MathDB

A8

Part of 2020 Princeton University Math Competition

Problems(4)

2020 PUMaC Algebra A8

Source:

1/1/2022
Let ana_n be the number of unordered sets of three distinct bijections f,g,h:{1,2,...,n}→{1,2,...,n}f, g, h : \{1, 2, ..., n\} \to \{1, 2, ..., n\} such that the composition of any two of the bijections equals the third. What is the largest value in the sequence a1,a2,...a_1, a_2, ... which is less than 20212021?
algebra
2020 PUMaC Combinatorics A8

Source:

1/1/2022
Let f(k)f(k) denote the number of triples (a,b,c)(a, b, c) of positive integers satisfying a+b+c=2020a + b + c = 2020 with (k−1)(k - 1) not dividing a,ka, k not dividing bb, and (k+1)(k + 1) not dividing cc. Find the product of all integers kk in the range 3 \le k \le 20 such that (k+1)(k + 1) divides f(k)f(k).
combinatorics
2020 PUMaC Geometry A8

Source:

12/31/2021
A1A2A3A4A_1A_2A_3A_4 is a cyclic quadrilateral inscribed in circle Ω\Omega, with side lengths A1A2=28A_1A_2 = 28, A2A3=123A_2A_3 =12\sqrt3, A3A4=283A_3A_4 = 28\sqrt3, and A4A1=8A_4A_1 = 8. Let XX be the intersection of A1A3,A2A4A_1A_3, A_2A_4. Now, for i=1,2,3,4i = 1, 2, 3, 4, let ωi\omega_i be the circle tangent to segmentsAiX A_iX, Ai+1XA_{i+1}X, and Ω\Omega, where we take indices cyclically (mod 44). Furthermore, for each ii, say ωi\omega_i is tangent to A1A3A_1A_3 at XiX_i , A2A4A_2A_4 at YiY_i , and Ω\Omega at TiT_i . Let P1P_1 be the intersection of T1X1T_1X_1 and T2X2T_2X_2, and P3P_3 the intersection of T3X3T_3X_3 and T4X4T_4X_4. Let P2P_2 be the intersection of T2Y2T_2Y_2 and T3Y3T_3Y_3, and P4P_4 the intersection of T1Y1T_1Y_1 and T4Y4T_4Y_4. Find the area of quadrilateral P1P2P3P4P_1P_2P_3P_4.
geometry
2020 PUMaC NT A8

Source:

1/1/2022
What is the smallest integer a0a_0 such that, for every positive integer nn, there exists a sequence of positive integers a0,a1,...,an−1,ana_0, a_1, ..., a_{n-1}, a_n such that the first n−1n-1 are all distinct, a0=ana_0 = a_n, and for 0≤i≤n−10 \le i \le n -1, aiai+1a_i^{a_{i+1}} ends in the digits 0ai‾\overline{0a_i} when expressed without leading zeros in base 1010.
number theory