$show=home

2014 Issue 447

  1. Find all the integer solutions of the following equation \[1+x+x^{2}+x^{3}=y^{2}.\]
  2. Let $x,y,z$ be three coprime positive integers satisfying \[(x-z)(y-z)=z^{2}.\] Prove that $xyz$ is a perfect square. 
  3. Solve the following equation \[\frac{1}{\sqrt{3x}}+\frac{1}{\sqrt{9x-3}}=\frac{1}{\sqrt{5x-1}}+\frac{1}{\sqrt{7x-2}}.\]
  4. Given a circle $(O,R)$ and a chord $AB$ with the distance from $O$ is $d$ ($0<d<R$). Two circles $(I)$, $(K)$ are externally tangent at $C$, are both tangent to $AB$ and are internally tangent with $(O)$ ($I$ and $K$ are in the same half-plane determined by the line through $AB$). Find the locus of the points $C$ which vary when $(I)$ and $(K)$ vary.
  5. Find all positive integers $a$ and $b$ so that both equations $x^{2}-2ax-3b=0$ and $x^{2}-2bx-3a=0$ have positive integer solution.
  6. Find all positive real numbers $x,y,z$ satisfying system of equations \[\begin{cases} \dfrac{1}{x+1}+\dfrac{1}{y+1}+\dfrac{1}{z+1} & =1\\ xyz(x+y+z)(x+1)(y+1)(z+1 & =1296 \end{cases}.\]
  7. Given a tetrahedron $ABCD$ and the lengths of its sides $AB=BD=DC=x$, $BC=CA=AD=y$. Prove that \[ \frac{3}{5}<\frac{x}{y}<\frac{5}{3}.\]
  8. Find the maximum value of the expression \[P=|(a^{2}-b^{2})(b^{2}-c^{2})(c^{2}-a^{2})|,\] in which $a,b,c$ are nonnegative numbers satisfying $a+b+c=\sqrt{5}$.
  9. Solve equation \[x^{4}+ax^{3}+bx^{2}+2ax+4\] given $9(a^{2}+b^{2})=16$.
  10. Find all pairs of positive integers $(a,b)$ satisfying the following properties: $4a+1$ and $4b-1$ are coprime and $a+b$ is a divisor of $16ab+1$.
  11. Given two sequences \[a_{1}=0,\,a_{2}=16,\,a_{3}=18,\,a_{n+2}=8a_{n}+6a_{n-1}\] and \[b_{1}=3,\,b_{2}=19,\,b_{3}=69,\,b_{n+2}=3b_{n+1}+5b_{n}-b_{n-1}\] for $n\geq2$. Prove that \begin{align*} b_{n} & =C_{n}^{0}a_{n}+C_{n}^{1}a_{n-1}+\ldots+C_{n}^{n-1}a_{1}+3C_{n}^{n},\\ a_{n} & =C_{n}^{0}b_{n}-C_{n}^{1}b_{n-1}+C_{n}^{2}b_{n-2}-\ldots+(-1)^{n-1}C_{n}^{n-1}b_{1}+(-1)^{n}3C_{n}^{n}. \end{align*}
  12. Given a triangle $ABC$ and its circumscribed circle $(O)$. The points $A_{1},B_{1}$and $C_{1}$ are on the sides $BC,CA$ and $AB$ respectively. The circumscribed circles $(AB_{1}C_{1})$, $(BC_{1}A_{1})$, and $(CA_{1}B_{1})$ intersect $(O)$ at $A_{2},B_{2}$ and $C_{2}$ respectively. Find the positions of $A_{1},B_{1}$ and $C_{1}$ so that $\dfrac{S_{A_{1}B_{1}C_{1}}}{S_{A_{2}B_{2}C_{2}}}$ is minimal.

$type=three$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: 2014 Issue 447
2014 Issue 447
Mathematics & Youth
https://www.molympiad.org/2017/08/mathematics-and-youth-magazine-problems_79.html
https://www.molympiad.org/
https://www.molympiad.org/
https://www.molympiad.org/2017/08/mathematics-and-youth-magazine-problems_79.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