In a triangle with , denote its incenter and midpoint of by , , respectively. Let be the reflected point of wrt . Let the lines and meet at . Suppose that . Prove that .

Triangle ABC has incircle touching the sides at D, E, F as shown.
AD, BE, CF concurrent at Gergonne point G.
BG and CG cuts the incircle at X and Y, respectively.
AG cuts the incircle at K.
Prove: K, X, D, Y form a harmonic quadrilateral. (KX/KY = DX/DY)

Let be the midpoint of side in , and is such that the quadrilateral is cyclic and the circumcircle of is tangent to the line . If is the second common point of the line and the circumcircle of , determine the ratio .

Determine all pairs of positive integers such that is divisible by .

Let triangle be triangle with orthocenter and circumcircle . A point lies on line . intersects the circumcircle of triangle again at . intersects circumcircle of triangle again at and intersects circumcircle of triangle again at . Let be the intersection of with . Let lies on such that . Prove that
1. the ratio and the angle between and do not depend on the choices of .
2. .

Proposed by: k12byda5h

Let be an isosceles triangle with , and let be its incenter. Incircle touches sides at , respectively. Foot of altitudes from to are , respectively. Rays intersect at , respectively. Prove that touches incircle at .

A function from the positive integers to the positive integers is called Canadian if it satisfies for all pairs of positive integers and .

Find all positive integers such that for all Canadian functions .

Let be a triangle with circumcircle . Arbitrary points on respectively. Circumcircle of triangle intersects at . intersects at . cuts and at respectively. Construct parallelogram . Prove that are collinear.

Two squirrels, Bushy and Jumpy, have collected 2021 walnuts for the winter. Jumpy numbers the walnuts from 1 through 2021, and digs 2021 little holes in a circular pattern in the ground around their favourite tree. The next morning Jumpy notices that Bushy had placed one walnut into each hole, but had paid no attention to the numbering. Unhappy, Jumpy decides to reorder the walnuts by performing a sequence of 2021 moves. In the -th move, Jumpy swaps the positions of the two walnuts adjacent to walnut .

Prove that there exists a value of such that, on the -th move, Jumpy swaps some walnuts and such that .

Prove that for every positive integer there exists a positive integer , such that each of the numbers have at least one block of in their decimal representation.

For example, the numbers 4202213 and 544202212022 have at least one block of in their decimal representation.

Let be real numbers such that . Prove that

Is there an infinite sequence of prime numbers such that for every is satisfied? Explain the answer.

If a,b,c are positive real numbers, such that a+b+c=1. Prove that:
(b+c)(a+c)/(a+b)+ (b+a)(a+c)/(c+b)+(b+c)(a+b)/(a+c)>= Sqrt.(6(a(a+c)+b(a+b)+c(b+c)) +3