## BDMO 2018 National Olympiad: Problemsets

nahin munkar
Posts: 81
Joined: Mon Aug 17, 2015 6:51 pm
Location: banasree,dhaka

### BDMO 2018 National Olympiad: Problemsets

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?

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 ?

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 ?

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 (33.41 KiB) Viewed 3304 times

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?

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!

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 .

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 (19.49 KiB) Viewed 3304 times
# Mathematicians stand on each other's shoulders. ~ Carl Friedrich Gauss

nahin munkar
Posts: 81
Joined: Mon Aug 17, 2015 6:51 pm
Location: banasree,dhaka

### Re: BDMO 2018 National Olympiad: Problemsets

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?

Problem 2
If the average of first $n$ positive integers is $2018$, then find the value of n ?

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)

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?

Problem 5
$3x^2 + y^2 = 108$ ;
Determine all the positive integer values of $x$ and $y$.

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°.

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) ?

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 (8.6 KiB) Viewed 3303 times

Problem 9
Find the number of positive integers that are divisors of at least one of $10^{10}$, $12^{12}$, $15^{15}$.

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 $\angle AOC$ .
# Mathematicians stand on each other's shoulders. ~ Carl Friedrich Gauss

nahin munkar
Posts: 81
Joined: Mon Aug 17, 2015 6:51 pm
Location: banasree,dhaka

### Re: BDMO 2018 National Olympiad: Problemsets

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

Where $x$ is real number.

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 $AB=?$

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?

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 $m$?

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.

Problem 6
Find all integer solution ($m,n$) for the following equation:
$3(m^2+n^2)-7(m+n)=-4$

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.

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
Attachments Questions - BdMO 2018 National Secondary And Higher Secondary.pdf
# Mathematicians stand on each other's shoulders. ~ Carl Friedrich Gauss

nahin munkar
Posts: 81
Joined: Mon Aug 17, 2015 6:51 pm
Location: banasree,dhaka

### Re: BDMO 2018 National Olympiad: Problemsets

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

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 $AB=?$

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 $S$ ?

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 $m$?

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.

Problem 6
Find all integer solution ($m,n$) for the following equation:
$3(m^2+n^2)-7(m+n)=-4$

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

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