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1998 DMM Individual Round - Duke Math Meet

Source:

February 15, 2024
DMMalgebrageometrynumber theorycombinatorics

Problem Statement

p1. Find the greatest integer nn such that nlog⁡104n \log_{10} 4 does not exceed log⁡101998\log_{10} 1998.
p2. Rectangle ABCDABCD has sides AB=CD=12/5AB = CD = 12/5, BC=DA=5BC = DA = 5. Point PP is on ADAD with ∠BPC=90o\angle BPC = 90^o. Compute BP+PCBP + PC.
p3. Compute the number of sequences of four decimal digits (a,b,c,d)(a, b, c, d) (each between 00 and 99 inclusive) containing no adjacent repeated digits. (That is, each digit is distinct from the digits directly before and directly after it.)
p4. Solve for tt, −π/4≤t≤π/4-\pi/4 \le t \le \pi/4 : sin⁡3t+sin⁡2tcos⁡t+sin⁡tcos⁡2t+cos⁡3t=62\sin^3 t + \sin^2 t \cos t + \sin t \cos^2 t + \cos^3 t =\frac{\sqrt6}{2}
p5. Find all integers nn such that n−3n - 3 divides n2+2n^2 + 2.
p6. Find the maximum number of bishops that can occupy an 8×88 \times 8 chessboard so that no two of the bishops attack each other. (Bishops can attack an arbitrary number of squares in any diagonal direction.)
p7. Points A,B,CA, B, C, and DD are on a Cartesian coordinate system with A=(0,1)A = (0, 1), B=(1,1)B = (1, 1), C=(1,−1)C = (1,-1), and D=(−1,0)D = (-1, 0). Compute the minimum possible value of PA+PB+PC+PDPA + PB + PC + PD over all points PP.
p8. Find the number of distinct real values of xx which satisfy (x−1)(x−2)(x−3)(x−4)(x−5)(x−6)(x−7)(x−8)(x−9)(x−10)+(12⋅32⋅52⋅72⋅92)/210=0.(x-1)(x-2)(x-3)(x-4)(x-5)(x-6)(x-7)(x-8)(x-9)(x-10)+(1^2 \cdot 3^2\cdot 5^2\cdot 7^2\cdot 9^2)/2^{10} = 0.
PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.
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