Exotic Arithmetic, 2020 Invariants
1. One hundred blue points and 100 different red points in the plane are
given no three of which are collinear. Is it possible to pair up the blue
and red points so that none of the 100 segments connecting the pairs
intersect?
2. (2018 Purple Comet) Let a, b, c and d be real numbers such that a2 +
b2 + c2 + d2 = 3a + 8b + 24c + 37d = 2018. Evaluate 3b + 8c + 24d + 37a.
3. There are three sizes of boxes available, large, medium, and small.
There are 11 large boxes. Inside some of these boxes are placed 8
medium boxes and the other large boxes are left empty. Inside some
of the medium boxes are places 8 small boxes and the other medium
boxes are left empty. After this there are 102 empty boxes altogether.
How many boxes are there in all?
4. Allison has an incredible coin machine. When she puts in a nickel, it
gives back five pennies, and when she puts in a penny, it gives back five
nickels. If she starts with just one penny, is it possible that she will
ever have the same number of pennies and nickels?
5. The first quadrant is decomposed into squares for the following game.
Some of these squares are occupied by counters. A position with coun-
ters may be transformed to another position according to the following
rule: if the neighboring squares to the right and above a counter are
both free, it is possible to remove the counter and replace it with coun-
ters at both these free squares. The goal is to have all the shaded
squares free of counters. Is it possible to reach this goal if the initial
position has just one counter in the lower left hand corner?
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Exotic Arithmetic, 2020 Invariants
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6. On the island of Camelot live 45 chameleons 13 of which are grey, 15
of which are brown, and 17 of which are crimson. If two chameleons
of different colors meet, they both simultaneously change to the third
color. Is it possible that they will all eventually be the same color?
7. There are 1997 nonzero real numbers written on a blackboard. An
operation consists of choosing any two of these, a and b, erasing them,
and writing a + 2b and b − a2 in their places. Prove that no sequence of
operations can return the set of numbers to the original set.
8. You have three piles of stones containing 5, 49, and 51 stones. You can
join any two piles together into one pile and you can divide a pile with
an even number of stones into two piles of equal size. Can you ever
achieve 105 piles each with one stone?
9. A collection S of numbers is defined as follows:
(a) 2 is in S,
(b) if n is in S, then 2n is also in S, and
(c) if n is in S, then n + 7 is also in S.
(d) no other numbers belong to S.
What is the smallest number larger than 2004 which is NOT in S?
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Exotic Arithmetic, 2020 Invariants
10. Define a function f as follows:
(
n − 12 if n > 25
f (n) =
2n n ≤ 25
How many of the first 1000 positive integers have the property that
f k (n) = f ◦ f ◦ . . . ◦ f (n) = 16?
11. The numbers 1 to 100 are written on a board. Select a pair of them
a and b, and replace them with a + b + ab. After 99 moves of this
type, there is just one number left. What can it be? Must it be that
number. For this operation and those below, check for associativity
and commutativity.
(a) a ⊗ b = a + b + 1.
ab
(b) a ⊗ b = a+b
.
√
(c) a ⊗ b = a2 + b 2 .
(d) a ⊗ b = log(10a + 10b ).
(e) a ⊗ b = (a − 1)(b − 1) + 1.
(f) a ⊗ b = (a + 1)(b + 1) − 1.
(g) a ⊗ b = gcd(a + 1, b + 1) − 1.
12. A game is played with tokens according to the following rule. In each
round, the player with the most tokens gives one token to each of the
other players and also places one token in the discard pile. The game
ends when some player runs out of tokens. Players A, B, and C start
with 15, 14, and 13 tokens, respectively. How many rounds will there
be in the game?
