(use in a problem for junior) to practice. Can anyone share to me?
be an acute angled scalene triangle with circumcircle 
.
and 
be the circumcenter and orthocenter of 
respectively. Let 
and 
be points on segments 
and 
Prove that 
and 
intersect on the circumcircle of 
.
grid with 
rows and 
columns, and each square of the grid is filled with a non-negative integer. Let 
be the average of all the numbers in the grid. Prove that if 
the there exist two identical rows (meaning same numbers in the same order).
and 
, then we can deduce 
\le 2 $ ?
satisfying 
.![\[
0 \le (x^2 + 1)(y^2 + 1) + 4(x - 1)(y - 1) \le 164.
\]](http://latex.artofproblemsolving.com/4/2/d/42d0b88388462f16dd99d928b94024e689ba7c2e.png)
be a sequence defined by 
.
if 
.
that satisfy 
with 
,
and area 
.
.
and extend to meet 
and 
is the diameter of 
of 
,
of 
Y by Adventure10, Mango247
Bisectors of angle 
and externalangle 
of trapezoid 
with bases 
and 
intersect at point 
, and the bisector of angle 
and external angle 
intersect at point 
. Prove that the midpoint of the segment 
is equidistant from the lines 
and 
.
and externalangle 
of trapezoid 
with bases 
and 
intersect at point 
, and the bisector of angle 
and external angle 
intersect at point 
. Prove that the midpoint of the segment 
is equidistant from the lines 
and 
.This post has been edited 1 time. Last edited by parmenides51, Jul 29, 2019, 2:56 PM
Reason: source
Reason: source
