MathDB

1

An expectation inequality

{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^{\text{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)}.

Proposed by Tamás Móri

11

1

Again a Riemannian metric

(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.
Proposed by Tran Quoc Binh

10

1

A Riemannian metric on R^n

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,\cdots, n

then

{x}

is a point on

{g(a,b)}

and conversely.
Proposed by Lajos Tamássy and Dávid Kertész

8

1

A condition which implies linearity

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}}.

Proposed by Zoltán Daróczy

9

1

A different kind of power mean inequality

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}?

Proposed by Maksa Gyula and Zsolt Páles

7

1

Another take on Cauchy's functional equation

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.
Proposed by Zoltán Boros

6

1

A commutative algebra

Let

{\mathcal A}

be a

{C^{\ast}}

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.
Proposed by Lajos Molnár

5

1

A subalgebra having a special property

A subalgebra

\mathfrak{h}

of a Lie algebra

\mathfrak g

is said to have the

\gamma

property with respect to a scalar product

{\langle \cdot,\cdot \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.
Proposed by Péter Nagy Tibor

4

1

An Abelian Group of size n

Let

A

be an Abelian group with

n

elements. Prove that there are two subgroups in

\text{GL}(n,\Bbb{C})

, isomorphic to

S_n

, whose intersection is isomorphic to the automorphism group of

A

.
Proposed by Zoltán Halasi

3

1

Find all such n

Find for which positive integers

n

the

A_n

alternating group has a permutation which is contained in exactly one

2

-Sylow subgroup of

A_n

.
Proposed by Péter Pál Pálfy

1

Strong lower bound on sumset |A+qA|

Let

q

be a positive integer. Prove there exists a constant

C_q

such that the following inequality holds for any finite set

A

of integers:

|A+qA|\ge (q+1)|A|-C_q.

Proposed by Antal Balog.

2

1

Diophantine equation with no solutions

Prove there exists a constant

k_0

such that for any

k\ge k_0

, the equation

a^{2n}+b^{4n}+2013=ka^nb^{2n}

has no positive integer solutions

a,b,n

.
Proposed by István Pink.

2013 Miklós Schweitzer

Subcontests

An expectation inequality

Again a Riemannian metric

A Riemannian metric on R^n

A condition which implies linearity

A different kind of power mean inequality

Another take on Cauchy's functional equation

A commutative algebra

A subalgebra having a special property

An Abelian Group of size n

Find all such n

Strong lower bound on sumset |A+qA|

Diophantine equation with no solutions