MathDB

Isogonal lines at the intersection of two circles

Source: RMO Delhi 2016, P3

10/11/2016

Two circles

C_1

and

C_2

intersect each other at points

A

and

B

. Their external common tangent (closer to

B

) touches

C_1

at

P

and

C_2

at

Q

. Let

C

be the reflection of

B

in line

PQ

. Prove that

\angle CAP=\angle BAQ

.

geometry

number twice the square of sum of digits in decimal

Source: RMO Mumbai 2016, P3

10/11/2016

For any natural number

n

, expressed in base

10

, let

S(n)

denote the sum of all digits of

n

. Find all natural numbers

n

such that

n=2S(n)^2

.

number theoryinequalities

All roots integers for a polynomial

Source: RMO Maharashtra and Goa 2016, P3

10/11/2016

Find all integers

k

such that all roots of the following polynomial are also integers:

f(x)=x^3-(k-3)x^2-11x+(4k-8).

number theoryalgebrapolynomial

Number Theory

Source: RMO 2016 Karnataka Region P3

10/16/2016

Let

a,b,c,d,e,d,e,f

be positive integers such that

\dfrac a b < \dfrac c d < \dfrac e f

. Suppose

af-be=-1

. Show that

d \geq b+f

.

number theory

S(n) is sum of digits of n: n^3 = 8S(n)^3+6S(n)n+1

Source: RMO Hyderabad 2016 , P3 .

10/12/2016

For any natural number

n

, expressed in base

10

, let

S(n)

denote the sum of all digits of

n

. Find all positive integers

n

such that

n^3 = 8S(n)^3+6S(n)n+1

.

number theory

RMO 2016 ,Q3

Source: Oct 23,2016

10/25/2016

The precent ages in years of two brothers

A

and

B

,and their father

C

are three distinct positive integers

a ,b

and

c

respectively .Suppose

\frac{b-1}{a-1}

and

\frac{b+1}{a+1}

are two consecutive integers , and

\frac{c-1}{b-1}

and

\frac{c+1}{b+1}

are two consecutive integers . If

a+b+c\le 150

, determine

a,b

and

c

.

number theory

2016 Chandigarh RMO (ad + bc) divides each of a, b, c,d

Source:

8/9/2019

a, b, c, d

are integers such that

ad + bc

divides each of

a, b, c

and

d

. Prove that

ad + bc =\pm 1

number theorydividesdivisor

3

Problems(7)