## BdMO National Olympiad 2015:Problemsets

Discussion on Bangladesh Mathematical Olympiad (BdMO) National
samiul_samin
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Joined: Sat Dec 09, 2017 1:32 pm

### BdMO National Olympiad 2015:Problemsets

Primary

Problem 1
Write down all the prime numbers in the range of $1$ to $50$.

Problem 2
Four people $A, B, C$ and $D$ have an average monthly income of $10000$ taka. First three of them have an average monthly income of $12000$ taka. Average income of first two of them is $15000$ taka. Find the monthly income of $B, C$ and $D$ if $A$ has a monthly income of $20000$ taka?

Problem 3
In the following figures a rectangular piece of paper $ABCD$ has been folded several times. First, the side $CD$ was made to fall on the line $AD$. Point $E$ in figure $(ii)$ represents the point $C$ after folding. In the next figure the portion $BF$ was made to fall on $EF$. Lastly, the side $AG$ was made to fall on $GH$. Find the lengths of $GJ, IJ, IE, ED, EH$ and $HF$. It is given that$AB = 8$ and $BC = 15$. Screenshot_2019-02-21-12-07-53-1.png (23.48 KiB) Viewed 835 times

Problem 4
A circus party has the same number of lions as tigers. You asked to the owner of the circus the number of lions and tigers. He gave you the following information:
(i) An elephant is enough to feed all the tigers and lions in the circus.
(ii) Eighteen deers produce the same amount of meat as an elephant does.
(iii) A lion eats twice as much as a tiger.
(iv) One buffalo is enough to feed a lion and a tiger.
(v) A tiger will eat exactly the same amount of meat a deer has.
Find the number of tigers and lions in that circus party.

Problem 5 .
Surjo is four years old and he is learning to write numbers. His math notebook looks like a square grid with $20$ rows and $20$ columns. He usually writes the numbers from top to bottom and when one column is finished he starts writing along the next column. One day he starts writing the numbers from left to right (along the rows). How many of the numbers will be placed in exactly the same place where it would have appeared if he had written along the columns?

Problem 6
In the following figure $BKLGNM$, $CMNHPO$ and $DOPIRQ$ are regular hexagons (all six sides of each hexagon are equal and so are the angles). $BKLGNM$ has an area of $24$ square units. What is the area of the rectangle $AFJE$? Screenshot_2019-02-21-12-07-49-1.png (12.88 KiB) Viewed 835 times
Last edited by samiul_samin on Thu Feb 21, 2019 11:06 pm, edited 2 times in total.

samiul_samin
Posts: 1004
Joined: Sat Dec 09, 2017 1:32 pm

### Re: BdMO National Olympiad 2015:Problemsets

Junior

Problem 1.
A small country has a very simple language. People there have only two letters and all their words have exactly seven letters. Calculate the number maximum of words people use in that country?

Problem 2
In the following figures, the larger circles are identical and so are the smaller ones. In $(i)$ the circles have a common center and the lines $AD$ and $BC$ divide both the circles in four equal halves. The larger circle has an area of $100$ square meters. Find the area of the shaded region in figure$(ii)$. Screenshot_2019-02-21-12-08-05-1.png (15.15 KiB) Viewed 833 times
Problem 3.
A circus party has the same number of lions as tigers. You asked to the owner of the circus the number of lions and tigers. He gave you the following information:
(i) An elephant is enough to feed all the tigers and lions in the circus.
(ii) Eighteen deers produce the same amount of meat as an elephant does.
(iii) A lion eats twice as much as a tiger.
(iv) One buffalo is enough to feed a lion and a tiger.
(v) A tiger will eat exactly the same amount of meat a deer has.
Find the number of tigers and lions in that circus party.

Problem 4
In the following figure $BKLGNM$, $CMNHPO$ and $DOPIRQ$ are regular hexagons (all six sides of each hexagon are equal and so are the angles). $BKLGNM$ has an area of $24$ square units. What is the area of the rectangle $AFJE$? Screenshot_2019-02-21-12-07-49-1.png (12.88 KiB) Viewed 833 times

Problem 5

In a party, boys shake hands with girls but each girl shake hands with everyone else .If there are $40$ handshakes , find out the number of boys and girls in the party ?

Problem 6.
$ABCD$ is a parallelogram where $\angle{ACB}=80^{\circ}$ $\angle{ACD}=20^{\circ}$ ,$P$
is a point on $AC$ such that,$\angle{ABP}=20^{\circ}$ and $Q$ is a point on $AB$ such that $\angle{ACQ}=30^{\circ}$ .Find the magnitude of the angle determined by the lines $CD$ and $PQ$.
Last edited by samiul_samin on Thu Feb 21, 2019 11:07 pm, edited 1 time in total.

samiul_samin
Posts: 1004
Joined: Sat Dec 09, 2017 1:32 pm

### Re: BdMO National Olympiad 2015:Problemsets

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?

Problem 2

How many pairs of integers ($m,n$) satisfy the equation $m+n=mn$ ?

Problem 3
Given a positive integer $n$,let $p(n)$ be the product of the non- zero digits of $n$. If $n$ has only one digit then $p(n)=n$.
Let $S=p(1)+p(2)+p(3)+\ldots+p(999)$.What is the largest prime factor of $S$?

Problem 4

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 participants recorder 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 number of handshakes at the party with PROOF .(when two participants shook hands with each other ,this will be counted as one handshake )

Problem 5
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.

Problem 6
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 $m+n$?

Problem 7
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 $\frac {AO}{OA'}\times \frac {BO}{OB'}\times \frac {CO}{OC'}$.
Last edited by samiul_samin on Thu Feb 21, 2019 11:07 pm, edited 1 time in total.

samiul_samin
Posts: 1004
Joined: Sat Dec 09, 2017 1:32 pm

### Re: BdMO National Olympiad 2015:Problemsets

Higher 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?

Problem 2
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?

Problem 3

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 $p(x)$?

Problem 4

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 participants recorder 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 number of handshakes at the party with PROOF .(when two participants shook hands with each other ,this will be counted as one handshake )

Problem 5
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.

Problem 6
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 $m+n$?

Problem 7
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 $\frac {AO}{OA'}\times \frac {BO}{OB'}\times \frac {CO}{OC'}$.
Last edited by samiul_samin on Thu Feb 21, 2019 11:08 pm, edited 1 time in total.

samiul_samin
Posts: 1004
Joined: Sat Dec 09, 2017 1:32 pm

### Re: BdMO National Olympiad 2015:Problemsets

Solution
More discussion
This topic is mainly for showcasing the problems. Please use individual topics on each problem for discussion (the problem numbers are the links).
Happy Problem Solving   samiul_samin
Posts: 1004
Joined: Sat Dec 09, 2017 1:32 pm

### Re: BdMO National Olympiad 2015:Problemsets

samiul_samin wrote:
Thu Feb 21, 2019 1:41 pm
Higher 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?

Problem 2
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?

Problem 3

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 $p(x)$?

Problem 4

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 participants recorder 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 number of handshakes at the party with PROOF .(when two participants shook hands with each other ,this will be counted as one handshake )

Problem 5
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.

Problem 6
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 $m+n$?

Problem 7
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 $\frac {AO}{OA'}\times \frac {BO}{OB'}\times \frac {CO}{OC'}$.
BLUNDER   