1. Let 
be a point in the interior of a triangle 
, and let 
be the point of intersection of the line 
and the side 
of the triangle, of the line 
and the side 
, and of the line 
and the side 
, respectively. Prove that the area of the triangle must be 
if the area of each of the triangles 
and 
is 
.
be a point in the interior of a triangle , and let be the point of intersection of the line and the side of the triangle, of the line and the side , and of the line and the side , respectively. Prove that the area of the triangle must be if the area of each of the triangles and is . 
2. Into each box of a 
square grid, a real number greater than or equal to 
and less than or equal to is inserted. Consider splitting the grid into 
non-empty rectangles consisting of boxes of the grid by drawing a line parallel either to the horizontal or the vertical side of the grid. Suppose that for at least one of the resulting rectangles the sum of the numbers in the boxes within the rectangle is less than or equal to , no matter how the grid is split into such rectangles. Determine the maximum possible value for the sum of all the numbers inserted into the boxes.
square grid, a real number greater than or equal to and less than or equal to is inserted. Consider splitting the grid into non-empty rectangles consisting of boxes of the grid by drawing a line parallel either to the horizontal or the vertical side of the grid. Suppose that for at least one of the resulting rectangles the sum of the numbers in the boxes within the rectangle is less than or equal to , no matter how the grid is split into such rectangles. Determine the maximum possible value for the sum of all the numbers inserted into the boxes. 
4. Let be an acute triangle. Denote by 
the foot of the perpendicular line drawn from the point 
to the side , by 
the midpoint of , and by 
the orthocenter of . Let 
be the point of intersection of the circumcircle 
of the triangle and the half line 
, and 
be the point of intersection (other than ) of the line 
and the circle . Prove that 
.
be an acute triangle. Denote by the foot of the perpendicular line drawn from the point to the side , by the midpoint of , and by the orthocenter of . Let be the point of intersection of the circumcircle of the triangle and the half line , and be the point of intersection (other than ) of the line and the circle . Prove that .
must hold.
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