MathDB

P,Q

Source: Iran 2005

8/27/2005

Suppose

P,Q\in \mathbb R[x]

that

deg\ P=deg\ Q

and

PQ'-QP'

has no real root. Prove that for each

\lambda \in \mathbb R

number of real roots of

P

and

\lambda P+(1-\lambda)Q

are equal.

functionratioalgebra proposedalgebra

ABC

Source: Iran 2005

8/27/2005

Suppose in triangle

ABC

incircle touches the side

BC

at

P

and

\angle APB=\alpha

. Prove that :

\frac1{p-b}+\frac1{p-c}=\frac2{rtg\alpha}

geometryIrantrigonometryTrigonometric Equations

a_n

Source: Iran 2005

8/29/2005

k

is an integer. We define the sequence

\{a_n\}_{n=0}^{\infty}

like this:

a_0=0,\ \ \ a_1=1,\ \ \ a_n=2ka_{n-1}-(k^2+1)a_{n-2}\ \ (n \geq 2)

p

is a prime number that p\equiv 3(\mbox{mod}\ 4) a) Prove that a_{n+p^2-1}\equiv a_n(\mbox{mod}\ p) b) Prove that a_{n+p^3-p}\equiv a_n(\mbox{mod}\ p^2)

number theory proposednumber theory

Coins

Source: Iran 2005

9/1/2005

a) Year 1872 Texas 3 gold miners found a peice of gold. They have a coin that with possibility of

\frac 12

it will come each side, and they want to give the piece of gold to one of themselves depending on how the coin will come. Design a fair method (It means that each of the 3 miners will win the piece of gold with possibility of

\frac 13

) for the miners. b) Year 2005, faculty of Mathematics, Sharif university of Technolgy Suppose

0<\alpha<1

and we want to find a way for people name

A

and

B

that the possibity of winning of

A

is

\alpha

. Is it possible to find this way? c) Year 2005 Ahvaz, Takhti Stadium Two soccer teams have a contest. And we want to choose each player's side with the coin, But we don't know that our coin is fair or not. Find a way to find that coin is fair or not? d) Year 2005,summer In the National mathematical Oympiad in Iran. Each student has a coin and must find a way that the possibility of coin being TAIL is

\alpha

or no. Find a way for the student.

probabilitygeometrygeometric transformationreflectionalgebrapolynomialPutnam

Proteins

Source: Iran 2005

9/21/2005

Suppose we have some proteins that each protein is a sequence of 7 "AMINO-ACIDS"

A,\ B,\ C,\ H,\ F,\ N

. For example

AFHNNNHAFFC

is a protein. There are some steps that in each step an amino-acid will change to another one. For example with the step

NA\rightarrow N

the protein

BANANA

will cahnge to

BANNA

("in Persian means workman"). We have a set of allowed steps that each protein can change with these steps. For example with the set of steps: \\ 1)\ AA\longrightarrow A\\ 2)\ AB\longrightarrow BA\\ 3)\ A\longrightarrow \mbox{null} Protein

ABBAABA

will change like this:

\\ ABB\underline{AA}BA\\ \underline{AB}BABA\\ B\underline{AB}ABA\\ BB\underline{AA}BA\\ BB\underline{AB}A\\ BBB\underline{AA}\\ BBB\underline{A}\\ BBB

You see after finite steps this protein will finish it steps. Set of allowed steps that for them there exist a protein that may have infinitely many steps is dangerous. Which of the following allowed sets are dangerous? a)

NO\longrightarrow OONN

b)

\left\{\begin{array}{c}HHCC\longrightarrow HCCH\\ CC\longrightarrow CH\end{array}\right.

c) Design a set of allowed steps that change

\underbrace{AA\dots A}_{n}\longrightarrow\underbrace{BB\dots B}_{2^{n}}

d) Design a set of allowed steps that change

\underbrace{A\dots A}_{n}\underbrace{B\dots B}_{m}\longrightarrow\underbrace{CC\dots C}_{mn}

You see from

c

and

d

that we acn calculate the functions

F(n)=2^{n}

and

G(M,N)=mn

with these steps. Find some other calculatable functions with these steps. (It has some extra mark.)

functioncombinatorics proposedcombinatorics

4

Problems(5)