by Davdav1232, May 8, 2025, 8:24 PM 
be an isosceles triangle with 
. Let 
be a point on 
. Let 
be a point inside the triangle such that 
and![\[
CL \cdot BD = BL \cdot CD.
\]](http://latex.artofproblemsolving.com/1/6/7/16771838c95f86e92f79b8d049e46ab473e6287d.png)
Prove that the circumcenter of triangle 
lies on line 
.by EeEApO, May 8, 2025, 6:08 PM 
questions, each with 
answer choices. A participant who answers all questions correctly will receive a gift. To ensure that at least one member of my family answers all questions correctly, how many family members need to take the quiz?
, each of our twin children also begins to have twins every year. If this pattern continues, how many years will it take for my family to grow large enough to have the required number of members to guarantee winning the quiz gift?by TBazar, May 8, 2025, 6:57 AM 
triangle with 
. We take 
, 
point on AC, AB respectively such that 
, 
. 
, 
lines intersect at point 
. If 
, find 
by Com10atorics, Jun 7, 2021, 2:43 PM 
be positive real numbers such that 
. Prove that
.by ayan_mathematics_king, Jul 28, 2019, 3:46 AM 
, prove that 
where 
by adihaya, Mar 30, 2016, 2:40 PM 
of functions by 
for 
. Prove that each 
is a polynomial with integer coefficients.by mattilgale, Jan 29, 2007, 8:40 AM 
be a convex pentagon such that![\[ \angle BAC = \angle CAD = \angle DAE\qquad \text{and}\qquad \angle ABC = \angle ACD = \angle ADE.
\]](http://latex.artofproblemsolving.com/d/3/d/d3d9a82f4318190298a8f008d417a364e03f1fca.png)
The diagonals 
and 
meet at 
. Prove that the line 
bisects the side 
.by darij grinberg, Jan 19, 2005, 11:00 AM 
, 
, 
, ... be an infinite sequence of real numbers satisfying the equation 
for all 
, where and are two different positive reals., , , ... be bounded?by darij grinberg, May 18, 2004, 8:25 PM 
, 
, and 
are fixed on a line in this order. Let 
be a circle passing through and whose center does not lie on the line 
. Denote by the intersection of the tangents to at and . Suppose meets the segment 
at 
. Prove that the intersection of the bisector of 
and the line does not depend on the choice of .
The oldest, shortest words — "yes" and "no" — are those which require the most thought.
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be positive real numbers such that 
Prove that:
April 2021