In a trapezium , the sides and are parallel and the angles and are acute. Show that it is possible to divide the triangle into 4 disjoint triangle and the triangle into 4 disjoint triangles such that the triangles and are congruent for all .

I have noticed thousands upon thousands of inequalities that you have posted to HSO and was wondering where you get the inspiration, imagination, and even the validation that such inequalities are true? Also, what do you find particularly appealing and important about specifically inequalities rather than other branches of mathematics? Thank you :)

Find all functions such that

Triangle has side lengths , , and . Select a point inside , and construct the incenters of , , and and denote them as , , . What is the maximum area of the triangle ?

Problem. Given be non-negative real numbers such that Prove that

A rectangle with odd integer side lengths is divided into small rectangles with integer side lengths. Prove that there is at least one among the small rectangles whose distances from the four sides of are either all odd or all even.

Proposed by Jeck Lim, Singapore

A number is interesting if 2018 divides (the number of positive divisors of ). Determine all positive integers such that there exists an infinite arithmetic progression with common difference whose terms are all interesting.

Given a fixed circle with its center . There are two fixed points and one moving point on . The midpoint of the line segment is . is a fixed point on . Line intersects at , and line intersects at .

Find all the fixed points such that is always tangent to when varies.
Hint

Find all pairs of positive integers such that:

Let G be a grid of size m*n.

We have 2 dominoes in flowers and not connected like here
IMAGE
Determine a necessary and sufficient condition on m and n, so that G can be covered with these 2 kinds of dominoes.