Math Bustle

Divisional MathOlympiad's Questions

Bangladesh National MathOlympiad Questions

Higher secondary 2011

Higher secondary 2010

Problem 1:
Let S=11+22+33++20102010 . What is the remainder when S is divided by 2?
viewtopic.php?f=13&t=620

Problem 2:
Isosceles triangle ABC is right angled at B and AB=3. A circle of unit radius is drawn with its centre on any of the vertices of this triangle. Find the maximum value of the area of that part of the triangle that is not shared by the circle.
viewtopic.php?f=13&t=622

Problem 3:
A series is formed in the following manner:
A(1)=1
A(n)=f(m) numbers of f(m) followed by f(m) numbers of 0.
m is the number of digits in A(n−1).
Find A(30). Here f(m) is the remainder when m is divided by 9.
viewtopic.php?f=13&t=621

Problem 4:
Given a point P inside a circle , two perpendicular chords through P divide into distinct regions a b c d  clockwise such that a contains the centre of .
Prove that

[a]+[c][b]+[d]

Where [x] = area of x.
viewtopic.php?f=13&t=623

Problem 5:
How many regular polygons can be constructed from the vertices of a regular polygon with 2010 sides? (Assume that the vertices of the 2010-gon are indistinguishable)
viewtopic.php?f=13&t=625

Problem 6:
a and b are two positive integers both less than 2010; a=b . Find the number of ordered pairs (ab)  such that a2+b2 is divisible by 5. Find a+b so that a2+b2 is maximum.
viewtopic.php?f=13&t=624

Problem 7:
Let ABC be a triangle with ACAB: Let P be the intersection point of the perpendicular bisector of BC and the internal angle bisector of CAB: Let X and Y be the feet of the perpendiculars from P to lines AB and AC respectively. Let Z be the intersection point of lines and XYBC: Determine the value of BZ/ZC

viewtopic.php?f=13&t=626

Problem 8:
Find all prime numbers p and integers a and b (not necessarily positive) such that pa+pb is the square of a rational number.
viewtopic.php?f=13&t=628

Problem 9:
Find the number of odd coefficients in expansion of (x+y)2010.
viewtopic.php?f=13&t=627

Problem 10:
a1a2akan is a sequence of distinct positive real numbers such that a1a2ak and akak+1an. A Grasshopper is to jump along the real axis, starting at the point O and making n jumps to the right of lengths a1a2an respectively. Prove that, once he reaches the rightmost point, he can come back to point O by making n jumps to the left of lengths a1a2an in some order such that he never lands on a point which he already visited while jumping to the right. (The only exceptions are point O and the rightmost point)

 Higher secondary 2009:

 Higher Secondary 2008:

 Higher Secondary 2007:

Math Club Questions. Likewise BdMO

A1.
Let ABC be a triangle.The length of AB=x, BC=y, CA=z. Hence, D & E are the midpoint of AB and AC respectively.Let ABC turn around BC axis. Express the volume of the 3d object which was made by ADE triangle with x, y, z.

A2.
Let ABCD be a quadrangle. The length of AB=a, BC=b, CD=c. Express the largest & least area of ABCD with a,b,c.

B1.
Factor it:x^6+4x^5+5x^4+4x^3+x^2.

B2.
Let ABC be a triangle. D,E,F are the midpoints on AB,BC & CA respectively. G & H are points on internally AD & AF such that DF=EG=EH. Prove that D,E,F,H,G are concyclic.

B3.
A football team has 3,8,8 & 6 goalkeepers,midfielders & strikers respectively. There are three ways to choose play style 4-3-3, 4-4-2, 5-4-1. How many ways are there to choose 11 players for the coach?