Common Estimates and Inequalities
Read this topicExplanations 1
- Inequalities What are some common inequalities? Trivial Inequality: x2≥0.x^2 \geq 0.x2≥0. Equality holds if x=0x=0x=0. AM-GM: a1+a2+...+an2≥a1a2...ann\frac{a_1+a_2+...+a_n}{2} \…
By Simran Khunger, Darren Huang, Alex Federspiel
Problems 3
- Solve for the possible integer solution(s) for this equation. x2+xy+y2=(x+y3+1)3.x^2+xy+y^2 = \left(\frac{x+y}{3}+1\right)^3.x2+xy+y2=(3x+y+1)3.
- Find all solutions to (m2+n)(m+n2)=(m−n)3(m^2+n)(m + n^2)= (m - n)^3(m2+n)(m+n2)=(m−n)3, where mmm and nnn are non-zero integers.
- Find all integer solutions (x,y)(x,y)(x,y) to 7x2−13xy+7y23=∣x−y∣+1.\sqrt[3]{7x^2-13xy+7y^2}=|x-y|+1.37x2−13xy+7y2=∣x−y∣+1.