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BdMO Online Forum • View topic - BdMO National Olympiad 2013: Problemsets

BdMO National Olympiad 2013: Problemsets

For discussing Olympiad Level Number Theory problems
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BdMO National Olympiad 2013: Problemsets

Post Number:#1  Unread postby BdMO » Fri Jan 10, 2014 1:56 am

Bangladesh National Mathematical Olympiad 2013: Primary

Problem $1$:
A group of $7$ women takes $7$ days to make $7$ Nokshikatha. How many days will a group of $5$ women take for making $5$ Nokshikatha?
http://www.matholympiad.org.bd/forum/vi ... =13&t=2905

Problem $2$:
Nazia's mobile phone has a strange problem. Each time she sends an SMS, it is also sent to all the existing numbers of her call list. The actual recipient of the SMS is then added to her call list.. At some point, Nazia deleted her call list. The next SMS she sent was the $16^{th}$ SMS sent from her mobile phone. How many numbers were there in her call list before she deleted her call list?
http://www.matholympiad.org.bd/forum/vi ... =13&t=2906

Problem $3$:
A cube-shaped room has six walls (floor, roof and east, west, north, south walls). A grasshopper is sitting at the south-west corner of the floor. The grasshopper needs to go to the north-east corner of the roof by jumping upward, northward or eastward and in each jump it goes one-third of the room's length. If the grasshopper gets $5$ points for each upward jump, $3$ points for each eastward jump and $1$ point for each northward jump, then what is the difference of the maximum and the minimum number of points it can have when it reaches its destination?
http://www.matholympiad.org.bd/forum/vi ... =13&t=2907

Problem $4$:
The English alphabets are arranged in $3$ rows in a Keyboard. Now somebody presses one key in the first row in such a way that there are same number of keys on both sides of that key in that row. Now a second person presses a key in the second row in the same way and a third person also does the same in the third row. Show that it is impossible.
http://www.matholympiad.org.bd/forum/vi ... =13&t=2908

Problem $5$:
For any two numbers $x$ and $y$, the absolute value of $x$ and $y$ is defined as $|x-y| = $ difference between the numbers $x$ and $y$. For example, $|5-2| = 3, |3-9| = 6$. Let $a_1, a_2, a_3, ... ... ..., a_n$ be a sequence of numbers such that each term in the sequence is larger than the previous term.
Let $S = |a_1 - a_2| + |a_2 - a_3|+ … + |a_{n-1} - a_n|$. What is the minimum number of numbers that you need to know from the sequence in order to find $S$?
http://www.matholympiad.org.bd/forum/vi ... =13&t=2909

Problem $6$:
A polygon is an area bounded by three or more edges in a plane. For example a triangle has three edges, a quadrilateral has four edges, a pentagon has five edges and in this way they are given names according to their number of edges. A regular polygon is a polygon that has edges of equal lengths and all of it's angles are equal as well.
Your father is going to set tiles on your room’s floor. He asked your choice of tiles(polygon). But there are some conditions.
(a) You can choose only one type of regular polygon.
(b) When tiles will be set, there must be no gap among them.
(c) No overlapping is allowed.
(d) You should not be concerned about your room size, rather the above three conditions.
Which type of the regular polygon can you use to draw such a figure that satisfies the above conditions? Write down the logic of not using other regular polygons rather than the one/s you chose. Here is a sample picture for you.
Primary 6.JPG
Primary 6.JPG (12.93 KiB) Viewed 28533 times

http://www.matholympiad.org.bd/forum/vi ... =13&t=2910

Problem $7$:
Arefin, Farhad, Himu, Mahdi, Rachi, Sadia and Tusher are seven friends who live in Gulshan by the side of same linear road. Distances of others home from Tusher home are given below. Tusher’s home is at the starting point of that road. They want to meet at the same place on that road every evening for gossiping. Find a place on that road so that the sum of distances of that place from everyone’s home is minimum. Write down your logic and distance of that place from Tusher’s home.
Primary 7.JPG
Primary 7.JPG (37.41 KiB) Viewed 28533 times

http://www.matholympiad.org.bd/forum/vi ... =13&t=2911

Problem $8$:
There are some boys and girls in a class. Every boy is friends with exactly three girls, and every girl is friends with exactly three boys. If there are $13$ boys in the class, how many girls are there? (Assume that friendship is mutual, i.e. if $A$ is friend of $B$ then $B$ is also friend of $A$.)
http://www.matholympiad.org.bd/forum/vi ... =13&t=2912
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2013 national primary_complete.pdf
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Re: BdMO National Olympiad 2013: Problemsets

Post Number:#2  Unread postby BdMO » Fri Jan 10, 2014 1:58 am

Bangladesh National Mathematical Olympiad 2013: Junior

Problem $1$:
Nazia’s mobile phone has a strange problem. Each time she sends an SMS, it is also sent to all the existing numbers of her call list. The actual recipient of the SMS is then added to her call list.. At some point, Nazia deleted her call list. The next SMS she sent was the $16^{th}$ SMS sent from her mobile phone. How many numbers were there in her call list before she deleted her call list?
viewtopic.php?f=13&t=2906

Problem $2$:
Two isosceles triangles are possible with area of 120 square unit and length of edges integers. One of these two triangles has sides of lengths $17,17$ and $16$. Determine the length of edges of second one.
[Hint: In $\triangle ABC$ if $AB=AC$ and $AD$ is perpendicular to $BC$ then $BD=CD$.]
viewtopic.php?f=13&t=2913

Problem $3$:
$ABC$ is a triangle where $AN$ is perpendicular to $CB$ and $BM$ perpendicular to $AC$. The length of $BC$ is $10$, that of $AC$ is $12$ and that of $AN$ is $6$. Find the length of $BM$.
viewtopic.php?f=13&t=2914

Problem $4$:
Let $a$ be an integer divisible by $2$ but not divisible by $4$. What is the largest positive integer $n$ such that $2^n$ divides $a^{2012} + a^{2013} + ........ + a^{3012}$?
viewtopic.php?f=13&t=2915

Problem $5$:
There are some boys and girls in a class. Every boy is friends with exactly three girls, and every girl is friends with exactly three boys. If there are $13$ boys in the class, how many girls are there? (Assume that friendship is mutual, i.e. if $A$ is friend of $B$ then $B$ is also friend of $A$.)
viewtopic.php?f=13&t=2912

Problem $6$:
You have $1, 2, 3, 4, 5, 6, 7, 8, 9$ kg weights in your home. You have only one piece of each of the weights. You also have a balance. If you put a weight on the left side of that balance, its weight becomes twice. Every time you choose three weights to put on the balance in such a manner that the balance remains in equilibrium. On every turn, the total weight the balance can carry gets reduced by $3$ kg. First time the balance may carry at most $20$ kg in total. In how many ways can you keep weights on balance?
viewtopic.php?f=13&t=2916

Problem $7$:
A positive integer is called “Fantabulous” if there is another fantabulous positive integer smaller than it. Find the number of fantabulous integers.
viewtopic.php?f=13&t=2917

Problem $8$:
$ABCD$ is a rectangle where $AB= \sqrt{2}$ and $BC= \sqrt6$ . $P$ and $Q$ are two points on $AC$ such that $AP=CQ$. From $P$ and $Q$, two perpendicular $PR$ and $QS$ are drawn on $BC$ and $AD$ respectively. If $PRQS$ is a rhombus and $PR=CQ$, then find the length of $PQ$.
viewtopic.php?f=13&t=2918

Problem $9$:
The ratio of GCD and LCM of two integers is $1: 36$ and sum of the integers is $5460$. What is the difference between these two integers?
viewtopic.php?f=13&t=2919

Problem $10$:
There is a point $O$ inside $\triangle ABC$. Join $A,O; B,O$ and $C,O$ and extend those lines. They will intersect $BC, AC$ and $AB$ at points $D, E$ and $F$ respectively. $AF:FB = 4:3$ and area of $\triangle BOF$ and $\triangle BOD$ is $60$ and $70$ square units respectively. Find the triangle with the largest area among $\triangle AOF, \triangle AOE, \triangle COE$ and $\triangle COD$ and write down the area of it.
viewtopic.php?f=13&t=2920
Attachments
2013 national junior_complete.pdf
(215.1 KiB) Downloaded 590 times
BdMO
 
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Re: BdMO National Olympiad 2013: Problemsets

Post Number:#3  Unread postby BdMO » Fri Jan 10, 2014 2:00 am

Bangladesh National Mathematical Olympiad 2013 : Secondary

Problem 1:
If $f: \mathbb R \mapsto \mathbb R$ is a function such that $f(x)=-f(-x)=f(x+1)$ for all real $x$, then what is the value of $f(2013)$?
viewtopic.php?f=13&t=2921

Problem 2:
A polygon is called degenerate if one of its vertices falls on a line that joins its neighboring two vertices. In a pentagon $ABCDE$, $AB=AE$, $BC=DE$, $P$ and $Q$ are midpoints of $AE$ and $AB$ respectively. $PQ||CD$, $BD$ is perpendicular to both $AB$ and $DE$. Prove that $ABCDE$ is a degenerate pentagon.
viewtopic.php?f=13&t=2922

Problem 3:
Let $ABCDEF$ be a regular hexagon with $AB=7$. $M$ is the midpoint of $DE$. $AC$ and $BF$ intersect at $P$, $AC$ and $BM$ intersect at $Q$, $AM$ and $BF$ intersect at $R$. Find the value of $[APB]+[BQC]+[ARF]-[PQMR]$. Here $[X]$ denotes the area of polygon $X$.
viewtopic.php?f=13&t=2923

Problem 4:
$ABCD$ is a quadrilateral where $\angle B=\angle D=90^{\circ}$. $E$ and $F$ are two points on $BD$ such that $AE$ is perpendicular to $BD$ and $CF||AE$. Prove that, $DE=BF$.
viewtopic.php?f=13&t=2924

Problem 5:
$ABCD$ is a paddy field of trapezoidal shape. Growth of paddy has been uniform everywhere in the field. Farmers are cutting the paddy and piling it in the nearest edge ($AB$, $BC$, $CD$ or $DA$). What is the portion of the total paddy that is piled up in the side $CD$? It is given that, $\angle DAB=\angle ABC=120^{\circ}$, $\angle BCD=\angle CDA=60^{\circ}$, $AB=BC=50$ units.
viewtopic.php?f=13&t=2925

Problem 6:
There are some boys and girls in a class. Every boy knows exactly $r$ girls, and every girl knows exactly $r$ boys. Show that there are an equal number of boys and girls in the class. (Assume that knowing is mutual, i.e. if $A$ knows $B$ then $B$ knows $A$.)
viewtopic.php?f=13&t=2926

Problem 7:
$ABCD$ is s quadrilateral. $AB||CD$. $P$ is a point on $AB$ and $Q$ is a point on $CD$. A line parallel to $AB$ intersects $AD$, $BC$, $CP$, $DP$, $AQ$, $BQ$ at points $M, N, X, Y, R, S$ respectively. Prove that $MX+NY=RS$.
viewtopic.php?f=13&t=2927

Problem 8:
There are $n$ cities in the country. Between any two cities there is at most one road. Suppose that the total number of roads is $n$. Prove that there is a city such that starting from there it is possible to come back to it without ever travelling the same road twice.
viewtopic.php?f=13&t=2928

Problem 9:
If there exists a prime number $p$ such that $p+2q$ is prime for all positive integer $q$ smaller than $p$, then $p$ is called an "awesome prime". Find the largest "awesome prime" and prove that it is indeed the largest such prime.
viewtopic.php?f=13&t=2929

Problem 10:
Six points $A$, $B$, $C$, $D$, $E$, $F$ are chosen on a circle anticlockwise. None of $AB$, $CD$, $EF$ is a diameter. Extended $AB$ and $DC$ meet at $Z$, $CD$ and $FE$ at $X$, $EF$ and $BA$ at $Y$. $AC$ and $BF$ meets at $P$, $CE$ and $BD$ at $Q$ and $AE$ and $DF$ at $R$. If $O$ is the point of intersection of $YQ$ and $ZR$, find the angle $XOP$.
viewtopic.php?f=13&t=2930
Attachments
2013 national secondary_complete.pdf
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Re: BdMO National Olympiad 2013: Problemsets

Post Number:#4  Unread postby BdMO » Fri Jan 10, 2014 2:02 am

Bangladesh National Mathematical Olympiad 2013 : Higher Secondary

Problem 1:
A polygon is called degenerate if one of its vertices falls on a line that joins its neighboring two vertices. In a pentagon $ABCDE$, $AB=AE$, $BC=DE$, $P$ and $Q$ are midpoints of $AE$ and $AB$ respectively. $PQ||CD$, $BD$ is perpendicular to both $AB$ and $DE$. Prove that $ABCDE$ is a degenerate pentagon.
viewtopic.php?f=13&t=2922

Problem 2:
Let $g$ be a function from the set of ordered pairs of real numbers to the same set such that $g(x, y)=-g(y, x)$ for all real numbers $x$ and $y$. Find a real number $r$ such that $g(x, x)=r$ for all real numbers $x$.
viewtopic.php?f=13&t=2931

Problem 3:
$ABCDEF$ be a regular hexagon with $AB=7$. $M$ is the midpoint of $DE$. $AC$ and $BF$ intersect at $P$, $AC$ and $BM$ intersect at $Q$, $AM$ and $BF$ intersect at $R$. Find the value of $[APB]+[BQC]+[ARF]-[PQMR]$. Here $[X]$ denotes the area of a polygon $X$.
viewtopic.php?f=13&t=2923

Problem 4:
If the fraction $\frac{a}{b}$ is greater than $\frac{31}{17}$ in the least amount while $b<17$, find $\frac{a}{b}$.
viewtopic.php?f=13&t=2932

Problem 5:
Let $x>1$ be an integer such that for any two positive integers $a$ and $b$, if $x$ divides $ab$ then $x$ either divides $a$ or divides $b$. Find with proof the number of positive integers that divide $x$.
viewtopic.php?f=13&t=2933

Problem 6:
There are $n$ cities in the country. Between any two cities there is at most one road connecting them. Suppose that the total number of roads is $n$. Prove that there is a city such that starting from there it is possible to come back to it without ever travelling the same road twice.
viewtopic.php?f=13&t=2928

Problem 7:
If there exists a prime number $p$ such that $p+2q$ is prime for all positive integer $q$ smaller than $p$, then $p$ is called an "awesome prime". Find the largest "awesome prime" and prove that it is indeed the largest such prime.
viewtopic.php?f=13&t=2929

Problem 8:
$ABC$ is an acute angled triangle. Perpendiculars drawn from its vertices on the opposite sides are $AD$, $BE$ and $CF$. The line parallel to $DF$ through $E$ meets $BC$ at $Y$ and $BA$ at $X$. $DF$ and $CA$ meet at $Z$. Circumcircle of $XYZ$ meets $AC$ at $S$. Given, $\angle B=33^{\circ}$ find the angle $\angle FSD$ with proof.
viewtopic.php?f=13&t=2934

Problem 9:

Six points $A$, $B$, $C$, $D$, $E$, $F$ are chosen on a circle anticlockwise. None of $AB$, $CD$, $EF$ is a diameter. Extended $AB$ and $DC$ meet at $Z$, $CD$ and $FE$ at $X$, $EF$ and $BA$ at $Y$. $AC$ and $BF$ meets at $P$, $CE$ and $BD$ at $Q$ and $AE$ and $DF$ at $R$. If $O$ is the point of intersection of $YQ$ and $ZR$. Find the angle $XOP$.
viewtopic.php?f=13&t=2930

Problem 10:
$X$ is a set of $n$ elements. $P_m(X)$ is the set of all $m$ element subsets (i.e. subsets that contain exactly $m$ elements) of $X$. Suppose $P_m(X)$ has $k$ elements. Prove that the elements of $P_m(X)$ can be ordered in a sequence $A_1, A_2,...A_i,...A_k$ such that it satisfies the two conditions: (A) each element of $P_m(X)$ occurs exactly once in the sequence, (B) for any $i$ such that $0<i<k$, the size of the set $A_i \cap A_{i+1}$ is $m-1$.
viewtopic.php?f=13&t=2935
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2013 national higher secondary_complete.pdf
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Re: BdMO National Olympiad 2013: Problemsets

Post Number:#5  Unread postby *Mahi* » Fri Jan 10, 2014 2:09 am

This topic is mainly for showcasing the problems. I am locking this post to avoid any cluttering here. Please use individual topics on each problem for discussion (the link is just below the problem description)

Special thanks to bristy1588, Labib and kfoozminus for helping with LaTeXing the problemset.

Good luck for BdMO 2014!
Please read Forum Guide and Rules before you post.

Use $L^AT_EX$, It makes our work a lot easier!

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