Sunday

November 29, 2015
Posted by **help** on Thursday, April 1, 2010 at 4:08pm.

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

- math -
**PROMYS**, Thursday, May 13, 2010 at 10:51pmNow now..... no using outside sources!! If you continue to do this, we will track your IP address to find out who you are and disqualify you...

- math -
**Anonymous**, Friday, February 17, 2012 at 11:34amFind a polynomial with integer coefficients for which √ √ is a root. That is find such that for some non-negative integer , and integers

and(√ √) .

- math -
**etovey**, Sunday, February 23, 2014 at 10:30pmpromys uk tracking is not so easy especially not with a subponea

- math -
**etovey**, Thursday, March 13, 2014 at 10:03amand it's illegal to buddy