MathDB

7

Part of 2022 JHMT HS

Problems(5)

Inequality with Equality Condition

Source:

8/8/2024
Find the least positive integer NN such that there exist positive real numbers a1,a2,…,aNa_1,a_2,\dots,a_N such that \sum_{k=1}^{N}ka_k=1   \text{and}   \sum_{k=1}^{N}\frac{a_k^2}{k}\leq \frac{1}{2022^2}.
inequalitiesalgebra2022
Area Optimization with Equality Constraint

Source:

8/8/2024
Two rays emanate from the origin OO and form a 45∘45^\circ angle in the first quadrant of the Cartesian coordinate plane. For some positive numbers XX, YY, and SS, the ray with the larger slope passes through point A=(X,S)A = (X, S), and the ray with the smaller slope passes through point B=(S,Y)B = (S, Y). If 6X+6Y+5S=6006X + 6Y + 5S = 600, then determine the maximum possible area of △OAB\triangle OAB.
geometryoptimizationslope2022
Expectation of Spider Web Strand Intersections

Source:

8/8/2024
A spider sits on the circumference of a circle and wants to weave a web by making several passes through the circle's interior. On each pass, the spider starts at some location on the circumference, picks a destination uniformly at random from the circumference, and travels to that destination in a straight line, laying down a strand of silk along the line segment they traverse. After the spider does 20222022 of these passes (with each non-initial pass starting where the previous one ended), what is the expected number of points in the circle's interior where two or more non-parallel silk strands intersect?
expected value2022
Closed-Form Expression for Improper Integral

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8/9/2024
Let aa be the unique real number xx satisfying xex=2xe^x = 2. Find a closed-form expression for ∫a∞x+1x(xex)11−1 dx. \int_{a}^{\infty} \frac{x + 1}{x\sqrt{(xe^x)^{11} - 1}}\,dx. You may express your answer in terms of elementary operations, functions, and constants.
calculusintegration2022
Convex Heptagon and Circular Sectors

Source:

8/8/2024
Let HOPKINSHOPKINS be an irregular convex heptagon (i.e., its angles and side lengths are all distinct, with the angles all having measure less than 180∘180^{\circ}) with area 18761876 such that all of its side lengths are greater than 55, OP=20OP=20, and KI=22KI=22. Arcs with radius 22 are drawn inside HOPKINSHOPKINS with their centers at each of the vertices and their endpoints on the sides, creating circular sectors. Find the area of the region inside HOPKINSHOPKINS but outside the sectors.
geometry2022