AoPS Academy
[/quote]
, do not answer with "
is a solution" only. Either you post any kind of proof or at least something unexpected (like "
is the smallest solution). Someone that does not see that is a solution of the above without your post is completely wrong here, this is an IMO-level forum.
be the orthocenter of acute triangle 
, let 
be the foot of the altitude from 
to 
, and let 
be the reflection of across 
. Suppose that the circumcircle of triangle 
intersects line at two distinct points 
and 
. Prove that is the midpoint of 
.
, let 
be the circumcentre, be the orthocentre, and 
be the centroid.
satisfy the following condition:
is constant.
be an acute triangle inscribed in a circle of center . If the altitudes 
intersect at and the circumcenter of 
is 
, prove that 
is a parallelogram.
such that among any 
positive real numbers 
with 
, there exist three that are the side lengths of an acute triangle.
be the set of integers, and let 
be a function. Prove that there are infinitely many integers 
such that the function 
defined by 
is not bijective. is bijective if for every integer 
, there exists exactly one integer 
such that 
.
is the center of the circle. If three angles 
and 
be equal, find 
points on the plane. No three of them lie on the same straight line and no four lie on the same circle. Prove that it is possible to split these points into pairs and cover each pair of points with a circle containing no other points.
with 
, the foot of altitudes from 
to the sides 
are 
, respectively. is the orthocenter. 
is the midpoint of segment . Lines 
and 
intersect at 
. Let the tangents drawn to circumcircle 
from 
and intersect at 
. Prove that 
are colinear
triangle with 
. We take , 
point on AC, AB respectively such that 
, 
. 
, 
lines intersect at point . If 
, find 
be a convex quadrilateral with ![\[\angle ADC = 90^\circ, \ \ \angle BCD = \angle ABC > 90^\circ, \mbox{ and } AB = 2CD.\]](http://latex.artofproblemsolving.com/b/4/1/b416d5c0b29c3a69a3e87cd22b6000aa8e70d456.png)
The line through 
, parallel to 
, intersects the external angle bisector of 
at point 
. Prove that the angles 
, 
, 
, 
, 
can be divided into two groups, so that the angles in each group have a sum of 
. be a point outside circle 
centered at . Tangents from to touch at 
. Line 
intersects bigger arc at .The line drawn from parallel to 
intersects 
at 
. Ray 
intersects small arc at . Prove that the circumcircle of 
is tangent to .
![[asy]
/* AMC8 P17 2024, revised by Teacher David */
unitsize(29pt);
import math;
add(grid(3,3));
pair [] a = {(0.5,0.5), (0.5, 1.5), (0.5, 2.5), (1.5, 2.5), (2.5,2.5), (2.5,1.5), (2.5,0.5), (1.5,0.5)};
for (int i=0; i<a.length; ++i) {
pair x = (1.5,1.5) + 0.4*dir(225-45*i);
draw(x -- a[i], arrow=EndArrow());
}
label("$K$", (1.5,1.5));
[/asy]](http://latex.artofproblemsolving.com/b/8/7/b873b72677375174654958acad429fa4affaefb6.png)
so its E
side is 
answer is 
answer choice 
