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# Triangle Problem | PRMO-2018 | Problem No-24

Try this beautiful Problem on Trigonometry from PRMO -2018.You may use sequential hints to solve the problem.

Try this beautiful Trigonometry Problem based on Triangle from PRMO -2018, Problem 24.

## Triangle Problem – PRMO 2018- Problem 24

If $\mathrm{N}$ is the number of triangles of different shapes (i.e. not similar) whose angles are all integers (in degrees), what is $\mathrm{N} / 100$ ?

,

• $15$
• $22$
• $27$
• $32$
• $37$

Trigonometry

Triangle

Integer

## Suggested Book | Source | Answer

Pre College Mathematics

#### Source of the problem

Prmo-2018, Problem-24

#### Check the answer here, but try the problem first

$27$

## Try with Hints

#### First Hint

Given that $\mathrm{N}$ is the number of triangles of different shapes. Therefore the different shapes of triangle the angles will be change . at first we have to find out the posssible orders of the angles that the shape of the triangle will be different…

Now can you finish the problem?

#### Second Hint

case 1 : when $x \geq 1$ & $y \geq 3 \geq 1$
$$x+y+z=180$$
$={ }^{179} \mathrm{C}_{2}=15931$
Case 2 : When two angles are same
$$2 x+y=180$$
1,1,178
2,2,176
$\vdots$
89,89,2

#### Solution

But we have one case $60^{\circ}, 60^{\circ}, 60^{\circ}$
$$\text { Total }=89-1=88$$
Such type of triangle $=3(88)$
When 3 angles are same $=1(60,60,60)$
So all distinct angles’s triangles
$$\begin{array}{l} =15931-(3 \times 88)-1 \ \neq 3 ! \ =2611 \end{array}$$
Now, distinct triangle $=2611+88+1$
$=2700 \ N=2700 \ \frac{N}{100}=27 \$

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