2023 EGMO

Part of EGMO

Subcontests

(5)

1

a MID(point) geo problem

We are given an acute triangle

ABC

D

be the point on its circumcircle such that

AD

is a diameter. Suppose that points

K

L

lie on segments

AB

AC

, respectively, and that

DK

DL

are tangent to circle

AKL

. Show that line

KL

passes through the orthocenter of triangle

ABC

1

mods with a twist

We are given a positive integer

s \ge 2

. For each positive integer

k

, we define its twist

k’

as follows: write

k

as+b

a, b

are non-negative integers and

b < s

k’ = bs+a

. For the positive integer

n

, consider the infinite sequence

d_1, d_2, \dots

d_1=n

d_{i+1}

is the twist of

d_i

for each positive integer

i

. Prove that this sequence contains

1

if and only if the remainder when

n

s^2-1

1

s

1

Bananagrams

k

be a positive integer. Lexi has a dictionary

\mathbb{D}

consisting of some

k

-letter strings containing only the letters

A

B

. Lexi would like to write either the letter

A

B

in each cell of a

k \times k

grid so that each column contains a string from

\mathbb{D}

when read from top-to-bottom and each row contains a string from

\mathbb{D}

when read from left-to-right. What is the smallest integer

m

\mathbb{D}

contains at least

m

different strings, then Lexi can fill her grid in this manner, no matter what strings are in

\mathbb{D}

1

A Familiar Point

ABC

be a triangle with circumcircle

\Omega

S_b

S_c

respectively denote the midpoints of the arcs

AC

AB

that do not contain the third vertex. Let

N_a

denote the midpoint of arc

BAC

BC

A

I

be the incenter of

ABC

\omega_b

be the circle that is tangent to

AB

and internally tangent to

\Omega

S_b

\omega_c

be the circle that is tangent to

AC

and internally tangent to

\Omega

S_c

. Show that the line

IN_a

, and the lines through the intersections of

\omega_b

\omega_c

\Omega

1

Sequence Gets Ratio’d

n \ge 3

positive real numbers

a_1, a_2, \dots, a_n

1 \le i \le n

b_i = \frac{a_{i-1} + a_{i+1}}{a_i}

(here we define

a_0

a_n

a_{n+1}

a_1

). Assume that for all

i

j

1

n

a_i \le a_j

b_i \le b_j

a_1 = a_2 = \dots = a_n