$show=home

2018 Issue 497

  1. Find all positive integers $n$ such that there exist prime numbers $p, q$ satisfying $$1 !+2 !+3 !+\ldots+n !=p^{2}+q^{2}+5895$$ Notice that $n !=1.2 .3 \ldots n$. 
  2. Given an acute angle $A B C$ with the altitude $A H,$ the median $B M,$ the angle bisector $C K$. Show that if $HMK$ is an equilateral triangle then so is $A B C$. 
  3. Suppose that $n$ is a positive integer so that $3^{n}+7^{n}$ is divisible by $11$. Find the remainder in the divison of $2^{n}+17^{n}+2018^{n^{2}}$ by $11$ 
  4. Given a circle $(O)$ and suppose that $B C$ is a fixed chord of $(O) .$ Let $A$ be a point moving on the major arc $B C ;$ and $M,$ $N$ respectively the midpoints of $A B$ and $A C$. Show that each of the altitudes $M M^{\prime}$ and $N N^{\prime}$ of $\triangle A M N$ contains some fixed point. 
  5. Solve the equation $$13 x^{2}-5 x-13=(16 x-11) \sqrt{2 x^{2}-3}$$
  6. Solve the system of equations $$\begin{cases}\sqrt{4 x^{2}+5}+\sqrt{4 y^{2}+5}&=6|x y| \\ 2^{\dfrac{1}{x^{8}}+\dfrac{2}{y^{4}}}+2^{\dfrac{1}{y^{8}}+\dfrac{2}{x^{4}}}&=16 \end{cases}.$$
  7. Show that the following inequality holds for all positive numbers $a, b, c$ $$a^{2}+b^{2}+c^{2} \geq \frac{1}{2}(a b+b c+c a) + \sqrt{\frac{2(a+b+c)\left(a^{3} b^{3}+b^{3} c^{3}+c^{3} a^{3}\right)}{(a+b)(b+c)(c+a)}}$$
  8. Given triangle $A B C$ with the incenter $I$, the centroid $G$. There lines $A G$, $B G$ and $CG$ respectively intersect the circumcircle of $A B C$ at $A_{2}$, $B_{2}$, $C_{2}$. Show that $$G A_{2}+G B_{2}+G C_{2} \geq I A+I B+I C.$$
  9. Given continuous functions $f,g:[a, b] \rightarrow[a, b]$ such that $$f(g(x))=g(f(x)) \quad \forall x \in[a, b]$$ where $a, b$ are real numbers. Show that the equation $f(x)=g(x)$ has at least one solution.
  10. Find all pairs of integers $(m, n)$ so that both $m^{5} n+2019$ and $m n^{5}+2019$ are cubes of integers.
  11. Let $\quad S=\{1,2,3, \ldots, 2019\}$ and suppose that $S_{1}, S_{2}, \ldots, S_{410}$ are subsets of $S$ such that for every $i \in S$, there exist exactly $40$ of them containing i. Prove that there exist different indices $j, k, l \in\{1,2, \ldots, 410\}$ such that $\left|S_{j} \cap S_{k} \cap S_{l}\right| \leq 1$
  12. Given a cyclic quadrilateral $A B C D$ with $E$ is the intersection between $A C$ and $B D .$ The circumcircles of $E A D$ and $E B C$ intersect at another point $F$ (besides $E$). The perpendicular bisectors of $B D$, $A C$ respectively intersect $F B$, $F A$ at $K$, $L$. Show that $K L$ goes through the midpoint of $A B$.

$type=three$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: 2018 Issue 497
2018 Issue 497
Mathematics & Youth
https://www.molympiad.org/2020/09/2018-issue-497.html
https://www.molympiad.org/
https://www.molympiad.org/
https://www.molympiad.org/2020/09/2018-issue-497.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