AoPS Academy
(use in a problem for junior) to practice. Can anyone share to me?
be an acute angled scalene triangle with circumcircle 
. Let 
and 
be the circumcenter and orthocenter of 
respectively. Let 
and 
be points on segments 
and , respectively, such that
Prove that 
and 
intersect on the circumcircle of 
.
+1 w
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 
. Prove that:![\[
0 \le (x^2 + 1)(y^2 + 1) + 4(x - 1)(y - 1) \le 164.
\]](http://latex.artofproblemsolving.com/4/2/d/42d0b88388462f16dd99d928b94024e689ba7c2e.png)
be a sequence defined by 
. Then the product of four consecutive terms in can be written as the product of two terms in . Find 
if 
.
that satisfy 
with 
, 
and area 
. The circle with diameter 
meets 
at 
. Let the orthocenter of the triangle be , connect 
and extend to meet at 
such that 
and 
is the diameter of . Draw the circumcircle 
of 
, chord 
of is tangent to and it passes through , compute .
denote four points in space such that at most one of the distances 
is greater than 
. Determine the maximum value of the sum of the six distances.
.
,
,
and 
would have to be the midpoint of 
.
and 
.
,
,
,
within the given space.
as large as possible, so we notice that they are diagonals our rhombus, and that 
and 
are the legs of a right triangle with hypotenuse 
.
.
and 
are equilateral! We have two equilateral triangles of side 
.
,