PROMYS 2016
Application Problem Set
http://www.promys.org
Please attempt each of the following problems. Though they can all be solved with no
more than a standard high school mathematics background, most of the problems require
considerably more ingenuity than is usually expected in high school. You should keep in
mind that we do not expect you to find complete solutions to all of them. Rather, we are
looking 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 mathematical
proofs to support the surviving ones.
• Can you solve special cases of a problem, or state and solve simpler but related problems?
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 or
articles 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 be
good strategy to devote most of your time to a small selection of problems which you find
especially interesting. Be sure to tell us about progress you have made on problems not yet
completely solved. For each problem you solve, please justify your answer clearly
and tell us how you arrived at your solution.
1. If (x + 1)1000 is multiplied out, how many of the coefficients are odd? How many are
not divisible by 3? by 5? Can you generalize?
2. In how many ways can 2016 be written as a sum a0 + a1 · 2 + ... + ak · 2k , if the ai are
allowed to take the values 0, 1, 2 or 3?
3. The numbers 3, 5 and 7 are all prime, and form a triple of evenly spaced numbers with
spacing 2. Can you find any other triples of evenly spaced prime numbers with spacing
2? Can you find all such triples? Find more examples of triples of evenly spaced prime
numbers with other spacings. Can you find four evenly spaced prime numbers? What
can you say about the possible spacings? What’s the smallest possible largest prime
number in such a set of four? What about longer groups of evenly spaced primes?
What’s the smallest possible largest prime number for a group of length 5, or 6, or . . .?
4. Show that there are no positive integers n for which n4 + 2n3 + 2n2 + 2n + 1 is a perfect
square. Are there any positive integers n for which n4 + n3 + n2 + n + 1 is a perfect
square? If so, find all such n.
5. The repeat of a positive integer is obtained by writing it twice in a row (so, for example,
the repeat of 2016 is 20162016). Is there a positive integer whose repeat is a perfect
square? If so, how many such positive integers can you find?
6. According to the Journal of Irreproducible Results, any obtuse angle is a right angle!
C
D
x
B
A
P
Here is their argument. Given the obtuse angle x, we make a quadrilateral ABCD
with ∠DAB = x, and ∠ABC = 90◦ , and AD = BC. Say the perpendicular bisector
to DC meets the perpendicular bisector to AB at P . Then P A = P B and P C =
P D. So the triangles P AD and P BC have equal sides and are congruent. Thus
∠P AD = ∠P BC. But P AB is isosceles, hence ∠P AB = ∠P BA. Subtracting, gives
x = ∠P AD − ∠P AB = ∠P BC − ∠P BA = 90◦ . This is a preposterous conclusion –
just where is the mistake in the “proof” and why does the argument break down there?
7. Let us say that a function f from the set of rational numbers to the set of non-zero
rational numbers is a rational matching if each rational number is paired with exactly
one non-zero rational number, in such a way that each non-zero rational number gets
exactly one rational partner. Such a function is said to be bijective. Can you find an
example of a rational matching? Is there an order-preserving rational matching? (That
is, a rational matching f with the property that whenever a < b, we have f (a) < f (b).)
8. The squares of an infinite chessboard are numbered as follows: in the zeroth row and
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 nor anywhere to
the left of it in the same row.
... ...
6
5
4
3
2
1
0
7
4
5
2
3
0
1
...
7
6
1
0
3
2
...
7
0
1
2
3
...
7 ...
6 7 ...
5 4 7 ...
4 5 6 ...
What number will appear in the 2016th row and 1601st column? Can you generalize?
9. Let P0 be an equilateral triangle of area 1. Each side of P0 is trisected, and the corners
are snipped off, creating a new polygon (in fact, a hexagon) P1 . What is the area of
P1 ? Now repeat the process to P1 – i.e. trisect each side and snip off the corners – to
obtain a new polygon P2 . What is the area of P2 ? Now repeat this process infinitely
often to create an object P∞ . What is the area of P∞ ?
10. The tail of a monstrously huge rabbit is tied to a pole in the ground by an infinitely
stretchy elastic cord. A flea sits on the pole watching the rabbit (hungrily). The rabbit
sees the flea, leaps into the air and lands one kilometer from the pole (with its tail
still attached to the pole by the elastic cord). The flea gives chase and leaps into the
air landing on the stretched elastic cord one centimeter from the pole. The monster
rabbit, seeing this, again leaps into the air and lands another kilometer away from the
pole (i.e., a total of two kilometers from the pole). Undaunted, the flea bravely leaps
into the air again, landing on the elastic cord one centimeter further along. Once again
the rabbit jumps another kilometer and the flea jumps another centimeter along the
cord. If this continues indefinitely, will the flea ever catch up to the rabbit? (Assume
the earth is flat and extends infinitely far in all directions.)
PROMYS 2016
Application Problem Set
http://www.promys.org
Please attempt each of the following problems. Though they can all be solved with no
more than a standard high school mathematics background, most of the problems require
considerably more ingenuity than is usually expected in high school. You should keep in
mind that we do not expect you to find complete solutions to all of them. Rather, we are
looking 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 mathematical
proofs to support the surviving ones.
• Can you solve special cases of a problem, or state and solve simpler but related problems?
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 or
articles 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 be
good strategy to devote most of your time to a small selection of problems which you find
especially interesting. Be sure to tell us about progress you have made on problems not yet
completely solved. For each problem you solve, please justify your answer clearly
and tell us how you arrived at your solution.
1. If (x + 1)1000 is multiplied out, how many of the coefficients are odd? How many are
not divisible by 3? by 5? Can you generalize?
2. In how many ways can 2016 be written as a sum a0 + a1 · 2 + ... + ak · 2k , if the ai are
allowed to take the values 0, 1, 2 or 3?
3. The numbers 3, 5 and 7 are all prime, and form a triple of evenly spaced numbers with
spacing 2. Can you find any other triples of evenly spaced prime numbers with spacing
2? Can you find all such triples? Find more examples of triples of evenly spaced prime
numbers with other spacings. Can you find four evenly spaced prime numbers? What
can you say about the possible spacings? What’s the smallest possible largest prime
number in such a set of four? What about longer groups of evenly spaced primes?
What’s the smallest possible largest prime number for a group of length 5, or 6, or . . .?
4. Show that there are no positive integers n for which n4 + 2n3 + 2n2 + 2n + 1 is a perfect
square. Are there any positive integers n for which n4 + n3 + n2 + n + 1 is a perfect
square? If so, find all such n.
5. The repeat of a positive integer is obtained by writing it twice in a row (so, for example,
the repeat of 2016 is 20162016). Is there a positive integer whose repeat is a perfect
square? If so, how many such positive integers can you find?
6. According to the Journal of Irreproducible Results, any obtuse angle is a right angle!
C
D
x
B
A
P
Here is their argument. Given the obtuse angle x, we make a quadrilateral ABCD
with ∠DAB = x, and ∠ABC = 90◦ , and AD = BC. Say the perpendicular bisector
to DC meets the perpendicular bisector to AB at P . Then P A = P B and P C =
P D. So the triangles P AD and P BC have equal sides and are congruent. Thus
∠P AD = ∠P BC. But P AB is isosceles, hence ∠P AB = ∠P BA. Subtracting, gives
x = ∠P AD − ∠P AB = ∠P BC − ∠P BA = 90◦ . This is a preposterous conclusion –
just where is the mistake in the “proof” and why does the argument break down there?
7. Let us say that a function f from the set of rational numbers to the set of non-zero
rational numbers is a rational matching if each rational number is paired with exactly
one non-zero rational number, in such a way that each non-zero rational number gets
exactly one rational partner. Such a function is said to be bijective. Can you find an
example of a rational matching? Is there an order-preserving rational matching? (That
is, a rational matching f with the property that whenever a < b, we have f (a) < f (b).)
8. The squares of an infinite chessboard are numbered as follows: in the zeroth row and
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 nor anywhere to
the left of it in the same row.
... ...
6
5
4
3
2
1
0
7
4
5
2
3
0
1
...
7
6
1
0
3
2
...
7
0
1
2
3
...
7 ...
6 7 ...
5 4 7 ...
4 5 6 ...
What number will appear in the 2016th row and 1601st column? Can you generalize?
9. Let P0 be an equilateral triangle of area 1. Each side of P0 is trisected, and the corners
are snipped off, creating a new polygon (in fact, a hexagon) P1 . What is the area of
P1 ? Now repeat the process to P1 – i.e. trisect each side and snip off the corners – to
obtain a new polygon P2 . What is the area of P2 ? Now repeat this process infinitely
often to create an object P∞ . What is the area of P∞ ?
10. The tail of a monstrously huge rabbit is tied to a pole in the ground by an infinitely
stretchy elastic cord. A flea sits on the pole watching the rabbit (hungrily). The rabbit
sees the flea, leaps into the air and lands one kilometer from the pole (with its tail
still attached to the pole by the elastic cord). The flea gives chase and leaps into the
air landing on the stretched elastic cord one centimeter from the pole. The monster
rabbit, seeing this, again leaps into the air and lands another kilometer away from the
pole (i.e., a total of two kilometers from the pole). Undaunted, the flea bravely leaps
into the air again, landing on the elastic cord one centimeter further along. Once again
the rabbit jumps another kilometer and the flea jumps another centimeter along the
cord. If this continues indefinitely, will the flea ever catch up to the rabbit? (Assume
the earth is flat and extends infinitely far in all directions.)
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