides of a right triangle,$a$is the hypotenuse. Prove that the equation$-a x^{2}+b x+c=0$has two distinct roots$x_{1}$,$x_{2}$such that$-\sqrt{2}<x_{1}<x_{2}<\sqrt{2}$. 6. Consider a convex polygon with$2009$vertices. A scissor is used to cut along all of its diagonals, thereby dividing the original polygon into smaller convex polygons. How many vertices are there of a resulting polygon with the greatest number of edges? 7. Determine the smallest value of the expression $$T=\frac{a b+b c}{a^{2}-b^{2}+c^{2}}+\frac{b c+c a}{b^{2}-c^{2}+a^{2}}+\frac{c a+a b}{c^{2}-a^{2}+b^{2}}$$ where$a$,$b$,$c$are three sides of a triangle$A B C$and$a b c=1$. 8. Let$A B C D . A_{1} B_{1} C_{1} D_{1}$be a cube. On its three skew edges, choose three points$M$,$N$,$P$. Determine the positions of the points$M$,$N$,$P$such that the triangle$M N P$has a) The smallest perimeter possible. b) The smallest area possible. ### Issue 384 1. Replace the distinct letters by distinct numbers such that the following expression becomes a true equality $$\mathrm{VE}+\mathrm{TRUONG}+\mathrm{SA}=22 \times 12 \times 2009.$$ 2. It is well-known that the two right triangles whose side lengths are positive integers$(5,12,13)$and$(6,8,10)$possess additional property that the area of each triangle equals its perimeter. Are there other triangles with similar properties? 3. Suppose given$1003$nonzero rational numbers in which any quadruple form a proportion. Prove that at least$1000$numbers are equal. 4. Let$x$,$y$,$z$be non-negative real numbers such that$x+y+z=1$. Find the maximum valuc of the following expression $$P=(x+2 y+3 z)(6 x+3 y+2 z).$$ 5. Let$A B C D$be a cyclic quadrilateral, inscribed in a circle$(O)$. The angle bisectors of$B A D$and$B C D$meet at a point$K$on the diagonal$B D$. Let$Q$be the second intersection point (different from$A$) of$A P$and the circle$(O)$;$M$and$N$be respectively the midpoints of$B D$and$C P$. The line through$C$and parallel to$A D$meets$A M$at$P$. Prove that a)$S_{A BQ}=S_{A D Q}$. b)$DN$is perpendicular to$C P$. 6. For each natural number$n,$let$p(n)$be its largest odd divisor. Determine the sum $$\sum_{n=2006}^{4012} p(n)$$ 7. Solve for$x$$$\sqrt{3 x-2}=-4 x^{2}+21 x-22$$ 8. Let$A B C$be a triangle whose circumcircle is$(O)$and such that$A C<A B$. The tangent lines to$(O)$at$B$,$C$intersect at$T$. The line through$A$and perpendicular to$A T$meet$B C$at$S$. Choose the points$B_{1}$and$C_{1}$on$S T$such that$T B_{1}=T C_{1}=T B$and that$C_{1}$lies between$S$and$T$. Prove that$A B C$and$A B_{1} C_{1}$are similar. ### Issue 385 1. Let$S(n)$be denote the sum of the digits of$n$. Find a positive integer$n$such that$S(n)=n^{2}-2009 n+11$. 2. Inside a square$A B C D$thoose two points$P$,$Q$such that$B P$and$D Q$are parallel and$B P^{2}+D O^{2}=P Q^{2}$. Find the measure of the angle$P A Q$. 3. Compare$2008$with the sum$S$of$2009$terms $$S=\frac{2008+2007}{2009+2008}+\frac{2008^{2}+2007^{2}}{2009^{2}+2008^{2}}+\ldots +\frac{2008^{2009}+2007^{2009}}{2009^{2009}+2008^{2009}}.$$ 4. Let$a$,$b$,$c$be three positive numbers such that$a+b+c=1$. Find the least value of the expression $$P=\frac{9}{1-2(a b+b c+c a)}+\frac{2}{a b c}.$$ 5. Let$B$,$C$be two fixed points on the circle$(\omega)$such that$B C$does not pass through the center of$(\omega) .$On the major arc$B C$, choose a point$A$differs from$B$and$C$. Another point$M$moves on the line segment$B C$. The lines passing through$M$and parallel to$A B$,$A C$intersect$A C$and$A B$at$F$and$E$, respectively. When$M$moves on the line segment$B C$and for each point$A$, let$x$be the least possible length of$EF$. Find the positions of$A$and$M$such that$x$is greatest possible. 6. Find the greatest and the least value of the expression $$A=\frac{y^{2}}{25}+\frac{t^{2}}{144}$$ where$x$,$y$,$z$,$t$satisfy the system of equations $$\begin{cases}x^{2}+y^{2}+2 x+4 y-20 &=0 \\ t^{2}+z^{2}-2 t-143 &= 0\\ x t+y z-x+t+2 z-61 &\geq 0\end{cases}$$ 7. Consider the sequence$\left(u_{n}\right)$defined as follow $$u_{0}=9,\, u_{1}=161,\quad u_{n}=18 u_{n-1}-u_{n-2},\,\forall n=2,3, \ldots$$ Prove that for any$n$,$\dfrac{u_{n}^{2}-1}{5}$is always a perfect square. 8. Let$P$be an arbitrary point inside a given triangle$A B C$. Let$A'$,$B'$,$C'$be the orthogonal projection of$P$on$B C$,$C A$,$A B$respectively. Let$I$be the incenter and$r$be the inradius of the triangle$A B C$. Find the least value of the expression $$P A^{\prime}+P B^{\prime}+P C^{\prime}+\frac{P I^{2}}{2 r}$$ ### Issue 386 1. Find the last two decimal digits of the sum$2008^{2009}+2009^{2008}$2. Let$ABC$be an isosceles right triangle with the right angle at$A$. Let$G$be a point on$A B$such that$A G=\dfrac{1}{3} A B$, let$M$be the midpoint of$B C$and$E$be the foot of the altitude from$M$to$CG$. The two lines$M G$and$A C$meet at$D$. Prove that$D E=B C$. 3. Find all integer solutions of the equation $$4 x^{4}+2\left(x^{2}+y^{2}\right)^{2}+x y(x+y)^{2}=132$$ 4. Let$a$,$b$and$c$satisfy the conditions$a \leq b \leq c$and $$a+b+c=\frac{1}{a}+\frac{1}{b}+\frac{1}{c}.$$ Find the least value of the expression$P=a b^{2} c^{3}$. 5. A triangle$A B C$inscribed in a circle centered at$O$, radius$R$,$A D$is its anglebisector. Let$E$,$F$be the circumeenters pf the triangles$A B D$and$A C D$. respectively. Given that$B C=a$. Determine the area of the quadrilateral$A E O F$. 6. Let$A B C$be a triangle with incenter$I$such that its centroid$G$lies inside$(I)$. Let$a$,$b$,$c$be the lengths of the sides$B C$,$A C$,$A B$, respectively. Find the greatest and least value of the following expression $$P=\frac{a^{2}+b^{2}+c^{2}}{a b+b c+c a}$$ 7. Let$x$,$y$,$z$be positive numbers satisfying $$x^{2}+y^{2}+z^{2}=\frac{1-16 x y z}{4}.$$ Find the least value of the expression $$S=\frac{x+y+z+4 x y z}{1+4 x y+4 y z+4 z x}.$$ 8. Solve for$x$$$2^{x}+5^{x}=2-\frac{x}{3}+44 \log _{2}\left(2+\frac{131 x}{3}-5^{x}\right)$$ 9. Let$A B C$be a right triangle, right angle at$A$,$M$is the midpoint of$B C$. Construct a right angle$P M Q$with$P \in A B$,$Q \in A C$. Prove that $$P Q^{2} \geq A P \cdot C Q+A Q \cdot B P$$ 10. Let$a_{n}$be the last non-zero digit (counting from left to right) when expressing$n !$in the decimal number system. Is the sequence$\left(a_{n}\right)$for$n=1.2 .3 \ldots$periodic? (That is, there exist the positive integers$T$and$N$such that$a_{i+T}=a_{i} \forall i \geq N$). 11. Given$n$non-negative numbers$a_{1}, a_{2}, \ldots, a_{n}(n \geq 3)$satisfying $$a_{1}^{2}+a_{2}^{2}+\ldots+a_{n}^{2}=1.$$ Prove that $$\frac{1}{\sqrt{3}}\left(a_{1}+a_{2}+\ldots+a_{n}\right) \geq a_{1} a_{2}+a_{2} a_{3}+\ldots+a_{n} a_{1}.$$ When does equality occur? 12. Let$\left(x_{n}\right)(n=1,2, \ldots)$be a sequence given by $$x_{1}=a \ (a>1),\, x_{2}=1,\quad x_{n+2}=x_{n}-\ln x_{n},\,\forall n \in \mathbb{N}^{*}.$$ Put$\displaystyle S_{n}=\sum_{k=5}^{n-1}(n-k) \ln \sqrt{x_{2 k-1}}(n \geq 2)$. Find$\displaystyle\lim_{n \rightarrow \infty}\left(\frac{S_{n}}{n}\right)$. ### Issue 387 1. Prove that there exists a$2009$-digits multiple of$2007$so that it is formed by four digits$0,2,7$and$9$and the sum of its digits is$7209$. 2. Let$a_{1}, a_{2}, \ldots, a_{n}$be$n$odd integers$(n>2007)$satisfying the following condition $$a_{1}^{2}+a_{2}^{2}+\ldots+a_{2005}^{2}=a_{2006}^{2}+a_{2007}^{2}+\ldots+a_{n}^{2}.$$ Determine the least possible value of$n$and construct an example of such a collection$\left(a_{1}, a_{2}, \ldots, a_{n}\right)$for the smallest value found above. 3. Determine the value of$S$, given that $$S=\frac{2^{3}-1}{2^{3}+1} \times \frac{3^{3}-1}{3^{3}+1} \times \ldots \times \frac{2009^{3}-1}{2009^{3}+1}$$ 4. Let$A B C$be a right triangle with right angle at$A$and$A C>A B$. Choose a poind$D$on$A C$such that$A B=A D .$Let$E$be the foot of the altitude from$D$onto$B C .$The line passing through$A$and parallel to$B C$meets$D E$at$H .$Prove that $$A H<\frac{\sqrt{2}}{2} A C.$$ 5. Let$A B$be a fixed chord of a given circle$(O)$and$E$is a point moving on$A B$(but distinct from$A$and$B$). From$E$, draw another chord$C D$.$P$and$Q$are two points on the rays$D A$and$D B$respectively such that$P$is the reflection of$Q$through$E$. Prove that the circle$(I)$passing through$C$and touches$P Q$at$E$always passes through a fixed point. 6. Of all pentagons whose sum of the squares of its diagonals equals$1,$which one has the smallest possible sum of cube of its sides. 7. Solve the system of equations $$\begin{cases}\sqrt{2 x}+2 \sqrt{6-x}-y^{2} &=2 \sqrt{2} \\ \sqrt{2 x}+2 \sqrt{6-x}+2 \sqrt{2} y &=8+\sqrt{2}\end{cases}$$ 8. Prove that $$\frac{\sin A}{\tan \frac{B}{2}}+\frac{\sin B}{\tan \frac{C}{2}}+\frac{\sin C}{\tan \frac{A}{2}} \geq \frac{9}{2}$$ where$A$,$B$,$C$are the measures of the angles of a triangle. When does equality occur?. 9. In a triangle$A B C$, let$P$be a point such that$P A=P B+P C$.$R$is the midpoint of the chord$A B$(the one that contains$P$) of the circumcircle of the triangle$A B P$and$S$is the midpoint of the chord$A C$(containing$P$) of the circumcircle of the triangle$A C P$. Prove that circumcircles of the triangles$B P S$and$CPR$touch each other. 10. Does there exist a function$\mathbb{R} \rightarrow \mathbb{R}$such that •$f$is continuous on$\mathbb{R}$; and •$f(x+2008)(f(x)+\sqrt{2009})=-2010, \forall x \in \mathbb{R} . ?$11. Let$\left(u_{n}\right)(n=1,2, \ldots)$be a sequence given by $$u_{1}=2,1;\quad u_{n+1}=\frac{-2 u_{n}^{2}+5 u_{n}}{-u_{n}^{2}+2 u_{n}+1},\,\forall n=1,2, \ldots$$ Find$\displaystyle\lim_{n\to\infty} \left(u_{1}+u_{2}+\ldots+u_{n}\right)$. 12.$A B C$and$A^{\prime} B^{\prime} C^{\prime}$are two triangles in a given plane whose inradii are$r, r^{\prime}$respectively. Let$R'$be the radius of the circumcircle of the triangle$A^{\prime} B^{\prime} C^{\prime} .$Prove that $$\left(\sin \frac{B^{\prime}}{2}+\sin \frac{C^{\prime}}{2}\right) P A +\left(\sin \frac{C^{\prime}}{2}+\sin \frac{A^{\prime}}{2}\right) P B +\left(\sin \frac{A^{\prime}}{2}+\sin \frac{B^{\prime}}{2}\right) P C \geq \frac{6 r r^{\prime}}{R^{\prime}}$$ for any point$P$in the same plane. When does equalitiy occur? ### Issue 388 1. Prove that a number of the form$(\overline{33 \ldots 3})^{2}$with$k(k>0)$digits$3$can always be written as the difference between of the number whose digits are 1 and the one whose digits are$2$. 2. Find all real numbers$a$such that$|3 a-2| \leq 1$and$A=\dfrac{3 a-1}{4 a^{4}+a^{2}}$is an integer. 3. Find the greatest value of the expression $$P=3 \sqrt{x}+8 \sqrt{y}$$ where$x, y$are two non-negative numbers satisfying$17 x^{2}-72 x y+90 y^{2}-9=0$4. Solve the equation $$\sqrt{x-2}+\sqrt{4-x}+\sqrt{2 x-5}=2 x^{2}-5 x$$ 5. From a point$A$outside a given circle witlı center$O$, construct two tangent lines$A B$and$A C$with$B, C$are the tangency points. On$O B$, choose$N$such that$B N=2 O N$. The perpendicular bisector of the line segment$C N$meets$O A$at$M .$Determine the ratio$\dfrac{A M}{A O}$6. Find the least value of the expression $$A=\frac{1}{a^{4}(b+1)(c+1)}+\frac{1}{b^{4}(c+1)(a+1)}+\frac{1}{c^{4}(a+1)(b+1)}$$ where$a, b, c$are positive real numbers satisfying$a b c=1$. 7. Let$a, b$be two real numbers in the interval$(0 ; 1) .$A sequence$\left(u_{n}\right)(n=0,1,2, \ldots)$is given by $$u_{0}=a, u_{1}=b,\quad u_{n+2}=\frac{1}{2010} \cdot u_{n+1}^{4}+\frac{2009}{2010} \sqrt{u_{n}},\,\forall n \in \mathbb{N}.$$ Prove that$\left(u_{n}\right)$has a finite limit, and find that limit. 8. Let$l_a$,$l_b$,$l_c$be respectively the lengths of the interior angle-bisectors from the three vertices$A$,$B$,$C$of a triangle$A B C$. Let$R$be its circumradius. Prove that $$\frac{l_{a}+l_{b}+l_{c}}{R} \leq 2\left(\cos \frac{A}{2} \cos \frac{B}{2}+\cos \frac{B}{2} \cos \frac{C}{2}+\cos \frac{C}{2} \cos \frac{A}{2}\right).$$ When does the equality occur? 9. Let$A B$be a fixed chord on a given circle$(O) .$A point$C$moves on the circle and let$M$be a point on the zigzag$A C B$(consisting of two line segments$A C$and$C B)$such that it divides this zigzag into two parts of equal length. Find the locus of$M$when$C$. moves around the circle$(O)$. 10. Let$m, n, d$be positive integers such that$m<n$and$\gcd(d, m)=1$,$\gcd(d, n)=1$. (Here,$\gcd(a, b)$denotes the greatest common divisor of$a$and$b$.) Prove that in an arithmetic progression of$n$terms with common difference$d$, there always exist two distinct terms whose product is a multiple of$m n$. 11. A function$f: \mathbb{R} \rightarrow \mathbb{R}$is given with the following properties •$f(0)=1$•$f(x) \leq 1,\, \forall x \in \mathbb{R}$•$\displaystyle f\left(x+\frac{11}{24}\right)+f(x)=f\left(x+\frac{1}{8}\right)+f\left(x+\frac{1}{3}\right)$Put$\displaystyle F(x)=\sum_{n=0}^{2009} f(x+n)$. Find$F(2009)$1. Given two positive degrees polynomials with real coefficients $$P(x)=x^{n}+a_{1} x^{n-1}+\ldots+a_{n-1} x+a_{n}$$ and $$Q(x)=x^{m}+b_{1} x^{m-1}+\ldots+b_{m-1} x+b_{m}$$ where$Q(x)$has exactly$m$real roots and$P(x)$is divisible by$Q(x)$. Prove that if there exists a$k \in\{1,2, \ldots, m\}$such that$\left|b_{k}\right|>C_{m}^{k} .2009^{k},$then there also exists an$i \in\{1,2, \ldots, n\}$such that$|a,|>2008$### Issue 389 1. Prove that the number of decimal digits in$2008^{2009}+2^{2009}$and$2008^{2009}$are equal. 2. Find the least value of the expression$A=1-x y,$where$x$and$y$are real numbers satisfying the following condition $$x^{2009}+y^{2009}=2 x^{1004} y^{1004}$$ 3. Does there exist a positive integer number$n$such that$n^{6}+26^{n}=21^{2009} ?$4. Let$A B C$be a right triangle with right angle at$A .$On the sides$A B, B C$and$C^{\prime} A$, choose$D, E$and$F$respectively such that$D E \perp B C$and$D E=D F$.$M$is the midpoint of EF. Prove that$\widehat{B C M}=\widehat{B F E}$. 5. Let$x, y, z$be real numbers in the interval$(0 ; 1)$. Prove that $$\frac{1}{x(1-y)}+\frac{1}{y(1-z)}+\frac{1}{z(1-x)} \geq \frac{3}{x y z+(1-x)(1-y)(1-z)}.$$ When does the equality occur? 6. Solve the equation $$\sqrt{x^{2}+4 x+3}+\sqrt{4 x^{2}-9 x-3} -\sqrt{3 x^{2}-2 x+2}+\sqrt{2 x^{2}-3 x-2}$$ 7. Let$A_{1} A_{2} \ldots A_{n}$be a convex polygon$(n \geq 3)$circumscribed around the circle centered at$J$. Prove that for any point$M$, $$\sum_{i=1}^{n} \cos \frac{A_{i}}{2}\left(M A_{i}-J A_{i}\right) \geq 0$$ 8. Find all functions$f: \mathbb{R} \rightarrow \mathbb{R}$satifying the condition $$f(x y+f(z))=\frac{x f(y)+y f(x)}{2}+z$$ for all$x, y, z$in$\mathbb{R}$. 9. Let$a, b, c$be positive numbers. Prove that $$\frac{a^{2}}{b}+\frac{b^{2}}{c}+\frac{c^{2}}{a} \geq \sqrt{a^{2}-a b+b^{2}}+\sqrt{b^{2}-b c+c^{2}}+\sqrt{c^{2}-c a+a^{2}}.$$ 10. Let$A B C$be an acute triangle and the altitudes$A D, B E, C F$meet at$H .$On$D E$choose a point$K$such that$D K=D H$. On$D H,$choose a point$I$such that$\widehat{I K D}=90^{\circ} .$Prove that the circle centered at$I$with radius$I K$touches the circle whose diameter is$B C$. 11. Let$\left(u_{n}\right)$be a sequence of positive numbers. Put$S_{n}=u_{1}^{3}+u_{2}^{3}+u_{3}^{3}+\ldots+u_{n}^{3}$for$n=1,2, \ldots$. Assume that $$u_{n+1} \leq\left(\left(S_{n}-1\right) u_{n}+u_{n-1}\right) \frac{1}{S_{n+1}},$$ for any$n=2,3, \ldots$Find$\displaystyle\lim_{n\to\infty} u_{n}$12. Let$p$be a prime number and$m, n, q$be natural numbers satisfying$2 \leq n \leq m$and$(p, q)=1 .$Prove that$C_{q p^{n}}^{n}$is divisible by$p^{n+n+1}$(The binomial coefficient$C_{n}^{k}$is the number of ways of picking$k$unordered elements from a set of$n$elements). ### Issue 390 1. Find all triples$(a, b, c)$of positive integers such that$(a+b+c)^{2}-2 a+2 b$is a perfect square. 2. Given a triangle$A B C$with$A B=A C$and$\widehat{B A C}=80^{\circ} .$Choose a point$I$inside the triangle so that$\widehat{I A C}=10^{\circ}$,$\widehat{I C A}=20^{\circ} .$Find the measure of the angle$\widehat{C B I}$. 3. Let$G$and$I$be respectively the centroid and the incenter of a given triangle$A B C .$Prove that if$A B^{2}-A C^{2}=2\left(I B^{2}-I C^{2}\right)$then$G I$is parallel to$B C$. 4. Solve the equation $$\left(x^{2}+1\right)\left|x^{2}+2 x-1\right|+6 x\left(1-x^{2}\right)=\left(x^{2}+1\right)^{2}$$ 5. Let$\left(a_{n}\right)$be a sequence given by $$a_{1}=1,\quad a_{n+1}=\sqrt{a_{n}\left(a_{n}+1\right)\left(a_{n}+2\right)\left(a_{n}+3\right)+2},\,\forall n \in \mathbb{N}^{*}.$$ Compare$\dfrac{1}{2}$with the sum $$S=\frac{1}{a_{1}+2}+\frac{1}{a_{2}+2}+\frac{1}{a_{3}+2}+\ldots+\frac{1}{a_{2009}+2}.$$ 6. Let$a, b, c$be positive real numbers such that$a+b+c=6 .$Prove that $$\frac{a}{\sqrt{b^{3}+1}}+\frac{b}{\sqrt{c^{3}+1}}+\frac{c}{\sqrt{a^{3}+1}} \geq 2$$ 7. Find all triangles whose inradius equal 3 and the side lengths form the first three terms of an arithmetic progression with common difference$d$distinct from$0 .$8. Let$A B C$be a triangle with$A B=3 R$,$B C=R \sqrt{7}, C A=2 R .$Let$M$be an arbitrary point on the spherical surface$(C ; R) $. Find the least value of$M A+2 M B$. 9. There are$294$people in a meeting. Those who are acquainted shake hands with each other. Knowing that if$A$shakes hands with$B$then one of them shakes hands at most 6 times. What is the greatest number of possible handshakes? 10. Given a positive integer$m$, find all functions$f: \mathbb{N} \rightarrow \mathbb{N}$such that for every$x, y \in \mathbb{N}$we have • If$f(x)=f(y)$then$x=y$•$f(f(f(\ldots)))=x+y$. Here,$f$appears$m$times on the left hand side. 11. Let$f(x)$be a continuous function on the closed interval$[0 ; 1],$and differentiable on the open interval$(0 ; 1)$such that$f(0)=0f(1)=1 .$Prove that for two arbitrary real numbers$k_{1}, k_{2},$there exist two distinct numbers$a, b$in the open interval$(0 ; 1)$such that $$\frac{k_{1}}{f(a)}+\frac{k_{2}}{f(b)}=k_{1}+k_{2}.$$ 12. Let$A B C$be a triangle with orthocenter$H$. Prove that the common tangent, distinct from$A H,$of the incircles of the triangles$A B H$and$A C H$passes through the midpoint of$B C$.  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,... 