MathDB

R9

Part of 2021 Stanford Mathematics Tournament

Problems(1)

2021 SMT Guts Round 9 - final- p33-36 - Stanford Math Tournament

Source:

2/11/2022
p33. Lines ℓ1\ell_1 and ℓ2\ell_2 have slopes m1m_1 and m2m_2 such that 0<m2<m10 < m_2 < m_1. ℓ1′\ell'_1 and ℓ2′\ell'_2 are the reflections of ℓ1\ell_1 and ℓ2\ell_2 about the line ℓ3\ell_3 defined by y=xy = x. Let A=ℓ1∩ℓ2=(5,4)A = \ell_1 \cap \ell_2 = (5, 4), B=ℓ1∩ℓ3B = \ell_1 \cap \ell_3, C=ℓ1′∩ℓ2′C = \ell'_1 \cap \ell'_2 and D=ℓ2∩ℓ3D = \ell_2 \cap \ell_3. If 4−5m1−5−4m1=m2\frac{4-5m_1}{-5-4m_1} = m_2 and (1+m12)(1+m22)(1−m1)2(1−m2)2=41\frac{(1+m^2_1)(1+m^2_2)}{(1-m_1)^2(1-m_2)^2} = 41, compute the area of quadrilateral ABCDABCD.
p34. Suppose S(m,n)=∑i=1m(−1)iinS(m, n) = \sum^m_{i=1}(-1)^ii^n. Compute the remainder when S(2020,4)S(2020, 4) is divided by S(1010,2)S(1010, 2).
p35. Let NN be the number of ways to place the numbers 1,2,...,121, 2, ..., 12 on a circle such that every pair of adjacent numbers has greatest common divisor 11. What is N/144N/144? (Arrangements that can be rotated to yield each other are the same).
p36. Compute the series ∑n=1∞(−1)n−1(2n2)=1(22)−1(42)+1(62)−1(82)−1(102)+1(122)+...\sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{{2n \choose 2}} =\frac{1}{{2 \choose 2}} - \frac{1}{{4 \choose 2}} +\frac{1}{{6 \choose 2}} -\frac{1}{{8 \choose 2}} -\frac{1}{{10 \choose 2}}+\frac{1}{{12 \choose 2}} +...
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.
algebrageometrynumber theorycombinatoricsStanford Math TournamentSMT