BDMO 2018 National Olympiad: Problemsets

Discussion on Bangladesh Mathematical Olympiad (BdMO) National
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nahin munkar
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BDMO 2018 National Olympiad: Problemsets

Unread post by nahin munkar » Tue Jan 08, 2019 4:43 pm

Bangladesh Mathematical Olympiad 2018 : Primary

problem 1
The average age of the $5$ people in a Room is $30$. The average age of the $10$ people in another Room is $24$. If the two groups are combined, what is the average age of all the people?
http://matholympiad.org.bd/forum/viewto ... =13&t=5372

Problem 2
Two-thirds of the people in a room sat in three-fourths of the chairs. The rest of the people remained standing. If there were $6$ empty chairs, how many people were there in the room ?
http://matholympiad.org.bd/forum/viewto ... =13&t=5373

Problem 3
The area of a rectangle is $120$. All the lengths of the sides of this rectangle are integer, what can be the lowest possible perimeter of this rectangle ?
http://matholympiad.org.bd/forum/viewto ... =13&t=5374

Problem 4
In square $ABCE$, $AF=3FE$ & $CD=3ED$. What is the ratio of the area of triangle $\triangle BFD$ and square $ABCE$ ?
Capturep4.PNG
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http://matholympiad.org.bd/forum/viewto ... =13&t=5375

Problem 5
A singles tournament had six players. Each player played every other player only once, with no ties. If Pritom won $3$ games, Monisha won $2$ games, Sadman won $3$ games, Richita won $3$ games and Somlota won $2$ games, how many games did Zubayer win?
http://matholympiad.org.bd/forum/viewto ... =13&t=5376

Problem 6
An even number is called a ‘good’ even number if its first and last digits are also an even number. How many $3$-digit ‘good’ even numbers are there? Try to find their number without writing all such number!
http://matholympiad.org.bd/forum/viewto ... =13&t=5377

Problem 7
abc is a three digit number. abc is appended to abc to create a six digit number abcabc. If abcabc is divisible by abc $\times$ abc, then determine the abc .
http://matholympiad.org.bd/forum/viewto ... =13&t=5378

Problem 8
From $1$ to $6$ nodes, one can go only right side and downward. From $7$ to $11$ nodes, one can go right side or along the diagonal. If you start from $1$, in how many ways can you reach $12$ ?
Capturep8.PNG
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http://matholympiad.org.bd/forum/viewto ... =13&t=5379
# Mathematicians stand on each other's shoulders. ~ Carl Friedrich Gauss

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nahin munkar
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Re: BDMO 2018 National Olympiad: Problemsets

Unread post by nahin munkar » Tue Jan 08, 2019 4:59 pm

Bangladesh Mathematical Olympiad 2018: Junior

Problem 1
The area of a rectangle is $240$. All the lengths of the sides of this rectangle are integer, what can be the lowest possible perimeter of this rectangle?
http://matholympiad.org.bd/forum/viewto ... =13&t=5366

Problem 2
If the average of first $n$ positive integers is $2018$, then find the value of n ?
http://matholympiad.org.bd/forum/viewto ... =13&t=5367

Problem 3
Shoumo, Oindry and Esha take turns counting from $1$ to one more than the last number said by the last person. Shoumo starts by saying “$1$”, so Oindry follows by saying “$1,2$”, Esha follows by saying “$1,2,3$”. Shoumo then says “$1,2,3,4$”, and so on. What is the $50^{th}$ number said ?
(For example the numbers $3,4$ said by Shoumo last time here are ninth and tenth numbers)
http://matholympiad.org.bd/forum/viewto ... =13&t=5368

Problem 4
The squares of three positive numbers add up to $2018$. The biggest of these three numbers is the sum of the smaller two. If the difference between the smaller two numbers is $2$, what is the difference between the cubes of the smaller two numbers?
http://matholympiad.org.bd/forum/viewto ... =13&t=5369

Problem 5
$3x^2 + y^2 = 108$ ;
Determine all the positive integer values of $x$ and $y$.
http://matholympiad.org.bd/forum/viewto ... =13&t=5370

Problem 6
Given $8$ lines on a plane and no two of them are parallel. Prove that, at least two of them form an angle less than 23°.
http://matholympiad.org.bd/forum/viewto ... =13&t=5371

Problem 7
All possible $4$ digit numbers are created using $5,6,7,8$ and then sorted from smallest to largest. In the same manner, all possible $4$ digit numbers are created using $3,4,5,6$ and then sorted from smallest to largest. Then first number of the second type is subtract from first number of the first type, second number of the second type is subtract from second number of the first type and so on. What will be the summation of these difference (subtraction results) ?
http://matholympiad.org.bd/forum/viewto ... =13&t=5380

Problem 8
In triangle $\triangle ABC$, $AB=10$, $CA=12$. The bisector of $\angle B$ intersects $CA$ at $E$, and the bisector of $\angle C$ intersects $AB$ at $D$. $AM$ and $AN$ are the perpendiculars to $CD$ and $BE$ respectively. If $MN=4$, then find $BC$.
junior8.PNG
junior8.PNG (8.6 KiB) Viewed 1068 times
http://matholympiad.org.bd/forum/viewto ... =13&t=5381

Problem 9
Find the number of positive integers that are divisors of at least one of $10^{10}$, $12^{12}$, $15^{15}$.
http://matholympiad.org.bd/forum/viewto ... =13&t=5382

Problem 10
$\triangle ABC$ is an equilateral triangle. $D$ and $E$ is such a point that, $AD:CD=1:2$ and $BE:AE=1:2$. If $O$ is the intersection point of $BD$ and $CE$ find $ \angle AOC$ .
http://matholympiad.org.bd/forum/viewto ... =13&t=4242
# Mathematicians stand on each other's shoulders. ~ Carl Friedrich Gauss

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nahin munkar
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Re: BDMO 2018 National Olympiad: Problemsets

Unread post by nahin munkar » Tue Jan 08, 2019 5:21 pm

Bangladesh Mathematical Olympiad 2018: Secondary

Problem 1
Solve:
$x^2(2-x)^2=1+2(1-x)^2$

Where $x$ is real number.
http://matholympiad.org.bd/forum/viewto ... =13&t=4238

Problem 2
$AB$ is a diameter of a circle and $AD$ & $BC$ are two tangents of that circle.$AC$ & $BD$ intersect on a point of the circle.$AD=a$ & $BC=b$.If $a\neq b$ then $AB=?$
http://matholympiad.org.bd/forum/viewto ... =13&t=4232

Problem 3
ফারিহা $4$ টি fair ছক্কা নিক্ষেপ করল কিন্তু ছক্কাগুলো নাজিয়াকে দেখাল না।ছক্কায় পাওয়া নাম্বার গুলার গুনফল $144$. নাম্বার গুলার যোগফল $S$। যেখানে $14\leq S\leq 18$| নাজিয়া বলল নাম্বার গুলার যোগফল এদের মাঝে এক টির সমান হওয়া সম্ভব না।{$14 ,15, 16, 17, 18$} এই range এর মঝে কোনটি সেই অসম্ভব নাম্বার?
http://matholympiad.org.bd/forum/viewto ... =13&t=4236


Problem 4
After drawing $m$ lines on a plane, Sabbir got exactly $200$ different intersection points on that plane. What is the lowest value of $m$?
http://matholympiad.org.bd/forum/viewto ... =13&t=4229

Problem 5
Four circles are drawn with the sides of quadrilateral $ABCD$ as diameters. The two circles passing through $A$ meet again at $A′$ . The two circles passing through $B$ meet again at $B′$ . The two circles passing through $C$ meet again at $C′$. The two circles passing through D meet again at $D′$. Suppose, $ A′, B′, C′, D′ $ are all distinct. Is the quadrilateral $A′B′C′D′$ similar to $ABCD$ ? Show with proof.
http://matholympiad.org.bd/forum/viewto ... =13&t=5364

Problem 6
Find all integer solution ($m,n$) for the following equation:
$3(m^2+n^2)-7(m+n)=-4$
http://matholympiad.org.bd/forum/viewto ... =13&t=4228

Problem 7
The vertices of a regular nonagon ($9$-sided polygon) are labeled with the digits $1$ through $9$ in such away that the sum of the numbers on every three consecutive vertices is a multiple of 3. Two acceptable arrangements are considered to be indistinguishable if one can be obtained from the other by rotating the nonagon in the plane. Find the number of distinguishable acceptable arrangements.
http://matholympiad.org.bd/forum/viewto ... =13&t=5365

Problem 8
একটা টুর্ণামেন্ট খেলা হচ্ছে $n$ জনের মধ্যে। সবাই প্রত্যেকের সাথে একবার করে খেলে। কোনো খেলায় ড্র হয় না। একটি সংখ্যা $k$ কে $n$-good বলা হবে যদি এমন কোনো টুর্ণামেন্ট থাকে যাতে করে সে টুর্ণামেন্ট এ যেকোনো $k$ জনের জন্য এমন একজন প্লেয়ার থাকে যে সেই $k$ জনের সবাইকে হারিয়েছে।
a) প্রমাণ করতে হবে $n \geq 2^{k+1}-1$
b) এমন সব $n$ বের করতে হবে যেন $2$ একটা $n$-good হয়
http://matholympiad.org.bd/forum/viewto ... =13&t=4237
Attachments
Questions - BdMO 2018 National Secondary And Higher Secondary.pdf
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# Mathematicians stand on each other's shoulders. ~ Carl Friedrich Gauss

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nahin munkar
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Re: BDMO 2018 National Olympiad: Problemsets

Unread post by nahin munkar » Tue Jan 08, 2019 6:00 pm

Bangladesh Mathematical Olympiad 2018: Higher Secondary

Problem 1
Solve:
$x^2(2-x)^2=1+2(1-x)^2$
Where $x$ is real number.

http://matholympiad.org.bd/forum/viewto ... =13&t=4238

Problem 2
$AB$ is a diameter of a circle and $AD$ & $BC$ are two tangents of that circle.$AC$ & $BD$ intersect on a point of the circle.$AD=a$ & $BC=b$.If $a\neq b$ then $AB=?$
http://matholympiad.org.bd/forum/viewto ... =13&t=4232

Problem 3
Urmi rolls four fair six-sided dice. She doesn’t see the results. Her friend Ipshita tells her that the product of the numbers is $144$. Ipshita also says the sum of the dice, $S$ satisfies $14\leq S\leq 18$ . Urmi tells Ipshita that $S$ cannot be one of the numbers in the set {$14,15,16,17,18$} if the product is $144$. Which number in the range {$14,15,16,17,18$} is an impossible value for $S$ ?
http://matholympiad.org.bd/forum/viewto ... =13&t=4236

Problem 4
After drawing $m$ lines on a plane, Sabbir got exactly $200$ different intersection points on that plane. What is the lowest value of $m$?
http://matholympiad.org.bd/forum/viewto ... =13&t=4229

Problem 5
Four circles are drawn with the sides of quadrilateral $ABCD$ as diameters. The two circles passing through $A$ meet again at $A′$ . The two circles passing through $B$ meet again at $B′$ . The two circles passing through $C$ meet again at $C′$. The two circles passing through D meet again at $D′$. Suppose, $ A′, B′, C′, D′ $ are all distinct. Is the quadrilateral $A′B′C′D′$ similar to $ABCD$ ? Show with proof.
http://matholympiad.org.bd/forum/viewto ... =13&t=5364

Problem 6
Find all integer solution ($m,n$) for the following equation:
$3(m^2+n^2)-7(m+n)=-4$
http://matholympiad.org.bd/forum/viewto ... =13&t=4228

Problem 7
Evaluate:
$\int^{\pi/2}_0 \frac{\cos^4x + \sin x \cos^3 x + \sin^2x\cos^2x + \sin^3x\cos x}{\sin^4x + \cos^4x + 2\ sinx\cos^3x + 2\sin^2x\cos^2x + 2\sin^3x\cos x} dx$

http://matholympiad.org.bd/forum/viewto ... =13&t=5384

Problem 8
একটা টুর্ণামেন্ট খেলা হচ্ছে $n$ জনের মধ্যে। সবাই প্রত্যেকের সাথে একবার করে খেলে। কোনো খেলায় ড্র হয় না। একটি সংখ্যা $k$ কে $n$-good বলা হবে যদি এমন কোনো টুর্ণামেন্ট থাকে যাতে করে সে টুর্ণামেন্ট এ যেকোনো $k$ জনের জন্য এমন একজন প্লেয়ার থাকে যে সেই $k$ জনের সবাইকে হারিয়েছে।
a) প্রমাণ করতে হবে $n \geq 2^{k+1}-1$
b) এমন সব $n$ বের করতে হবে যেন $2$ একটা $n$-good হয়
http://matholympiad.org.bd/forum/viewto ... =13&t=4237


N.B. All the problems (only except P7) of higher secondary pset is identical to the secondary problems of BdMO 2018.
# Mathematicians stand on each other's shoulders. ~ Carl Friedrich Gauss

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