## BdMO 2011 National Olympiad: Problemsets

Moon
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### BdMO 2011 National Olympiad: Problemsets

Problem 1:
Find the value of $1+\frac{1}{1+\dfrac{1}{1+1}}$
viewtopic.php?f=13&t=668

Problem 2:
$2, 11$ and $2011$ are prime numbers. What will be the GCD of $2\times 2011$ and $11 \times 2011$?
viewtopic.php?f=13&t=669

Problem 3:
There is a basket in front of you that has $100$ balls numbered from $1$ to $100$ (each ball has a different number). You are asked to pick balls from this basket randomly without looking. After picking how many balls can you be sure that you have chosen at least one even-numbered ball?
viewtopic.php?f=13&t=670

Problem 4:
To form a triangle, the sum of lengths of any two sides has to be greater than the length of the other. Suppose that we want to form a triangle with sides $a, 31$ and $a + 1$ where a is an integer (or whole number) greater than $1$. Find the minimum value of $a$.
viewtopic.php?f=13&t=671

Problem 5:
We say that a number has a 'square root' if we find another integer which gives us the first number if multiplied by itself. For example, $4$ has a square root since , but $5$ has no square root. Out of the $100$ numbers from $1$ to $100$, how many integers have square roots?
viewtopic.php?f=13&t=672

Problem 6:
In the diagram, the same square has been represented in two ways. If $a = 5$ and $b = 12$ in this diagram, find the value of $c$. (the white rectangle in the second diagram is a square)
viewtopic.php?f=13&t=673

Problem 7:
You can very easily check that sum of three consecutive integers is three times the second number. Similarly, sum of five consecutive numbers is five times the third number, sum of seven consecutive numbers is seven times the fourth number and so on. What is the smallest even number that can be expressed as sum of three consecutive integers, sum of five consecutive integers and also sum of eleven consecutive integers?
viewtopic.php?f=13&t=674

Problem 8:
If $3$ and $4$ individually divides a number, then also divides that number. For example, all of $3, 4$ and $12$ divide $48$. But if a number is divisible by $3$ and $6$ individually it may or may not be divisible by $3 \times 6=18$ . For example, both of $54$ and $60$ are divisible by $3$ and $6$ individually, though only $54$ is divisible by $18$. Can you explain why this happens?
viewtopic.php?f=13&t=675

Problem 9:
In the first round of a chess tournament, each player plays against every other player exactly once. A player gets $3, 1$ or $-1$ points respectively for winning, drawing or losing a match. After the end of the first round, it is found that the sum of the scores of all the players is $20$. How many players were there in the tournament?
viewtopic.php?f=13&t=676

Problem 10:
$A, B, C, D, E, F$ are six children of different ages in the range of 11 to 16. It is known that C and F always speak truth whereas among the rest two are truthful and the other two lie. When they are asked about their ages, they replied as follows-

A: The sum of the ages of the other five is an even number.
B: A is the eldest.
C: The sum of the ages of the other five is divisible by 5.
D: E is elder than A by two years.
E: The sum of the ages of A, B, D, E is an odd number.
F: The sum of the ages of the other five is divisible by 5.
Find out: (i) What is the sum of the ages of C and F? (ii) Which two among these six children lie? (iii) What are the ages of A and E?
viewtopic.php?f=13&t=677
Attachments BdMO 2011_National_Primary.pdf
"Inspiration is needed in geometry, just as much as in poetry." -- Aleksandr Pushkin

learn how to write equations, and don't forget to read Forum Guide and Rules.

Moon
Posts: 751
Joined: Tue Nov 02, 2010 7:52 pm
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### Re: BdMO 2011 National Olympiad: Problemsets

Problem 1:
We say that a number has a 'square root' if we find another integer which gives us the first number if multiplied by itself. For example, $4$ has a square root since $2 \times 2=4$, but $5$ has no square root. Out of the $100$ numbers from $1$ to $100$, how many integers have square roots?
viewtopic.php?f=13&t=672

Problem 2:
$\triangle ABC$ is a scalene triangle. $D$ is the midpoint of $BC$. $\triangle ADC$ is equilateral and $\triangle ADB$ is isosceles. Find the angles of $\triangle ABC$.
viewtopic.php?f=13&t=680

Problem 3:
$A, B, C, D, E, F$ are six children of different ages in the range of 11 to 16. It is known that C and F always speak truth whereas among the rest two are truthful and the other two lie. When they are asked about their ages, they replied as follows-

A: The sum of the ages of the other five is an even number.
B: A is the eldest.
C: The sum of the ages of the other five is divisible by 5.
D: E is elder than A by two years.
E: The sum of the ages of A, B, D, E is an odd number.
F: The sum of the ages of the other five is divisible by 5.
Find out: (i) What is the sum of the ages of C and F? (ii) Which two among these six children lie? (iii) What are the ages of A and E?
viewtopic.php?f=13&t=677

Problem 4:
Consider the numbers $1, 2, 3, \cdots 160$. What is the maximum number of numbers you can choose from this list so that no two numbers differ by $4$? Show the logic behind your answer.
viewtopic.php?f=13&t=681

Problem 5:
Let, $A=211$ and $B=106^{211}$, which one is larger? Show logic.
($n!$ denotes the product of all the integers from $1$ to $n$. That means $n! =1\times 2 \times 3 \times 4 \times \cdots \times n$. For example $5!=1\times 2 \times 3 \times 4 \times 5 =120$.)
viewtopic.php?f=13&t=682

Problem 6:
$E$ is the midpoint of side $BC$ of rectangle $ABCD$. $A$ point $X$ is chosen on $BE$. $DX$ meets extended $AB$ at $P$. Find the position of $X$ so that the sum of the areas of $\triangle BPX$ and $\triangle DXC$ is maximum with proof.
viewtopic.php?f=13&t=683

Problem 7:
Consider a sequence of four integers so that the GCD of first three is the same as the LCM of the last three. How many such sequences exist so that the sum of the numbers is $2011$?
viewtopic.php?f=13&t=684

Problem 8:
(a) Prove that in a circle, the line connecting the center and the midpoint of a chord intersects the chord at right angles.
(b) Suppose the points $A, B, C, D$ and $E$ are chosen counter-clockwise on a circle. $X$ is a point inside the circle, and $AX=CX=EX$. Prove that $BX=DX$.
viewtopic.php?f=13&t=685

Problem 9:
$p$ is a prime and sum of the numbers from $1$ to $p$ is divisible by all primes less or equal to $p$. Find the value of $p$ with proof.
viewtopic.php?f=13&t=686

Problem 10:
After counting the number of new MOVers, Munir Hasan came with a peculiar plan of getting introduced to everyone. He declared that everyone will stand in a circle and starting from Munir Hasan pairs will be formed towards right. After each member has introduced himself to the one in his pair, members in the pair will interchange their positions and a similar grouping will be done. The process will be repeated until someone has been paired with the same person for the second time. Then the group will be split up, Munir Hasan will form a subgroup with members not introduced with each other and the entire process will be repeated. Pairing will always start from Munir Hasan. It was seen that Munir Hasan needed to split his group $10$ times. What was the number of new MOVers?
viewtopic.php?f=13&t=687
Attachments BdMO 2011_National_Junior.pdf
"Inspiration is needed in geometry, just as much as in poetry." -- Aleksandr Pushkin

learn how to write equations, and don't forget to read Forum Guide and Rules.

Moon
Posts: 751
Joined: Tue Nov 02, 2010 7:52 pm
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### Re: BdMO 2011 National Olympiad: Problemsets

Problem 1:
There are $2011$ mathematicians in a party. It is known that, Mahbub, the host of the party (who is also a mathematician) knows all other mathematicians. Two mutually unacquainted mathematicians will become friend of each other eventually after the party if they have a common friend/acquaintance (who will introduce them to each other of course). After the end of the party how many pairs of mathematicians will be left who are not yet introduced to each other?
viewtopic.php?f=13&t=688

Problem 2:
In the first round of a chess tournament, each player plays against every other player exactly once. A player gets $3, 0 or -1$ points respectively for winning, drawing or losing a match. After the end of the first round, is it possible that the sum of the scores of all the players is $21$? State your answer with logic.
viewtopic.php?f=13&t=690

Problem 3:
In triangle $ABC$, the incircle touches $AB, BC$ and $CA$ at $D, E$ and $F$. Show that if $\triangle ABC$ and $\triangle DEF$ are similar then $\triangle ABC$ is equilateral.
viewtopic.php?f=13&t=689

Problem 4:
$E$ is the midpoint of side $BC$ of rectangle $ABCD$. $A$ point $X$ is chosen on $BE$. $DX$ meets extended $AB$ at $P$. Find the position of $X$ so that the sum of the areas of $\triangle BPX$ and $\triangle DXC$ is maximum with proof.
viewtopic.php?f=13&t=683

Problem 5:
In a scalene triangle $ABC$ with $\angle A = 90^{\circ}$, the tangent line at $A$ to its circumcircle meets line $BC$ at $M$ and the incircle touches $AC$ at $S$ and $AB$ at $R$. The lines $RS$ and $BC$ intersect at $N$ while the lines $AM$ and $SR$ intersect at $U$. Prove that the triangle $UMN$ is isosceles.
viewtopic.php?f=13&t=692

Problem 6:
$p$ is a prime and sum of the numbers from $1$ to $p$ is divisible by all primes less or equal to $p$. Find the value of $p$ with proof.
viewtopic.php?f=13&t=693

Problem 7:
Consider a group of $n > 1$ people. Any two people of this group are related by mutual friendship or mutual enmity. Any friend of a friend and any enemy of an enemy is a friend. If $A$ and $B$ are friends/enemies then we count it as $1$ friendship/enmity. It is observed that the number of friendships and number of enmities are equal in the group. Find all possible values of $n$.
viewtopic.php?f=13&t=694

Problem 8:
Bhaskaracharya has set up a strange study group. Any member of that group has exactly one immediate teacher (the teacher who teaches him) except for Bhaskaracharya himself, although teacher of teacher is also respected as a teacher. As the chancellor of the group, Bhaskaracharya is not taught by anybody. No two members of that group can be teachers of each other. The study group operates in a pairs where each pair consists of one member and his immediate teacher. If such a pairing is possible, is it unique? Justify your answer.
viewtopic.php?f=13&t=695

Problem 9:
Prove that $\sqrt{\sqrt{3}+\sqrt{2}}+\sqrt{\sqrt{3}-\sqrt{2}}$
is irrational.
viewtopic.php?f=13&t=696

Problem 10:
The repeat of a natural number is obtained by writing it twice in a row (for example, the repeat of $123$ is $123123$). Find a positive integer (if any) whose repeat is a perfect square.
viewtopic.php?f=13&t=697
Attachments BdMO 2011_National_Secondary.pdf
"Inspiration is needed in geometry, just as much as in poetry." -- Aleksandr Pushkin

learn how to write equations, and don't forget to read Forum Guide and Rules.

Moon
Posts: 751
Joined: Tue Nov 02, 2010 7:52 pm
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### Re: BdMO 2011 National Olympiad: Problemsets

Problem 1:
Prove that for any non-negative integer $n$ the numbers $1, 2, 3, ..., 4n$ can be divided in tow mutually exclusive classes with equal number of members so that the sum of numbers of each class is equal.
viewtopic.php?f=13&t=709

Problem 2:
In the first round of a chess tournament, each player plays against every other player exactly once. A player gets $3, 1$ or $-1$ points respectively for winning, drawing or losing a match. After the end of the first round, it is found that the sum of the scores of all the players is $90$. How many players were there in the tournament?
viewtopic.php?f=13&t=708

Problem 3:
$E$ is the midpoint of side $BC$ of rectangle $ABCD$. $A$ point $X$ is chosen on $BE$. $DX$ meets extended $AB$ at $P$. Find the position of $X$ so that the sum of the areas of $\triangle BPX$ and $\triangle DXC$ is maximum with proof.
viewtopic.php?f=13&t=683

Problem 4:
Which one is larger 2011! or, $(1006)^{2011}$? Justify your answer.
viewtopic.php?f=13&t=707

Problem 5:
In a scalene triangle $ABC$ with $\angle A = 90^{\circ}$, the tangent line at $A$ to its circumcircle meets line $BC$ at $M$ and the incircle touches $AC$ at $S$ and $AB$ at $R$. The lines $RS$ and $BC$ intersect at $N$ while the lines $AM$ and $SR$ intersect at $U$. Prove that the triangle $UMN$ is isosceles.
viewtopic.php?f=13&t=706

Problem 6:
$p$ is a prime and sum of the numbers from $1$ to $p$ is divisible by all primes less or equal to $p$. Find the value of $p$ with proof.
viewtopic.php?f=13&t=693

Problem 7:
Consider a group of $n > 1$ people. Any two people of this group are related by mutual friendship or mutual enmity. Any friend of a friend and any enemy of an enemy is a friend. If $A$ and $B$ are friends/enemies then we count it as $1$ friendship/enmity. It is observed that the number of friendships and number of enmities are equal in the group. Find all possible values of $n$.
viewtopic.php?f=13&t=694

Problem 8:
$ABC$ is a right angled triangle with $\angle A = 90^{\circ}$ and $D$ be the midpoint of $BC$. A point $F$ is chosen on $AB$. $CA$ and $DF$ meet at $G$ and $GB \parallel AD$. $CF$ and $AD$ meet at $O$ and $AF = FO$. $GO$ meets $BC$ at $R$. Find the sides of $ABC$ if the area of $GDR$ is $\dfrac{2}{\sqrt{15}}$
viewtopic.php?f=13&t=704

Problem 9:
The repeat of a natural number is obtained by writing it twice in a row (for example, the repeat of $123$ is $123123$). Find a positive integer (if any) whose repeat is a perfect square.
viewtopic.php?f=13&t=703

Problem 10:
Consider a square grid with $n$ rows and $n$ columns, where $n$ is odd (similar to a chessboard). Among the $n^2$ squares of the grid, $p$ are black and the others are white. The number of black squares is maximized while their arrangement is such that horizontally, vertically or diagonally neighboring black squares are separated by at least one white square between them. Show that there are infinitely many triplets of integers $(p, q, n)$ so that the number of white squares is $q^2$.
viewtopic.php?f=13&t=702
Attachments BdMO 2011_National_Higher_Secondary.pdf
"Inspiration is needed in geometry, just as much as in poetry." -- Aleksandr Pushkin

learn how to write equations, and don't forget to read Forum Guide and Rules.

FahimFerdous
Posts: 176
Joined: Thu Dec 09, 2010 12:50 am

### Re: BdMO 2011 National Olympiad: Problemsets

Many many thanks Moon vaia. the arrivals
Posts: 41
Joined: Tue Dec 21, 2010 10:17 pm

### Re: BdMO 2011 National Olympiad: Problemsets

i have solved one its number 4.
i have proved (1006)^2011 is greater.(just counting power)
number 1 is rather easy.i did it by induct on n.but i am not sure wheather it is correct.
however
n=1 the statement can be varified as follow 1+4=2+3
suppose for some m we can fulfil the statement for 4m numbers.
now for m+1
the numbers are 1,2,3.....4m,4m+1,4m+2,4m+3,4m+4
but 4m+1+4m+4=4m+2+4m+3
so prove the result
but number 8 is quite perplexing.however i have proved this ratios.
DR=(a/6) and 2DR=RC
c=(12/(root 15)]*(1/b)
a=[root(159/15)]b
but still cant determine the length. women of purity are for men of purity and hence men of purity are for women of purity - THE HOLY QURAN

FahimFerdous
Posts: 176
Joined: Thu Dec 09, 2010 12:50 am

### Re: BdMO 2011 National Olympiad: Problemsets

I've solved 8 problems frm Secondary and 7 problems frm H.Secondary. There are 4 more problems I can't solve as the 10th frm secondary and the 9th frm h.sec are the same one. I don't dare to touch the 10th frm h.sec as it's combinatorics and I'm awful at this. Now Moon vaia, I'm not actually sure If I understood the 8th problem of secondary. I will write in my next post what I understood and please anyone tell me if I'm right or wrong.

Moon
Posts: 751
Joined: Tue Nov 02, 2010 7:52 pm
All the reports have been noticed, and the problem set is correct now. Thanks for helping me correct all the problems. 