$show=home

2014 Issue 443

  1. $21$ distinct integers are chosen so that the sum of any subset of $11$ numbers among them is always greater than the sum of the remaining $10$. If one of them is $101$, and the largest number is $2014$, find the other $19$ numbers.
  2. In a triangle $ABC$ where $\widehat{BAC}=40^{0}$ and $\widehat{ABC}=60^{0}$, point $D$ and $E$ are chosen on the sides $AC$ and $AB$ respectively such that $\widehat{CBD}=40^{0}$ and $\widehat{BCE}=70^{0}$. $BD$ and $CE$ intersect at point $F$. Prove that $AF$ is perpendicular to $BC$.
  3. Solve the following system of equations \[\begin{cases} 2\sqrt{2x}-\sqrt{y} & =1\\ \sqrt[3]{8x^{3}+y^{3}} & =\sqrt[3]{2}(\sqrt{x}+\sqrt{y}-1) \end{cases}.\]
  4. In a triangle $ABC$, points $E,D$ on the sides $AB$ and $AC$ respectively such that $\widehat{ABD}=\widehat{ACE}$. The circumcircle of triangle $ADB$ meets $CE$ at $M$ and $N$. The circumcircle of triangle $AEC$ meets $BD$ at $I$ and $K$. Prove that the points $M,I,N,K$ lie on a circle.
  5. Prove that for all positive rel numbers $a,b,c$ the following inequality holds \[\frac{a^{2}}{a+b}+\frac{b^{2}}{b+c}+\frac{c^{2}}{c+a}\geq\frac{\sqrt{2}}{4}(\sqrt{a^{2}+b^{2}}+\sqrt{b^{2}+c^{2}}+\sqrt{c^{2}+a^{2}}).\]
  6. Determine all real solutions of the equation \[(x^{5}+x-1)^{5}+x^{5}=2.\]
  7. Let $M$ be a point inside a given triangle $ABC$ and let $x,y,z$ denote the distance from $M$ onto $BC,CA,AB$ respectively. Prove that $\widehat{BAM}=\widehat{CBM}=\widehat{ACM}$ if and only if \[\frac{bx}{c}=\frac{cy}{a}=\frac{az}{b}\] where $BC=a$, $CA=b$, $AB=c$.
  8. Let $x,y,z$ be theree arbitrary numbers from the interval $[0,1]$. Determine the maximum value of $P$, where \[P=\frac{x}{y+z+1}+\frac{y}{z+x+1}+\frac{z}{z+y+1}+(1-x)(1-y)(1-z).\]

$type=three$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

2006,1,2007,12,2008,12,2009,12,2010,12,2011,12,2012,12,2013,12,2014,12,2015,12,2016,12,2017,12,2018,11,2019,12,2020,12,2021,4,Anniversary,4,
ltr
item
Mathematics & Youth: 2014 Issue 443
2014 Issue 443
Mathematics & Youth
https://www.molympiad.org/2017/08/mathematics-and-youth-magazine-problems_97.html
https://www.molympiad.org/
https://www.molympiad.org/
https://www.molympiad.org/2017/08/mathematics-and-youth-magazine-problems_97.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