[/quote]
,
and 
Prove that
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 
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.
Y by chessgocube, centslordm
The following method of approximate measurement is known for distances. Suppose, for example, that the observer is on the river bank at point 
in order to measure its width. To do this, he fixes point 
on the opposite bank so that the angle between the shoreline and the line 
is close to the line. Then the observer pulls forward the right hand with the raised thumb, closes left eye and aligns the raised finger with point . Next, opens the left eye, closes right and estimates the distance between the point on the opposite bank to which the finger points, and point . Multiply this distance by 
and get the approximate value of the distance to point , ie the width of the river. Justify this method of measuring distance.
in order to measure its width. To do this, he fixes point 
on the opposite bank so that the angle between the shoreline and the line 
is close to the line. Then the observer pulls forward the right hand with the raised thumb, closes left eye and aligns the raised finger with point . Next, opens the left eye, closes right and estimates the distance between the point on the opposite bank to which the finger points, and point . Multiply this distance by 
and get the approximate value of the distance to point , ie the width of the river. Justify this method of measuring distance.This post has been edited 1 time. Last edited by parmenides51, May 22, 2022, 7:45 PM
