MathDB

10

Part of 2020 BMT Fall

Problems(5)

BMT Algebra #10 - Crazy Trigonometric Sum

Source:

10/11/2020
For k≥1k\ge 1, define ak=2ka_k=2^k. Let S=∑k=1∞cos⁡−1(2ak2−6ak+5(ak2−4ak+5)(4ak2−8ak+5)).S=\sum_{k=1}^{\infty}\cos^{-1}\left(\frac{2a_k^2-6a_k+5}{\sqrt{(a_k^2-4a_k+5)(4a_k^2-8a_k+5)}}\right). Compute ⌊100S⌋\lfloor 100S\rfloor.
Bmtalgebrafloor function
BMT 2020 Fall - Geometry 10

Source:

12/30/2021
Let EE be an ellipse where the length of the major axis is 2626, the length of the minor axis is 2424, and the foci are at points RR and SS. Let AA and BB be points on the ellipse such that RASBRASB forms a non-degenerate quadrilateral, lines RARA and SBSB intersect at PP with segment PRPR containing AA, and lines RBRB and ASAS intersect at Q with segment QRQR containing BB. Given that RA=ASRA = AS, AP=26AP = 26, the perimeter of the non-degenerate quadrilateral RPSQRP SQ is m+nm +\sqrt{n}, where mm and nn are integers. Compute m+nm + n.
geometry
2020 BMT Team 10

Source:

1/9/2022
How many integers 100≤x≤999100 \le x \le 999 have the property that, among the six digits in ⌊280+x100⌋\lfloor 280 + \frac{x}{100} \rfloor and xx, exactly two are identical?
floor functionnumber theory
2020 BMT Individual 10

Source:

1/9/2022
Given that pp and p4+34p^4 + 34 are both prime numbers, compute pp.
number theory
2020 BMT Discrete #10

Source:

3/10/2024
Let ψ(n)\psi (n) be the number of integers 0≤r<n0 \le r < n such that there exists an integer xx that satis es x2+x≡rx^2 + x \equiv r (mod nn). Find the sum of all distinct prime factors of ∑i=04∑j=04ψ(3i5j).\sum^4_{i=0}\sum^4_{j=0} \psi(3^i5^j).
number theory