MathDB

6

Part of 2023 Stanford Mathematics Tournament

Problems(3)

SMT 2023 Algebra #6

Source:

5/3/2023
What is the area of the figure in the complex plane enclosed by the origin and the set of all points 1z\tfrac{1}{z} such that (1−2i)z+(−2i−1)z‾=6i(1-2i)z+(-2i-1)\overline{z}=6i?
SMT 2023 Discrete #6

Source:

5/3/2023
We say that an integer x∈{1,…,102}x\in\{1,\dots,102\} is <spanclass=′latex−italic′>square−ish</span><span class='latex-italic'>square-ish</span> if there exists some integer nn such that x≡n2+n(mod103)x\equiv n^2+n\pmod{103}. Compute the product of all <spanclass=′latex−italic′>square−ish</span><span class='latex-italic'>square-ish</span> integers modulo 103103.
SMT 2023 Geometry #6

Source:

8/9/2023
Let ABC be a triangle and ω1\omega_1 its incircle. Let points DD and EE be on segments ABAB, ACAC respectively such that DEDE is parallel to BCBC and tangent to ω1\omega_1 . Now let ω2\omega_2 be the incircle of △ADE\vartriangle ADE and let points FF and GG be on segments AD,AD, AEAE respectively such that F G is parallel to DEDE and tangent to ω2\omega_2. Given that ω2\omega_2 is tangent to line AFAF at point X and line AGAG at point YY , the radius of ω1\omega_1 is 6060, and 4(AX)=5(FG)=4(AY),4(AX) = 5(F G) = 4(AY), compute the radius of ω2\omega_2.
geometry