MathDB

a bound for partitions of a number

Source: Iran 3rd round 2011-combinatorics exam-p3

9/4/2011

Suppose that

p(n)

is the number of partitions of a natural number

n

. Prove that there exists

c>0

such that

P(n)\ge n^{c \cdot \log n}

.
proposed by Mohammad Mansouri

logarithmscombinatorics proposedcombinatorics

number of solutions to the equation

Source: Iran 3rd round 2011-number theory exam-p3

9/5/2011

Let

k

be a natural number such that

k\ge 7

. How many

(x,y)

such that

0\le x,y<2^k

satisfy the equation

73^{73^x}\equiv 9^{9^y} \pmod {2^k}

?
Proposed by Mahyar Sefidgaran

modular arithmeticnumber theory proposednumber theory

O,I,D collinear if D,X,Y collinear

Source: Iran 3rd round 2011-geometry exam-p3

9/6/2011

In triangle

ABC

,

X

and

Y

are the tangency points of incircle (with center

I

) with sides

AB

and

AC

respectively. A tangent line to the circumcircle of triangle

ABC

(with center

O

) at point

A

, intersects the extension of

BC

at

D

. If

D,X

and

Y

are collinear then prove that

D,I

and

O

are also collinear.
proposed by Amirhossein Zabeti

geometrycircumcircleratiotrigonometryprojective geometrygeometry proposed

a bound for f(i)

Source: Iran 3rd round 2011-algebra exam-p3

9/7/2011

We define the polynomial

f(x)

in

\mathbb R[x]

as follows:

f(x)=x^n+a_{n-2}x^{n-2}+a_{n-3}x^{n-3}+.....+a_1x+a_0

Prove that there exists an

i

in the set

\{1,....,n\}

such that we have

|f(i)|\ge \frac{n!}{\dbinom{n}{i}}

.
proposed by Mohammadmahdi Yazdi

algebrapolynomialsearchalgebra proposed

unfixed tetragon

Source: Iran 3rd round 2011-final exam-p3

9/11/2011

We have connected four metal pieces to each other such that they have formed a tetragon in space and also the angle between two connected metal pieces can vary. In the case that the tetragon can't be put in the plane, we've marked a point on each of the pieces such that they are all on a plane. Prove that as the tetragon varies, that four points remain on a plane.
proposed by Erfan Salavati

geometry proposedgeometry

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Problems(5)