MathDB

10

Part of 2023 Harvard-MIT Mathematics Tournament

Problems(4)

2023 HMMT Geometry #10

Source:

3/1/2023
Triangle ABCABC has incenter II. Let DD be the foot of the perpendicular from AA to side BCBC. Let XX be a point such that segment AXAX is a diameter of the circumcircle of triangle ABCABC. Given that ID=2ID = 2, IA=3IA = 3, and IX=4IX = 4, compute the inradius of triangle ABCABC.
HMMTgeometry
2023 Combinatorics #10

Source:

2/28/2024
Let x0=x101=0x_0 = x_{101} = 0. The numbers x1,x2,...,x100x_1, x_2,...,x_{100} are chosen at random from the interval [0,1][0, 1] uniformly and independently. Compute the probability that 2xi≥xi−1+xi+12x_i \ge x_{i-1} + x_{i+1} for all i=1,2,...,100.i = 1, 2,..., 100.
combinatorics
HMMT Feb 2023 team p10

Source:

2/20/2023
One thousand people are in a tennis tournament where each person plays against each other person exactly once, and there are no ties. Prove that it is possible to put all the competitors in a line so that each of the 998998 people who are not at an end of the line either defeated both their neighbors or lost to both their neighbors.
2023 Algebra NT #10 \zeta= e^{2\pi i/99}

Source:

2/28/2024
Let ζ=e2πi/99\zeta= e^{2\pi i/99} and ωe2πi/101\omega e^{2\pi i/101}. The polynomial x9999+a9998x9998+...+a1x+a0x^{9999} + a_{9998}x^{9998} + ...+ a_1x + a_0 has roots ζm+ωn\zeta^m + \omega^n for all pairs of integers (m,n)(m, n) with 0≤m<990 \le m < 99 and 0≤n<1010 \le n < 101. Compute a9799+a9800+...+a9998a_{9799} + a_{9800} + ...+ a_{9998}.
algebracomplex numbers