MathDB

3

Part of 2018 Stanford Mathematics Tournament

Problems(3)

a shape in the Cartesian coordinate plane with area greater than 1

Source: 2018 SMT - Stanford Math Tournament , Team Round, Proof Question 3

1/26/2022
Show that if A A is a shape in the Cartesian coordinate plane with area greater than 1 1, then there are distinct points (a,b)(a, b), (c,d)(c, d) in AA where a−c=2x+5ya - c = 2x + 5y and b−d=x+3yb - d = x + 3y where x,yx, y are integers.
analytic geometryareasnumber theory
SMT 2018 Geometry #3 area

Source:

8/8/2023
Let ABCABC be a triangle and DD be a point such that AA and DD are on opposite sides of BCBC. Give that ∠ACD=75o\angle ACD = 75^o, AC=2AC = 2, BD=6BD =\sqrt6, and ADAD is an angle bisector of both △ABC\vartriangle ABC and △BCD\vartriangle BCD, find the area of quadrilateral ABDCABDC.
geometry
SMT 2018 Geometry Tiebreaker #3

Source:

8/8/2023
A triangle has side lengths of 77, 88, and 99. Find the radius of the largest possible semicircle inscribed in the triangle.
geometry