MathDB

4

Part of 2016 Tuymaada Olympiad

Problems(2)

Equation dio

Source: Tuymaada 2016, P4, Senior

7/20/2016
For each positive integer kk find the number of solutions in nonnegative integers x,y,zx,y,z with x≤y≤zx\le y \le z of the equation 8k=x3+y3+z3−3xyz8^k=x^3+y^3+z^3-3xyz
number theory proposedDiophantine Equationsnumber theoryTuymaada 2016
Three variable inequality for juniors

Source: Tuymaada 2016. Junior League/P4

7/22/2016
Non-negative numbers aa, bb, cc satisfy a2+b2+c2≥3a^2+b^2+c^2\geq 3. Prove the inequality (a+b+c)3≥9(ab+bc+ca). (a+b+c)^3\geq 9(ab+bc+ca).
inequalities