2023 APMO

Part of APMO

Subcontests

(5)

1

R+ FE with arbitrary constant

c>0

be a given positive real and

\mathbb{R}_{>0}

be the set of all positive reals. Find all functions

f \colon \mathbb{R}_{>0} \to \mathbb{R}_{>0}

such that f((c+1)x+f(y))=f(x+2y)+2cx \textrm{for all } x,y \in \mathbb{R}_{>0}.

1

APMO 2023 P5

n

line segments on the plane, no three intersecting at a point, and each pair intersecting once in their respective interiors. Tony and his

2n - 1

friends each stand at a distinct endpoint of a line segment. Tony wishes to send Christmas presents to each of his friends as follows: First, he chooses an endpoint of each segment as a “sink”. Then he places the present at the endpoint of the segment he is at. The present moves as follows :

\bullet

If it is on a line segment, it moves towards the sink.

\bullet

When it reaches an intersection of two segments, it changes the line segment it travels on and starts moving towards the new sink. If the present reaches an endpoint, the friend on that endpoint can receive their present. Prove that Tony can send presents to exactly

n

2n - 1

1

APMO Number Theory

Find all integers

n

n \geq 2

\dfrac{\sigma(n)}{p(n)-1} = n

\sigma(n)

denotes the sum of all positive divisors of

n

p(n)

denotes the largest prime divisor of

n

1

APMO 2023 Problem 3

ABCD

be a parallelogram. Let

W, X, Y,

Z

be points on sides

AB, BC, CD,

DA

, respectively, such that the incenters of triangles

AWZ, BXW, CYX,

DZY

form a parallelogram. Prove that

WXYZ

is a parallelogram.

1

APMO 2023 Problem 1

n \geq 5

be an integer. Consider

n

squares with side lengths

1, 2, \dots , n

, respectively. The squares are arranged in the plane with their sides parallel to the

x

y

axes. Suppose that no two squares touch, except possibly at their vertices. Show that it is possible to arrange these squares in a way such that every square touches exactly two other squares.