[/quote]
,
.
points are collinear in Euclidean plane, such that Shining cannot find a group of three points such that their circumcircle contains none of the other remaining points. Is he always able to do so?
and 
Prove that
![$$ a+b+kc^2\geq\frac{3\sqrt[3]{k}}{4}$$](http://latex.artofproblemsolving.com/5/5/b/55be0378562556d1aa4400bef11d1510ff722854.png)
Where 
be an infinite sequence of positive integers. Prove that there exists a unique integer 
such that![\[a_n < \frac{a_0+a_1+a_2+\cdots+a_n}{n} \leq a_{n+1}.\]](http://latex.artofproblemsolving.com/9/4/f/94fc5a51b7e68588123c5b527fe75183bc4c4937.png)
.
such that
.
denote the number of positive integers less than 
that are relatively prime to 
for which the equation 
has at least 
solutions in 
is a rational number.
and other permutations.
with real coefficients for which there exists a polynomial 
with real coefficients such that for all real 
,
of points in the plane is balanced if, for any two different points 
and 
in 
.
.
,
,
divides 
if and only if 
?
be a sequence of real numbers, and let 
with 
is good if there exists some 
with 
such that 
,
be the set of all good indices. Prove that 
.
such that for any positive real numbers 
,
.
,
is a point on the side 
, so that vertex 
is located on side 
,
,
is a point on side 
.
.
Y by
A point 
is chosen on the side 
of a rectangle 
. From the vertex 
, the perpendicular 
is dropped to the segment 
. The segments and divide the rectangle into three parts such that each of them has the inscribed circle (see figure). Prove that if the circles tangent to are equal then the third circle is also equal to them.
is chosen on the side 
of a rectangle 
. From the vertex 
, the perpendicular 
is dropped to the segment 
. The segments and divide the rectangle into three parts such that each of them has the inscribed circle (see figure). Prove that if the circles tangent to are equal then the third circle is also equal to them.Attachments:
This post has been edited 1 time. Last edited by QueenArwen, Mar 11, 2025, 10:52 AM
