MathDB

Three-variable inequality

Source: Turkish TST 2012 Problem 3

3/26/2012

For all positive real numbers

a, b, c

satisfying

ab+bc+ca \leq 1,

prove that

a+b+c+\sqrt{3} \geq 8abc \left(\frac{1}{a^2+1}+\frac{1}{b^2+1}+\frac{1}{c^2+1}\right)

inequalitiesinequalities proposedalgebra

A game in a 1xm board

Source: Turkish TST 2012 Problem 6

3/26/2012

Two players

A

and

B

play a game on a

1\times m

board, using

2012

pieces numbered from

1

to

2012.

At each turn,

A

chooses a piece and

B

places it to an empty place. After

k

turns, if all pieces are placed on the board increasingly, then

B

wins, otherwise

A

wins. For which values of

(m,k)

pairs can

B

guarantee to win?

LaTeXcombinatorics proposedcombinatorics

Find a subset of Z+ satisfying a property

Source: Turkish TST 2012 Problem 9

3/26/2012

Let

\mathbb{Z^+}

and

\mathbb{P}

denote the set of positive integers and the set of prime numbers, respectively. A set

A

is called

S-\text{proper}

where

A, S \subset \mathbb{Z^+}

if there exists a positive integer

N

such that for all

a \in A

and for all

0 \leq b <a

there exist

s_1, s_2, \ldots, s_n \in S

satisfying

b \equiv s_1+s_2+\cdots+s_n \pmod a

and

1 \leq n \leq N.

Find a subset

S

of

\mathbb{Z^+}

for which

\mathbb{P}

is

S-\text{proper}

but

\mathbb{Z^+}

is not.

modular arithmeticnumber theoryprime numbersnumber theory proposed

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