Prove the following identity:
\[
\sum_{k=0}^{n-1} \binom{z}{k} \frac{x^{n-k}}{n-k} = \sum_{k=1}^n \binom{z-k}{n-k} \frac{(x+1)^k -1}{k}.
\]

Show that if \(X\) is a Poisson random variable with parameter \(\mu\), there exists a constant \(c>0\) such that for \(t>\mu+1\), \(\mathbb{P}(X-\mu \geq t)\geq ce^{-2t\log (1+(t+1)/\mu)}\).

Denote \(P = \{(x, y, z) \in \mathbb{R^3}: 10< x,y,z <31\}\). Suppose a function \(f (v): \mathbb{R^3} \to \mathbb{R_{\geq 0}}\) satisfies:
(a) \(f(\lambda v) = \lambda^{25} f(v)\) for all \(v\in P\) and \(0<\lambda \in \mathbb{R}\),
(b) \(f(v+w) \geq f(v)\) for every \(v, w \in P\),
(c) \(f (v)\) is locally bounded.
Show that \(f (v)\) is locally Lipschitz in \(P\).

Show that any set of d + 2 points in R^d can be partitioned into two sets whose convex hulls intersect.

Each punch can be centered anywhere in the plane and removes all points within distance 1 from its center. What is the minimum number of punches needed to remove every point in the annulus between the circles of radius 7 and 10 (with the same center)? Describe your construction. The person with the smallest number of punches earns +4, and the next four best answers earn +3.

Let \( X_1, X_2, \ldots \) be an infinite sequence of standard normal random variables which are not necessarily independent. Show that there exists a universal constant \( C \) such that \(\mathbb{E} \left[ \max_i \frac{|X_i|}{\sqrt{1 + \log i}} \right] \leq C\).

Let \(P\) be a regular \(2n\)-gon. A perfect matching is a partition of vertex points into \(n\) unordered pairs; each pair represents a chord drawn inside \(P\). Two chords are said to “intersect” if they have a nonempty intersection.

Let \(X\) be the (random) number of intersection points (formed by intersecting chords) in a perfect matching chosen uniformly at random from the set of all possible matchings. Note that more than two chords can intersect at the same point, and in this case this intersection point is just counted once. Compute \(\lim_{n\rightarrow \infty} \frac{\mathbb E[X]}{n^2}\).

For given \(a, b \in \mathbb{R}\) and \(c \in \mathbb{Z}\), find all function \(f: \mathbb{R} \to \mathbb{R}\) which is continuous at 0 and satisfies
\[
f(ax) = f(bx) + x^c \quad \forall x\in \mathbb{R}\setminus \{0\}.
\]