MathDB

4

Part of 2022 JHMT HS

Problems(5)

Periodicity of Remainder Sequence

Source:

8/8/2024
For an integer aa and positive integers nn and kk, let fk(a,n)f_k(a, n) be the remainder when aka^k is divided by nn. Find the largest composite integer n≤100n\leq 100 that guarantees the infinite sequence f1(a,n),f2(a,n),f3(a,n),…,fi(a,n),… f_1(a,n),f_2(a,n),f_3(a,n),\ldots,f_i(a,n),\ldots to be periodic for all integers aa (i.e., for each choice of aa, there is some positive integer TT such that fk(a,n)=fk+T(a,n)f_k(a,n) = f_{k+T}(a,n) for all kk).
number theory2022
Cyclic Hexagon

Source:

8/8/2024
Hexagon ARTSCIARTSCI has side lengths AR=RT=TS=SC=42AR=RT=TS=SC=4\sqrt2 and CI=IA=102CI=IA=10\sqrt2. Moreover, the vertices AA, RR, TT, SS, CC, and II lie on a circle K\mathcal{K}. Find the area of K\mathcal{K}.
geometrycircumcircle2022
Rectangle Partitioned by Curves

Source:

8/9/2024
Consider the rectangle in the coordinate plane with corners (0,0)(0, 0), (16,0)(16, 0), (16,4)(16, 4), and (0,4)(0, 4). For a constant x0∈[0,16]x_0 \in [0, 16], the curves \{(x, y) : y = \sqrt{x} \,\text{ and }\, 0 \leq x \leq 16\}   \text{and}   \{(x_0, y) : 0 \leq y \leq 4\} partition this rectangle into four 2D regions. Over all choices of x0x_0, determine the smallest possible sum of the areas of the bottom-left and top-right 2D regions in this partition. (The bottom-left region is {(x,y):0≤x<x0  and  0≤y<x}\{(x, y) : 0 \leq x < x_0 \,\text{ and }\, 0 \leq y < \sqrt{x}\}, and the top-right region is {(x,y):x0<x≤16  and  x<y≤4}\{(x, y) : x_0 < x \leq 16 \,\text{ and }\, \sqrt{x} < y \leq 4\}.)
geometryrectangleoptimizationcalculus2022
Range Contained in Subset

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8/8/2024
For a nonempty set AA of integers, let range A=max⁡A−min⁡A\mathrm{range} \, A=\max A-\min A. Find the number of subsets SS of {1,2,3,4,5,6,7,8,9,10} \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\} such that range S\mathrm{range} \, S is an element of SS.
combinatorics2022
Product and Sum of Digits

Source:

8/8/2024
For a positive integer nn, let p(n)p(n) denote the product of the digits of nn, and let s(n)s(n) denote the sum of the digits of nn. Find the sum of all positive integers nn satisfying p(n)s(n)=8p(n)s(n)=8.
number theory2022