MathDB

again an easy diophantine equation 3^x=y2^x+1

Source: Romanian IMO TST 2005 - day 1, problem 1

3/31/2005

Solve the equation

3^x=2^xy+1

in positive integers.

inductionmodular arithmeticLaTeXnumber theory proposednumber theory

convex polygon with 4n+2 sides

Source: Romanian IMO TST 2005 - day 2, problem 1

4/1/2005

Prove that in any convex polygon with

4n+2

sides (

n\geq 1

) there exist two consecutive sides which form a triangle of area at most

\frac 1{6n}

of the area of the polygon.

geometrygeometry proposed

concurrent lines in hexagon

Source: Romanian IMO TST 2005 - day 3, problem 1

4/19/2005

Let

A_0A_1A_2A_3A_4A_5

be a convex hexagon inscribed in a circle. Define the points

A_0'

,

A_2'

,

A_4'

on the circle, such that A_0A_0' \parallel A_2A_4, A_2A_2' \parallel A_4A_0, A_4A_4' \parallel A_2A_0 . Let the lines

A_0'A_3

and

A_2A_4

intersect in

A_3'

, the lines

A_2'A_5

and

A_0A_4

intersect in

A_5'

and the lines

A_4'A_1

and

A_0A_2

intersect in

A_1'

. Prove that if the lines

A_0A_3

,

A_1A_4

and

A_2A_5

are concurrent then the lines

A_0A_3'

,

A_4A_1'

and

A_2A_5'

are also concurrent.

trigonometrygeometrytrapezoidgeometry proposed

easy functional equation f(x+a)=f(x)-x

Source: Romanian IMO TST 2005 - day 4, problem 1

4/23/2005

Let

a\in\mathbb{R}-\{0\}

. Find all functions

f: \mathbb{R}\to\mathbb{R}

such that

f(a+x) = f(x) - x

for all

x\in\mathbb{R}

. Dan Schwartz

functioninductionalgebra proposedalgebra

2004 queens on a 2004 x 2004 chess table

Source: Romanian IMO TST 2005 - day 5, problem 1

4/24/2005

On a

2004 \times 2004

chess table there are 2004 queens such that no two are attacking each other\footnote[1]{two queens attack each other if they lie on the same row, column or direction parallel with on of the main diagonals of the table}. Prove that there exist two queens such that in the rectangle in which the center of the squares on which the queens lie are two opposite corners, has a semiperimeter of 2004.

geometryrectanglemodular arithmeticcombinatorics proposedcombinatorics

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Problems(5)