BdMO National Junior :Problem Collection

[samiul_samin]

Bangladesh National Mathematical Olympiad 2007 :Junior

Problem 1 :
If $\dfrac {10}{2}=4$,then $5\times 2=?$
[The base of a number may not be $10$ ]
viewtopic.php?f=19&t=453

Problem 2 :
In this figure there is a square inscribed in another square.Using this figure derive the Pythagorean Theorem.
viewtopic.php?f=13&t=5624

Problem 3:
A man has $4$ childern.The age of the first child is a square number.By multiplying the digits of this square number you will get the age of second child and by summing up the digits you will get the age of third child.If you add the digits of the age of the second child,you will get the age of fourth child.If the difference of age of two consequetive childern is not more than $25$ years,then find the ages if $4$ children.
viewtopic.php?f=13&t=5650

Problem 4:
If $log_{(x+3)}(x^2+15)=2$ then $x=?$
viewtopic.php?f=13&t=5628

Problem 5:
Find the difference between the non shaded area ($ABDF$ & $FCE$) of thre triangles.Instead of triangles,if there were circles or any other shape,what would be the result -comment on that.
viewtopic.php?f=13&t=5648

Screenshot_2019-02-24-10-46-54-1.png

Problem 6:
Mathematics,English and Bangla classes started on the very first day of a month.Mathematics class schedule is $1,3,5,7,9,...$ the schedule for English is $1,4,7,10,13,...$.and for Bangla it is $1,5,9,13,17...$ .In next $3$ months how many time you have to attend all three of these classes at same day?Suppose all the months are of $30$ days.
viewtopic.php?f=13&t=5651

Problem 7:
A ball is thrown upward vertically to a height of $650$ meters from ground.Each $2$ times it hits the ground it bounces $\dfrac 25$ of the height it fell in the previous stage.How much the ball will travel before it stops?
viewtopic.php?f=13&t=5617


Problem 8:
Draw a square which has area that is three times the area of another given square.
viewtopic.php?f=13&t=5626


Problem 9:
$\sqrt {-1}$ is called the imaginary number '$i$'.Using this can you find the value of $\dfrac {1+i}{1-i}$?
viewtopic.php?f=13&t=5625

Problem 10:
Find the sum of first $20$ terms of the series $4+7+13+25+...$
What is the sum of first $n$ terms?
viewtopic.php?f=13&t=5627

Problem 11:

If $a,b,c$ are the sides of a triangle such that $a^2+b^2+c^2=ab+bc+ca$. Prove that the triangle is equilateral.
viewtopic.php?f=19&t=2590

Problem 12:
Two circles of equal radius intersect each other at point $C$ and $D$.The centers of the two circles are poont $A$ and $B$ respectively.If their radius is $10$ and area of $ABC$ is $40$.
Then find the distance $x$ between $A$ and $B$.

Screenshot_2019-02-24-11-03-51-1.png

viewtopic.php?f=13&t=5649

Problem 13:
A drawer in a darkened room contains $100$ black socks, $80$ blue socks, $60$ red socks and $40$ purple socks. A youngster selects socks one at a time from the drawer but is unable to see the color of the socks drawn. What is the smallest number of socks that must be selected to guarantee that the selection contains at least $10$ pairs? [A pair of socks = $2$ socks of the same color.]
viewtopic.php?f=13&t=5610

[samiul_samin]

Bangladesh National Mathematical Olympiad 2008 :Junior

Problem 1:
You are given unlimited supply of $1×1,2×2,3×3,4×4,6×6$ square.Find a set of $10$ squares whose areas add to $48$.
viewtopic.php?f=13&t=5632

Problem 2:
Faria sets of her bike to Nazia's house.At exactly the same time,Nazia sets off to Faria's house along the same straight road in her car.A while later,they pass each other (neither spotting the other) and shortly after ,Nazia arrives at Faria's house and find that she is not there.Nazia waits for $22$ minutes and then heads back along the same road,arriving at her own place at exactly the same time as Faria.Faria traveled at the same speed the whole time whereas Nazia traveled $4$ times as fast as Faria on the way of Faria's house and $5$ times as fast on the way back .How many time did it take Faria to reach Nazia's house?
viewtopic.php?f=13&t=5652


Problem 3:
Find the $10^{th}$ term of the following sequence.What is the $n^{th}$ term?
\[3,8,17,32,57,...\]
viewtopic.php?f=13&t=5630

Problem 4:
$p$ is a prime number and given that $p>3$.What be the remainder if $p^2$ divided by $12$?
viewtopic.php?f=13&t=5629

Problem 5:
If $a,b,c$ are the sides of a triangle such that $a^2+b^2+c^2=ab+bc+ca$. Prove that the triangle is equilateral.
viewtopic.php?f=19&t=2590

Problem 6:
The diagonals of a rectangle exceeds the length by $2$ $cm$.If the width of the triangle is $10$ $cm$,find the length.
viewtopic.php?f=13&t=5644


Problem 7:
A good number is the sum of a two digit number,with distinct digits,and its reverse.For example $37+73=110$ is good.How many good numbers are perfect squares?
viewtopic.php?f=13&t=5631

Problem 8:
Find four prime numbers less than 100 which are factor of \[
3^{32}-2^{32}\]
viewtopic.php?f=19&t=1420


Problem 9:
In the convex quadrlateral $ABCD$,points $M,N$ lie on side $AB$ such that $AM=MN=NB$ and points $P,Q$ lie on $CD$ such that $CP=PQ=QD$.Prove that $AMPC=\dfrac 13 $Area of $ABCD$.
viewtopic.php?f=13&t=5645

Problem 10:
In the triangle $ABC$,points $D,E,F$ are on sides $AB,BC,CA$ respectively with $AD=DB,CE=3BE,AF=2CF$.If the area of triangle $ABC$ is $480$ $cm^2$,then find the area of the triangle $DEF$.
viewtopic.php?f=13&t=5646

[samiul_samin]

Bangladesh National Mathematical Olympiad 2009:Junior

Problem 1:
Some students are visiting a stable.At a certain time,it was counted that there are $71$ heads and $228$ legs were there.How many students were there at the time of counting?
viewtopic.php?f=13&t=5655

Problem 2:
In how many ways it is possible to form two -digit prime number with the digits $2,3,5,8 ?$
Write all of them.
viewtopic.php?f=13&t=5656

Problem 3:
The nine squares of a $3\times 3$ checkerboard must be painted so that each row,each column and each of the two digonals have no two squares of the same colour.What is the least number of colours needed?
viewtopic.php?f=13&t=5623

Problem 4:
\[x^2-8xy+9y^2-16y+10\]Find the least possible value of the expression.\[(x,y)\in R\]
viewtopic.php?f=19&t=1579

Problem 5:
In a $N\times N$ grid,Abir picks three lattice points as vertexes of a triangle.Surprisingly,he always chooses the $(0,0)$ point.What is the largest area of the triangle Abir can draw.
viewtopic.php?f=13&t=2988

Problem 6:
A square is divided into three pieces of equal area as shown.The distance between the parallel lines is $1$ $cm$.What is the area of the square?

Screenshot_2019-02-24-10-55-35-1.png

viewtopic.php?f=13&t=5647

Problem 7:
Express $\dfrac {7}{26}$ as $\dfrac {1}{a}+\dfrac {1}{b}$
($a,b$ both are positive integers).
viewtopic.php?f=13&t=5642

Problem 8:
There are three points in plane.One can draw as many parallelograms as possible keeping those three points as the three vertices of the parallelogram.Find the difference between the area of parallelogram having the maximum perimeter possible and the parallelogram having the minmum perimeter possible.
viewtopic.php?f=13&t=5643

Problem 9:
One angle of a triangle is twise of another angle of the same triangle. An angle of this triangle is $120 ^{\circ}$. The bisector of the second largest triangle intersects its opposite side at point $D$. The distance of $D$ from the vertex containing the largest angle is $10$ $cm$. If the length of the largest side of this triangle is $2x$, then a relation like the following is true:
\[x^4-C_3x^3-C_2x^2-C_1x+1875=0\]
Find the value of $C_1,C_2$ and $C_3$ analytically.
viewtopic.php?f=13&t=577

Problem 10:
In a strange language there are only two letters,a and b,and it is postulated that the letter a is a word.Furthermore all additional words are formed according to the following rules:
$(i)$ Given any word ,a new word can be formed from it by adding one b at the right hand end.
$(ii)$ If in any word a sequence aaa appears ,a new word can be formed by replacing aaa by the letter b.
$(iii)$ If in any word a sequence bbb appears ,a new word can be formed by omitting bbb.
$(iv)$ Given any word,a new word can be formed by writing down the sequence that constitutes the given word twice.
For example $(iv)$, aa is a word and by $(iv)$ again aaaa is a word,Hence by $(ii)$ ba is a word,and by $(i)$, bab is also a word.Again by $(i)$ babb is a word,and so by $(iv)$ babbbabb is also a word.Finally ,by $(iii)$ we fond that baabb is a word.
Prove that in this language baabaabaa is not a word.
viewtopic.php?f=13&t=5653

[samiul_samin]

Bangladesh National Mathematical Olympiad 2010:Junior

Problem 1:
The three digit number $1*3$ is divided by $11$.Find with proof the missing digit(represented by the asterik).
viewtopic.php?f=13&t=5633

Problem 2:
A rectangle and a square have the same area,find with proof,which one has a greater perimeter?
viewtopic.php?f=13&t=5639

Problem 3:
One day Tom was playing with numbers.He wrote $11$ fractions using all natural numbers from $1$ to $22$ exactly once-either as numerator or as denomator.How many of these fractions ,at most,are integers?
viewtopic.php?f=13&t=5635

Problem 4:
Find the smallest number,divisible by $13$ ,such that the remainder is $1$ when divided by $4,6$ or $9$.
viewtopic.php?f=13&t=5636

Problem 5:
Find all pairs of positive integers $(m,n)$ which satisfy $m^{3}+1331=n^{3}$
viewtopic.php?f=13&t=2983

Problem 6:
Nonte has been given all the numbers from $1$ to $40$ & Fonte has been given all the odd numbers from $1$ to $100$.They have to take any two given numbers so that the summation of those $2$ numbers is divisible by $3$.He,who will make the numbers divisible by $3$ in most ways,will be announced winner.Who will be the winner and why?
viewtopic.php?f=13&t=5638

Problem 7:
There are $25$ points on a plane,no three of which lie on a line.Find the minimum number of lines needed to separate them from one another?
viewtopic.php?f=13&t=5637

Problem 8:
Trapezium is any quadrilateral two opposite sides of which are parallel and another two are not.The diagonal of isosceles trapezium divides it into two isosceles triangle.Find the angles of the trapezium.
viewtopic.php?f=13&t=5640

Problem 9:
Tom and Jerry have $14$ tiles in total.Of them $8$ are colured blue and $6$ are colured red.They want to arrange them in a straight line such that between any two red tiles there at least one blue tile.How many possible ways are there of arranging them in line?
viewtopic.php?f=13&t=5634

Problem 10:
$ABCD$ is a quadrilateral.$P,Q$ and $R$ are the midpoints of $AB,BC$ and $CD$ respectively.If $PQ=3,QR=4$ and $PR=5$;find the area of $ABCD$.
viewtopic.php?f=13&t=5641

[samiul_samin]

This topic is for showcasing the problems. Please use individual topics on each problem for discussion (the link is just below the problem description).

[otis]

Your solution is quite complicated so I can't fully understand the steps you take to solve this problem.
slope