$show=home

2015 Issue 453

  1. Find all natural numbers $x,y,z$ such that $x!+y!+z!=\overline{xyz}$ and $\overline{xyz}$ is a three-digit number. Recall that $n!=1.2.3\ldots n$ and $0!=1$.
  2. Given a triangle $ABC$ with $\widehat{BAC}=50^{0}$, $\widehat{ABC}=60^{0}$. On the sides $AB$ and $BC$, choose $D$ and $E$ respectively such that $\widehat{ACD}=\widehat{CAE}=30^{0}$. Find the angle $\widehat{CDE}$.
  3. Given three positive numbers $a,b,c$. Find the maximum value of the expression \[T=\frac{a+b+c}{(4a^{2}+2b^{2}+1)(4c^{2}+3)}.\]
  4. Suppose that $ABC$ is an acute triangle. Its altitudes $AD$, $BE$ and $CF$ are concurrent at the point $H$. Let $K$ be a point on the side $DC$ and choose $S$ on $HK$ such that $AS\perp HK$. Let $I$ be the intersection between $EF$ and $AH$. Prove that $SH$ is the angle bisector of the angle $\widehat{DSI}$.
  5. Let $a,b,c$ be positive numbers satisfying \[a^{4}+b^{4}+c^{4}\leq2(a^{2}b^{2}+b^{2}c^{2}+c^{2}a^{2}).\] Prove that at lease one of the following equations has no solution \[ax^{2}+2bx+2c=0,\] \[bx^{2}+2cx+2a=0,\] \[cx^{2}+2ax+2b=0.\]
  6. Solve the system of equations \[\begin{cases} x^{3}+y^{3} & =1\\ x^{5}+y^{5} & =1 \end{cases}.\]
  7. Given a quadrilateral pyramid $S.ABCD$ whose base $ABCD$ is a parallelogram. A point $M$ varies on the side $AB$ ($M$ is different from $A$ and $B$) Let $(\alpha)$ be a plane which goes through $M$ and is parallel to $SA$ and $BD$. Determine the cross section of $S.ABCD$ determined by $(\alpha)$ and find the position for $M$ so that the cross section has maximal area.
  8. Solve the equation \[3\sin^{3}x+2\cos^{3}x=2\sin x+\cos x.\]
  9. Find all pairs of positive integers $(a,b)$ so that $4a+1$ and $4b-1$ are coprime and $a+b$ is a divisor of $16ab+1$.
  10. Given a polynomial \[P(x)=a_{n}x^{n}+\ldots+a_{1}x+a_{0}\quad(a_{n}\ne0,\,n\geq2)\] with integer coefficients. Prove that there are infinitely many integers $k$ such that $P(x)+k$ cannot be factorized as a product of two positive degree polynomials with integral coefficients.
  11. Suppose that the funtion $f:\mathbb{N\to\mathbb{N}}$ is onto and the funtion $g:\mathbb{N}\to\mathbb{N}$ is one-to-one. Assume furthermore that $f(x)\geq g(n)$ for all $n\in\mathbb{N}$. Show that \[f(n)=g(n),\,\forall n\in\mathbb{N}.\]
  12. Given a triangle $ABC$ inscribed in a circle $(O)$. $M$ and $N$ are two fixed points on $(O)$ so that $MN\parallel BC$. Let $P$ vary on the line $AM$. The line through $P$ which is parallel to $BC$ intersects $CA$, $AB$ at $E,F$ respectively. The circumscribed circle of the triangle $NEF$ intersects $(O)$ at the second point $Q$ (different from $N$). Prove that the line $PQ$ always goes through a fixed point when $P$ moves.

$type=three$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: 2015 Issue 453
2015 Issue 453
Mathematics & Youth
https://www.molympiad.org/2017/08/mathematics-and-youth-magazine-problems_77.html
https://www.molympiad.org/
https://www.molympiad.org/
https://www.molympiad.org/2017/08/mathematics-and-youth-magazine-problems_77.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