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

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Try this beautiful Problem from Singapore Mathematics Olympiad, 2012 based on Functional Equations.

Let L denote the minimum value of the quotient of a 3- digit number formed by three distinct divided by the sum of its digits.Determine \(\lfloor 10L \rfloor \).

- 105
- 150
- 102
- 200

Functional Equation

Max and Min Value

But try the problem first...

Answer: 105

Source

Suggested Reading

Singapore Mathematical Olympiad, 2012

Challenges and Thrills - Pre - College Mathematics

First hint

If you got stuck at first only here is the hint to begin with :

Anyway a three digit number we can be expressed as 100x + 10 y +z depending on the place values. and if we do minimize it :

F(x y z) = \(\frac {100x + 10y + z}{x + y + z}\)

Lets consider that for distinct x , y , z, F(x , y , Z) has the minimum value when x<y<z.

Again we can assume,

\( 0 < a < b < c \leq 9\)

Note ,

F(x,y,z) = \(\frac {100 x + 10 y + z }{x +y + z}\) = 1 + \(\frac {99 x + 9 y }{x+y+z}\)

Try the rest of the sum...............

Second Hint

From the last hint we can say

F(x y z ) is minimum when c = 9 (say)

F(x y 9) = 1+ \(\frac {99x +9y }{x+y+9} = 1 + \frac {9(x + y + 9) + 90 x - 81}{x+y +9} = 10 + \frac {9(10x -9)}{x+y +9}\)

Try to do for the next case for minimum value when b = 8..................

Final Step

In the last hint we can do the next step which is b= 8:

F(x 8 9) = 10 + \(\frac {9(10x -9)}{x+17}= 10 + \frac {90(a+17)- 1611}{x + 17} = 100 - \frac {1611}{x +17} \)

which has the minimum value of x = 1 and so 10 L = 105.(answer)

- https://www.cheenta.com/problem-on-area-of-triangle-smo-2010-problem-32/
- https://www.youtube.com/watch?v=XOrePzJWFiE&t=25s

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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