MathDB

7

Part of 2021 Harvard-MIT Mathematics Tournament.

Problems(4)

2021 Algebra/NT #7: "Complex" system of complexes

Source:

5/30/2021
Suppose that xx, yy, and zz are complex numbers of equal magnitude that satisfy x+y+z=−32−i5x+y+z = -\frac{\sqrt{3}}{2}-i\sqrt{5} and xyz=3+i5.xyz=\sqrt{3} + i\sqrt{5}. If x=x1+ix2,y=y1+iy2,x=x_1+ix_2, y=y_1+iy_2, and z=z1+iz2z=z_1+iz_2 for real x1,x2,y1,y2,z1x_1,x_2,y_1,y_2,z_1 and z2z_2 then (x1x2+y1y2+z1z2)2(x_1x_2+y_1y_2+z_1z_2)^2 can be written as ab\tfrac{a}{b} for relatively prime positive integers aa and bb. Compute 100a+b.100a+b.
algebra
2021 Combo #7: function sets

Source:

5/30/2021
Let S={1,2,…,2021}S = \{1, 2, \dots , 2021\}, and let F\mathcal{F} denote the set of functions f:S→Sf : S \rightarrow S. For a function f∈F,f \in \mathcal{F}, let Tf={f2021(s):s∈S},T_f =\{f^{2021}(s) : s \in S\}, where f2021(s)f^{2021}(s) denotes f(f(⋯(f(s))⋯ ))f(f(\cdots(f(s))\cdots)) with 20212021 copies of ff. Compute the remainder when ∑f∈F∣Tf∣\sum_{f \in \mathcal{F}} |T_f| is divided by the prime 20172017, where the sum is over all functions ff in F\mathcal{F}.
functionCombo
2021 Geo #7: Rotate?

Source:

5/30/2021
Let OO and AA be two points in the plane with OA=30OA = 30, and let Γ\Gamma be a circle with center OO and radius rr. Suppose that there exist two points BB and CC on Γ\Gamma with ∠ABC=90∘\angle ABC = 90^{\circ} and AB=BCAB = BC. Compute the minimum possible value of ⌊r⌋.\lfloor r \rfloor.
geometryrotation
2021 Team #7

Source:

6/27/2021
In triangle ABCABC, let MM be the midpoint of BCBC and DD be a point on segment AMAM. Distinct points YY and ZZ are chosen on rays CA→\overrightarrow{CA} and BA→\overrightarrow{BA} , respectively, such that ∠DYC=∠DCB\angle DYC=\angle DCB and ∠DBC=∠DZB\angle DBC=\angle DZB. Prove that the circumcircle of ΔDYZ\Delta DYZ is tangent to the circumcircle of ΔDBC\Delta DBC.
geometrytangencyCeva s theorem