MathDB

7

Part of 2022 Harvard-MIT Mathematics Tournament

Problems(4)

2022 Algebra/NT #7

Source:

3/11/2022
Let (x1,y1)(x_1, y_1), (x2,y2)(x_2, y_2), (x3,y3)(x_3, y_3), (x4,y4)(x_4, y_4), and (x5,y5)(x_5, y_5) be the vertices of a regular pentagon centered at (0,0)(0, 0). Compute the product of all positive integers k such that the equality x1k+x2k+x3k+x4k+x5k=y1k+y2k+y3k+y4k+y5kx_1^k+x_2^k+x_3^k+x_4^k+x_5^k=y_1^k+y_2^k+y_3^k+y_4^k+y_5^k must hold for all possible choices of the pentagon.
algebra
2022 Team 7 f(x)^2 - f(y)f(z) = x(x + y + z)(f(x) + f(y) + f(z))

Source:

3/14/2022
Find, with proof, all functions f:R−{0}→Rf : R - \{0\} \to R such that f(x)2−f(y)f(z)=x(x+y+z)(f(x)+f(y)+f(z))f(x)^2 - f(y)f(z) = x(x + y + z)(f(x) + f(y) + f(z)) for all real x,y,zx, y, z such that xyz=1xyz = 1.
algebrafunctional
2022 Geometry 7

Source:

3/14/2022
Point PP is located inside a square ABCDABCD of side length 1010. Let O1O_1, O2O_2, O3O_3, O4O_4 be the circumcenters of PABP AB, PBCP BC, PCDP CD, and PDAP DA, respectively. Given that PA+PB+PC+PD=232P A+P B +P C +P D = 23\sqrt2 and the area of O1O2O3O4O_1O_2O_3O_4 is 5050, the second largest of the lengths O1O2O_1O_2, O2O3O_2O_3, O3O4O_3O_4, O4O1O_4O_1 can be written as ab\sqrt{\frac{a}{b}}, where aa and bb are relatively prime positive integers. Compute 100a+b100a + b.
geometry
2022 Combinatorics 7

Source:

3/18/2022
Let S={(x,y)∈Z2∣0≤x≤11,0≤y≤9}S = \{(x, y) \in Z^2 | 0 \le x \le 11, 0\le y \le 9\}. Compute the number of sequences (s0,s1,...,sn)(s_0, s_1, . . . , s_n) of elements in SS (for any positive integer n≥2n \ge 2) that satisfy the following conditions: ∙\bullet s0=(0,0)s_0 = (0, 0) and s1=(1,0)s_1 = (1, 0), ∙\bullet s0,s1,...,sns_0, s_1, . . . , s_n are distinct, ∙\bullet for all integers 2≤i≤n2 \le i \le n, sis_i is obtained by rotating si−2s_{i-2} about si−1s_{i-1} by either 90o90^o or 180o180^o in the clockwise direction.
combinatorics