MathDB

9

Part of 2023 AIME

Problems(2)

*Fun* Polynomial Counting!

Source: 2023 AIME I #9

2/8/2023
Find the number of cubic polynomials p(x)=x3+ax2+bx+cp(x) = x^3 + ax^2 + bx + c, where aa, bb, and cc are integers in {−20,−19,−18,…,18,19,20}\{-20, -19,-18, \dots , 18, 19, 20\}, such that there is a unique integer m≠2m \neq 2 with p(m)=p(2)p(m) = p(2).
AMCAIMEAIME Ialgebrapolynomial
Geo Geo Geo

Source: 2023 AIME II / 9

2/16/2023
Circles ω1\omega_1 and ω2\omega_2 intersect at two points PP and QQ, and their common tangent line closer to PP intersects ω1\omega_1 and ω2\omega_2 at points AA and BB, respectively. The line parallel to line ABAB that passes through PP intersects ω1\omega_1 and ω2\omega_2 for the second time at points XX and YY, respectively. Suppose PX=10,PY=14,PX = 10, PY = 14, and PQ=5PQ = 5. Then the area of trapezoid XABYXABY is mnm\sqrt{n} where mm and nn are positive integers and nn is not divisible by the square of any prime. Find m+nm + n.
AMCAIMEAIME II