## Nice characterization side-lengths of a triangle

Find the greatest ${k}$ such that ${a,b,c>0}$ and ${kabc > a^3+b^3+c^3}$ implies that ${a,b,c}$ are the side-lengths of a triangle.

## Best approximation of a certain square root

Let ${\lambda}$ be a real number such that the inequality

$\displaystyle 0 < \sqrt{2002}-\frac{a}{b} < \frac{\lambda}{ab}$

holds for an infinity of pairs ${(a,b)}$ of natural numbers. Prove that ${\lambda\geq 5}$.

## Miklos Schweitzer 2013 Problem 7

November 22, 2013 1 comment

Problem 7. Suppose that ${f: \Bbb{R} \rightarrow \Bbb{R}}$ is an additive function (that is ${f(x+y) = f(x)+f(y)}$ for all ${x, y \in \Bbb{R}}$) for which ${x \mapsto f(x)f(\sqrt{1-x^2})}$ is bounded of some nonempty subinterval of ${(0,1)}$. Prove that ${f}$ is continuous.

Miklos Schweitzer 2013

Categories: Algebra, Analysis, Geometry

## Continuity in Geometry

Here are a few interesting geometry problems which use continuity problems in their solutions.

Pb 1. Consider three parallel lines in the plane ${d_1,d_2,d_3}$. Prove that there exist points ${A_i\in d_i}$ such that the triangle ${A_1A_2A_3}$ is equilateral.

Pb 2. Consider a triangle ${ABC}$. Prove that ${ABC}$ is equilateral if and only if for every point ${M}$ in the plane we can construct a triangle with sides ${MA,MB,MC}$.

## Miklos Schweitzer 2013 Problem 8

November 9, 2013 1 comment

Problem 8. Let ${f : \Bbb{R} \rightarrow \Bbb{R}}$ be a continuous and strictly increasing function for which

$\displaystyle f^{-1}\left(\frac{f(x)+f(y)}{2}\right)(f(x)+f(y)) =(x+y)f\left(\frac{x+y}{2}\right)$

for all ${x,y \in \Bbb{R}}$ (${f^{-1}}$ denotes the inverse of ${f}$). Prove that there exist real constants ${a \neq 0}$ and ${b}$ such that ${f(x)=ax+b}$ for all ${x \in \Bbb{R}}$.

Miklos Schweitzer 2013

## Miklos Schweitzer 2013

November 9, 2013 2 comments

Problem 1. Let ${q \geq 1}$ be an integer. Prove that there exists an integer ${C_q}$ such that for any finite set ${A}$ containing only integers we have

$\displaystyle |A+qA| \geq (q+1)|A|-C_q.$

(${A+qA}$ is the set of integers of the form ${a+qa'}$ where ${a,a' \in A}$)

Problem 2. Prove that there is a constant ${k_0}$ such that the diophantine equation

$\displaystyle a^{2n}+b^{4n}+2013=ka^n b^{2n}$

has no positive integer solutions ${(a,b,n)}$ for ${k \geq k_0}$. (here I guess $k$ needs to be an integer, or else the conclusion is clearly false)

Problem 3. What are the numbers ${n}$ for which the ${A_n}$ alternating group has a permutation which is contained in exactly one ${2}$-Sylow subgroup of ${A_n}$?

Problem 4. Let ${A}$ be an Abelian group with ${n}$ elements. Prove that there are two subgroups in ${GL(n,\Bbb{C})}$, isomorphic to ${S_n}$, whose intersection is isomorphic to the automorphism group of ${A}$.

Problem 5. A subalgebra ${\mathfrak h}$ of a Lie algebra ${\mathfrak g}$ is said to have tha ${\gamma}$ property with respect to a scalar product ${\langle .,.\rangle}$ given on ${\mathfrak g}$ if ${X \in \mathfrak{h}}$ implies ${\langle [X,Y],X\rangle =0}$ for all ${Y \in \mathfrak g}$. Prove that the maximum dimension of ${\gamma}$-property subalgebras of a given ${2}$ step nilpotent Lie algebra with respect to a scalar product is independent of the selection of the scalar product.

Problem 6. Let ${\mathcal A}$ be a ${C^*}$ algebra with a unit element and let ${\mathcal A_+}$ be the cone of the positive elements of ${\mathcal A}$ (this is the set of such self adjoint elements in ${\mathcal A}$ whose spectrum is in ${[0,\infty)}$. Consider the operation

$\displaystyle x \circ y =\sqrt{x}y\sqrt{x},\ x,y \in \mathcal A_+.$

Prove that if for all ${x,y \in \mathcal A_+}$ we have

$\displaystyle (x\circ y)\circ y = x \circ (y \circ y),$

then ${\mathcal A}$ is commutative.

Problem 7. Suppose that ${f: \Bbb{R} \rightarrow \Bbb{R}}$ is an additive function (that is ${f(x+y) = f(x)+f(y)}$ for all ${x, y \in \Bbb{R}}$) for which ${x \mapsto f(x)f(\sqrt{1-x^2})}$ is bounded of some nonempty subinterval of ${(0,1)}$. Prove that ${f}$ is continuous. [Solution]

Problem 8. Let ${f : \Bbb{R} \rightarrow \Bbb{R}}$ be a continuous and strictly increasing function for which

$\displaystyle f^{-1}\left(\frac{f(x)+f(y)}{2}\right)(f(x)+f(y)) =(x+y)f\left(\frac{x+y}{2}\right)$

for all ${x,y \in \Bbb{R}}$ (${f^{-1}}$ denotes the inverse of ${f}$). Prove that there exist real constants ${a \neq 0}$ and ${b}$ such that ${f(x)=ax+b}$ for all ${x \in \Bbb{R}}$. [Solution]

Problem 9. Prove that there is a function ${f: (0,\infty) \rightarrow (0,\infty)}$ which is nowhere continuous and for all ${x,y \in (0,\infty)}$ and any rational ${\alpha}$ we have

$\displaystyle f\left( \left(\frac{x^\alpha+y^\alpha}{2}\right)^{\frac{1}{\alpha}}\right)\leq \left(\frac{f(x)^\alpha +f(y)^\alpha }{2}\right)^{\frac{1}{\alpha}}.$

Is there such a function if instead the above relation holds for every ${x,y \in (0,\infty)}$ and for every irrational ${\alpha}$.

Problem 10. Consider a Riemannian metric on the vector space ${\Bbb{R}^n}$ which satisfies the property that for each two points ${a,b}$ there is a single distance minimising geodesic segment ${g(a,b)}$. Suppose that for all ${a \in \Bbb{R}^n}$, the Riemannian distance with respect to ${a}$, ${\rho_a : \Bbb{R}^n \rightarrow \Bbb{R}}$ is convex and differentiable outside of ${a}$. Prove that if for a point ${x \neq a,b}$ we have

$\displaystyle \partial_i \rho_a(x)=-\partial_i \rho_b(x),\ i=1..n$

then ${x}$ is a point on ${g(a,b)}$ and conversely.

Problem 11. (a) Consider an ellipse in the plane. Prove that there exists a Riemannian metric which is defined on the whole plane, and with respect to which the ellipse is a geodesic. Prove that the Gaussian curvature of any such Riemannian metric takes a positive value.

(b) Consider two nonintersecting, simple closed smooth curves in the plane. Prove that if there is a Riemmanian metric defined on the whole plane and the two curves are geodesics of that metric, then the Gaussian curvature of the metric vanishes somewhere.

Problem 12. There are ${n}$ tokens in a pack. Some of them (at least one, but not all) are white and the rest are black. All tokens are extracted randomly from the pack, one by one, without putting them back. Let ${X_i}$ be the ratio of white tokens in the pack before the ${i}$-th extraction and let

$\displaystyle T =\max \{ |X_i-X_j| : 1 \leq i \leq j \leq n\}.$

Prove that ${\Bbb{E}(T) \leq H(\Bbb{E}(X_1))}$, where ${H(x)=-x\ln x -(1-x)\ln(1-x)}$.

(Thanks again to Eles Andras for the translation of the problems from Hungarian to English)

## How can everybody (probably) get the right change?

We are all annoyed when we buy something and the seller doesn’t have the right change. You have a few possibilities: leave the change (not a real option…), go change your money yourself, or wait for another buyer. Below a simplistic situation is presented, where ${100}$ people wait in line to buy cinema tickets. The price is ${5}$ euros, and the customers all have bills of ${5}$ and ${10}$ euros (to be more precise ${40}$ people have ${10}$ euros bills and ${60}$ people have ${5}$ euros bills). Note that there are more people with ${5}$ euros bills than those with ${10}$ euros bills, so in the end everybody could receive change, even if the cashier has no ${5}$ euro bills in the beginning. Still, the problem is that each client would like to pay for his ticket and receive the change on the spot, without waiting. In order to do that the cashier would need to have some number of ${5}$ euro bills in the beginning.
The worst case is when all ${40}$ clients with ${10}$ euros bills are first in line, case in which the cashier would need ${40}$ bills of ${5}$ euros for everyone to be happy. We can ask for a more weak condition, namely that the probability that everyone is satisfied is big, let’s say ${95\%}$. How many ${5}$ euros bills would the cashier need to fulfill this goal? How many bills does he need to reach a probability of ${99\%}$?