Consider a function \(f: \{1,2,\dots, n\}\rightarrow \mathbb{R}\) satisfying the following for all \(1\leq a,b,c \leq n-2\) with \(a+b+c\leq n\).

\[ f(a+b)+f(a+c)+f(b+c) – f(a)-f(b)-f(c)-f(a+b+c) \geq 0 \text{ and } f(1)=f(n)=0.\]

Prove or disprove this: all such functions \(f\) always have only nonnegative values on its domain.

Acknowledgement: This problem arises during a research discussion between June Huh, Jaehoon Kim and Matt Larson.

Does there exist a nontrivial subgroup \(G\) of \( GL(10, \mathbb{C}) \) such that each element in \(G\) is diagonalizable but the set of all the elements of \(G\) is not simultaneously diagonalizable?

Can we find a sequence \(a_i, i=0,1,2,…\) with the following property: for each given integer \(n\geq 0\), we have \[\lim_{L\to +\infty}\sum_{i=0}^L 2^{ni} |a_i|\leq 23^{(n+11)^{10}} \quad \text{ and }\quad \lim_{L\to +\infty}\sum_{i=0}^L 2^{ni} a_i = (-1)^n ?\]

Let \(N\) be the number of ordered tuples of positive integers \( (a_1,a_2,\ldots, a_{27} )\) such that \( \frac{1}{a_1} + \frac{1}{a_2} + \cdots +\frac{1}{a_{27}} = 1\). Compute the remainder of \(N\) when \(N\) is divided by \(3\).

Let \(f(x) = x^4 + (2-a)x^3 – (2a+1)x^2 + (a-2)x + 2a\) for some \(a \geq 2\). Draw two tangent lines of its graph at the point \((-1,0)\) and \((1,0)\) and let \(P\) be the intersection point. Denote by \(T\) the area of the triangle whose vertices are \((-1,0), (1,0)\) and \(P\). Let \(A\) be the area of domain enclosed by the interval \([-1,1]\) and the graph of the function on this interval. Show that \(T \leq 3A/2.\)