MathDB

1

Exponential sequence converges in fractional parts

a_1 = 2022

and

a_{n+1} = a_n + e^{-a_n}

for

n \geq 1

. Prove that there exists a positive real number

r

for which the sequence

\{ra_1\}, \{ra_{10}\}, \{ra_{100}\}, . . .

converges.
Note:

\{x \} = x - \lfloor x \rfloor

denotes the part of

x

after the decimal point.
Proposed by Ethan Tan

5

2

Continuous function with arbitrarily many intersections

Let

[0, 1]

be the set

\{x \in \mathbb{R} : 0 \leq x \leq 1\}

. Does there exist a continuous function

g : [0, 1] \to [0, 1]

such that no line intersects the graph of

g

infinitely many times, but for any positive integer

n

there is a line intersecting

g

more than

n

times?
Proposed by Ethan Tan

Expected area on a clock

A clock has an hour, minute, and second hand, all of length

1

. Let

T

be the triangle formed by the ends of these hands. A time of day is chosen uniformly at random. What is the expected value of the area of

T

?
Proposed by Dylan Toh

4

2

No two cycles share an edge

Let

\mathcal{G}

be a simple graph with

n

vertices and

m

edges such that no two cycles share an edge. Prove that

2m < 3n

.
Note: A simple graph is a graph with at most one edge between any two vertices and no edges from any vertex to itself. A cycle is a sequence of distinct vertices

v_1, \dots, v_n

such that there is an edge between any two consecutive vertices, and between

v_n

and

v_1

.
Proposed by Ethan Tan

Sum of divisors perfect square

Do there exist infinitely many positive integers

m

such that the sum of the positive divisors of

m

(including

m

itself) is a perfect square?
Proposed by Dylan Toh

3

2

Fibonacci hops in the Euclidean plane

Bugs Bunny plays a game in the Euclidean plane. At the

n

-th minute

(n \geq 1)

, Bugs Bunny hops a distance of

F_n

in the North, South, East, or West direction, where

F_n

is the

n

-th Fibonacci number (defined by

F_1 = F_2 =1

and

F_n = F_{n-1} + F_{n-2}

for

n \geq 3

). If the first two hops were perpendicular, prove that Bugs Bunny can never return to where he started.
Proposed by Dylan Toh

Erasing random numbers on a blackboard

The numbers

1, 2, \dots , n

are written on a blackboard and then erased via the following process: [*] Before any numbers are erased, a pair of numbers is chosen uniformly at random and circled. [*] Each minute for the next

n -1

minutes, a pair of numbers still on the blackboard is chosen uniformly at random and the smaller one is erased. [*] In minute

n

, the last number is erased.
What is the probability that the smaller circled number is erased before the larger?
Proposed by Ethan Tan

2

Derivative greater than function

Let

f: \mathbb{R} \to \mathbb{R}

be a differentiable function such that

f'(x) > f(x)>0

for all real numbers

x

. Show that

f(8) > 2022f(0)

.
Proposed by Ethan Tan

Tetrahedron volume at least 1

Show that if the distance between opposite edges of a tetrahedron is at least

1

, then its volume is at least

\frac{1}{3}

.
Proposed by Simeon Kiflie

1

2

Two lines divide square

Two straight lines divide a square of side length

1

into four regions. Show that at least one of the regions has a perimeter greater than or equal to

2

.
Proposed by Dylan Toh

RIP Atlantis

The city of Atlantis is built on an island represented by

[ -1, 1]

, with skyline initially given by

f(x) = 1 - |x|

. The sea level is currently

y=0

, but due to global warming, it is rising at a rate of

0.01

a year. For any position

-1 < x < 1

, while the building at

x

is not completely submerged, then it is instantaneously being built upward at a rate of

r

per year, where

r

is the distance (along the

x

-axis) from this building to the nearest completely submerged building. How long will it be until Atlantis becomes completely submerged?
Proposed by Ethan Tan

ICMC 6

Subcontests

Exponential sequence converges in fractional parts

Continuous function with arbitrarily many intersections

Expected area on a clock

No two cycles share an edge

Sum of divisors perfect square

Fibonacci hops in the Euclidean plane

Erasing random numbers on a blackboard

Derivative greater than function

Tetrahedron volume at least 1

Two lines divide square

RIP Atlantis