The Art and Craft of Problem Solving

(Ann) #1
1.3 A PROBLEM SAMPLER 9

1.3.7 (AIME 19 94) Find the positive integer n for which

llog 2 1j + llog 2 2j + llog 2 3j + ... + llog 2 nj = 1994 ,


where lx j denotes the greatest integer less than or equal to x. (For example, l n j = 3.)
1.3.8 (AIME 19 94) For any sequence of real numbers A = (a 1 ,a 2 , a 3 , ... ), define LlA
to be the sequence (a 2 - ai , a 3 - a 2 , a 4 - a 3 , ... ) whose nth tenn is an+ 1 - an. Suppose
that all of the tenns of the sequence Ll (LlA) are 1, and that a 19 = a9 4 = O. Find a I.
1.3.9 (USAMO 19 89) The 20 members of a local tennis club have scheduled exactly
14 two-person games among themselves, with each member playing in at least one
game. Prove that within this schedule there must be a set of six games with 12 distinct
players.
1.3. 10 (USAMO 19 95) A calculator is broken so that the only keys that still work
are the sin, cos, tan, sin-I, cos-I, and tan-I buttons. The display initially shows
O. Given any positive rational number q, show that pressing some finite sequence of
buttons will yield q. Assume that the calculator does real number calculations with
infinite precision. All functions are in tenns of radians.
1.3.11 (Russia 19 95) Solve the equation

cos( cos( cos( cosx))) = sin(sin( sin(sinx))).

1.3.12 (IMO 19 76) Detennine, with proof, the largest number that is the product of
positive integers whose sum is 19 76.

1.3.13 (Putnam 19 78) Let A be any set of 20 distinct integers chosen from the arith­

metic progression 1,4,7, ... , 100. Prove that there must be two distinct integers in A

whose sum is 10 4.
1.3.14 (Putnam 19 94) Let (an) be a sequence of positive reals such that, for all n,
an :S a 2 n + a 2 n+ I· Prove that :L:= 1 an diverges.
1.3.15 (Putnam 19 94) Find the positive value of m such that the area in the first quad­
rant enclosed by the ellipse x^2 / 9 + y^2 = 1, the x-axis, and the line y = 2x 13 is equal to
the area in the first quadrant enclosed by the ellipse x^2 / 9 + y^2 = 1, the y-axis, and the

liney = mx.

1.3. 16 (Putnam 19 90) Consider a paper punch that can be centered at any point of the
plane and that, when operated, removes from the plane precisely those points whose
distance from the center is irrational. How many punches are needed to remove every
point?


Open-Ended Problems
These are mathematical questions that are sometimes vaguely worded, and possibly
have no actual solution (unlike the two types of problems described above). Open­
ended problems can be very exciting to work on, because you don't know what the
outcome will be. A good open-ended problem is like a hike (or expedition!) in an
uncharted region. Often partial solutions are all that you can get. (Of course, partial

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