is 
and 
.
to 
is 
and 
.
and 
is 
is 
.
is tangent to the 
with 
and 
.
and 
be the midpoints of sides 
,
and 
be points on segment 
and 
,
is the angle bisector of 
and 
is the angle bisector of 
.
is parallel to 
if and only if 
is chosen. Prove that there exists a prime number that divides infinitely many terms of the sequence 
,![\[
b_k = a^{k^k} \cdot 2^{2^k - k} + 1.
\]](http://latex.artofproblemsolving.com/1/7/5/1751a60482d729a36c71b77ac9c978e724f40da0.png)
be a 
makes a fixed angle with the tangent to 
be a rectangular hyperbola centered at the origin, scaled such that it is tangent to the logarithmic spiral at some point.
such that every positive integer appears exactly once and the product of the first 
terms is a perfect 
power ?
be non-negative real numbers such that ![\[b+c\leqslant a+1,\quad c+a\leqslant b+1,\quad a+b\leqslant c+1.\]](http://latex.artofproblemsolving.com/b/9/e/b9e86e898c0536b7323a03611d5bdbf679caa710.png)
Prove that 
called ‘nice’ if and only if 
and 
are both perfect squares. How many positive integers 
such that 
are composites for all nice numbers 
decide whether there exist pairwise distinct positive integers 
such that for every 
,
divides 
.
of positive integers with the following property: all of the terms of the sequence 
have a greatest common divisor 
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
