What is the NO-SHORTCUT approach for learning great Mathematics?

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Try this beautiful problem from PRMO, 2018 based on Angles in a circle.

Let AB be a chord of circle with centre O. Let C be a point on the circle such that \(\angle ABC\) = \(30^{\circ}\) and

O lies inside the triangle ABC. Let D be a point on AB such that \(\angle DCO\) = \(\angle OCB\) = \(20^{\circ}\). Find the

measure of \(\angle CDO\) in degrees.

- \(75^{\circ}\)
- \(80^{\circ}\)
- \(60^{\circ}\)

Geometry

Circle

Angle

But try the problem first...

Answer:$80$

Source

Suggested Reading

PRMO-2018, Problem 8

Pre College Mathematics

First hint

We have to find out the \(\angle CDO\).If we can find out the value of \(\angle ODC\) & \(\angle COD\)...then we can easily find out the value of \(\angle CDO\)

Can you now finish the problem ..........

Second Hint

For \(\angle ODC\) & \(\angle COD\) ,we have to find all the angles of the triangles using cyclic property and given data.such as OB=OC,so \(\angle OBC=\angle OCB\).\(\angle ABC=30\) So the \(\angle ABO=30-20\)

Can you finish the problem........

Final Step

Given \(\angle OCB = 20^{\circ}\)

\(\angle OBC = 20^{\circ}\)[as OB=OC ,radius of the circle]

\(\angle OBA =\angle ABC -\angle OBC=30^{\circ}-20^{\circ}=10^{\circ}\)

\(\angle OAB = 10^{\circ}\)[as OB=OA,radius of the circle]

\(\angle BOC =180-\angle OBC-\angle OCB=180-20-20=140\),

Now \(\angle BOC =140^{\circ} \Rightarrow \angle A = 70°\)[since an arc subtends double angle compare to circumference]

\(\angle OAC=\angle BAC -\angle BAO=70-10= 60^{\circ}\)

\(\angle ACD = 40^{\circ}\)

Now C is circumcenter of \(\triangle AOD\)

as \(\angle OCD = 2\angle OAD\)

\(\angle AOD =\frac{1}{2}\angle OAD = 20^{\circ}\)

\(\angle DOC = \angle AOD + \angle AOC\)

= 20 + 60

= 80

\(\angle ODC = 180 – (\angle DOC + \angle OCD)\)

= 180 – (80 + 20)

= 80°

- https://www.youtube.com/watch?v=i_pmSwUO4LA
- https://www.cheenta.com/combination-of-cups-problem-prmo-2018-problem-11/

What to do to shape your Career in Mathematics after 12th?

From the video below, let's learn from Dr. Ashani Dasgupta (a Ph.D. in Mathematics from the University of Milwaukee-Wisconsin and Founder-Faculty of Cheenta) how you can shape your career in Mathematics and pursue it after 12th in India and Abroad. These are some of the key questions that we are discussing here:

- What are some of the best colleges for Mathematics that you can aim to apply for after high school?
- How can you strategically opt for less known colleges and prepare yourself for the best universities in India or Abroad for your Masters or Ph.D. Programs?
- What are the best universities for MS, MMath, and Ph.D. Programs in India?
- What topics in Mathematics are really needed to crack some great Masters or Ph.D. level entrances?
- How can you pursue a Ph.D. in Mathematics outside India?
- What are the 5 ways Cheenta can help you to pursue Higher Mathematics in India and abroad?

Cheenta has taken an initiative of helping College and High School Passout Students with its "Open Seminars" and "Open for all Math Camps". These events are extremely useful for students who are really passionate for Mathematic and want to pursue their career in it.

Cheenta is a knowledge partner of Aditya Birla Education Academy

Advanced Mathematical Science. Taught by olympians, researchers and true masters of the subject.

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