AoPS Academy
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 
.
be an acute-angled triangle with 
. Let 
be the orthocenter of triangle 
, and let 
be the midpoint of the side 
. Let 
be a point on the side 
and 
a point on the side 
such that 
and the points , , are on the same line. Prove that the line 
is perpendicular to the common chord of the circumscribed circles of triangle and triangle 
.
, 
, 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 .
and 
be two points inside triangle 
such that![\[ \angle MAB = \angle NAC\quad \mbox{and}\quad \angle MBA = \angle NBC.
\]](http://latex.artofproblemsolving.com/2/0/4/20410178c70693881f348c67454f01057a3fc40c.png)
![\[ \frac {AM \cdot AN}{AB \cdot AC} + \frac {BM \cdot BN}{BA \cdot BC} + \frac {CM \cdot CN}{CA \cdot CB} = 1.
\]](http://latex.artofproblemsolving.com/b/b/6/bb64f80eccb4634b9c9e15cade2f7c0b87b60923.png)
the set of all positive real numbers.
which have the property:![\[f(x)f(y)=2f(x+yf(x))\]](http://latex.artofproblemsolving.com/4/f/f/4ffbde35ff082585a188c1a4368d39338d1b07a4.png)
and 
.
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?
, 
the following inequality holds:
When does equality hold?
with 
, the foot of altitudes from 
to the sides 
are 
, respectively. is the orthocenter. is the midpoint of segment . Lines 
and 
intersect at 
. Let the tangents drawn to circumcircle 
from and intersect at 
. Prove that 
are colinear
be an integer and let 
be positive integers. In a group of 
people, some games of chess are played. Two people can play each other at most once. Prove that it is possible for the following two conditions to hold at the same time:
.
with 
, there is someone who has played exactly 
games of chess.
be an acute triangle with circumcenter 
and orthocenter , satisfying . The tangent line at to the circumcicle of intersects in . Let 
be the midpoint of 
. Prove that 
.
be midpoint of side . It is well-known that 
is parallelogram. is orthocenter of triangle 
. Since 
and 
, we deduce that 
. 
and triangle 
, where is the midpoint of side .
