Wrong Solution

Source: Very interesting problem taken from Tanya Khovanova’s Math Blog

Problem:
I found this cute problem in the Russian book Sharygin Geometry Olympiad by Zaslavsky, Protasov and Sharygin.
Find numbers p and q that satisfy the equation: x2 + px + q = 0.
The book asks you to find a mistake in the following solution:
By Viète’s formulae we get a system of equations p + q = − p, pq = q. Solving the system we get two solutions: p = q = 0 and p = 1, q = −2.
What is wrong with this solution?

Update (26-05-2011)
Solution:
Posted by Sudeep Kamath (EECS PhD Student, UC Berkeley, EE IITB 2008 Alumnus) in comments and of course on Tanya Khovanova’s Math Blog


Comments

  1. shaan8:37 PM, May 23, 2011

    here, we are looking for the coefficients "p" & "q" most probably,
    i think the question DOES NOT that
    P & q ARE THE SOLUTIONS OF GIVEN QUADRATIC..

    ReplyDelete
  2. Sudeep10:37 PM, May 23, 2011

    Very nice problem.

    This approach misses out the case when p=q, so that you only need to enforce one root of the equation to be p. This gives the additional solution p=q=-1/2.

    ReplyDelete
  3. Pratik Poddar2:46 PM, May 26, 2011

    @Shaan. I did not understand your comment.

    @Sudeep. \m/ \m/ I could not solve it, but was happy_max when I read the solution :) Thanks.

    ReplyDelete
  4. satpal10:48 AM, May 11, 2012

    but when you put p=q=-1/2 in the equation the roots of the quadratic are coming out to be -1/2 and 1 which are distinct and we need the equal roots..

    ReplyDelete
  5. Pratik Poddar4:55 AM, May 25, 2012

    @satpal..
    When you put p = q = -1/2,
    The equation becomes 2x^2 - x - 1 = 0 and -1/2 is a root of this equation. The question never says that p and q are the only roots of the equation. :)

    Its just a terminology difference. I would not have put this if this was not in Russian Olympiad book :)

    ReplyDelete

Post a Comment

Popular posts from this blog

Buying Dimsums

Source: Alok Goyal (Stellaris VP, Ex-Helion VC) puzzle blog Problem: A fast food restaurant sells dimsums in boxes of 7 and 3. What’s the greatest number of dimsums a person cannot buy. Generalize it for p and q where p and q are relatively prime. I loved the puzzle. Hope you enjoy it too.
Read more

Fraction Brainteaser

Source: Sent to me by Gaurav Sinha Problem: Siddhant writes a Maths test and correctly answers 5 out of 6 Arithmetic questions and 20 out of 28 Geometry questions.  In total, Siddhant scores 25 out of 34.  Vaibhav writes another Maths test and correctly answers 20 out of 25 Arithmetic questions and 6 out of 9 Geometry questions. in total, Vaibhav scores 26 out of 34. Note that a) Vaibhav scores more than Siddhant b) Siddhant score better than Vaibhav in both individual topics -  5/6 > 20/25 and 20/28 > 6/9 How is it possible? 
Read more

Consecutive Heads

Let's say A keep tossing a fair coin, until he get 2 consecutive heads, define X to be the number of tosses for this process; B keep tossing another fair coin, until he get 3 consecutive heads, define Y to be the number of the tosses for this process. 1) Calculate P{X>Y} 2) What's the expected value of X 3) What's the expected value of Y This is probably the hardest puzzle I have ever put on my blog. Hence, I will post its solution in the post directly rather than on comment. Solution: 1) (Solved by me finally after 13 months :)) Make a state diagram. Let the state be (a,b) where a is the number of consecutive heads streak "A" is on currently and b is the number of consecutive heads streak "B" is on currently. So, (0,3) (1,3) are final accepted states and (2,0) (2,1) (2,2) (2,3) are failure states. If you get tails, your contribution to the state reaches to "0" f(State) = P(X>Y | "State" configuration initially) ...
Read more