Problems(3)
max no of several identical matches (2022 Novosibirsk Oral Geo Oly 7.7)
Source:
9/2/2023
Vera has several identical matches, from which she makes a triangle. Vera wants any two sides of this triangle to differ in length by at least matches, but it turned out that it is impossible to add such a triangle from the available matches (it is impossible to leave extra matches). What is the maximum number of matches Vera can have?
geometrycombinatorial geometrycombinatorics
intersection of perp. diagonals lies on XY (2022 Novosibirsk Oral Geo Oly 8.7)
Source:
9/2/2023
The diagonals of the convex quadrilateral intersect at the point . The points and are symmetrical to the point with respect to the midpoints of the sides and , respectively. It is known that . Prove that the point of intersection of the perpendicular bisectors of the diagonals of the quadrilateral lies on the line .
geometryconcurrencyconcurrent
equidistant points from midpoint,orthocenter (2022 Novosibirsk Oral Geo Oly 9.7)
Source:
9/3/2023
Altitudes and of an acute-angled triangle intersect at point . A straight line passing through point parallel to line intersects the circumscribed circles of triangles and at points and , respectively. Prove that points and are equidistant from the midpoint of segment .
geometryorthocenterequal segments