MathDB

Phi of consecutive numbers are power of 2

Source: India Postal Set 1 P1 2016

1/18/2017

Let

n

be an odd positive integer such that

\varphi (n)

and

\varphi (n+1)

are both powers of

2

(here

\varphi(n)

denotes Euler’s totient function). Prove that

n+1

is a power of

2

or

n = 5

.

number theory

Angle BDC is 24 degrees

Source: India Postal Set 2 P1 2016

1/18/2017

Let

ABCD

be a convex quadrilateral in which \angle BAC = 48^{\circ}, \angle CAD = 66^{\circ}, \angle CBD = \angle DBA.Prove that

\angle BDC = 24^{\circ}

.

geometryangleAngle Chasing

Operation on polynomials

Source: India Postal Set 3 P1 2016

1/18/2017

If the polynomials

f(x)

and

g(x)

are written on a blackboard then we can also write down the polynomials

f(x)\pm g(x), f(x)g(x), f(g(x))

and

cf(x)

, where

c

is an arbitrary real constant. The polynomials

x^3 - 3x^2 + 5

and

x^2 - 4x

are written on the blackboard. Can we write a nonzero polynomial of the form

x^n - 1

after a finite number of steps? Justify your answer.

polynomialalgebracombinatoricsblackboard

Partition of positive reals

Source: India Postal Set 4 P1 2016

1/18/2017

The set of all positive real numbers is partitioned into three mutually disjoint non-empty subsets:

\mathbb R^+ = A \cup B\cup C

and

A \cap B = B \cap C = C \cap A = \emptyset

whereas none of

A, B, C

is empty. [*] Show that one can choose

a \in A, b \in B

and

c \in C

such that

a,b, c

are the sides of a triangle. [*] Is it always possible to choose three numbers from three different sets

A,B,C

such that these three numbers are the sides of a right-angled triangle?

set theoryinequalitiesalgebra

Angles of ABC in decagon

Source: India Postal Set 6 P1 2016

1/18/2017

Let

A_1A_2A_3\cdots A_{10}

be a regular decagon and

A=A_1A_4\cap A_2A_5, B=A_1A_6\cap A_2A_7, C=A_1A_9\cap A_2A_{10}.

Find the angles of the triangle

ABC

.

geometry

Sum and product equal to 6

Source: India Postal Set 5 P 1 2016

1/18/2017

Show that there are infinitely many rational triples

(a, b, c)

such that

a + b + c = abc = 6.

number theoryrational number

1

Problems(6)