MathDB

Sequence

Source: Iran 2005

8/27/2005

Suppose

\{x_n\}

is a decreasing sequence that

\displaystyle\lim_{n \rightarrow\infty}x_n=0

. Prove that

\sum(-1)^nx_n

is convergent

limitalgebra proposedalgebra

SOAB)+S(OAC)=2S(OBC)

Source: Iran 2005

8/27/2005

Suppose

O

is circumcenter of triangle

ABC

. Suppose

\frac{S(OAB)+S(OAC)}2=S(OBC)

. Prove that the distance of

O

(circumcenter) from the radical axis of the circumcircle and the 9-point circle is

\frac {a^2}{\sqrt{9R^2-(a^2+b^2+c^2)}}

geometrycircumcircleEulerpower of a pointradical axisgeometry proposed

m=a^2+a+1

Source: Iran 2005

8/29/2005

Let

a\in\mathbb N

and

m=a^2+a+1

. Find the number of

0\leq x\leq m

that:x^3\equiv1(\mbox{mod}\ m)

number theory proposednumber theory

Vectors

Source: Iran 2005

9/1/2005

n

vectors are on the plane. We can move each vector forward and backeard on the line that the vector is on it. If there are 2 vectors that their endpoints concide we can omit them and replace them with their sum (If their sum is nonzero). Suppose with these operations with 2 different method we reach to a vector. Prove that these vectors are on a common line

vectorgeometry proposedgeometry

Sets in R^n

Source: Iran 2005

9/21/2005

We define a relation between subsets of

\mathbb R ^n

.

A \sim B\Longleftrightarrow

we can partition

A,B

in sets

A_1,\dots,A_n

and

B_1,\dots,B_n

(i.e

\displaystyle A=\bigcup_{i=1} ^n A_i,\ B=\bigcup_{i=1} ^n B_i, A_i\cap A_j=\emptyset,\ B_i\cap B_j=\emptyset

) and

A_i\simeq B_i

. Say the the following sets have the relation

\sim

or not ? a) Natural numbers and composite numbers. b) Rational numbers and rational numbers with finite digits in base 10. c)

\{x\in\mathbb Q|x<\sqrt 2\}

and

\{x\in\mathbb Q|x<\sqrt 3\}

d)

A=\{(x,y)\in\mathbb R^2|x^2+y^2<1\}

and

A\setminus \{(0,0)\}

geometrygroup theoryabstract algebrageometric transformationGausstopologyabsolute value

2

Problems(5)