s.t 
be a parallelogram with 
A point 
is chosen on the extension of ray 
past 
The circumcircle of 
meets the segment 
again at 
The circumcircle of triangle 
meets the segment 
at 
Prove that lines 
are concurrent.
numbers 
are written on the blackboard . An allowed operation consists of choosing two different numbers 
and 
of the blackboard such that 
,
. Prove that there will be two numbers on the board after some minutes, such that one is divisible by the other.
- Greatest common divisor of numbers 
and 
,
.
be an equilateral triangle of side length 
.
is chosen on side 
such that the length of 
is 
.
and 
are chosen on sides 
and 
.
to their corresponding opposite vertices. The intersections of these line segments form a triangle, labeled 
.
is an equilateral triangle with side length 
.
.
.
Let 
.![$$\sqrt[3]{\frac{a}{b( a+2)}} + \sqrt[3]{\frac{b}{a(2b+1)}} \geq \frac {2}{\sqrt[3]{3}} $$](http://latex.artofproblemsolving.com/2/f/e/2fe89d457c77a842e3ea12956d64ba0b9d853433.png)
such that 
.
.
Y by Mango247
A set 
of points in the 
-dimensional space is called interesting, if for any plane there exist at least 
points in outside this plane. For which minimal 
any interesting set contains an interesting subset with at most points?
(IMC 2013.9)
of points in the 
-dimensional space is called interesting, if for any plane there exist at least 
points in outside this plane. For which minimal 
any interesting set contains an interesting subset with at most points?(IMC 2013.9)
This post has been edited 1 time. Last edited by parmenides51, Mar 15, 2021, 9:29 PM
