Alice & Bob Corner 2014-2015
Problems curated by Padraig Condon
Figure 1: http://xkcd.com/356
Problems submitted by memebers attributed where necessary
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Problem Solvers Week 1
1. A monk walks up a mountain, taking 12 hours to complete the trip. He
then sleeps for 12 hours and then descends from the top taking 12 hours
more. Prove at some point on both days he was at exactly the same height
at exactly the same time.
2. Prove that
1
1
1
1
1
+
+
+ ··· +
=
log2 N
log3 N
log4 N
log100 N
log100! N
3. Which of the expressions,
(1 + x2 x3 )1000 or (1 x2 + x3 )1000
will have the larger coefficient for x20 after expansion and collecting of
terms?
R 1 ln x
4. 0 1+x
2 dx
5. The integer A consists of 666 threes, and the integer B has 666 sixes.
What digits appear in the product A ⇥ B?
6. Prove the following interesting property of the natural numbers; For all
numbers N there exist integers x and y such that xy + y x = N as in the
case where N is 17 we have 23 + 32 = 17.
7. Which integers have the following property? If the final digit is deleted,
the integer is divisible by the new number.
8. Bonus Problem (Daron Anderson)
Identify the following (finite) integer sequence:
7, 9, 10, 12, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32,
33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 47, 48, 49, 50, 52, 54, 55, 59, 64
A clue would be to think about the Japanese cartoons you might know.
Problem Solvers Week 2
1. (Aran Nolan)
Suppose we choose two random numbers a and b and then we raise a
to the power of b to get a new number. We add up all of the digits in
this number to get a newer number. Then we add up all of the digits in
this newer number to get a new newer number. We continue this process
until we are left with a number that is just a single digit. What is the
probability that digit is 1? For example
73829
29
11
2
2. How can we put 100 apples into two bags such that one bag has twice as
many as the other bag. None of the apples can be harmed.
3. Explain the following partition of the nonnegative integers; A = {0, 3, 6, 8, 9 . . . },
B = {1, 4, 7, 11, 14 . . . }, C = {2, 5, 10, 12 . . . }.
What are the relative sizes of these sets?
4. (IV Paper 1992)
Let a1 , a2 , a3 ... be the sequence of positive integers that have no “9”s in
their decimal representation. Show that this series
1
1
1
+
+
a1
a2
a3
converges.
5. What is the most number of parts we can divide the plane into with n
lines?
R⇡
6. Compute the integral 02 cos2 (cos x) + sin2 (sin x)dx
7. Prove there exist two positive irrational numbers s and t such that st is
rational.
8. On average how many times must one roll a die until all of the digits have
come up at least once?
9. Define a bijection between (0,1] and (0,1).
10. If we start filling in real entries to an n x n matrix (to any slot), at what
point can we stop and guarantee that no matter what remaining entries
are filled in, the determinant will be nonzero? Can we beat n+1
2 ?
11. (Emma Howard)
Suppose we have a bunny placed at some integer along the real line. It
takes some integer jump j to the left or to the right. And then continues
taking that jump size in the same direction. Every time it takes one jump,
we get to drop a bomb somewhere on the number line. Can we hit the
bunny with a finite number of bombs?
12. (Tim Murphy)
There is a rabbit is in the middle of a circular pond. A poacher is on the
edge of the pond. The poacher can run 4-times as fast as the rabbit can
swim. Can the rabbit get away ?
13.
Rb
a
x
b
ea
ex
x
dx
14. Derive the determinant of the Vandermonde matrix
0
1
1
1
···
1
1
B a1
a2
···
an
an+1 C
B
C
B ..
..
.
.. C
..
..
B .
C
.
.
.
B n 1
C
n 1
n 1
n A
@a
a
·
·
·
a
a
n
n
1
2
an1
an2
···
ann
ann+1
without using induction.
15. (Vladimir Dotsenko)
Show that the system of equations
z1 + 2z2 + ... + nzn = 0,
z1 2 + 2z2 2 + ... + nzn 2 = 0,
...
n
n
z1 + 2z2 + ... + nzn n = 0.
has the only complex solution z1 = · · · = zn = 0.
16. Alice and Bob play the following game with a monic polynomial with zero
coe↵cient equal to 1; They take it in turns to fill in the coefficients of
all the x powers between 2 and n-1. If the resulting polynomial has no
real root, Alice wins; otherwise, Bob wins. If Alice goes first, who has a
winning strategy?
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Problem Solvers Week 3
1. (Mark Allen)
If a rabbit in the centre of a circular pond can swim with speed v, what
is the maximum speed a poacher on the edge of the pond could have so
that the rabbit can still escape?
2. On average how often does one need to roll a die before a 6 comes up?
3. How likely is it that a drunkard’s walk (stumbles left or right with probability half) beginning at 0 on the number line will reach ±2n before
returning to 0?
4. (Karl Downey)
What is the average number of random steps a bug on a vertex of a cube
must take to get to the opposite vertex?
5. What’s the most likely; Rolling 6 dice and getting 1 or more sixes, rolling
12 dice and getting 2 or more, or rolling 18 dice and getting 3 or more?
Explain.
6. Alice and Bob play the following game based on the total roll of two
standard dice. Alice says that a 12 will be rolled first. Bob says that two
consecutive 7s will be rolled first. They keep rolling until one of them
wins. Who is more likely to win?
7. What is the largest number that can be written in standard mathematical
notation, using only three decimal digits and no other symbols?
Hint; Reference the works of James Joyce and Donald Knuth
8. 1, z, z 2 , and z 3 (all distinct) lie on the same circle on the complex plane.
What is the centre of that circle?
9. Can we rearrange the digits in a power of two to get a di↵erent power of
two?
10. Prove sin x  x for all x.
11. Give an intuitive proof as to why the derivative of sin x is cos x.
12. Consider the equation xn = e in a finite group G. Prove the number of
elements that satisfy this equation must be a divisor of the order of the
group.
13. The smallest distance between any two of six towns is d kilometres. The
largest distance
between any two of the towns is D kilometres. Show
p
that D
3. Assume the land is flat.
d
R1 2
14. 0 xlog x1 dx
15.
16.
R1 p
3
0
R 1R 1
0 0
2x3
3x2
1
1
xy
x + 1dx
dxdy
17. (Sean Di✏ey) The positive integers are partitioned into two sets:
f(1), f(2), f(3) ...
g(1), g(2), g(3) ...
With f(1) < f(2) < ...
And g(1) < g(2) < ...
And g(n) = f(f(n)) + 1
Find f(240)
18. (Keith Glennon)
Construct an explicit one-parameter family of homotopic curves between
the top half and bottom half of the unit circle, and a continuous oneparameter family of Riemannian metrics such that at any fixed value of
the parameter the homotopic curve is a geodesic for that metric
2
Problem Solvers Week 4
1. A family has 2 children. One of them is a boy. What is the probability
the other is a boy?
2. A family has 2 children. One of them is a boy called Dante. What is the
probability the other is a boy?
3. (Conor McMeel)
Suppose we have an n-sided die, and you keep rolling until you roll a
number thats less than or equal to a previous roll. Whats the expected
number of times you’ll roll? What happens as n goes to infinity?
4. Suppose there is an election. Alice and Bob run against each other. Alice
wins with a votes. Bob only got b votes. Suppose we take the voting
ballots out of the ballot box at random and add up their scores as we go
along, what is the probability at some point they’ll be drawing?
Pi
5. Show k=0 ki = 2i
6. (Owen Ward)
Consider a 3 ⇥ 3 matrix of 1s and -1s. The probability an entry is 1 is p.
What is the probability that the matrix is singular and for what p is it
minimised?
P1
1
7.
n=0 1+n2
8.
3n
3
=3
n
1
+6n
n
2
+ n2
9. (Ewan Dalby)
Can you find a permutation of the natural numbers
P1
(n)
sum over n of n=1 1(n) = c⇡ where c is rational
(n) such that the
Problem Solvers Week 5
1. Suppose a rectangular closed box has surface area S and volume V. Prove
that
S V 2/3
and hence or otherwise prove that, for closed boxes of given surface area,
the one with the largest volume is a cube.
2. (Conor Davis)
Alice and Bob have 8 barrels of wine for a party that they are hosting
tomorrow. Eve puts poison into one of their barrels (they do not know
which one). Bob has 3 pet mice which can drink from the barrels. If a
mouse drinks from the poisoned barrel it will die just before the party
tomorrow. Using the mice to determine which barrels are not poisoned,
what is the maximum number of barrels that Alice and Bob can safely
serve at their party tomorrow?
3. I choose an integer from 0 through 15. You ask me 7 yes or no questions. I
answer them all, but I am allowed to lie once. (I needn’t but I am allowed
to.) Determine my number!
4. Compute
lim sin sin . . . sin n,
{z
}
n!1 |
n compositions
5. Prove that for each positive integer n the following fraction is in lowest
terms
26n + 3
65n + 8
6. Using a wooden ball, a sheet of paper, a compass, a ruler, and a pencil, see
if you can draw a line segment on the paper equal to the ball’s diameter.
7. Batter A has a higher hitting batting average than batter B for the first
half of the season. A also has a higher batting average than B for teh
second hal↵ of the season. Does it follow that A has a better batting
average than B for teh whole season?
8. (Putnam Exam, 1968)
Prove that ⇡ < 22/7.
(Hint: Prove that 22/7 - < ⇡ =
R1
0
[x4 (1
x)4 /(1 + x2 )]dx
9. (IVs 1999)
Evaluate
f (m, n) =
as a function of m and n only.
Z
1
xm (1
x)n dx
0
10. Devise an experiment which uses only tosses of a fair coin, but which
has success probability 1/3. Do the same for any success probability p,
0  p  1.
2
Problem Solvers Week 6
1. I choose a number which is either 1, 2 or 3. You are allowed ask me one
question which I will answer truthfully as Yes, No, or I don’t know. Can
you determine my number?
p
2. Prove that n 2 is irrational for all n
3 (Hint, consider using Fermat’s
Last Theorem)
3. If a bottle, partly filled with liquid, has a round, square, or rectangular
bottom which is flat, can you find its volume using only a ruler? You may
not add or pour out liquid.
4. At each plane lattice point there is placed a positive number in such a way
that each is the average of its four nearest neighbours. Show that all the
numbers are the same.
5. (Superbrain, 2009)
Whilst on holiday you go to observe a famous statue of height s on top of
a pedestal of height p. Your eyes are a height h above the ground, where
h < p. What distance l should you stand from the pedestal to ensure the
best view? (Part of the question is to work out what is meant by the best
view!)
6. A rectangle is divided into many smaller rectangles, each of which has the
property that at least one pair of sides has integer length. Prove that the
original rectangle also has this property.
7. (Superbrain 2014)
Given seven dwarves of distinct heights, in how many ways can they be
lined up so that the changes in height alternate (i.e. smaller, larger,
smaller, larger, smaller, larger, or instead larger, smaller, larger, smaller,
larger, smaller)?
1
8. Let x be a real number. Which is greater sin(cos(x)) or cos(sin(x))?
9. Estimate the size of the largest subset of {1, 2, 3, ..., n} which doesn’t contain any term equal to double another term.
2
Problem Solvers Week 7
1. Is it more efficient to put a square into a circle, or a circle into a square?
By efficient we mean occupy a greater percentage of the space on o↵er.
2. Is it more efficient to put an n-gon into a m-gon, or a m-gon into a n-gon,
where m < n?
3. Suppose Alice leaves Dublin for Galway at the same time Bob leaves Galway for Dublin. They travel at constant speed along the same route and
meet each other at noon. If Alice reaches Galway at 4pm and Bob reaches
Dublin at 9pm what time did they set out?
4. What is the largest (in terms of area) equilateral triangle we can fit into
a square?
5. (Tim Murphy)
Using non-overlapping circles of any size, is there a limit to the density
with which you can fill a given square?
6. Find a way to draw a line that spilits 100,000 random points in the plane
into two groups, each of 50,000.
7. (Brian Tyrrell)
Find all cases where reversing the digits of a2 gives 2a, for a 2 Z.
What about the general case for an ?
(find all cases where reversing the digits of an gives na, for a 2 Z.)
8. Pick a random number between 0 and 1. Then keep picking random
numbers so long as they keep decreasing, and stop when the trend is
broken. On average how many numbers will you have picked?
9. What is the smallest positive value of 33m 7n , where m and n are positive
integers?
10. For a positive integer n, let f(n) denote the number of ways of expressing
n as a sum of exact nonnegative powers of 3. So for exammple f(4) = 2,
since we can write
4 = 30 + 30 + 30 + 30 or 4 = 30 + 31
1
Prove that for any positive integer n,
f (3n) = f (3n
1) + f (n)
11. How many digits does the number 125100 have?
12. Find all functions F(x) : R ! R having the property that for any x1 and
x2 the following inequality holds:
F (x1 )
F (x2 )  (x1
x2 ) 2
13. There are n cities in a given country (where n > 3), with each pair of cities
connected by a road not passing through any other city. Show that the
number of routes between any pair of cities that does not revisit any city
on the way is b(n 2)!ec, the integer part of (n - 2)!e.
2
Problem Solvers Week 8
1. If there are 2n teams in a football tournament, and it’s done by knockout,
how many matches are there in the tournament?
2. Prove in every round robin tournament (everbody plays everybody) there
is always a player p who for any other player q, has either beaten q or
beaten someone who has beaten q.
3. Can you divide the face of a watch with 2 straight lines so that the sums
of the numbers in each part are equal?
4. If you’re trapped in a circular cage with somebody who can run at precisely
the same speed as you, can you always catch them?
5. For what positive x is the x-th root of x the greatest?
6. Alice, Bob and Eve compete in a 100m race and run at constant speeds.
Alice beats Bob by 10 metres and Bob beats Eve by 10 metres. How many
metres does Alice beat Eve by?
7. Prove
✓ ◆
✓ ◆
✓ ◆
✓ ◆
n
n
n
n
+2
+3
+ ··· + n
= n2n
1
2
3
n
1
8. In a lab, 2n students are present for a class, and the teacher needs to pair
them up for performing a certain experiment. How many ways are there
to do that?
9. Show that
(2n)!
2n n!
is an integer.
10. Show that
(kn)!
kn n!
is an integer.
11. Two men are located at opposite ends of a mountain range, at the same
elevation. If the mountain range never drops below this starting elevation,
is it possible for the two men to walk along the mountain range and reach
each other’s starting place, while always staying at the same elevation?
12. What is the maximum number of spots you can paint onto a football so
that all the spots are the same distance from each other?
13. Prove that
lim n2
n!1
Z
1/n
xx+1 dx = 1/2
0
1
Problem Solvers Week 9
1. If a set of positive integers has sum n, what is the biggest its product can
be?
2. Prove that all positive integers except the powers of two can be written
as the sum of at least two consecutive positive integers
3. What is the expected number of coin flips to get three consecutive heads?
4. You have 3 nested triangles, each circumscribed inside the larger. The
edges of the largest are parallel to those of the smallest. Show that the
area of the largest squared added to the area of the smallest squared is
equal to the area of the middle one squared.
5. A palindrome is a number or word that is the same when read forward and
backward, for example, ”176671” and ”civic.” Can the number obtained
by writing the numbers from 1 to n in order (for some n > 1 ) be a
palindrome?
6. Show that for any positive integer n, there exists a positive multiple of n
that contains only the digits 7 and 0.
7. Given a right angled triangle and a finite set of points inside it, prove that
there is a path connecting all the points so that the sum of the squares of
the lines segments is bounded by the length of the hypotenuse
8. Estimate the largest collection of triples one can choose from n elements
so that no two of them overlap in more than one element
9. Let f (x) = 1/(1 x) Let f r represent r iterations of f. What is f 1999 (2000)?
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