MathDB

9

Part of 2022 JHMT HS

Problems(5)

Pair of Sequences and Recurrences

Source:

8/8/2024
Let {an}n=011\{ a_n \}_{n=0}^{11} and {bn}n=011\{ b_n \}_{n=0}^{11} be sequences of real numbers. Suppose a0=b0=−1a_0 = b_0 = -1, a1=b1a_1 = b_1, and for all integers n∈{2,3,…,11}n \in \{2, 3, \ldots, 11\}, \begin{align*} a_n & = a_{n-1} - (11 - n)^2(1 - (11 - (n - 1))^2)a_{n-2}   \text{and} \\ b_n & = b_{n-1} - (12 - n)^2(1 - (12 - (n - 1))^2)b_{n-2}. \end{align*} If b11=2a11b_{11} = 2a_{11}, then determine the value of a1a_1.
algebra2022
Sphere Point Angle and Expected Value

Source:

8/8/2024
Let BB and DD be two points chosen independently and uniformly at random from the unit sphere in 3D space centered at a point AA (this unit sphere is the set of all points in R3\mathbb{R}^3 a distance of 11 away from AA). Compute the expected value of sin⁡2∠DAB\sin^2\angle DAB.
geometry3D geometrysphereexpected valueprobability2022
Ratio Involving Convex Quadrilateral Side Lengths

Source:

8/8/2024
In convex quadrilateral KALEKALE, angles ∠KAL\angle KAL, ∠AKL\angle AKL, and ∠ELK\angle ELK measure 110∘110^\circ, 50∘50^\circ, and 10∘10^\circ, respectively. Given that KA=LEKA = LE and that KL‾\overline{KL} and AE‾\overline{AE} intersect at point XX, compute the value of KX2AL⋅EX\tfrac{KX^2}{AL\cdot EX}.
geometry2022
Functional Equation and Integral

Source:

8/9/2024
There is a unique continuous function ff over the positive real numbers satisfying f(4)=1f(4) = 1 and 9−(f(x))4=x2(f(x))2−2xf(x) 9 - (f(x))^4 = \frac{x^2}{(f(x))^2} - 2xf(x) for all positive xx. Compute the value of ∫0140(f(x))3 dx\int_{0}^{140} (f(x))^3\,dx.
algebrafunctional equationcalculusintegration2022
Equilateral Hexagon Inside Triangle

Source:

8/8/2024
In △PQR\triangle{PQR}, PQ=4PQ=4, PR=5PR=5, and QR=6QR=6. Assume that an equilateral hexagon ABCDEFABCDEF is able to be drawn inside △PQR\triangle{PQR} so that AB‾\overline{AB} is parallel to QR‾\overline{QR}, CD‾\overline{CD} is parallel to PQ‾\overline{PQ}, EF‾\overline{EF} is parallel to RP‾\overline{RP}, BC‾\overline{BC} lies on RP‾\overline{RP}, DE‾\overline{DE} lies on QR‾\overline{QR}, and AF‾\overline{AF} lies on PQ‾\overline{PQ}. Find the area of hexagon ABCDEFABCDEF.
geometry2022