MMO2025 Round 4

Let $s_i=t_i^2/4$. Since $t_i^3\le 2t_i^2=8s_i$, so for $0\le s_1,\dots ,s_n\le 1$, it suffices to prove $$\frac{s_1}{s_2+1}+\frac{s_2}{s_3+1}+\cdots +\frac{s_n}{s_1+1}\le \frac{n}{2}.$$ Now for any $0\le x,y\le 1$, we consider: $$(1+y)(1-y+x)=2x+(1-y)(1+y-x)\ge 2x$$ which implies: $$\frac{x}{y+1}\le \frac{1-y+x}{2}.$$

Applying this to the set $\{s_i\}$ cyclically: $$\sum _i\frac{s_i}{s_{i+1}+1}\le \frac{n}{2}+\frac{1}{2}\sum _i(s_i-s_{i+1})=\frac{n}{2}$$ so the inequality is proved. Equality holds for each $i$ when $s_{i+1}=1$ or $(s_i,s_{i+1})=(1,0)$, i.e., when the sequence $\{t_1,t_2,\dots ,t_n,t_1\}$ consists only of 0 and 2, and contains no consecutive 0's.

Remark. Refer to the solution of problem E5.