Question

According to the Journal of Irreproducible Results, any obtuse angle is a right angle! Here is
their argument.
Given the obtuse angle x, we make a quadrilateral ABCD with
DAB = x, and
ABC =
90◦, andAD = BC. Say the perpendicular bisector toDC meets the perpendicular bisector to
AB at P. ThenPA = PB andPC = PD. So the trianglesPADandPBC have equal sides
and are congruent. Thus
PAD =
PBC. But PAB is isosceles, hence
PAB =
PBA.
Subtracting, gives x =
PAD−
PAB =
PBC −
PBA = 90◦. This is a preposterous
conclusion – just where is the mistake in the “proof” and why does the argument break down
there?
5. Consider a rectangular array of dots with an even number of rows and an even number of
columns. Color the dots, each one red or blue, in such a way so that in each row half the
dots are red and half are blue, and also in each column half are red and half are blue. Now,
whenever two points of the same color are adjacent (in a row or column), join them by an edge
of that color. Show that the number of red edges is the same as the number of blue edges.
6. The squares of an infinite chessboard are numbered as follows: in the first row and first column
we put 0, and then in every other square we put the smallest non-negative integer that does not
appear anywhere below it in the same column or anywhere to the left of it in the same row.
What number will appear in the 1000th row and 700th column? Can you generalize?
7. Let’s agree to say that a positive integer is prime-like if it is not divisible by 2, 3, or 5. How
many prime-like positive integers are there less than 100? less than 1000? A positive integer
is very prime-like if it is not divisible by any prime less than 15. How many very primelike
positive integers are there less than 90000? Without giving an exact answer, can you
say approximately how many very prime-like positive integers are less than 1010? less than
10100? Explain your reasoning as carefully as you can.
8. Find a positive integer m such that
1
2m is a perfect square and
1
3m is a perfect cube. Can you
find a positive integer n for which
1
2n is a perfect square,
1
3n is a perfect cube and
1
5n is a
perfect fifth

Answer 1

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Answer 2

Find a polynomial with integer coefficients for which √ √ is a root. That is find such that for some non-negative integer , and integers
and(√ √) .