MathDB

7

Part of 2023 Stanford Mathematics Tournament

Problems(3)

SMT 2023 Algebra #7

Source:

5/3/2023
Consider a sequence F0=2F_0=2, F1=3F_1=3 that has the property Fn+1Fn−1−Fn2=(−1)n⋅2F_{n+1}F_{n-1}-F_n^2=(-1)^n\cdot2. If each term of the sequence can be written in the form a⋅r1n+b⋅r2na\cdot r_1^n+b\cdot r_2^n, what is the positive difference between r1r_1 and r2r_2?
SMT 2023 Discrete #7

Source:

5/3/2023
Let SS be the number of bijective functions f:{0,1,…,288}→{0,1,…,288}f:\{0,1,\dots,288\}\rightarrow\{0,1,\dots,288\} such that f((m+n)(mod17))f((m+n)\pmod{17}) is divisible by 1717 if and only if f(m)+f(n)f(m)+f(n) is divisible by 1717. Compute the largest positive integer nn such that 2n2^n divides SS.
SMT 2023 Geometry #7

Source:

8/9/2023
Triangle ABCABC has AC=5AC = 5. DD and EE are on side BCBC such that ADAD and AEAE trisect ∠BAC\angle BAC, with DD closer to BB and DE=32DE =\frac32, EC=52EC =\frac52 . From BB and EE, altitudes BFBF and EGEG are drawn onto side ACAC. Compute CFCG−AFAG\frac{CF}{CG}-\frac{AF}{AG} .
geometry