[/quote]
,
Prove that 
,
,
,
can be expressed by a sum of positive integers 
'![\[n=a_1+a_2+\dots+a_m,\]](http://latex.artofproblemsolving.com/6/c/2/6c2a979a5ff6c40bf2d7152c8c32088fc36f848b.png)
where 
,
,
,
and 
.
equally spaced points on a circular track 
of circumference 
in some order, each label used once. Initially, Bunbun the Bunny begins at 
.
,
,
,
to 
,
of 
is a diameter of 
with the property that ![\[f(x-f(y))=f(f(x))-f(y)-1\]](http://latex.artofproblemsolving.com/f/2/5/f25cc1e8ae0be1fd02b347fd94be4fab88af1d46.png)
holds for all 
.
be the set of positive integers. Find all functions 
such that![\[ m^2 + f(n) \mid mf(m) +n \]](http://latex.artofproblemsolving.com/2/f/4/2f409d1de993f1af8fd839bb8e9f87a57e1b8608.png)
and 
board with a number written in each cell. Every two neighbour rows sum up to at least 
,
.
square is divided into unit squares. Is it possible to fill each unit square with a number 
in such a way that, whenever one places the tile so that it fully covers nine unit squares, the tile will cover nine different numbers?
with no parallel sides is inscribed in a circle 
.
are inscribed in triangles 
,
and 
,
,
,
.
and the number of distinct prime divisors of 
.
be a positive integer. Prove that there exist infinitely many positive integers 
and 
does not divide 
for any positive integers 
satisfying 
.
such that for any 
:
![\[
\begin{cases}
x - y z = 1,\\[2pt]
y - z x = 2,\\[2pt]
z - x y = 4.
\end{cases}
\]](http://latex.artofproblemsolving.com/b/2/d/b2d94e49491a71f86109103f985fe3f238cf4d95.png)
such that 
for all 
Y by PikaPika999, cursed_tangent1434
On triangle 
, draw exterior equilateral triangles on sides 
and 
to obtain 
and 
, respectively. Let 
be the intersection of the altitude through 
and the median through 
. Let 
be the intersection of the altitude through 
and line 
. Let 
be the intersection of the median through and the line 
. Prove that , , and lie on a common line.
![[asy]
size(481.245,258.036);
label("$A$", (-1.5,7.8));
label("$B$", (5.2, 0.7));
label("$C$", (-5.3,0.7));
label("$B'$", (-9.1,7.4));
label("$C'$", (7.8,9.8));
path line = (0,1)--(-1.40861,7.40089);
draw(line, gray+linewidth(1.5)+solid );
path line = (-5,1)--(1.7957,4.20044);
draw(line, gray+linewidth(1.5)+solid );
path line = (-1.40861,1)--(-1.40861,7.40089);
draw(line, orange+linewidth(1.5)+solid );
path line = (-2.60567,5.26737)--(5,1);
draw(line, orange+linewidth(1.5)+solid );
path line = (-8.74763,7.31068)--(5,1);
draw(line, blue+linewidth(1.5)+solid );
path line = (-5,1)--(7.33903,9.75046);
draw(line, blue+linewidth(1.5)+solid );
path line = (5,1)--(-1.40861,7.40089);
draw(line, black+linewidth(2.5)+solid );
path line = (-8.74763,7.31068)--(-1.40861,7.40089);
draw(line, black+linewidth(2.5)+solid );
path line = (-1.40861,7.40089)--(7.33903,9.75046);
draw(line, black+linewidth(2.5)+solid );
path line = (-8.74763,7.31068)--(-5,1);
draw(line, black+linewidth(2.5)+solid );
path line = (-5,1)--(-1.40861,7.40089);
draw(line, black+linewidth(2.5)+solid );
path line = (-5,1)--(5,1);
draw(line, black+linewidth(2.5)+solid );
path line = (5,1)--(7.33903,9.75046);
draw(line, black+linewidth(2.5)+solid );
path line = (-10.3821,3.4813)--(8.86775,3.62203);
draw(line, rgb(0, 0.666, 0)+linewidth(1)+dashed);
dot((0.436645,3.5604), linewidth(3)+solid);
dot((-1.40861,3.54691), linewidth(3)+solid);
dot((-0.561846,3.5531), linewidth(3)+solid);
path frame = (-10.3821,-0.0943394)--(-10.3821,10.2271)--(8.86775,10.2271)--(8.86775,-0.0943394)--cycle;
clip(frame);
[/asy]](//latex.artofproblemsolving.com/4/6/9/469fd874cf0fb80a65093eb5bb02d383b06b4bcb.png)
, draw exterior equilateral triangles on sides 
and 
to obtain 
and 
, respectively. Let 
be the intersection of the altitude through 
and the median through 
. Let 
be the intersection of the altitude through 
and line 
. Let 
be the intersection of the median through and the line 
. Prove that , , and lie on a common line.![[asy]
size(481.245,258.036);
label("$A$", (-1.5,7.8));
label("$B$", (5.2, 0.7));
label("$C$", (-5.3,0.7));
label("$B'$", (-9.1,7.4));
label("$C'$", (7.8,9.8));
path line = (0,1)--(-1.40861,7.40089);
draw(line, gray+linewidth(1.5)+solid );
path line = (-5,1)--(1.7957,4.20044);
draw(line, gray+linewidth(1.5)+solid );
path line = (-1.40861,1)--(-1.40861,7.40089);
draw(line, orange+linewidth(1.5)+solid );
path line = (-2.60567,5.26737)--(5,1);
draw(line, orange+linewidth(1.5)+solid );
path line = (-8.74763,7.31068)--(5,1);
draw(line, blue+linewidth(1.5)+solid );
path line = (-5,1)--(7.33903,9.75046);
draw(line, blue+linewidth(1.5)+solid );
path line = (5,1)--(-1.40861,7.40089);
draw(line, black+linewidth(2.5)+solid );
path line = (-8.74763,7.31068)--(-1.40861,7.40089);
draw(line, black+linewidth(2.5)+solid );
path line = (-1.40861,7.40089)--(7.33903,9.75046);
draw(line, black+linewidth(2.5)+solid );
path line = (-8.74763,7.31068)--(-5,1);
draw(line, black+linewidth(2.5)+solid );
path line = (-5,1)--(-1.40861,7.40089);
draw(line, black+linewidth(2.5)+solid );
path line = (-5,1)--(5,1);
draw(line, black+linewidth(2.5)+solid );
path line = (5,1)--(7.33903,9.75046);
draw(line, black+linewidth(2.5)+solid );
path line = (-10.3821,3.4813)--(8.86775,3.62203);
draw(line, rgb(0, 0.666, 0)+linewidth(1)+dashed);
dot((0.436645,3.5604), linewidth(3)+solid);
dot((-1.40861,3.54691), linewidth(3)+solid);
dot((-0.561846,3.5531), linewidth(3)+solid);
path frame = (-10.3821,-0.0943394)--(-10.3821,10.2271)--(8.86775,10.2271)--(8.86775,-0.0943394)--cycle;
clip(frame);
[/asy]](http://latex.artofproblemsolving.com/4/6/9/469fd874cf0fb80a65093eb5bb02d383b06b4bcb.png)
This post has been edited 4 times. Last edited by cooljoseph, Apr 6, 2025, 9:56 PM
Reason: Fixed first sentence
Reason: Fixed first sentence
