Try this beautiful Trigonometry Problem based on Triangle from 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 $\mathrm{N} / 100$ ?
,
Trigonometry
Triangle
Integer
Pre College Mathematics
Prmo-2018, Problem-24
\(27\)
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?
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
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 \
$
In 2026, the following Cheenta students have been successful for Indian Statistical Institute's M.Stat Entrance. They ranked within the first 50 in the entire country in these entrances. I.S.I. M.Stat Entrance
In 2026, the following Cheenta students have been successful for Indian Statistical Institute's B.Stat Entrance and Chennai Mathematical Institute's B.Sc. Math Entrance. They ranked within the first 200 in the entire country in these entrances. Most of these students attended the problem solving workshops regularly, which happen 5 days every week. CMI B.Sc. Math Entrance […]
In 2025, 8 students from Cheenta Academy cracked the prestigious Regional Math Olympiad. In this post, we will share some of their success stories and learning strategies. The Regional Mathematics Olympiad (RMO) and the Indian National Mathematics Olympiad (INMO) are two most important mathematics contests in India.These two contests are for the students who are […]
Cheenta Academy proudly celebrates the success of 27 current and former students who qualified for the Indian Olympiad Qualifier in Mathematics (IOQM) 2025, advancing to the next stage — RMO. This accomplishment highlights their perseverance and Cheenta’s ongoing mission to nurture mathematical excellence and research-oriented learning.