Bangladesh Mathematical Olympiad 2020 Final

Part of Bangladesh Mathematical Olympiad

Subcontests

(12)

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Bangladesh Mathematical Olympiad 2020

2^{2921}

581

digits and starts with a

4

2^n

's starts with a

4

0

is the last digit?

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Bangladesh Mathematical Olympiad 2020

q

is called 'Kowai' number if

q = p^2 + 10

q

p

p^2-2

p^2-8

p^3+6

are prime numbers. WE know that, at least one 'Kowai' number can be found. Find the summation of all 'Kowai' numbers.

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Bangladesh Mathematical Olympiad 2020

Sokal da tries to find out the largest positive integer n such that if n transforms to base-7, then it looks like twice of base-10.

156

is such a number because

(156)_{10}

(312)_7

\times

156. Find out Sokal da's number.

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Bangladesh Mathematical Olympiad 2020 Final

You have 2020 piles of coins in front Of you. The first pile contains 1 coin, the second pile contains 2 coins, the third pile contains 3 coins and so on. So, the 2020th pile contains 2020 coins. Guess a positive integer k, in which piles contain at least k coins, take away exact k coins from these piles. Find the minimum number of turns you need to take way all of these coins?

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Bangladesh Mathematical Olympiad 2020 Final

ABC

be a triangle where

\angle

\angle

C = 65. D is the mid-point of BC. Circumcircle of ACD and ABD cuts AB and AC at point F and E respectively. Center of circumcircle of AEF is O.

\angle

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Bangladesh Mathematical Olympiad 2020 Final

Tiham is trying to find 6 digit positive integers

PQRSTU

PQRSTU

are not necessarily distinct). But he only wants the numbers where the sum of the 3 digit number

PQR

, and the 3 digit number

STU

is divisible by 37. How many such numbers Tiham can find?

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Bangladesh Mathematical Olympiad 2020 Final

P

is taken inside the square

ABCD

BP + DP=25

CP - AP = 15

\angle

\angle

ADP. What is the radius of the circumcircle of

ABCD

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Bangladesh Mathematical Olympiad 2020 Final

For a positive real number

[x]

be its integer part. For example,

[2.711] = 2, [7] = 7, [6.9] = 6

z

is the maximum real number such that [

\frac{5}{z}

\frac{6}{z}

] = 7. Find the value of

20z

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Bangladesh Mathematical Olympiad 2020 Final

Once in a restaurant Dr. Strange found out that there were 12 types of food items from 1 to 12 on the menu. He decided to visit the restaurant 12 days in a row and try a different food everyday. 1st day, he tries one of the items from the first two. On the 2nd day, he eats either item 3 or the item he didn’t tried on the 1st day. Similarly, on the 3rd day, he eats either item 4 or the item he didn’t tried on the 2nd day. If someday he's not able to choose items that way, he eats the item that remained uneaten from the menu. In how many ways can he eat the items for 12 days?

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Bangladesh Mathematical Olympiad 2020 Final

Prottasha has a 10 sided dice. She throws the dice two times and sum the numbers she gets. Which number has the most probability to come out?

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Bangladesh Mathematical Olympiad 2020 Final

Consider rectangle

ABCD

E

is the mid-point of

AD

F

is the mid-point of

ED

CE

AB

G

BF

CD

H

point. We can write ratio of areas of

BCG

BCH

\frac{m}{n}

. Find the value of

10m + 10n + mn

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Bangladesh Mathematical Olympiad 2020 Final

A pair of positive integers

(m,n)

is called 'steakmaker' if they maintain the equation 1 + 2

^m

^2

. For which values of m and n, the pair

(m,n)

are steakmaker, find the sum of

mn