MathDB

Tie 3

Part of 2020 BMT Fall

Problems(4)

2020 BMT Algebra Tiebreakers 3

Source:

1/6/2022
Let xx and yy be integers from −10-10 to 1010, inclusive, with xy≠1xy \ne1. Compute the number of ordered pairs (x,y)(x, y) such that \left| \frac{x + y}{1 - xy} \right|\le 1.
algebra
BMT 2020 Fall - Geometry Tiebreaker 3

Source:

12/30/2021
In unit cube ABCDEFGHABCDEFGH (with faces ABCDABCD, EFGHEFGH and connecting vertices labeled so that AE‾\overline{AE}, BF‾\overline{BF}, CG‾\overline{CG}, DH‾\overline{DH} are edges of the cube), LL is the midpoint of GHGH. The area of △CAL\vartriangle CAL can be written in the form mn\frac{m}{n} , where mm and nn are relatively prime positive integers. Compute m+nm + n.
geometry
segment tangent to incircle of 5-12-13 triangl 2020 BMT Individual Tiebreakers 3

Source:

1/6/2022
△ABC\vartriangle ABC has AB=5AB = 5, BC=12BC = 12, and AC=13AC = 13. A circle is inscribed in △ABC\vartriangle ABC, and MNMN tangent to the circle is drawn such that MM is on AC‾\overline{AC}, NN is on BC‾\overline{BC}, and MN‾∥AB‾\overline{MN} \parallel \overline{AB}. The area of △MNC\vartriangle MNC is m/nm/n , where mm and nn are relatively prime positive integers. Find m+nm + n.
geometryincircleright triangle
2020 BMT Discrete Tiebreaker #3

Source:

3/10/2024
Three distinct integers a1a_1, a2a_2, a3a_3 between 11 and 2121, inclusive, are selected uniformly at random. The probability that the greatest common factor of ai−aja_i-a_j and 2121 is 77 for some positive integers ii and jj, where 1≤i≠j≤31 \le i \ne j \le3 , can be written in the form mn\frac{m}{n} , where mm and nn are relatively prime positive integers. Compute m+nm + n.
number theory