$show=home

2021 Issue 528

  1. Assume that $A$ is a natural number which has two prime factors $p$ and $q$ only. Let $S$ be the sum of all positive factors of $A$. Show that $S < 2 A$.
  2. Denote $a_{n}$ the integer which is closest to $\sqrt{n}$ $(n \in \mathbb{N}^{*})$, for example $$\sqrt{1}=1=a_{1} ; \quad \sqrt{2} \approx 1,4 \Rightarrow a_{2}=1 ; \quad \sqrt{3} \approx 1,7 \Rightarrow a_{3}=2 ; \ldots.$$ Compute $$\frac{1}{a_{1}}+\frac{1}{a_{2}}+\ldots+\frac{1}{2020}+\frac{1}{2021}.$$
  3. Find all $4$-tuples of positive integers $(x ; y, z, t)$ satisfying $$\begin{cases}x+y+z+t=54 \\ \dfrac{x z}{(x+y)(z+t)} =\dfrac{23}{104} \\ x>z, x+y>z+t \end{cases}.$$
  4. Given an acute triangle $A B C$ ($CA < CB$) with the orthocenter $H$. Let $D$ be the second intersection between $B C$ and the circumcircle of $A B H$. Denote $P$ the intersection between $D H$ and $A C$. Let $M$ be the second intersection between $A H$ and the circumcircle of $A P D$; and $N$ the second intersection between the circumcircle of $A B C$ and the circumcircle of $A P D$. Show that three points $C, M$, and $N$ are collinear.
  5. Given positive numbers $a, b, c$ satisfying $4 b^{2}+c^{2}=a c$. Find the minimum value of the expression $$P=\frac{2 a}{a+2 b}+\frac{b}{b+c}+\frac{c}{c+a} .$$
  6. Solve the equation $$\sin 3 x-\cos 3 x+\sin x+\cos x=\frac{1}{\sin 3 x+\cos x}-\frac{1}{\cos 3 x-\sin x} .$$
  7. Given positive numbers $a, b, c$ satisfying $a^{2}+b^{2}+c^{2} \geq 2(a b+b c+c a)$. Find the minimum value of the expression $$P=a+b+c+\frac{8}{a b c} .$$
  8. Given two circles $(O ; R)$ and $\left(O^{\prime} ; R^{\prime}\right)\left(R>R^{\prime}\right)$ which is internally tangent to each other at $A$. Let $M$ be a point on $\left(O^{\prime} ; R^{\prime}\right)(M \neq A) .$ The tangent of $\left(O^{\prime} ; R^{\prime}\right)$ at $M$ intersects $(O ; R)$ at $P$ and $Q .$ The circumcircle of $O^{\prime} P Q$ intersects $\left(O^{\prime} ; R^{\prime}\right)$ at $B$ and $C$. Show that $A B M C$ is a harmonic quadrilateral, i.e. a cyclic quadrilateral of which the products of opposite sides are equal.
  9. Given a triangle $A B C$ and $x, y, z$ positive numbers. Show that $$\frac{x}{y+z}(1+\cos A)+\frac{y}{z+x}(1+\cos B)+\frac{z}{x+y}(1+\cos C) \geq \frac{\sqrt{3}}{2}(\sin A+\sin B+\sin C) .$$
  10. Two boxes contain $25$ small white and black balls. From each box, pick randomly $1$ ball. Find the probability that we get $2$ balls with different colors assuming that the box with more balls has more black balls and the probability to get two black balls is $0,42$.
  11. Find all functions $f: \mathbb{R} \rightarrow(0+\infty)$ satisfying $$\frac{1}{2015} \leq\left(\frac{f(x)}{f(r)}\right)^{\frac{1}{(x-r)^{2}}} \leq 2015, \forall x \in \mathbb{R}, \forall r \in \mathbb{Q}, x \neq r$$
  12. Outside a triangle $A B C$, draw pairwise similar triangles $B C P$, $A C Q$, $A B R$, and $B A S$. Let $K$, $L$ respectively be the midpoints of $B C$ and $C A .$ Prove that two triangles $R P K$ and $S Q L$ have the same area.

$type=three$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: 2021 Issue 528
2021 Issue 528
Mathematics & Youth
https://www.molympiad.org/2021/08/2021-issue-528.html
https://www.molympiad.org/
https://www.molympiad.org/
https://www.molympiad.org/2021/08/2021-issue-528.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