AoPS Academy
is given in the plane. The incircle of triangle 
has center 
and is tangent to diagonal 
at 
. Let 
be the center of parallelogram . Show that 
and 
are parallel.
+1 w
is valid while 
is not)
of positive integers such that 
is an integer.
for which 
is a perfect square.
, and 
, respectively, located at the origin of the coordinate plane. In a move, Carina may move a pin to an adjacent lattice point at distance 
away. What is the least number of moves that Carina can make in order for triangle to have area 2021?
in the coordinate plane where 
and 
are both integers, not necessarily positive.)
be a scalene triangle with circumcircle 
and incenter . Ray 
meets 
at 
and meets again at 
; the circle with diameter 
cuts again at 
. Lines 
and 
meet at 
, and 
is the midpoint of 
. The circumcircles of 
and 
intersect at points 
and 
. Prove that passes through the midpoint of either 
or 
.-by- board of unit squares for some odd positive integer . We say that a collection of identical dominoes is a maximal grid-aligned configuration on the board if consists of 
dominoes where each domino covers exactly two neighboring squares and the dominoes don't overlap: then covers all but one square on the board. We are allowed to slide (but not rotate) a domino on the board to cover the uncovered square, resulting in a new maximal grid-aligned configuration with another square uncovered. Let 
be the number of distinct maximal grid-aligned configurations obtainable from by repeatedly sliding dominoes. Find the maximum value of as a function of .
that satisfy the equation
is 
. The incenter of the triangle is , and the intouch triangle is 
. Let 
be the orthocenter of 
. Prove that , , and are collinear.
,then the triangles 
and 
are similar homotetic with centered at 
( with parallel sides ).On the other hand 
are orthocenter and circumcenter of the triangle , 
are orthocenter and circumcenter of the triangle , so 
are collinear.
are positive real numbers and 
.
.
