MathDB

4

Part of 2023 Stanford Mathematics Tournament

Problems(3)

SMT 2023 Algebra #4

Source:

5/3/2023
If the sum of the real roots xx to each of the equations 22x−2x+1+1−1k2=02^{2x}-2^{x+1}+1-\frac{1}{k^2}=0 for k=2,3,…,2023k=2,3,\dots,2023 is NN, what is 2N2^N?
SMT 2023 Geometry #4

Source:

8/9/2023
Equilateral triangle △ABC\vartriangle ABC is inscribed in circle Ω\Omega, which has a radius of 11. Let the midpoint of BCBC be MM. Line AMAM intersects Ω\Omega again at point DD. Let ω\omega be the circle with diameter MDMD. Compute the radius of the circle that is tangent to BC on the same side of BCBC as ω\omega, internally tangent to Ω\Omega, and externally tangent to ω\omega.
geometry
SMT 2023 Discrete #4

Source:

5/3/2023
Michelle is drawing segments in the plane. She begins from the origin facing up the yy-axis and draws a segment of length 11. Now, she rotates her direction by 120∘120^\circ, with equal probability clockwise or counterclockwise, and draws another segment of length 11 beginning from the end of the previous segment. She then continues this until she hits an already drawn segment. What is the expected number of segments she has drawn when this happens?