MathDB

1

Solve this in 10 minutes!

ABC

and

A'B'C'

be two triangles so that the midpoints of

\overline{AA'}, \overline{BB'}, \overline{CC'}

form a triangle as well. Suppose that for any point

X

on the circumcircle of

ABC

, there exists exactly one point

X'

on the circumcircle of

A'B'C'

so that the midpoints of

\overline{AA'}, \overline{BB'}, \overline{CC'}

and

\overline{XX'}

are concyclic. Show that

ABC

is similar to

A'B'C'

.
Proposed by usjl

6

1

Sequence as Coefficients

Find all positive integers

n

and sequence of integers

a_0,a_1,\ldots, a_n

such that the following hold: 1.

a_n\neq 0

; 2.

f(a_{i-1})=a_i

for all

i=1,\ldots, n

, where

f(x) = a_nx^n+a_{n-1}x^{n-1}+\cdots +a_0

.
Proposed by usjl

2

1

Isogonal conjugate of the incircle

Let

I

be the incenter of triangle

ABC

, and let

\omega

be its incircle. Let

E

and

F

be the points of tangency of

\omega

with

CA

and

AB

, respectively. Let

X

and

Y

be the intersections of the circumcircle of

BIC

and

\omega

. Take a point

T

on

BC

such that

\angle AIT

is a right angle. Let

G

be the intersection of

EF

and

BC

, and let

Z

be the intersection of

XY

and

AT

. Prove that

AZ

,

ZG

, and

AI

form an isosceles triangle.
Proposed by Li4 and usjl.

G

1

Weird but easy concurrency

Let

ABC

be a triangle such that the angular bisector of

\angle BAC

, the

B

-median and the perpendicular bisector of

AB

intersect at a single point

X

. Let

H

be the orthocenter of

ABC

. Show that

\angle BXH = 90^{\circ}

.
Proposed by usjl

N

1

Combinatorial functional equation

For each positive integer

k

, define

r(k)

as the number of runs of

k

in base-

2

, where a run is a collection of consecutive

0

s or consecutive

1

s without a larger one containing it. For example,

(11100100)_2

has

4

runs, namely

111-00-1-00

. Also,

r(0) = 0

. Given a positive integer

n

, find all functions

f : \mathbb{Z} \rightarrow\mathbb{Z}

such that

\sum_{k=0}^{2^n-1} 2^{r(k)}f(k+(-1)^{k} x)=(-1)^{x+n}\text{ for all integer $x$.}

Proposed by YaWNeeT

C

1

Robots Guessing the Numbers on Their Foreheads

Dexter's Laboratory has

2024

robots, each with a program setup by Dexter. One day, his naughty sister Dee Dee intrudes and writes an integer in

\{1, 2, \dots, 113\}

on each of the robot's forehead. Each robot detects the numbers on all other robots' foreheads, and guess its own number base on its program, individually and simultaneously. Find the largest positive integer

k

such that Dexter can setup the programs so that, no matter how the numbers distribute, there are always at least

k

robots who guess their numbers right.
Proposed by sn6dh

2024 Taiwan TST Round 3

Subcontests

Solve this in 10 minutes!

Sequence as Coefficients

Isogonal conjugate of the incircle

Weird but easy concurrency

Combinatorial functional equation

Robots Guessing the Numbers on Their Foreheads