MathDB

Difference of digits one

Source: IMOTC 2015 Practice Test 1 Problem 2

7/11/2015

A

10

-digit number is called a

<span class='latex-italic'>cute</span>

number if its digits belong to the set

\{1,2,3\}

and the difference of every pair of consecutive digits is

1

. a) Find the total number of cute numbers. b) Prove that the sum of all cute numbers is divisibel by

1408

.

combinatorics

Largest proper divisor of a composite number

Source: IMOTC 2015 Practice Test 2 Problem 2

7/11/2015

For a composite number

n

, let

d_n

denote its largest proper divisor. Show that there are infinitely many

n

for which

d_n +d_{n+1}

is a perfect square.

number theory

Functional equation

Source: Indian Team Selection Test 2015 Day 2 Problem 2

7/11/2015

Find all functions from

\mathbb{N}\cup\{0\}\to\mathbb{N}\cup\{0\}

such that

f(m^2+mf(n))=mf(m+n)

, for all

m,n\in \mathbb{N}\cup\{0\}

.

algebrafunctional equationfunction

Iterative process on a pair of real numbers

Source: Indian Team Selection Test 2015 Day 4 Problem 2

7/11/2015

Let

A

be a finite set of pairs of real numbers such that for any pairs

(a,b)

in

A

we have

a>0

. Let

X_0=(x_0, y_0)

be a pair of real numbers(not necessarily from

A

). We define

X_{j+1}=(x_{j+1}, y_{j+1})

for all

j\ge 0

as follows: for all

(a,b)\in A

, if

ax_j+by_j>0

we let

X_{j+1}=X_j

; otherwise we choose a pair

(a,b)

in

A

for which

ax_j+by_j\le 0

and set

X_{j+1}=(x_j+a, y_j+b)

. Show that there exists an integer

N\ge 0

such that

X_{N+1}=X_N

.

algebra

Equivalent polynomials

Source: Indian Team Selection Test 2015 Day 1 Problem 2

7/11/2015

Let

f

and

g

be two polynomials with integer coefficients such that the leading coefficients of both the polynomials are positive. Suppose

\deg(f)

is odd and the sets

\{f(a)\mid a\in \mathbb{Z}\}

and

\{g(a)\mid a\in \mathbb{Z}\}

are the same. Prove that there exists an integer

k

such that

g(x)=f(x+k)

.

algebrapolynomial

2

Problems(5)