## BDMO Secondary National 2021 #1

Mehrab4226
Posts:230
Joined:Sat Jan 11, 2020 1:38 pm
BDMO Secondary National 2021 #1
পূর্ণসংখ্যার কতগুলো ক্রমজোড় $$(m, n)$$ আছে যেন $$m$$ আর $$n$$ কোনো একটা সমকোণী ত্রিভুজের অতিভুজ বাদে বাকি দুটো বাহুর দৈর্ঘ্য হয় এবং ত্রিভুজটার ক্ষেত্রফল $$80$$-এর চেয়ে বড় না এমন একটা মৌলিক সংখ্যার সমান হয়?

How many ordered pairs of integers $(m,n)$ are there such that $m$ and $n$ are the legs of a right triangle with an area equal to a prime number not exceeding $80$?
The Mathematician does not study math because it is useful; he studies it because he delights in it, and he delights in it because it is beautiful.
-Henri Poincaré

Pro_GRMR
Posts:46
Joined:Wed Feb 03, 2021 1:58 pm

### Re: BDMO Secondary National 2021 #1

Mehrab4226 wrote:
Sat Apr 10, 2021 12:31 pm
How many ordered pairs of integers $(m,n)$ are there such that $m$ and $n$ are the legs of a right triangle with an area equal to a prime number not exceeding $80$?
According to the question:
$\frac{1}{2}mn = p \leq 80$ Such that $p$ is prime.
So, $\frac{1}{2}mn \in \{2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79\}$
and $mn \in \{4, 6, 10, 14, 22, 26, \dots , 158\}$
$m | a$ Such that $a \in \{4, 6, 10, 14, 22, 26, \dots , 158\}$
There are $3$ divisors of $4$ and all the other numbers have $4$ divisors each. Because there are $22$ such numbers the result is $21 \times 4 + 3 = \boxed{87}$
"When you change the way you look at things, the things you look at change." - Max Planck

Mehrab4226
Posts:230
Joined:Sat Jan 11, 2020 1:38 pm

### Re: BDMO Secondary National 2021 #1

Pro_GRMR wrote:
Sat Apr 10, 2021 12:50 pm
Mehrab4226 wrote:
Sat Apr 10, 2021 12:31 pm
How many ordered pairs of integers $(m,n)$ are there such that $m$ and $n$ are the legs of a right triangle with an area equal to a prime number not exceeding $80$?
According to the question:
$\frac{1}{2}mn = p \leq 80$ Such that $p$ is prime.
So, $\frac{1}{2}mn \in \{2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79\}$
and $mn \in \{4, 6, 10, 14, 22, 26, \dots , 158\}$
$m | a$ Such that $a \in \{4, 6, 10, 14, 22, 26, \dots , 158\}$
There are $3$ divisors of $4$ and all the other numbers have $4$ divisors each. Because there are $22$ such numbers the result is $21 \times 4 + 3 = \boxed{87}$
Aren't $(m,n)$ and $(n,m)$ the same triangle?
The Mathematician does not study math because it is useful; he studies it because he delights in it, and he delights in it because it is beautiful.
-Henri Poincaré

Pro_GRMR
Posts:46
Joined:Wed Feb 03, 2021 1:58 pm

### Re: BDMO Secondary National 2021 #1

Mehrab4226 wrote:
Sat Apr 10, 2021 1:38 pm
Mehrab4226 wrote:
Sat Apr 10, 2021 12:31 pm
How many ordered pairs of integers $(m,n)$ are there such that $m$ and $n$ are the legs of a right triangle with an area equal to a prime number not exceeding $80$?
Aren't $(m,n)$ and $(n,m)$ the same triangle?
Yeah but the question specifically asked for ordered pairs.
Let me give an example:
Let's say $\frac{1}{2}mn=3$. These four ordered pairs satisfy such constraint $(1, 6), (6, 1), (2, 3), (3, 2)$. Here the Order of the integers matter because they are ordered.
"When you change the way you look at things, the things you look at change." - Max Planck

Mehrab4226
Posts:230
Joined:Sat Jan 11, 2020 1:38 pm

### Re: BDMO Secondary National 2021 #1

Pro_GRMR wrote:
Sat Apr 10, 2021 5:19 pm
Mehrab4226 wrote:
Sat Apr 10, 2021 1:38 pm
Mehrab4226 wrote:
Sat Apr 10, 2021 12:31 pm
How many ordered pairs of integers $(m,n)$ are there such that $m$ and $n$ are the legs of a right triangle with an area equal to a prime number not exceeding $80$?
Aren't $(m,n)$ and $(n,m)$ the same triangle?
Yeah but the question specifically asked for ordered pairs.
Let me give an example:
Let's say $\frac{1}{2}mn=3$. These four ordered pairs satisfy such constraint $(1, 6), (6, 1), (2, 3), (3, 2)$. Here the Order of the integers matter because they are ordered.
Amar bhul gese
The Mathematician does not study math because it is useful; he studies it because he delights in it, and he delights in it because it is beautiful.
-Henri Poincaré