AoPS Academy
[/quote]
, do not answer with "
is a solution" only. Either you post any kind of proof or at least something unexpected (like "
is the smallest solution). Someone that does not see that is a solution of the above without your post is completely wrong here, this is an IMO-level forum.
satisfied 
Find minimum and maximum of 
such that:
with all 
is real number
be the set of integers. Determine all functions 
such that, for all integers 
and 
, 
Proposed by Liam Baker, South Africa
be a positive integer. Find all real solutions 
to the system:![\[a_1^2 + a_1 - 1 = a_2\]](http://latex.artofproblemsolving.com/a/4/e/a4e26b83c929b82800576bac11e12de0c3aa1cf6.png)
![\[ a_2^2 + a_2 - 1 = a_3\]](http://latex.artofproblemsolving.com/9/9/5/995e8eec93828b82901b0639463f907f5ff95d08.png)
![\[\hspace*{3.3em} \vdots \]](http://latex.artofproblemsolving.com/e/e/1/ee15e1fefe2ad550d3c9b1c51207eaf66fbfa41f.png)
![\[a_{n}^2 + a_n - 1 = a_1\]](http://latex.artofproblemsolving.com/7/c/7/7c727d1f2044c1bf7c80981e1074bb3cde807b8f.png)
of positive integers is called central if for every positive integer , the arithmetic mean of the first 
terms of the sequence is equal to .
of positive integers such that for every central sequence 
there are infinitely many positive integers with 
.
be the circumcircle of triangle 
, 
is the midpoint of 
and 
be the second intersection of 
and . Tangent line of at intersects at 
, let 
be a transversal of and 
be intersections of 
with . Let 
be a tangent line of . Prove that 
is cyclic
be positive integers that satisfy the inequality![\[
\frac{a}{b} < \frac{x}{y} < \frac{c}{d}
\]](http://latex.artofproblemsolving.com/8/d/9/8d982d52e655546ac29039fa6e440e68821bfe7f.png)
with 
. If 
, determine the minimum value of 
.
. Let 
and be the feet of the altitudes from 
to and from 
to 
respectively.
and 
as the reflections of across lines 
and , respectively. Let 
be the circumcircle of 
. Denote by the second intersection of line 
with , and by 
the intersection of ray 
with .
is the circumcenter of , prove that , , and are collinear if and only if quadrilateral 
can be inscribed within a circle.
if this angle is acute. Find the minimal number of circles one must draw to be sure.
if this angle equals 
.(One may draw as many circles as one needs).
if this angle is acute. Find the minimal number of circles one must draw to be sure. if this angle equals .(One may draw as many circles as one needs).
if this angle is acute. Find the minimal number of circles one must draw to be sure. if this angle equals .(One may draw as many circles as one needs). seems to be easy...
Then you draw a circle with radius 
and centre at 
. If 
is a secant to the circle in the original way 
is drawn, it will be acute; if it is tangent then it will be 
and if it is obtuse then it will be a secant when you produce 
to meet it. I have not figured it out yet.
degree angle by at first finding out 
degree then 
degree and then degree.Then,we find out 
degree in the following way:we know,
=value of angle.
=length of arc
=radius
,and 
,we get,
.
degree,for we will get a value of .,with centre and radius ,we cut off an arc at .from ,with radius ,we cut off an arc E on the previous arc.We join 
.Then,
degree.Then by drawing a triangle with degree and degree as interior angles,We get exterior angle 
degree. degree,or,not.
). We can easily find out if 
is smaller, bigger or equal to 
. If it is smaller or equal then we are done. Let 
. We can find out if 
=
(note that 
).
One way to construct 
angle is 
or regular pentagon interior angle take away 
Let 
be the given angle, 
and let 
Perpendicular to 
at cuts circle 
at 
is midpoint of 
and circle 
cuts the ray 
at 
(golden ratio). 
is a rectangle and circle 
cuts at 
Circle 
cuts at 
is equilateral. Bisector of 
cuts at 
and bisector of 
cuts at 
Compare 
and 
Prove that
Prove that
Prove that
, but i think that your solution is correct.