lines
such that for all 
,
be a fixed integer. There are 
beads on a circular necklace. You wish to paint the beads using 
colors, such that among any 
consecutive beads every color appears at least once. Find the largest value of 
for which this task is 
possible.
,
,
,
denote the feet of the altitudes from 
,
,
,
denote the circumcenter of 
.
and 
denote the projections of 
,
.
such that 
and 
.
and 
lies on 
such that 
,
bisects 
.
Y by
Let 
and 
be monic quadratic trinomials, and let 
and 
be the vertices of the parabolas 
and 
, respectively. Let 
denote the minimum value of the function 
. It is known that the differences 
and 
are equal positive numbers. Find the angle between the line 
and the 
-axis.
and 
be monic quadratic trinomials, and let 
and 
be the vertices of the parabolas 
and 
, respectively. Let 
denote the minimum value of the function 
. It is known that the differences 
and 
are equal positive numbers. Find the angle between the line 
and the 
-axis.This post has been edited 1 time. Last edited by egxa, Apr 18, 2025, 5:25 PM
and 
are 
and 
.
for 
for 
,
which will cause contradiction.
,
when 
(by looking the convex parabola).
.![\[ P_1(mP_2) - mP_1 = P_2(mP_1) - mP_2. \]](http://latex.artofproblemsolving.com/c/e/e/cee4cbd48406c2d474c10777d5344e8f40dd47f8.png)
and 
are 
and 
.![\[ P_1(d) - P_1(a) = P_2(b) - P_2(c). \]](http://latex.artofproblemsolving.com/d/f/a/dfa1c480bf7343b84e00d80edd64b6b1e2081652.png)
Just expand and you will get![\[ d^2 - a^2 + (-2a) (d-a) = b^2 - c^2 + (-2c) (b-c) \]](http://latex.artofproblemsolving.com/1/d/a/1dadf3ff3c97b49f467c4d3c4f6fd1382b49496b.png)
which turns into![\[ (d - a)^2 = (b - c)^2 \]](http://latex.artofproblemsolving.com/d/d/c/ddc3f886c61a2f8e7a38732fae971eba7967462e.png)
Hence the slope of 
will be 
,
