Session 1 Problems

 

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Some Old Putnam Problems

 

 

from 1939:

 

            A3.  Find the cubic equation whose roots are the cubes of the roots of the equation

 

 

 

            B10.  Given the power series

 

 

in which

 

 

show that there is a relation of the form

 

 

in which p, q r are constants independent of n.  Find these constants and the sum of the power series.

 

 

from 1958:

 

            A3.  Real numbers are chosen at random from the closed interval  [0,1]. If, after choosing the n'th number, the sum of the numbers so chosen first exceeds 1, show that the expected or average value for n is e.

 

            B2. Prove that the product of four consecutive positive integers cannot be a perfect square or cube.

 

            B5.  Given an infinite number of points in the plane, prove that, if all the distances determined between them are integers, then the points are all in a straight line.

 


Some New Putnam Problems

 

 

from 1993:

 

            A4.  Let x1, x2, x3, . . . x19 be positive integers each of which is less than or equal to 93.  Let y1, y2, y3, . . . y93 be positive integers each of which is less than or equal to 19.  Prove that there exists a (nonempty) sum of some xi's equal to a sum of some yj's.

 

            B1.  Find the smallest positive integer n such that for every integer m, with

0 < m < 1993, there exists an integer k for which

 

 

 

 

from 1997:

 

            A2.  Players 1, 2, 3, Š, n are seated around a table and each has a single penny.  Player 1 passes a penny to Player 2, who then passes two pennies to Player 3.  Player 3 them passes one penny to Player 4, who passes two pennies to Player 5 and so on.  Players alternately pass one penny or two to the next player who still has some pennies.  A player who runs out of pennies drops out of the game.

            Find an infinite set of numbers n for which some player ends up with all n pennies.

 

            A4.  Let G be a group with identity e and let f  be a function from G to G such that

 

whenever

 

            Prove that there exists an element a of G such that the function

 

 

is a group homomorphism  (that is

 

 

for all x and y in G).


Some Non-Putnam Problems

 

 

from Steinhaus One Hundred Problems in Elementary Mathematics:

 

            1.  Consider a set of points in the plane.  We connect each point to the nearest point by a straight line.  (Assume that all distances are different so that there is no doubt as to which point is the nearest one.)

            Prove that the resulting figure does not contain any closed polygon or intersecting segments.

 

            2.  Models of polyhedra are made from flat diagrams drawn on pasteboard.  On the diagram , faces are adjacent, and one makes the model by folding the pasteboard along the lines of the diagram.  A regular tetrahedron, for example, has two different diagrams.

            How many diagrams has a cube?

 

 

from Yaglom Challenging Mathematical Problems with Elementary Solutions:

 

            1.  Prove that for even n the following numbers are perfect squares:

 

            a.  The number of different arrangements of bishops on and n ´ n chessboard such that no bishop attacks a square on which another lies, and the maximum number of bishops is used.

 

            b. The number of different arrangements of bishops on and n ´ n chessboard such that every square is attacked by some bishop, and the minimum number of bishops is used.

 

            2. A takes a piece of paper and marks it with either a + or -.  The probability that the mark is + is 1/3.  The paper is then passed from B to C to D.  Each changes the sign on the paper with probability 2/3.  At the end of this process the paper has a + sign.  What is the probability that A wrote + originally?

 

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