(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 
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 
such that 
and 
is the diameter of 
of 
,
of 
Y by Adventure10, Mango247, Mango247
Three lines 
are drawn through a point 
inside a triangle 
such that 
meets 
at 
and 
at 
and 
, 
meets 
at 
and 
at 
and 
, 
meets 
at 
and 
at 
and 
. Prove that the lines 
are uniquely determined by these conditions. Find point for which the triangles 
have the same area and show that this point is unique.
are drawn through a point 
inside a triangle 
such that 
meets 
at 
and 
at 
and 
, 
meets 
at 
and 
at 
and 
, 
meets 
at 
and 
at 
and 
. Prove that the lines 
are uniquely determined by these conditions. Find point for which the triangles 
have the same area and show that this point is unique.This post has been edited 1 time. Last edited by parmenides51, Sep 29, 2022, 9:34 AM
