When Problems Attack!

By Andrew Dudzik (remove NO and SPAM from the e-mail address)

#18: No comment.

Total Problems to Date: 18 (Submissions welcome!)



    1) The Problem From Heck (39th Republic Competition of Mathematics In Macedonia)

    Find the maximum value of if .



    2) The Super Death Problem (Andrew Dudzik)

    Prove: in as many ways as you can. Current record is 3, set by yours truly.



    3) The Problem From Hell (Andrew Dudzik)

    Find ,

    a) using an algebraic method of your choice, and

    b) using a simple combinatorial argument.



    4) The Return of the Son of the Problem From Hell (Andrew Dudzik)

    Find ,

    a) using an algebraic method of your choice, and

    b) using a simple combinatorial argument.



    5) The Evil Alien Vampire Problem From Jupiter That Sucked My Brain! (Variation on Putnam 1980)

    Find .



    6) The Problem That Shot Kennedy (Paul Zeitz)

    Solve , without using trignometric functions.



    7) Satan, Embodied in A Math Problem (Andrew Dudzik)

    Find . Good. Now find . Now find . Splendid.



    8) The I Know What You Did Last Problem (Variation on British Mathematical Olympiad 1983)

    Prove that if , and , then , for all n.



    9) The I Still Know What You Did Last Problem (Variation on British Mathematical Olympiad 1983)

    Prove that if positive integers, a, b, and m satisfy , , and , for all n, then m=5, a=2, b=3.



    10) The Problem That Could Solve You! (Andrew Dudzik)

    Find , and prove that the sum converges if and only if .



    11) The Problem With Society (Andrew Dudzik)

    Given that f is a continuous, nondecreasing function, defined on [0,1], with f(0)=0 and f(1)=1, prove:



    12) When Problems Attack! (Andrew Dudzik)

    Find real numbers m and b such thatis minimal. No calculus! (Hint: you could solve the problem using vectors in 50-dimensional space, but you'd be an idiot.)



13) The Terrible Horrible No Good Very Bad Problem (Anthony Phillips, on the Mandelbrot Page)

Suppose that m lines are situated in the plane so that each line intersects exactly k others. Let f(k) be the number of values of m for which this is possible. Find f(20), and, given that f(k)=3, findand prove that its value is uniquely determined.



14) The Problem That Ate Everybody (Andrew Dudzik)

Suppose we have k people (k odd), and some number of locks. Each lock is unique, but may have more than one key. Furthermore, a given group of these k people possesses the keys to open all the locks if and only if the group is a majority. Let the minimum number of locks for which such a configuration is possible be f(k). Evaluate:. (Hint: In case you don't know, the following is true: .)

15)The Bugblatter Problem of Traal (Andrew Dudzik)

We all know the formula . Prove the following useless generalization:

Let m be a positive integer. Then , where brackets denote the greatest integer function.


16)The Problem of Aaaaarrrrggghh! (Andrew Dudzik)

Find infinitely many solutions to. (Hint: First find infinitely many solutions to )



17)The Houston We Have a Problem(Andrew Dudzik)

Let , and , where brackets denote the greatest integer function. (Terminate the sequence before the first nonpositive aj.) Find all positive integers A such that the sequence contains no perfect squares.



18)Problem Child(Andrew Dudzik)

Letdenote the sum over all even permutations of the integers 0 through n. That is, the permutations that can be obtained from {0,1,2...n} by switching two adjacent numbers an even number of times. Prove or disprove:

(Hint: you have to want it.)





Problems credited to Paul Zeitz are from The Art and Craft Of Problem Solving.


This entire web page, including the math formulae, was created using StarOffice, the no longer free alternative to MS Office.


Last updated 7/19/04 -- e-mail address now actually works. Note that there have been no actual updates to this page since the year 2000 and it's likely to stay that way, but this page is linked to in enough places that I might as well keep it here for the purpose of nostalgia.

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