MathDB

3

Part of 2019 Philippine TST

Problems(3)

Infinite positive integer sequence with an eventual cubic lower bound

Source: 2019 Philippine IMO TST1 Problem 3

5/4/2022
Let a1,a2,a3,…a_1, a_2, a_3,\ldots be an infinite sequence of positive integers such that a2≠2a1a_2 \ne 2a_1, and for all positive integers mm and nn, the sum m+nm + n is a divisor of am+ana_m + a_n. Prove that there exists an integer MM such that for all n>Mn > M, we have an≥n3a_n \ge n^3.
number theorydivisorBoundinginfinite sequence
Find all triples such that a quadratic bound implies a constant function

Source: 2019 Philippine IMO TST2 Problem 3

5/4/2022
Determine all ordered triples (a,b,c)(a, b, c) of real numbers such that whenever a function f:R→Rf : \mathbb{R} \to \mathbb{R} satisfies ∣f(x)−f(y)∣≤a(x−y)2+b(x−y)+c|f(x) - f(y)| \le a(x - y)^2 + b(x - y) + c for all real numbers xx and yy, then ff must be a constant function.
quadraticsalgebrainequalitiesFunctional inequalityfunctionabsolute valueconstant
Circle with diameter AC - AB

Source: 2019 Philippine IMO TST3 Problem 3

5/4/2022
Given △ABC\triangle ABC with AB<ACAB < AC, let ω\omega be the circle centered at the midpoint MM of BCBC with diameter AC−ABAC - AB. The internal bisector of ∠BAC\angle BAC intersects ω\omega at distinct points XX and YY. Let TT be the point on the plane such that TXTX and TYTY are tangent to ω\omega. Prove that ATAT is perpendicular to BCBC.
geometryangle bisectortangentperpendicular