2024 CMIMC Team

Part of 2024 CMIMC

Subcontests

(10)

1

2024 Team Problem 10

ABCD

has side length

2

0 \leq r \leq 2

P_r

\overline{AB}

AP_r = r

\Sigma_r

is constructed with diagonal

\overline{DP_r}

\mathcal{R}

be the set of all points that are in both

\Sigma_0

\Sigma_2

\Sigma_r

for at least one value of

r

. Find the area of the convex hull of

\mathcal{R}

.
Proposed by Justin Hsieh

1

2024 Team Problem 9

Suppose we have a cubic polynomial

p(x)

p(0)=0,p(1)=1,

p(x)\leq \sqrt x

0\leq x \leq 1.

p(0.5)

is maximized. What is the sum of

p(0.25)+p(0.75)?

Proposed by Ishin Shah

1

2024 Team Problem 8

\frac{(1-\tan10^\circ)(1-\tan 20^\circ)(1-\tan30^\circ)(1-\tan40^\circ)}{(1-\tan5^\circ)(1-\tan 15^\circ)(1-\tan25^\circ)(1-\tan35^\circ)}.

Proposed by Connor Gordon

1

2024 Team Problem 7

In the national math league, there are

7

teams. Their season is a round robin format, where each team plays other. Find the number of ways the games could go such that they have equal number of wins.
Proposed by Ishin Shah

1

2024 Team Problem 6

Cyclic quadrilateral

ABCD

has circumradius

3

. Additionally,

AC = 3\sqrt{2}

AB/CD = 2/3

AD = BD

CD

.
Proposed by Justin Hsieh

1

2024 Team Problem 5

An ant is currently on a vertex of the top face on a 6-sided die. The ant wants to travel to the opposite vertex of the die (the vertex that is farthest from the start), and the ant can travel along edges of the die to other vertices that are on the top face of the die.
Every second, the ant picks a valid edge to move along, and the die randomly flips to an adjacent face. If the ant is on any of the bottom vertices after the flip, it is crushed and dies. What is the probability that the ant makes it to its target? (If the ant makes it to the target and the die rolls to crush it, it achieved its dreams before dying, so this counts.)
Proposed by Lohith Tummala

1

2024 Team Problem 4

Eric and Christina are playing a game with

n

stones. They alternate taking some number of stones from the pile, with Eric going first. The number of stones Eric takes from the pile must be a power of

3

(e.g. 1, 3, 9, 27, ...), while the number of stones Christina takes must be a power of

2

(e.g. 1, 2, 4, 8, ...). Whoever takes the last stone wins. Find the sum of all

1\leq n \leq 100

for which Eric has a winning strategy.
Proposed by Connor Gordon

1

2024 Team Problem 3

Define a function

f: \mathbb{N} \rightarrow \mathbb{N}

f(x)=(x+1)!-x!

. Find the number of positive integers

x<49

f(x)

f(49)

.
Proposed by David Tang

1

2024 Team Problem 2

\triangle ABC

be equilateral. Let

D

be the midpoint of side

AC,

DEFG

be a square such that

D, F, B

are collinear and

E,G

AB,CB

respectively. What fraction of the area of

\triangle ABC

is covered by square

DEFG?

Proposed by Lohith Tummala

1

2024 Team Problem 1

x

\sqrt{x + 1}+ \sqrt{x} = 5.

Proposed by Eric Oh