[/quote]
,
be real numbers so that 
and 
Show that
be a quadrilateral inscribed in a circle 
Let the tangent to 
at 
meet rays 
and 
at 
and 
respectively. A point 
is chosen inside 
so that 
and 
Let 
be a point on segment 
satisfying 
Prove that lines 
and 
are concurrent.
has 2 roots 
and equation 
has 2 roots called 
.
be an acute triangle with incentre 
and 
.
and 
intersect the circumcircle of 
and 
,
and 
such that 
and 
are parallelograms (with 
,
)
and 
.
,
, 
.
.
be a function satisfying![\[
f(f(x)) = x^2 - 6x + 6
\quad \text{for all} \quad x \in \mathbb{Z}.
\]](http://latex.artofproblemsolving.com/8/a/e/8aebed1e260316d675ca7d104d83fabcd8b5717e.png)
Given that![\[
f(i) < f(i+1) \quad \text{for} \quad i = 0, 1, 2, 3, 4, 5,
\]](http://latex.artofproblemsolving.com/9/f/9/9f92574f9fe14e2e30c3e51e194c661fbd4ece07.png)
find the value of![\[
f(0) + f(1) + f(2) + \cdots + f(6).
\]](http://latex.artofproblemsolving.com/4/0/7/4075d7efa78dda4c46a5cf764fb8852e8c0fb09b.png)
,![\[
\frac{a}{b} + \frac{b}{c} + \frac{c}{a}.
\]](http://latex.artofproblemsolving.com/3/2/b/32b41635c159921e11245369e66185ec69b0d785.png)
be reals such that 
and 
Let 
and 
be the side lengths of a triangle. Prove that![\[\frac{ab+bc+ca}{(a+b)^2} + \frac{ab+bc+ca}{(b+c)^2} + \frac{ab+bc+ca}{(c+a)^2} \leqslant \frac{85}{36}.\]](http://latex.artofproblemsolving.com/f/8/0/f80aed170f4f4bd78a2e881ce93b6b7ddfa03c22.png)
its orthocenter. The circle with diameter ![$[AC]$](http://latex.artofproblemsolving.com/0/9/3/0936990e6625d65357ca51006c08c9fe3e04ba0c.png)
cuts the circumcircle of triangle 
at 
.
and 
is the midpoint of the segment ![$[BH]$](http://latex.artofproblemsolving.com/a/9/f/a9fd89d48264ff1c3f9b77f62ef66f916978d99d.png)
of positive integers is called central if for every positive integer 
,
terms of the sequence is equal to 
,
,
,
of positive integers such that for every central sequence 
,
,
,
.
and 
Prove that
specifies the greatest common divisor of 
and 
.
be a point inside equilateral triangle 
,
.
in degrees.
Y by Mango247
Two pillars are at a distance of 
meters. The large pillar is meters high. A wire is stretched between the tops of these pillars. After a strong wind small the pillar fell towards the larger one. What happened to the wire: it broke, sagged, stayed taut? Explain the answer.
meters. The large pillar is meters high. A wire is stretched between the tops of these pillars. After a strong wind small the pillar fell towards the larger one. What happened to the wire: it broke, sagged, stayed taut? Explain the answer.This post has been edited 1 time. Last edited by parmenides51, May 11, 2021, 1:22 PM
