$show=home

2018 Issue 488

  1. Give a pentagon $ABCDE$. Assume that $BC$ is parallel to $AD$, $CD$ is parallel to $BE$, $DE$ is parallel to $AC$, and $AE$ is parallel to $BD$. Show that $AB$ is parallel to $CE$. 
  2. Prove that \[\cfrac{1+\frac{1}{3}+\frac{1}{5}+\ldots+\frac{1}{4035}}{1+\frac{1}{2}+\frac{1}{3}+\ldots+\frac{1}{2018}}>\frac{2019}{4036}.\]
  3. Given two triples $(a,b,c)$; $(x,y,z)$ none of them contains all $0$'s, such that \[a+b+c=x+y+z=ax+by+cz=0.\] Prove that the expression $P=\dfrac{(b+c)^{2}}{ab+bc+ca}+\dfrac{(y+z)}{xy+yz+zx}^{2}$ is a constant.
  4. Outside a triangle $ABC$, draw triangles $ABD$, $BCE$, $CAF$ such that $\widehat{ADB}=\widehat{BEC}=\widehat{CFA}=90^{0}$, $\widehat{ABD}=\widehat{CBE}=\widehat{CAF}=\alpha$. Prove that $DF=AE$.
  5. Show that the following sum is a positive integer \[\begin{align} S=&1+\frac{1}{2}+\frac{1}{3}+\ldots+\frac{1}{2017}+\left(1+\frac{1}{2}+\frac{1}{3}+\ldots+\frac{1}{2017}\right)^{2}+\\&+\left(\frac{1}{2}+\frac{1}{3}+\ldots+\frac{1}{2017}\right)^{2}+\ldots+\left(\frac{1}{2017}\right)^{2}.\end{align}\]
  6. Solve the system of equations \[\begin{cases}2x^{5}-2x^{3}y-x^{2}y+10x^{3}+y^{2}-5y & =0\\ (x+1)\sqrt{y-5}-y+3x^{2}-x+2 & =0\end{cases}.\]
  7. Prove that $x_{0}=\cos\dfrac{\pi}{21}+\cos\dfrac{8\pi}{21}+\cos\dfrac{10\pi}{21}$ is a solution of the equation \[4x^{3}+2x^{2}-7x-5=0.\]
  8. In any triangle $ABC$, show that \[\cos(A-B)+\cos(B-C)+\cos(C-A)\leq\frac{1}{2}\left(\dfrac{a+b}{c}+\frac{b+c}{a}+\frac{c+a}{b}\right).\]
  9. Given $6$ positive numbers $a,b,c$, $x,y,z$ and assume that $x+y+z=1$. Show that \[ax+by+cz\geq a^{x}b^{y}c^{z}.\]
  10. Given an infinite sequence of positive integers $a_{1}<a_{2}<\ldots a_{n}<\ldots$ such that $a_{i+1}-a_{i}\geq8$ for all $i=1,2,3,\ldots$. For each $n$, let $s_{n}=a_{1}+a_{2}+\ldots a_{n}$. Show that for each $n$, there are at lease two square numbers inside the half-open interval $[s_{n},s_{n+1})$.
  11. Given two positive sequences $(a_{n})_{n\geq0}$ and $(b_{n})_{n\geq0}$ which are determined as follows \[a_{0}=\sqrt{3},\,b_{0}=2,\quad a_{n}^{2}+1=b_{n}^{2},\,a_{n}+b_{n}=\frac{1+a_{n+1}}{1-a_{n+1}},\,\forall n\in\mathbb{N}.\] Show that they converges and find the limits.
  12. Given a triangle $ABC$ with $AB\ne AC$. A circle $(O)$ passing through $B$, $C$ intersects the line segments $AB$ and $AC$ at $M$ and $N$ respectively. Let $P$ be the intersection of $BN$ and $CM$. Let $Q$ be the midpoint of the arc $BC$ which does not contian $M$, $N$. Let $K$ be the incenter of $PBC$. Show that $KQ$ always goes through a fixed point when $(O)$ varies.

$type=three$c=3$source=random$title=oot$p=1$h=1$m=hide$rm=hide

Anniversary_$type=three$c=12$title=oot$h=1$m=hide$rm=hide

Name

2006,1,2007,12,2008,12,2009,12,2010,12,2011,12,2012,12,2013,12,2014,12,2015,12,2016,12,2017,12,2018,11,2019,12,2020,12,2021,6,Anniversary,4,
ltr
item
Mathematics & Youth: 2018 Issue 488
2018 Issue 488
Mathematics & Youth
https://www.molympiad.org/2018/03/mathematics-and-youth-magazine-problems_68.html
https://www.molympiad.org/
https://www.molympiad.org/
https://www.molympiad.org/2018/03/mathematics-and-youth-magazine-problems_68.html
true
8958236740350800740
UTF-8
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