BdMO 2012 National: Problem Sets

Discussion on Bangladesh Mathematical Olympiad (BdMO) National
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BdMO 2012 National: Problem Sets

Unread post by Moon » Sat Feb 11, 2012 10:34 pm

Bangladesh National Mathematical Olympiad 2012: Primary

Problem 1:
Find a three digit number so that when its digits are arranged in reverse order and added with the original number, the result is a three digit number with all of its digits being equal. In case of two digit numbers, here is an example: $23+32=55 $
viewtopic.php?f=13&t=1731

Problem 2:
Subrata writes a letter to Ruponti every day (in successive intervals of $24$ hours) from Korea. But Ruponti receives the letters in intervals of $25$ hours. What is the number of the letter Ruponti receives on the $25^{th}$ day?
viewtopic.php?f=13&t=1709

Problem 3:
Prove that, the difference between two prime numbers larger than $2$ can’t be a prime number other than $2$.
viewtopic.php?f=13&t=1723

Problem 4:
Write a number in a paper and hold the paper upside down. If what you get is exactly same as the number before rotation then that number is called beautiful. Example: $986$ is a beautiful number. Find out the largest $5$ digit beautiful number.
viewtopic.php?f=13&t=1732

Problem 5:
If a number is multiplied with itself thrice, the resultant is called its cube. For example: $3 × 3 × 3 = 27$, hence $27$ is the cube of $3$. If $1,\ 170$ and $387$ are added with a positive integer, cubes of three consecutive integers are obtained. What are those three consecutive integers?
viewtopic.php?f=13&t=1733

Problem 6:
Consider the given diagram. There are three rectangles shown here. Their lengths are $3,\ 4$ and $5$ units respectively, widths respectively $2,\ 3$ and $4$ units. Each small grid represents a square $1$ unit long and $1$ unit wide. Use these diagrams to find out the sum of the consecutive numbers from $1$ to $500$. (If you use some direct formula for doing so, you must provide its proof)
Primary 6.jpg
Figure for Problem 6
Primary 6.jpg (15.53 KiB) Viewed 26781 times
viewtopic.php?f=13&t=1734

Problem 7:
When Tanvir climbed the Tajingdong mountain, on his way to the top he saw it was raining $11$ times. At Tajindong, on a rainy day, it rains either in the morning or in the afternoon; but it never rains twice in the same day. On his way, Tanvir spent $16$ mornings and $13$ afternoons without rain. How many days did it take for Tanvir to climb the Tajindong mountain in total?
viewtopic.php?f=13&t=1719

Problem 8:
A magic box takes two numbers. If we can obtain the first numbers by multiplying the second number with itself several times then a green light on the box turns on. Otherwise, a red light turns on. For example, if you enter $16$ and $2$ then the green light turns on because $2×2×2×2 = 16$. But if you enter $18$ and $9$ then the red light turns on. If the two numbers are equal then the green light turns on. If the first number is $256$ then for how many different second numbers will the green light turn on?
viewtopic.php?f=13&t=1724

Problem 9:
Each room of the Magic Castle has exactly one door. The rooms are designed such that when you can go from one room to the next one through a door, the second room's length is equal to the first room's width, and the second room's width is half of the first room's width (see the figure). Each door can be used only once. Magic Prince has entered the castle and now needs to get out. To get out of each room, the prince needs time equal to the width of the room. The prince has to use each door to get out of the castle. Due to the blessings of a Sufi, the prince can become as small as he wants (so that he can go into even very small rooms). If the castle is a square of side length $20$ meters then how long will it take for the prince to get out?
Primary 9.jpg
Figure for Problem 9
Primary 9.jpg (15.18 KiB) Viewed 26781 times
viewtopic.php?f=13&t=1727

Problem 10:
Tusher chose some consecutive numbers starting from $1$. He noticed that the least common multiple of those numbers is divisible by $100$. What is the minimum number of numbers he chose?
viewtopic.php?f=13&t=1735
Attachments
2012 national primary_complete.pdf
Primary National 2012
(166.32 KiB) Downloaded 980 times
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Re: BdMO 2012 National: Problem Sets

Unread post by Moon » Sat Feb 11, 2012 10:37 pm

Bangladesh National Mathematical Olympiad 2012: Junior

Problem 1:
Subrata writes a letter to Ruponti every day (in successive intervals of $24$ hours) from Korea. But Ruponti receives the letters in intervals of $25$ hours. What is the number of the letter Ruponti receives on the $25^{th}$ day?
viewtopic.php?f=13&t=1709

Problem 2:
Prove that, the difference between two prime numbers larger than $2$ can’t be a prime number other than $2$.
viewtopic.php?f=13&t=1723

Problem 3:
When Tanvir climbed the Tajingdong mountain, on his way to the top he saw it was raining $11$ times. At Tajindong, on a rainy day, it rains either in the morning or in the afternoon; but it never rains twice in the same day. On his way, Tanvir spent $16$ mornings and $13$ afternoons without rain. How many days did it take for Tanvir to climb the Tajindong mountain in total?
viewtopic.php?f=13&t=1719

Problem 4:
A magic box takes two numbers. If we can obtain the first number by multiplying the second number with itself several times then a green light on the box turns on. Otherwise, a red light turns on. For example, if you enter $16$ and $2$ then the green light turns on because $2×2× 2×2 = 16$. But if you enter $18$ and $9$ then the red light turns on. If the two numbers are equal then the green light turns on. If the first number is $256$ then for how many different second numbers will the green light turn on?
viewtopic.php?f=13&t=1724

Problem 5:
$ABC$ is a right triangle with hypotenuse $AC$. $D$ is the midpoint of $AC$. $E$ is a point on the extension of $BD$. The perpendicular drawn on $BC$ from $E$ intersects $AC$ at $F$ and $BC$ at $G.$ (a) Prove that, if $DEF$ is an equilateral triangle then $\angle ACB = 30^0$. (b) Prove that, if $\angle ACB = 30^0$ then $DEF$ is an equilateral triangle.
viewtopic.php?f=13&t=1725

Problem 6:
In triangle $ABC$, $AB=7,\ AC=3,\ BC=9$. Draw a circle with radius $AC$ and center $A$. What is the distance from $B$ to the point on the circle that is furthest from $B$?
viewtopic.php?f=13&t=1726

Problem 7:
Each room of the Magic Castle has exactly one door. The rooms are designed such that when you can go from one room to the next one through a door, the second room's length is equal to the first room's width, and the second room's width is half of the first room's width (see the figure). Each door can be used only once. Magic Prince has entered the castle and now needs to get out. To get out of each room, the prince needs time equal to the width of the room. The prince has to use each door to get out of the castle. Due to the blessings of a Sufi, the prince can become as small as he wants (so that he can go into even very small rooms). If the castle is a square of side length $20$ meters then how long will it take for the prince to get out?
Junior 7.jpg
Figure for Problem 7
Junior 7.jpg (15.51 KiB) Viewed 26819 times
viewtopic.php?f=13&t=1727

Problem 8:
Find the total number of the triangles whose all the sides are integer and the longest side is of $100$ in length. If the similar clause is applied for the isosceles triangle then what will be the total number of triangles?
viewtopic.php?f=13&t=1720

Problem 9:
Given triangle $ABC$, the square $PQRS$ is drawn such that $P,\ Q$ are on $BC,\ R$ is on $CA$ and $S$ is on $AB$. Radius of the triangle that passes through $A,\ B,\ C$ is $R$. If $AB = c,\ BC = a,\ CA = b,$ Show that $\frac{AS}{SB}=\frac{bc}{2aR}$
viewtopic.php?f=13&t=1728

Problem 10:
The $n$-th term of a sequence is the least common multiple (l.c.m.) of the integers from $1$ to $n$. Which term of the sequence is the first one that is divisible by $100$?
viewtopic.php?f=13&t=1730
Attachments
2012 national junior_complete.pdf
National 2012: Junior
(160.6 KiB) Downloaded 754 times
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Re: BdMO 2012 National: Problem Sets

Unread post by Moon » Sat Feb 11, 2012 10:39 pm

Bangladesh National Mathematical Olympiad 2012: Secondary

Problem 1:
Subrata writes a letter to Ruponti every day (in successive intervals of $24$ hours) from Korea. But Ruponti receives the letters in intervals of $25$ hours. What is the number of the letter Ruponti receives on the $25^{th}$ day?
viewtopic.php?f=13&t=1709

Problem 2:
When Tanvir climbed the Tajingdong mountain, on his way to the top he saw it was raining $11$ times. At Tajindong, on a rainy day, it rains either in the morning or in the afternoon; but it never rains twice in the same day. On his way, Tanvir spent $16$ mornings and $13$ afternoons without rain. How many days did it take for Tanvir to climb the Tajindong mountain in total?
viewtopic.php?f=13&t=1719

Problem 3:
In a given pentagon $ABCDE$, triangles $ABC, BCD, CDE, DEA$ and $EAB$ all have the same area. The lines $AC$ and $AD$ intersect $BE$ at points $M$ and $N$. Prove that $BM = EN$.
viewtopic.php?f=13&t=1711

Problem 4:
Find the total number of the triangles whose all the sides are integer and longest side is of $100$ in length. If the similar clause is applied for the isosceles triangle then what will be the total number of triangles?
viewtopic.php?f=13&t=1720

Problem 5:
In triangle $ABC$, medians $AD$ and $CF$ intersect at point $G$. $P$ is an arbitrary point on $AC$. $PQ$ & $PR$ are parallel to $AD$ & $CF$ respectively. $PQ$ intersects $BC$ at $Q$ and $PR$ intersects $AB$ at $R$. If $QR$ intersects $AD$ at $M$ & $CF$ at $N$, then prove that area of triangle $GMN$ is $\frac{(A)}{8}$ where $(A)$ = area enclosed by $PQ, PR, AD, CF$.
viewtopic.php?f=13&t=1712

Problem 6:
Show that for any prime $p$, there are either infinitely many or no positive integer $a$, so that $6p$ divides $a^p + 1$. Find all those primes for which there exists no solution.
viewtopic.php?f=13&t=1714

Problem 7:
In an acute angled triangle $ABC$, $\angle A= 60^0$. Prove that the bisector of one of the angles formed by the altitudes drawn from $B$ and $C$ passes through the center of the circumcircle of the triangle $ABC$.
viewtopic.php?f=13&t=1715

Problem 8:
The vertices of a right triangle $ABC$ inscribed in a circle divide the circumference into three arcs. The right angle is at $A$, so that the opposite arc $BC$ is a semicircle while arc $AB$ and arc $AC$ are supplementary. To each of the three arcs, we draw a tangent such that its point of tangency is the midpoint of that portion of the tangent intercepted by the extended lines $AB$ and $AC$. More precisely, the point $D$ on arc $BC$ is the midpoint of the segment joining the points $D'$ and $D''$ where the tangent at $D$ intersects the extended lines $AB$ and $AC$. Similarly for $E$ on arc $AC$ and $F$ on arc $AB$. Prove that triangle $DEF$ is equilateral.
viewtopic.php?f=13&t=1721

Problem 9:
Consider a $n×n$ grid of points. Prove that no matter how we choose $2n-1$ points from these, there will always be a right triangle with vertices among these $2n-1$ points.
viewtopic.php?f=13&t=1722

Problem 10:
A triomino is an $L$-shaped pattern made from three unit squares. A $2^k \times 2^k$ chessboard has one of its squares missing. Show that the remaining board can be covered with triominoes.
viewtopic.php?f=13&t=1717
Attachments
2012 national secondary_complete.pdf
National 2012: Secondary
(148.74 KiB) Downloaded 838 times
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Re: BdMO 2012 National: Problem Sets

Unread post by Moon » Sat Feb 11, 2012 10:43 pm

Bangladesh National Mathematical Olympiad 2012: Higher Secondary

Problem 1:
Subrata writes a letter to Ruponti every day (in successive intervals of $24$ hours) from Korea. But Ruponti receives the letters in intervals of $25$ hours. What is the number of the letter Ruponti receives on the $25^{th}$ day?
viewtopic.php?f=13&t=1709

Problem 2:
Superman is taking part in a hurdle race with $12$ hurdles. At any stage he can jump across any number of hurdles lying ahead. For example, he can cross all $12$ hurdles in one jump or he can cross $7$ hurdles in the first jump, $1$ in the later and the rest in the third jump. In how many different ways can superman complete the race?
viewtopic.php?f=13&t=1710

Problem 3:
In a given pentagon $ABCDE$, triangles $ABC, BCD, CDE, DEA$ and $EAB$ all have the same area. The lines $AC$ and $AD$ intersect $BE$ at points $M$ and $N$. Prove that $BM = EN$.
viewtopic.php?f=13&t=1711

Problem 4:
Consider the pattern অআআইইইঈঈঈঈউউউউউ... When the part with $11$ ‘ঔ’s end, the pattern continues with $12$ ‘অ’s, $13$ ‘আ’s and so on. What is the $2012^{th}$ letter in this pattern?
viewtopic.php?f=13&t=1713

Problem 5:
In triangle $ABC$, medians $AD$ and $CF$ intersect at point $G$. $P$ is an arbitrary point on $AC$. $PQ$ & $PR$ are parallel to $AD$ & $CF$ respectively. $PQ$ intersects $BC$ at $Q$ and $PR$ intersects $AB$ at $R$. If $QR$ intersects $AD$ at $M$ & $CF$ at $N$, then prove that area of triangle $GMN$ is $\frac{(A)}{8}$ where $(A)$ = area enclosed by $PQ, PR, AD, CF$.
viewtopic.php?f=13&t=1712

Problem 6:
Show that for any prime $p$, there are either infinitely many or no positive integer $a$, so that $6p$ divides $a^p + 1$. Find all those primes for which there exists no solution.
viewtopic.php?f=13&t=1714

Problem 7:
In an acute angled triangle $ABC$, $\angle A= 60^0$. Prove that the bisector of one of the angles formed by the altitudes drawn from $B$ and $C$ passes through the center of the circumcircle of the triangle $ABC$.
viewtopic.php?f=13&t=1715

Problem 8:
A decision making problem will be resolved by tossing $2n + 1$ coins. If Head comes in majority one option will be taken, for majority of tails it’ll be the other one. Initially all the coins were fair. A witty mathematician replaced $n$ pairs of fair coins with $n$ pairs of biased coins, but in each pair the probability of obtaining head in one is the same the probability of obtaining tail in the other. Will this cause any favor for any of the options available? Justify with logic.
viewtopic.php?f=13&t=1716

Problem 9:
A triomino is an $L$-shaped pattern made from three unit squares. A $2^k \times 2^k$ chessboard has one of its squares missing. Show that the remaining board can be covered with triominoes.
viewtopic.php?f=13&t=1717

Problem 10:
Consider a function $f: \mathbb{N}_0\to \mathbb{N}_0$ following the relations:
  • $f(0)=0$
  • $f(np)=f(n)$
  • $f(n)=n+f\left ( \left \lfloor \dfrac{n}{p} \right \rfloor \right)$ when $n$ is not divisible by $p$
Here $p > 1$ is a positive integer, $\mathbb{N}_0$ is the set of all nonnegative integers and $\lfloor x \rfloor$ is the largest integer smaller or equal to $x$.
Let, $a_k$ be the maximum value of $f (n)$ for $0\leq n \leq p^k$. Find $a_k$.
viewtopic.php?f=13&t=1718
Attachments
2012 national higher secondary_complete.pdf
National Higher Secondary 2012
(159.93 KiB) Downloaded 861 times
Last edited by Zzzz on Sun Feb 12, 2012 8:37 am, edited 1 time in total.
Reason: link of problem 5
"Inspiration is needed in geometry, just as much as in poetry." -- Aleksandr Pushkin

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Re: BdMO 2012 National: Problem Sets

Unread post by Moon » Sun Feb 12, 2012 10:18 pm

Thanks to Zzzz for LaTeXing and posting Primary and Junior problems. :)
"Inspiration is needed in geometry, just as much as in poetry." -- Aleksandr Pushkin

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