MathDB

Two Quadratics have Integer Roots

Source: Indian RMO 2013 Paper 1 Problem 6

2/1/2014

Suppose that

m

and

n

are integers, such that both the quadratic equations

x^2+mx-n=0

and

x^2-mx+n=0

have integer roots. Prove that

n

is divisible by

6

.

quadraticsmodular arithmeticnumber theoryDiophantine equationnumber theory unsolved

More balanced weighings with 100 coins than with 99

Source: Indian RMO, Paper 2, Problem 6

12/11/2013

For a natural number

n

, let

T(n)

denote the number of ways we can place

n

objects of weights

1,2,\cdots, n

on a balance such that the sum of the weights in each pan is the same. Prove that

T(100) > T(99)

.

functionmodular arithmeticcombinatorics unsolvedcombinatorics

Indian RMO - Paper 3

Source: RMO - Problem 6

12/11/2013

Let

n \ge 4

be a natural number. Let A_1A_2 \cdots A_n be a regular polygon and

X = \{ 1,2,3....,n \}

. A subset

\{ i_1, i_2,\cdots, i_k \}

of

X

, with k \ge 3 and i_1 < i_2 < \cdots < i_k, is called a good subset if the angles of the polygon A_{i_1}A_{i_2}\cdots A_{i_k} , when arranged in the increasing order, are in an arithmetic progression. If

n

is a prime, show that a proper good subset of

X

contains exactly four elements.

calculusintegrationarithmetic sequencenumber theory unsolvednumber theory

Indian RMO - Paper -4[6]

Source:

12/12/2013

Suppose that the vertices of a regular polygon of

20

sides are coloured with three colours - red, blue and green - such that there are exactly three red vertices. Prove that there are three vertices

A,B,C

of the polygon having the same colour such that triangle

ABC

is isosceles.

combinatoricspolygon

Reciprocal Roots Lead to Integer Coefficients

Source: Indian RMO 2013 Mumbai Region Problem 6

2/1/2014

Let

P(x)=x^3+ax^2+b

and

Q(x)=x^3+bx+a

, where

a

and

b

are nonzero real numbers. Suppose that the roots of the equation

P(x)=0

are the reciprocals of the roots of the equation

Q(x)=0

. Prove that

a

and

b

are integers. Find the greatest common divisor of

P(2013!+1)

and

Q(2013!+1)

.

number theorypolynomialrootsfactorialgreatest common divisor

6

Problems(5)