MathDB

7

Part of 2022 South East Mathematical Olympiad

Problems(2)

no integers on the interval

Source: 2022 China Southeast Grade 10 P7

8/3/2022
Let a,ba,b be positive integers.Prove that there are no positive integers on the interval [b2a2+ab,b2a2+ab−1)\bigg[\frac{b^2}{a^2+ab},\frac{b^2}{a^2+ab-1}\bigg).
number theory
the construction of the sequence

Source: 2022 China Southeast Grade 11 P7

8/3/2022
Prove that for any positive real number λ\lambda,there are nn positive numbers a1,a2,⋯ ,an(n≥2)a_1,a_2,\cdots,a_n(n\geq 2),so that a1<a2<⋯<an<2nλa_1<a_2<\cdots<a_n<2^n\lambda and for any k=1,2,⋯ ,nk=1,2,\cdots,n we have gcd⁡(a1,ak)+gcd⁡(a2,ak)+⋯+gcd⁡(an,ak)≡0(modak)\gcd(a_1,a_k)+\gcd(a_2,a_k)+\cdots+\gcd(a_n,a_k)\equiv 0\pmod{a_k}
number theory