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TST2025 #0l6v

TST2025Hard

From module Olympiad Number Theory

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Understand the scale
Scale 0.5: Easiest math competition problems, often solvable even for early middle school students without experience. 1: Problems strictly for beginners, often at a moderate middle school level. 1.5: Problems for stronger beginner students, on the level of the middling problems in most middle school contests. 2: For motivated beginners, harder questions from the previous categories. 2.5: More advanced beginner problems, hardest questions from previous categories. 3: Early intermediate problems that require more creative thinking. 3.5: Tougher early intermediate problems that consistently stretch into higher-level creative thinking or conceptual knowledge. 4: Intermediate-level problems. 4.5: Upper intermediate problems approaching the upper bound for non-invitational sprint math competitions. 5: More difficult AIME problems or simple proof-based Olympiad-style problems. 6: High-level AIME-styled questions or introductory Olympiad-level questions. 7: Tougher Olympiad-level questions, may require more technical knowledge. 8: High-level Olympiad-level questions. 9: Expert Olympiad-level questions. 9.5: The hardest problems appearing on Olympiads which the strongest students could reasonably solve. 10: Historically notorious problems, generally unsuitable for even very hard competitions because they are exceedingly tedious, long, and difficult.
Techniques: modulo, size analysis, order analysis, inequalitiesQuadratic residuesDeterminantsComplex numbers in geometryQuadratic fields

Problem

Problem: Prove that there exists a real number ε>0\varepsilon > 0 such that there are infinitely many sequences of integers 0<a1<a2<⋯<a20250 < a_1 < a_2 < \dots < a_{2025} satisfying

gcd(a12+1,a22+1,…,a20252+1)>a20251+ε.\text{gcd}(a_1^2 + 1, a_2^2 + 1, \dots, a_{2025}^2 + 1) > a_{2025}^{1+\varepsilon}.
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