AoPS Academy
be a cyclic quadrilateral. Let 
be the midpoint of the arc 
of its circumcircle which does not contain 
or 
. Let the lines 
and 
meet at 
and the lines 
and 
meet at 
. Prove that the lines 
and are parallel.
be an acute triangle with circumcenter 
and orthocenter 
. Let 
denote the circumcircle of triangle , and 
the midpoint of 
. The tangents to at 
and , and the line through perpendicular to line 
, determine a triangle whose circumcircle we denote by 
. Define 
and 
similarly., and are concurrent on line .
such that for all positive integers 
and 
,![\[f(m+nf(m))=f(n)^m+2024! \cdot m.\]](http://latex.artofproblemsolving.com/8/b/9/8b90a9326c0fa448a5a574809c184f43513f768f.png)
Jaedon Whyte
, and an odd prime 
not dividing any of them, and an integer 
such that if one considers the infinite sequence 
and looks at the highest power of that divides each of them, these powers are not all zero, and are all at most . Prove that there exists some 
(which may depend on 
) such that whenever divides an element of this sequence, the maximum power of that divides that element is exactly 
.
, 
, 
be an infinite sequence of positive integers. Suppose that there is an integer 
such that, for each 
, the number
is an integer. Prove that there is a positive integer such that 
for all 
. be a right triangle with 
. Point lies on the line 
such that is between and . Let 
be the midpoint of 
and let be the seconf intersection point of the circumcircle of 
and the circumcircle of 
. Prove that as varies, the line 
passes through a fixed point.
and 
intersect each other at 
and 
. The common tangent to two circles nearer to touch and at 
and 
respectively. Let 
and 
be the reflection of and respectively with respect to . The circumcircle of the triangle 
intersect circles and respectively at points 
and 
(both distinct from ). Show that the line 
is the second tangent to and .
to 
face-down on a table. On each move, the Detective can point to two cards and inquire if the numbers on them are consecutive. The Magician replies truthfully. After a finite number of moves, the Detective points to two cards. She wins if the numbers on these two cards are consecutive, and loses otherwise.
questions.
= 2∠AMP.
is the midpoint of the arc 
, containing 
, in the circumcircle of 
, and let 
be the -excircle's center. Assume that the external angle bisector of 
intersects at 
. Prove that 
is perpendicular to 
, where 
is the incenter of .
with 
and the circle 
of radius centered at 
Let be the midpoint of 
The line 
intersects a second time at 
Let be a point on such that 
Let be the intersection of 
and Prove that 
. Then, 
and 
are the projective varieties of some homogeneous degree 
polynomials in 
variables 
, 
, respectively, that are unique up to scaling. Similarly, the common tangents 
, 
, 
, and 
are the projective varieties of some homogeneous linear polynomials 
, 
, 
, and 
, respectively. Finally, the chords 
, 
, 
, and 
are again the projective varieties of some homogeneous linear polynomials 
, 
, 
, and 
.
, 
, 
, and 
, such that![\[
\mathcal E_1 = \lambda_{113} \ell_1\ell_3 + m_1^2 = \lambda_{124} \ell_2 \ell_4 + m_2^2
\quad \text{and} \quad
\mathcal E_2 = \lambda_{213} \ell_1\ell_3 + m_3^2 = \lambda_{224} \ell_2 \ell_4 + m_4^2,
\]](http://latex.artofproblemsolving.com/1/c/5/1c5d374bc04482be6e2d0f5478dabe5a23f36495.png)
since is in double-contact with 
with double-contact chord , etc. It follows that![\[ (m_1-m_2)(m_1+m_2) = \lambda_{124} \ell_2 \ell_4 - \lambda_{113} \ell_1\ell_3.\]](http://latex.artofproblemsolving.com/d/5/7/d57558a98c9ee6c687a89eab9ca483c9d270c1d9.png)
But the projective variety generated by the two sides is a degenerate conic formed by lines, by looking at the LHS, which also passes through the four intersection points of 
and 
, by looking at the RHS. But clearly it is neither nor , so it must be the last two sides of the complete quadrangle with the aforementioned 
points as vertices., we have that 
and 
have the same variety. By flipping signs, we can assume WLOG that 
and 
have the same variety, i.e. there exists some nonzero constant 
such that 
. Rewrite this as 
. polynomial,![\[ \mathcal M = (m_1 - \alpha m_3)^2 - \mathcal E_1 - \alpha^2 \mathcal E_3 = (m_2 - \alpha m_4)^2 - \mathcal E_1 - \alpha^2 \mathcal E_3\]](http://latex.artofproblemsolving.com/d/6/3/d63c942b38c2f0185fcdf11313d90af7e394043e.png)
passes through 
, 
, 
, 
, 
, 
, 
, and 
. Indeed by expansion, we have![\[ \mathcal M = - 2 \alpha m_1 m_3 - \lambda_{113} \ell_1\ell_3 - \alpha^2 \lambda_{213} \ell_1\ell_3 = (-2\alpha) m_1 m_3 + (-\lambda_{113} - \alpha^2 \lambda_{213}) \ell_1\ell_3, \]](http://latex.artofproblemsolving.com/a/6/4/a64e93ccf018d260ee68c7e26d56c18834406fe9.png)
so the variety of 
passes through the four intersection points of the varieties of 
and 
, namely, , , , and . Similarly, passes through the four intersection points of the varieties of 
and 
, so we are done. 
be two conics . If the four common tangents touch 
and 
then prove that :
and 