IP homework is way too hard

The second set of homework takes me two days with 16+ hours to finish and without leaving a logic hole.
The last two logic gaps I filled just now are
1. LP relaxation of TSP formulation with subtour elimination constrainted is strictly contained in TSP formulation with subtour elimination constrainted replaced by $u_i-u_j-nx_{ij} \leq n-1$. This invovle how to scale up the problem then scale down.
2. Find as many affine independent points as possible for $P={x \in R^+ | \sum^n_{i=1} x_{ij}=1 for i=1,...n and \sum^n_{j=1} x_{ij}=1 for j=1,...n}$. It would be easy to see the rules by beginning with n=3 and 4.

In homework 1 of IP class, the following question is challenging.
1. show that $max{x_1 − \sqrt(2) x_2 | 1 \leq x1 \leq \sqrt(2) x_2 and x_1, x_2 is integer}$ is feasible and bounded, but has no optimal solution. I use the contradiction to show that there is no optimal solution.