$show=home

2013 Issue 428

  1. Determine all triple of prime numbers $a,b,c$ (not necessarily distinct) such that \[abc<ab+bc+ca.\]
  2. Let $ABC$ be a right triangle, with right angle at $A$ and $AH$ is the altitude from $A$, $\widehat{ACB}=30^{0}$. Construct an equilateral triangle $ACD$ ($D$ and $B$ are in opposite side $AC$). K is the foot of the perpendicular line from $H$ onto $AC$. The line through $H$ and parallel to $AD$ meets $AB$ at $M$. Prove the points $D,K,M$ are colinear.
  3. Consider a $6\times6$ board of squares with 4 corner squares being deleted. Find the smallest number of squares that can be painted black given that among the 5 squares in any figure, there is at lease one black.
  4. Let $a,b,c$ be real numbers in the interval $[1,2]$. Prove the inequality \[a^{2}+b^{2}+c^{2}+3\sqrt[3]{(abc)^{2}}\geq2(ab+bc+ca).\]
  5. Let $ABC$ be a non-right triangle ($AB<AC$) with altitude $AH$. $E,F$ are the orthogonal projection of point $H$ onto $AB$ and $AC$ respectively. $EF$ meets $BC$ at $D$. Draw a semicircle with diameter $CD$ on the half-plane containing $A$ with edge $CD$. The line through $B$ and perpendicular to $CD$ meets the semicircle at $K$. Prove that $DK$ is tangent to the circumcircle of triangle $KEF$.
  6. Given that the equation \[ax^{3}-x^{2}+ax-b=0\quad(a\ne0,\,b\ne0)\] has three positive real roots. Determine the greatest value of the following expression \[P=\frac{11a^{2}-3\sqrt{3}ab-\frac{1}{3}}{9b-10(\sqrt{3}a-1)}.\]
  7. Solve the following system of equations \[\begin{cases} \sqrt{x-\frac{1}{4}}+\sqrt{y-\frac{1}{4}} & =\sqrt{3}\\ \sqrt{y-\frac{1}{16}}+\sqrt{z-\frac{1}{16}} & =\sqrt{3}\\ \sqrt{z-\frac{9}{16}}+\sqrt{x-\frac{9}{16}} & =\sqrt{3}\end{cases}.\]
  8. Let $a,b$ be real constants such that $ab>0$. Let $\{u_{n}\}$ be a sequence where $n=1,2,3,\ldots$ given by \[u_{1}=a,\quad u_{n+1}=u_{n}+bu_{n}^{2},\,\forall n\in\mathbb{N}^{*}.\] Determine the limit \[\lim_{n\to\infty}\left(\frac{u_{1}}{u_{2}}+\frac{u_{2}}{u_{3}}+\ldots+\frac{u_{n}}{u_{n+1}}\right).\]
  9. Find all positive integers $k$ with the property that there exists a polynomial $f(x)$ with integer coefficients of degree greater than 1 such that for all prime numbers $p$ and natural numbers $a,b$ if $p$ divides $(ab-k)$ then it also divides $(f(a)f(b)-k)$.
  10. Given $a_{i}\in[0,\alpha]$ ($i=\overline{1,n}$), ($\alpha>0$). Prove the inequality \[\prod_{i=1}^{n}(\alpha-a_{i})\leq\alpha^{n}\left(1-\sum_{i=1}^{n}\frac{a_{i}}{S_{i}+\alpha}\right)\] where ${\displaystyle S_{i}=\sum_{k=1}^{n}a_{k}-a_{i}}$ for all $i=\overline{1,n}$.
  11. Point $O$ is in the interior of triangle $ABC$. The ray $Ox$ parallel to $AB$ meets $BC$ at $D$, ray $Oy$ parallel to $BC$ meets $CA$ at $E$, ray $Oz$ parallel to $CA$ meets $AB$ at $F$. Prove that
    a) $3S_{DEF}\leq S_{ABC}$.
    b) $OD\cdot OE\cdot OF\leq27AB\cdot BC\cdot CA$.
  12. The circle $(O)$ and $(O')$ meet at points $A,B$. Point $C$ is fixed on $(O)$ and point $D$ is fixed on $(O')$. A moving point $P$ is on the opposite ray of ray $BA$. The circumcircles of traingles $PBC$, $PDB$ intersect $BD$, $BC$ at seconde points $E,F$ respectively. Prove that the midpoint of line segment $EF$ is always on a fixed segment $EF$ is always on a fixed straight line.

$type=three$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: 2013 Issue 428
2013 Issue 428
Mathematics & Youth
https://www.molympiad.org/2017/08/mathematics-and-youth-magazine-problems_16.html
https://www.molympiad.org/
https://www.molympiad.org/
https://www.molympiad.org/2017/08/mathematics-and-youth-magazine-problems_16.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