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 \(T \leq 3A/2.\)

Let \(f(t)=(t^{pq}-1)(t-1) \) and \(g(t)=(t^{p}-1)(t^q-1) \) where \(p\) and \(q\) are relatively prime positive integers. Prove that \(\frac{f(t)}{g(t)}\) can be written as a polynomial where it has just \(1\) or \(-1\) as coefficients. (For example, when \(p=2\) and \(q=3\), we have that \(\frac{f(t)}{g(t)} = t^2-t+1\).)

Let \(p\) be a prime number at least three and let \(k\) be a positive integer smaller than \(p\). Given \(a_1,\dots, a_k\in \mathbb{F}_p\) and distinct elements \(b_1,\dots, b_k\in \mathbb{F}_p\), prove that there exists a permutation \(\sigma\) of \([k]\) such that the values of \(a_i + b_{\sigma(i)}\) are distinct modulo \(p\).

Let \(S\) be a set of distinct \(20\) integers. A set \(T_A\) is defined as \(T_A:=\{ s_1+s_2+s_3 \mid s_1, s_2, s_3 \in S\}\). What is the smallest possible cardinality of \(T_A\)?

Let \(\mathbb{S}_n\) be the set of all permutations of \([n]=\{1,\dots, n\}\). For positive real numbers \(d_1,\dots, d_n\), prove \[ \sum_{\sigma\in \mathbb{S}_n} \frac{1}{ d_{\sigma(1)}(d_{\sigma(1)}+d_{\sigma(2)}) \dots (d_{\sigma(1)}+\dots + d_{\sigma(n)}) } = \frac{1}{d_1\dots d_n}.\]

Find a pair of nonisomorphic nonabelian groups so that their abelianizations are isomorphic and their commutator subgroups are perfect.