Let be the incentre of acute triangle with . The incircle of is tangent to sides , and at and , respectively. The line through perpendicular to meets at . Line meets again at . The circumcircles of triangle and meet again at .

Prove that lines and meet on the line through perpendicular to .

Proposed by Anant Mudgal, India

I have learned little about what bashing mean as i am planning to start geo, feels like its less effort required and doesnt need much knowledge about the synthetic division ones?
what do you guys recommend ? also state the major difference of them... especially of bashing pros and cons..

Solve this.

Let and Prove that

Positive integers integers are such that is prime and . Prove that there exists a positive integer such that .

A strictly decreasing function attaining all positive values and positive numbers are given. Numbers satisfy
for every . Prove that there exists a positive integer satisfying .

Convex quadrilateral is described on a circle , and is not a trapezius inscribed in a circle. Let the tangency points of and be respectively. A circle with a center , different from is tangent to the segement and lines . A circle with center , different from is tangent to segment and lines . A circle with center , different from is tangent to segment and lines . A circle with center , different from is tangent to segment and lines . Prove that the lines are concurrent.

For define be the product of first primes. Determine if there exists positive integers with the following property :


Proposed by Matko Ljulj

Let be reals such that and Prove thatLet be reals such that and Prove thatLet be reals such that and Prove that

A positive integer and a set consisting of disting positive integers smaller than are given. Prove that there exists a positive integer that can be written in the form , for in at least different ways.

Positive integer and colors are given. We will say that a set of points on a plane is , if it contains exactly 2 points of each color and if lines connecting every two points of the same color are pairwise distinct. Find, in terms of the least integer such that: in every set of points of a plane, no three of which are collinear, consisting of points of every color there exists a subset.

Is it true that?

A=1728400904217815186787639216753921417860004366580219212750904
024377969478249664644267971025952530803647043121025959018172048
336953969062151534282052863307398281681465366665810775710867856
720572225880311472925624694183944650261079955759251769111321319
421445397848518597584590900951222557860592579005088853698315463
815905425095325508106272375728975

B=2275643401548081847207782760491442295266487354750527085289354
965376765188468052271190172787064418854789322484305145310707614
546573398182642923893780527037224143380886260467760991228567577
953725945090125797351518670892779468968705801340068681556238850
340398780828104506916965606659768601942798676554332768254089685
307970609932846902