[/quote]
,
and 
Prove that
Prove that
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 
terms of the sequence is equal to 
of positive integers such that for every central sequence 
there are infinitely many positive integers 
.
be the circumcircle of triangle 
,
is the midpoint of 
and 
be the second intersection of 
and 
,
be a transversal of 
be intersections of 
with 
be a tangent line of 
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 
.
,
.
.
and 
to 
to 
respectively.
and 
as the reflections of 
and 
be the circumcircle of 
.
with 
the intersection of ray 
with 
is the circumcenter of 
can be inscribed within a circle.
,
,
at 
,
be the midpoints of 
and 
,
and 
lie on 
so that 
and 
.
and 
intersect on the circumcircle of 
.
,
lie on a line in this order and satisfy 
.
,
is acute, and let 
be its orthocentre. Points 
and 
,
and 
and 
and 
Y by
The axis of symmetry of a set on a plane is a straight line such that for any point from this set, a point symmetric to it with respect to this line also lies in this set.
1. The angle between two straight lines is eight degrees. Is there a polygon in which each of these lines is an axis of symmetry?
2.
is a non-convex quadrilateral. It is known that each of the triangles 
has an axis of symmetry. Prove that one of them is the axis of symmetry of the whole quadrilateral.
3. Is the statement of the previous item true if
is a convex quadrilateral?
4. The set of points on the plane has exactly
axes of symmetry. What is the smallest number of points this set can consist of?
1. The angle between two straight lines is eight degrees. Is there a polygon in which each of these lines is an axis of symmetry?
2.
is a non-convex quadrilateral. It is known that each of the triangles 
has an axis of symmetry. Prove that one of them is the axis of symmetry of the whole quadrilateral.3. Is the statement of the previous item true if
is a convex quadrilateral?4. The set of points on the plane has exactly
axes of symmetry. What is the smallest number of points this set can consist of?This post has been edited 1 time. Last edited by parmenides51, Feb 27, 2021, 4:05 PM
