Circles and are given. Let be the midpoint of the segment joining their centers, , be arbitrary points on , respectively such that . Find the locus of the midpoints of segments .
Proposed by: L Shatunov

Let , be the circumcenter and the incenter of an acute-angled scalene triangle ; , , be the touching points of its excircle with the side and the extensions of , respectively. Prove that if the orthocenter of the triangle lies on the circumcircle of , then it is symmetric to the midpoint of the arc with respect to .
Proposed by: P.Puchkov,E.Utkin

Let be a point inside a quadrilateral such that . Points , , are isogonally conjugated to with respect to the triangles , , , respectively. Prove that the diagonals of the quadrilaterals and concur.
Proposed by: G.Galyapin

Let be a quadrilateral such that the excircles and of triangles and touching their sides and respectively touch the extension of at the same point . The segment meets at point , and the line meets at and . Prove that one of angles and is right
Proposed by: I.Kukharchuk

Given triangle ABC. The tangent at ? to the circumcircle(ABC) intersects line BC at point T. Points D,E satisfy AD=BD, AE=CE, and ∠CBD=∠BCE<90 ∘ . Prove that D,E,T are collinear.

The real root of the polynomial can be written in the form
where and are positive integers. Find the value of .

Let be an acute triangle with , and its vertices lie on the circle . Let be the altitude from vertex . Let and be the feet of the perpendiculars from to the lines and , respectively. Let intersect the circle again at points and such that lies between and . Let the lines and intersect at point . Let be the midpoint of segment . Let intersect line at point .
a) Prove that .
b) Prove that line is parallel to line .
c)Let be the orthocenter of triangle , and let be the midpoint of segment . is the center (HBC). Let point lie on line such that ray is the angle bisector of . Prove that lines and intersect at a point lying on circle .

Three squares are drawn on the sides of triangle (i.e., the square on has as one of its sides and lies outside ). Show that the lines drawn from the vertices , , and to the centers of the opposite squares are concurrent.

IMAGE

Let be positive integers. Maximize if .

n is an integer. There are n integers such that they are larger or equal to 1, and less or equal to 6. Sum of them is larger or equal to 4n, while sum of their square is less or equal to 22n. Prove n is divisible by 5.

We are given in points in a plane and we connect some of them so that 10n^2 + 1 segments are drawn. We color these segments in 2 colors. Prove that we can find a monochromatic triangle.

I can solve only the first 4 problems of the last general round of the HMMT competition: https://hmmt-archive.s3.amazonaws.com/tournaments/2024/nov/gen/problems.pdf

As a prediction, would this mean I am good enough to qualify for AIME? How does the difficulty compare?

If , where , then maximum value of .

It is given that where is a natural number.What is the remainder when is divided by