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

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Try this beautiful problem from the PRMO, 2017 based on Roots and coefficients of equations.

Let a,b be integers such that all the roots of the equation \((x^{2}+ax+20)(x^{2}+17x+b)\)=0 are negetive integers, find the smallest possible values of a+b.

- is 107
- is 25
- is 840
- cannot be determined from the given information

Polynomials

Roots

Coefficients

But try the problem first...

Answer: is 25.

Source

Suggested Reading

PRMO, 2017, Question 4

Polynomials by Barbeau

First hint

\((x^{2}+ax+20)(x^{2}+17x+b)\)

where sum of roots \( \lt \) 0 and product \( \gt 0\) for each quadratic equation \(x^{2}\)+ax+20**=**0 and

\((x^{2}+17x+b)=0\)

\(a \gt 0\), \(b \gt 0\)

now using vieta's formula on each quadratic equation \(x^{2}\)+ax+20=0 and \((x^{2}+17x+b)=0\), to get possible roots of \(x^{2}\)+ax+20**=0** from product of roots equation \(20=(1 \times 20), (2 \times 10), (4 \times 5)\)

min a=4+5=9 from all sum of roots possible

Second Hint

again using vieta's formula, to get possible roots of \((x^{2}\)+17x+b)=0 from sum of roots equation \(17=-(\alpha + \beta) \Rightarrow (\alpha,\beta)=(-1,-16),(-2,-15),\)

\((-8,-9)\)

min b=(-1)(-16)=16 from all products of roots possible

Final Step

\((a+b)_{min}=a_{min}+b_{min}\)=9+16=25.

- https://www.cheenta.com/smallest-perimeter-of-triangle-aime-2015-question-11/
- https://www.youtube.com/watch?v=ST58GTF95t4&t=140s

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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