MathDB

1

Inter solution:$m^2+n^2=m^3$

BdMO National Higher Secondary Problem 3
Let

N

be the number if pairs of integers

(m,n)

that satisfies the equation

m^2+n^2=m^3

Is

N

finite or infinite?If

N

is finite,what is its value?

1

Paradox:Speaking lie;Telling truth!!!

BdMO National 2015 Secondary Problem 1.
A crime is committed during the hartal.There are four witnesses.The witnesses are logicians and make the following statement: Witness One said exactly one of the four witnesses is a liar.
Witness Two said exactly two of the four witnesses is a liar.
Witness Three said exactly three of the four witnesses is a liar.
Witness Four said exactly four of the four witnesses is a liar.
Assume that each of the statements is either true or false.How many of the winesses are liars?

3

1

Polinomial equation and modular arithmatic

Let

n

be a positive integer.Consider the polynomial

p(x)=x^2+x+1

. What is the remainder of

x^3

when divided by

x^2+x+1

.For what positive integers values of

n

is

x^{2n}+x^n+1

divisible by

p(x)

?
Post no:

100

6

1

Weird Trapezium problem

Trapezoid

ABCD

has sides

AB=92,BC=50,CD=19,AD=70

AB

is parallel to

CD

A circle with center

P

on

AB

is drawn tangent to

BC

and

AD

.Given that

AP=\dfrac mn

(Where

m,n

are relatively prime).What is

m+n

?

5

1

Bangladesh Mathematical Olympiad 2015, Secondary, P5

A tetrahedron is a polyhedron composed of four triangular faces. Faces

ABC

and

BCD

of a tetrahedron

ABCD

meet at an angle of

\pi/6

. The area of triangle

\triangle ABC

is

120

. The area of triangle

\triangle BCD

is

80

, and

BC = 10

. What is the volume of the tetrahedron? We call the volume of a tetrahedron as one-third the area of it's base times it's height.

7

1

Bangladesh Mathematical Olympiad 2015, Secondary, P7

In triangle

\triangle ABC

, the points

A', B', C'

are on sides

BC, AC, AB

respectively. Also,

AA', BB', CC'

intersect at the point

O

(they are concurrent at

O

). Also,

\frac {AO}{OA'}+\frac {BO}{OB'}+\frac {CO}{OC'} = 92

. Find the value of

\frac {AO}{OA'}\times \frac {BO}{OB'}\times \frac {CO}{OC'}

.

4

1

Bangladesh Mathematical Olympiad 2015, Secondary, P4

There are

36

participants at a BdMO event. Some of the participants shook hands with each other. But no two participants shook hands with each other more than once. Each participant recorded the number of handshakes they made. It was found that no two participants with the same number of handshakes made, had shaken hands with each other. Find the maximum possible number of handshakes at the party with proof. (When two participants shake hands with each other, this will be counted as one handshake.)

2015 Bangladesh Mathematical Olympiad

Subcontests

Inter solution:$m^2+n^2=m^3$

Paradox:Speaking lie;Telling truth!!!

Polinomial equation and modular arithmatic

Weird Trapezium problem

Bangladesh Mathematical Olympiad 2015, Secondary, P5

Bangladesh Mathematical Olympiad 2015, Secondary, P7

Bangladesh Mathematical Olympiad 2015, Secondary, P4