$show=home

2017 Issue 479

  1. Place the numbers from 1 to 20 on a circle such that the sum of any two numbers which are next to each other is a prime number.
  2. Given a right triangle $ABC$ with the right angle $A$ and $\widehat{ABC}<45^{0}$. On the haft plane determined by $AB$, containing $C$, choose two points $D$ and $E$ such that $BD=BA$, $\widehat{DBA}=90^{0}$, $\widehat{EBC}=\widehat{CBA}$, and $ED$ is perpendicular to $BD$. Prove that $BE=AC+DE$.
  3. Given two real numbers $x$ and $y$ such that $x^{2}+2xy+2y^{2}=1$. Find the maximum and minimum values of the expression \[P=x^{4}+y^{4}+(x+y)^{4}.\]
  4. Given a triangle $ABC$ with $AB<AC$. Let $(O)$ be the incircle of $ABC$. The side $BC$ is tangent to $(O)$ at $D$. Choose $I$ on $AD$ such that $OI$ is perpendicular to $AD$. The ray $IO$ intersect the perpendicular bisector of $BC$ at $K$. Prove that $BIKC$ is an inscribed quadrilateral.
  5. Solve the equation \[\frac{\sqrt{x^{2}+28x+4}}{x+2}+8=\frac{x+4}{\sqrt{x-1}}+2x.\]
  6. Given non-negative numbers $a,b$ and $c$ such that $ab+bc+ca=1$. Prove that \[\sqrt{3}\leq\sqrt{1+a^{2}}+\sqrt{1+b^{2}}+\sqrt{1+c^{2}}-a-b-c\leq2.\]
  7. Solve the system of equations \[\begin{cases} x^{8} & =21y+13\\ \frac{(x+y)^{25}}{2^{18}} & =(x^{3}+y^{3})^{3}(x^{4}+y^{4})^{4}\end{cases}.\]
  8. Given a triangle $ABC$ and $P$ is a point lying inside $ABC$. Let $E$ (resp. $AC$ and $AB$). Let $K$ be the perpendicular projection of $K$ on $BC$. On the side $AF$, choose an arbitrary point $M$. Assume that $N$ is the intersection between $MK$ and $PE$. Prove that $\widehat{KHM}=\widehat{KHN}$.
  9. Given a bijection $f:\mathbb{N}^{*}\to\mathbb{N}^{*}$. Prove that there exist positive intergers $a,b,c$ and $d$ such that $a<b<c<d$ and $f(x)+f(d)=f(b)+f(c)$.
  10. Given a sequence $(u_{n})$ whose terms are positive integers satisfying \[0\leq u_{m+n}-u_{m}-u_{n}\leq2\quad\forall m,n\geq1.\] Prove that there exist two positive numbers $a_{1},a_{2}$ such that \[[a_{1}n]+[a_{n}n]-1\leq u_{n}\leq[a_{1}n]+[a_{2}n]+1\] for all $n\leq2017$. ($[x]$ is the greatest integer which does not exceed $x$).
  11. Find all continuous functions $f:(0,+\infty)\to(0,\infty)$ such that \[f(x+2016)=f\left(\frac{x+2017}{x+2018}\right)+\frac{x^{2}+4033x+4066271}{x+2018},\forall x>0.\]
  12. Given a circle $(O)$ and two fiexd points $B,C$ on $(O)$. A point $A$ is moving on $(O)$ such that $ABC$ is always an acute triangle and $AB<AC$. Choose $D$ on the side $AC$ such that $AB=AD$. The line $BD$ intersects $(O)$ at $E$ which is different from $B$. The perpendicular projection of $E$ (resp. $D$) on $AC$ (resp. $AE$) is $H$ (resp. $M$). The circumcircle of $AMH$ intersects $(O)$ at $K$. Let $N$ be perpendicular projection of $K$ on $AB$.
    a) Prove that $MN$ goes through the midpoint $I$ of $BD$.
    b) The circumcircles of $BCD$ and $AMH$ intersect each other at $P$ and $Q$. Prove that the midpoint of $PQ$ always lies on a fixed line when $A$ is moving on $(O)$ in the given way.

$type=three$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: 2017 Issue 479
2017 Issue 479
Mathematics & Youth
https://www.molympiad.org/2017/06/mathematics-and-youth-magazine-problems_29.html
https://www.molympiad.org/
https://www.molympiad.org/
https://www.molympiad.org/2017/06/mathematics-and-youth-magazine-problems_29.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