2015 Issue 459

  1. Let $$\begin{align*} A & =\frac{1}{1.2^{2}}+\frac{2}{2.3^{2}}+\frac{1}{3.4^{2}}+\ldots+\frac{1}{49.50^{2}},\\ B & =\frac{1}{2^{2}}+\frac{1}{3^{2}}+\frac{1}{4^{2}}+\ldots+\frac{1}{50^{2}}.\end{align*}$$ Compare $A$ and $B$ with $\dfrac{1}{2}$.
  2. For all pairs of real numbers $(a,b)$ such that the polynomial \[A(x)=x^{2}-2ax+2a^{2}+b^{2}-5\] has solutions. Find the minimum value of the expression \[P=(a+1)(b+1).\]
  3. Suppose that $a_{1},a_{2},\ldots,a_{n}$ are different positive integers such that \[\frac{1}{a_{1}}+\frac{1}{a_{2}}+\ldots+\frac{1}{a_{n}}=1\] and also assume that the biggest number among them is equal to $2p$, where $p$ is some prime number. Determine $a_{i}$ ($i=\overline{1,n}$).
  4. Let $ABC$ be a isosceles right triangle ($AB=BC$) and let $O$ be the midpoint of $AC$. Through $C$ draw the line $d$ perpendicular to $BC$. Let $Cx$ be the opposite ray of the ray $CB$ and $M$ be an arbitrary point on $Cx$. Assume that $E$ is the intersection between $BE$ and $OM$. Prove that when $M$ varies on $Cx$, $I$ always belongs to a fixed curve.
  5. Solve the following equation \[x^{2}-2x=2\sqrt{2x-1}.\]
  6. Solve the following inequation \[\frac{2x^{3}+3x}{7-2x}>\sqrt{2-x}.\]
  7. Given an acute and non-isosceles triangle $ABC$ with the altitudes $AH$, $BE$, $CF$. Let $I$ be the incenter of $ABC$ (the center of the inscribed circle) and $R$ be the circumradius of $ABC$ (the radius of the circumscribed circle). Let $M,N$ and $P$ respectively be the midpoint of $BC,CA$ and $AB$. Assume that $K,J$ and $L$ respectively are the intersections between $MI$ and $AH$, $NI$ and $BE$, and $PI$ and $CF$. Prove that \[\frac{1}{HK}+\frac{1}{EJ}+\frac{1}{FL}>\frac{3}{R}.\]
  8. Let $a,b,c$ be the lengths of three sides of a triangle whose perimeter is equal to $3$. Find the minimum value of the expression \[T=a^{3}+b^{3}+c^{3}+\sqrt{5}abc.\]
  9. For every positive integer $n$ prove that the following numbers are perfect square \[ 10([(4+\sqrt{15})^{n}+1])([(4+\sqrt{15})^{n+1}]+1)-60,\] \[6([(4+\sqrt{15})^{n}+1])([(4+\sqrt{15})^{n+1}]+1)-60,\] \[15([(4+\sqrt{15})^{n}+1])^{2}-60,\] where $[x]$ is the integerpart of $x$.
  10. Let $f(x)=x^{3}-3x^{2}+1$.
    a) Find the number of different real solutions of the equation $f(f(x))=0$.
    b) Let $\alpha be$the maximal positive solution of $f(x)$.
    Prove that $[\alpha^{2020}]$ is divisible by $17$ (notice that $[x]$ is the integer part of $x$).
  11. Give the sequence $(u_{n})$ where $u_{1}=2$, $u_{2}=20$, $u_{3}=56$ and \[u_{n+3}=7u_{n+2}-11u_{n+1}+5u_{n}-3.2^{n},\quad\forall n\in\mathbb{N}^{*}.\] Find the remainder of the division $u_{2011}$ by $2011$.
  12. Consider any triangle $ABC$ and any line $d$. Let $A_{1},B_{1},C_{1}$ are the projections of $A,B,C$ onto $d$. It is a fact that the line through $A_{1}$ and perpendicular to $BC$, the line through $B_{1}$ and perpendicular to $AC$, and the line through $C_{1}$ and perpendicular to $AB$ are concurrent ar a point which is called the orthogonal pole of $d$ with respect to $ABC$. Prove that for any triangle and any point $P$ on its circumcircle, the Simon line corresponding to $P$ and the orthogonal pole of the Simon line with respect to the given triangle.




Mathematics & Youth: 2015 Issue 459
2015 Issue 459
Mathematics & Youth
Loaded All Posts Not found any posts VIEW ALL Readmore Reply Cancel reply Delete By Home PAGES POSTS View All RECOMMENDED FOR YOU LABEL ARCHIVE SEARCH ALL POSTS Not found any post match with your request Back Home Sunday Monday Tuesday Wednesday Thursday Friday Saturday Sun Mon Tue Wed Thu Fri Sat January February March April May June July August September October November December Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec just now 1 minute ago $$1$$ minutes ago 1 hour ago $$1$$ hours ago Yesterday $$1$$ days ago $$1$$ weeks ago more than 5 weeks ago Followers Follow THIS CONTENT IS PREMIUM Please share to unlock Copy All Code Select All Code All codes were copied to your clipboard Can not copy the codes / texts, please press [CTRL]+[C] (or CMD+C with Mac) to copy