5

Part of ICMC 6

Problems(2)

Continuous function with arbitrarily many intersections

Source: ICMC 2023 Round 1 P5

[0, 1]

\{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

n

times?
Proposed by Ethan Tan

college contestsreal analysiscontinuous functionfunctionICMC

Expected area on a clock

Source: ICMC 2023 Round 2 P5

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

1

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

geometrycombinatoricsICMCcollege contestsexpected valuearea of a triangle