PROMYS Europe 2015Application Problem Set

http://www.promys-europe.org

Please attempt each of the following problems. Though they can all be solved with nomore than a standard high school mathematics background, most of the problems requireconsiderably more ingenuity than is usually expected in high school. You should keep inmind that we do not expect you to find complete solutions to all of them. Rather, we arelooking to see how you approach challenging problems. Here are a few suggestions:

• Think carefully about the meaning of each problem.

• Examine special cases, either through numerical examples or by drawing pictures.

• Be bold in making conjectures.

• Test your conjectures through further experimentation, and try to devise mathematicalproofs to support the surviving ones.

• Can you solve special cases of a problem, or state and solve simpler but related prob-lems?

If you think you know the answer to a question, but cannot prove that your answer is correct,tell us what kind of evidence you have found to support your belief. If you use books orarticles in your explorations, be sure to cite your sources.

You may find that most of the problems require some patience. Do not rush through them.It is not unreasonable to spend a month or more thinking about the problems. It might begood strategy to devote most of your time to a small selection of problems which you findespecially interesting. Be sure to tell us about progress you have made on problems not yetcompletely solved. For each problem you solve, please justify your answer clearlyand tell us how you arrived at your solution.

1. Consider the sequence t0 = 3, t1 = 33, t2 = 333 , t3 = 3333

, . . . defined by t0 = 3 and

tn+1 = 3tn for n ≥ 0. What are the last two digits in t3 = 3333

? Can you say what thelast three digits are? Show that the last 10 digits of tk are the same for all k ≥ 10.

2. Show that there are no positive integers n for which n4 +2n3 +2n2 +2n+1 is a perfectsquare. Are there any positive integers n for which n4 + n3 + n2 + n + 1 is a perfectsquare? If so, find all such n.

3. The repeat of a positive integer is obtained by writing it twice in a row (so, for example,the repeat of 2005 is 20052005). Is there a positive integer whose repeat is a perfectsquare? If so, how many such positive integers can you find?

4. If (x + 1)1000 is multiplied out, how many of the coefficients are odd? How many arenot divisible by 3? by 5? Can you generalize?

5. According to the Journal of Irreproducible Results, any obtuse angle is a right angle!

x

A

P

B

CD

Here is their argument. Given the obtuse angle x, we make a quadrilateral ABCDwith ∠DAB = x, and ∠ABC = 90◦, and AD = BC. Say the perpendicular bisectorto DC meets the perpendicular bisector to AB at P . Then PA = PB and PC =PD. So the triangles PAD and PBC have equal sides and are congruent. Thus∠PAD = ∠PBC. But PAB is isosceles, hence ∠PAB = ∠PBA. Subtracting, givesx = ∠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?

6. Let us agree to say that a non-negative integer is “scattered” if its binary expansionhas no occurence of two ones in a row. For example, 37 is scattered but 45 is not, sincethe binary expansion of 37 is 100101 in which the ones are all separated by at leastone zero, while the binary expansion of 45 is 101101 which has two ones in successiveplaces. For an integer n ≥ 0, how many scattered non-negative integers are there lessthan 2n?

7. 10 people are to be divided into 3 committees, in such a way that every committeemust have at least one member, and no person can serve on all three committees. (Notethat we do not require everybody to serve on at least one committee.) In how manyways can this be done?

8. The squares of an infinite chessboard are numbered as follows: in the zeroth row andcolumn we put 0, and then in every other square we put the smallest non-negativeinteger that does not appear anywhere below it in the same column nor anywhere tothe left of it in the same row.

6

6

0 1 2 3 4 51 0 3 2 5 4 72 3 0 1 6 73 2 1 0 74 5 6 75 4 7

7

......

......

......

...

...

...

What number will appear in the 2014th row and 1928th column? Can you generalize?

9. Let’s agree to say that a positive integer is prime-like if it is not divisible by 2, 3, or5. How many prime-like positive integers are there less than 100? less than 1000? Apositive integer is very prime-like if it is not divisible by any prime less than 15. Howmany very prime-like positive integers are there less than 90000? Without giving anexact answer, can you say approximately how many very prime-like positive integersare less than 1010? less than 10100? Explain your reasoning as carefully as you can.

10. The tail of a giant wallaby is attached by a giant rubber band to a stake in the ground.A flea is sitting on top of the stake eyeing the wallaby (hungrily). The wallaby seesthe flea leaps into the air and lands one mile from the stake (with its tail still attachedto the stake by the rubber band). The flea does not give up the chase but leaps intothe air and lands on the stretched rubber band one inch from the stake. The giantwallaby, seeing this, again leaps into the air and lands another mile from the stake(i.e., a total of two miles from the stake). The flea is undaunted and leaps into theair again, landing on the rubber band one inch further along. Once again the giantwallaby jumps another mile. The flea again leaps bravely into the air and lands anotherinch along the rubber band. If this continues indefinitely, will the flea ever catch thewallaby? (Assume the earth is flat and continues indefinitely in all directions.)