Mathematics

Let $f(N)$ be the smallest number such that any $f(N)$ points in the plane, no three on a line, include $N$ points that are the vertices of a convex polygon. Prove or disprove that $f(N)=2^{N-2}+1$ for every $N\ge 3$.

Euler's totient $\phi (n)$ counts the integers in $1,\dots ,n$ that are coprime to $n$. When $n$ is prime, $\phi (n)=n-1$, which certainly divides $n-1$. Is there a composite number $n$ for which $\phi (n)$ divides $n-1$?

The Riemann zeta function is defined by $\zeta (s)={\sum }_{n\ge 1}{n}^{-s}$ for $\mathrm{Re}(s)>1$ and extended by analytic continuation to all complex $s\ne 1$. It vanishes at $s=-2,-4,-6,\dots$. Prove that every other zero of $\zeta$ has real part exactly $1/2$.

Prove that for every positive integer $n$ there is a prime number strictly between $n^2$ and $(n+1{)}^2$.

A Hadamard matrix of order $n$ is an $n\times n$ matrix $H$ with entries $\pm 1$ whose rows are mutually orthogonal, so that $H{H}^T=nI$. Such a matrix can exist only if $n$ is 1, 2 or a multiple of 4. Prove that a Hadamard matrix exists for every multiple of 4.

Let $N(a)$ be the number of times the integer $a>1$ appears in Pascal's triangle. For example, $N(3003)=8$. Is there a constant $M$ with $N(a)\le M$ for every $a>1$?

Prove that the regions of any map drawn in the plane can be colored with at most four colors so that any two regions sharing a border of positive length get different colors. Equivalently, prove that every planar graph is 4-colorable.

Prove that the equation $a^n+b^n=c^n$ has no solution in positive integers $a,b,c$ when $n$ is an integer greater than 2.

Prove that no packing of equal balls in three-dimensional space has average density greater than $\pi /\sqrt{18}\approx 0.7405$, the density of the familiar cannonball stacking (face-centered cubic packing).

The numbers $8=2^3$ and $9=3^2$ are consecutive perfect powers. Prove that they are the only such pair, that is, the only solution of $x^a-y^b=1$ in integers $x,y,a,b>1$ is $3^2-2^3=1$.

Prove that every closed (compact, without boundary) three-dimensional manifold in which every loop can be continuously shrunk to a point is homeomorphic to the three-dimensional sphere.

A Kakeya set (or Besicovitch set) in ${\mathbb{R}}^3$ is a set that contains a unit line segment in every direction. Such sets can have volume zero. Prove that every Kakeya set in ${\mathbb{R}}^3$ nevertheless has Hausdorff dimension 3.

A rigid planar shape must be moved around a right-angled corner in a hallway of width 1. What is the largest possible area of such a shape? Gerver's sofa has area about 2.2195. Prove that no shape with larger area can make the turn.

Let $x_1,x_2,x_3,\dots$ be any infinite sequence with every $x_i$ equal to $+1$ or $-1$. Prove that for every $C$ there are positive integers $k$ and $d$ with $\left|{\sum }_{i=1}^k{x}_{id}\right|>C.$

Find the exact value of ${\sum }_{n=1}^{\infty }\frac{1}{n^2}=1+\frac{1}{4}+\frac{1}{9}+\frac{1}{16}+\cdots$ and prove that your answer is correct.

Königsberg had four land masses, two river banks and two islands. One island was joined to each bank by two bridges, the other island to each bank by one bridge, and a seventh bridge joined the two islands. Is there a walk that crosses every bridge exactly once? More generally, which connected…

Using only a compass and an unmarked straightedge, can one construct in finitely many steps a square with exactly the same area as a given circle? Prove your answer.

A regular tetrahedron and a cube have the same volume. Can the tetrahedron be cut into finitely many polyhedral pieces that can be reassembled to form the cube? Prove your answer.

A floor is ruled with parallel lines a distance $d$ apart. A needle of length $\ell \le d$ is dropped at random. What is the probability that it lands across one of the lines?

On a game show there are three doors, one hiding a car and two hiding goats. You pick a door. The host, who knows where the car is, always opens one of the other two doors to show a goat and then offers you the chance to switch to the remaining closed door. Should you switch, and what is your…