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2012 BMT Tournament Round, Consolation Round - Berkley Math Tournament

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January 28, 2022
algebrageometrycombinatoricsnumber theoryBmtBerkeley Math Tournament

Problem Statement

p1. How many ways can we arrange the elements {1,2,...,n}\{1, 2, ..., n\} to a sequence a1,a2,...,ana_1, a_2, ..., a_n such that there is only exactly one aia_i, ai+1a_{i+1} such that ai>ai+1a_i > a_{i+1}?
p2. How many distinct (non-congruent) triangles are there with integer side-lengths and perimeter 20122012?
p3. Let ϕ\phi be the Euler totient function, and let S={x∣xϕ(x)=3}S = \{x| \frac{x}{\phi (x)} = 3\}. What is ∑x∈S1x\sum_{x\in S} \frac{1}{x}?
p4. Denote f(N)f(N) as the largest odd divisor of NN. Compute f(1)+f(2)+f(3)+...+f(29)+f(30)f(1) + f(2) + f(3) +... + f(29) + f(30).
p5. Triangle ABCABC has base ACAC equal to 218218 and altitude 100100. Squares s1,s2,s3,...s_1, s_2, s_3, ... are drawn such that s1s_1 has a side on ACAC and has one point each touching ABAB and BCBC, and square sks_k has a side on square sk−1s_{k-}1 and also touches ABAB and BCBC exactly once each. What is the sum of the area of these squares?
p6. Let PP be a parabola 6x2−28x+106x^2 - 28x + 10, and FF be the focus. A line ℓ\ell passes through FF and intersects the parabola twice at points P1=(2,−22)P_1 = (2,-22), P2P_2. Tangents to the parabola with points at P1,P2P_1, P_2 are then drawn, and intersect at a point QQ. What is m∠P1QP2m\angle P_1QP_2?
PS. You had better use hide for answers.
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