MathDB

Look at the exponent...

Source:

10/21/2010

Prove that the following statement is true for two natural nos.

m,n

if and only

v(m) = v(n)

where

v(k)

is the highest power of

2

dividing

k

.

\exists

a set

A

of positive integers such that

(i)

x,y \in \mathbb{N}, |x-y| = m \implies x \in A

or

y \in A

(ii)

x,y \in \mathbb{N}, |x-y| = n \implies x \not\in A

or

y \not\in A

number theory unsolvednumber theory

Nice and Simple

Source:

12/9/2010

For each

n\in \mathbb{N}

, let

S(n)

be the sum of all numbers in the set

\{ 1, 2, 3, \cdots , n \}

which are relatively prime to

n

.

(a)

Show that

2 \cdot S(n)

is not a perfect square for any

n

.

(b)

Given positive integers

m, n

, with odd

n

, show that the equation

2 \cdot S(x) = y^n

has at least one solution

(x, y)

among positive integers such that

m|x

.

number theoryrelatively primenumber theory unsolved

Find the locus

Source:

12/9/2010

Let

C_1 , C_2

be two circles in the plane intersecting at two distinct points. Let

P

be the midpoint of a variable chord

AB

of

C_2

with the property that the circle on

AB

as diameter meets

C_1

at a point

T

such that

P T

is tangent to

C_1

. Find the locus of

P

.

ratiogeometry unsolvedgeometry

Easy triplets of positive integers

Source:

10/22/2010

How many ordered triples

(a, b, c)

of positive integers are there such that none of

a, b, c

exceeds

2010

and each of

a, b, c

divides

a + b + c

?

number theory unsolvednumber theory

Inequality about square inside a triangle

Source:

12/9/2010

\triangle ABC

has semiperimeter

s

and area

F

. A square

P QRS

with side length

x

is inscribed in

ABC

with

P

and

Q

on

BC

,

R

on

AC

, and

S

on

AB

. Similarly,

y

and

z

are the sides of squares two vertices of which lie on

AC

and

AB

, respectively. Prove that

\frac 1x +\frac 1y + \frac 1z \le \frac{s(2+\sqrt3)}{2F}

inequalitiesgeometryCauchy Inequalityinequalities unsolved

4

Problems(5)