MathDB

1

Divisible FE

f:\mathbb{N}\to\mathbb{N}

satisfying that for all

m,n\in\mathbb{N}

, the nonnegative integer

|f(m+n)-f(m)|

is a divisor of

f(n)

.
Proposed by usjl

4

1

Cycle of Permutations

Let

k

be a positive integer, and set

n=2^k

,

N=\{1, 2, \cdots, n\}

. For any bijective function

f:N\rightarrow N

, if a set

A\subset N

contains an element

a\in A

such that

\{a, f(a), f(f(a)), \cdots\} = A

, then we call

A

as a cycle of

f

. Prove that: among all bijective functions

f:N\rightarrow N

, at least

\frac{n!}{2}

of them have number of cycles less than or equal to

2k-1

. Note: A function is bijective if and only if it is injective and surjective; in other words, it is 1-1 and onto.
Proposed by CSJL

6

1

Sum of Decimal Parts

For every positive integer

M \geq 2

, find the smallest real number

C_M

such that for any integers

a_1, a_2,\ldots , a_{2023}

, there always exist some integer

1 \leq k < M

such that

\left\{\frac{ka_1}{M}\right\}+\left\{\frac{ka_2}{M}\right\}+\cdots+\left\{\frac{ka_{2023}}{M}\right\}\leq C_M.

Here,

\{x\}

is the unique number in the interval

[0, 1)

such that

x - \{x\}

is an integer.
Proposed by usjl

1

An identity on the rationals

Let

\mathbb{Q}_{>1}

be the set of rational numbers greater than

1

. Let

f:\mathbb{Q}_{>1}\to \mathbb{Z}

be a function that satisfies

f(q)=\begin{cases} q-3&\textup{ if }q\textup{ is an integer,}\\ \lceil q\rceil-3+f\left(\frac{1}{\lceil q\rceil-q}\right)&\textup{ otherwise.} \end{cases}

Show that for any

a,b\in\mathbb{Q}_{>1}

with

\frac{1}{a}+\frac{1}{b}=1

, we have

f(a)+f(b)=-2

.
Proposed by usjl

C

1

Maximal Bipartite Graph without Perfect Matchings

There are

n

cities on each side of Hung river, with two-way ferry routes between some pairs of cities across the river. A city is “convenient” if and only if the city has ferry routes to all cities on the other side. The river is “clear” if we can find

n

different routes so that the end points of all these routes include all

2n

cities.
It is known that Hung river is currently unclear, but if we add any new route, then the river becomes clear. Determine all possible values for the number of convenient cities.
Proposed by usjl

G

2

A line that bisects two angles

Let

\Omega

be the circumcircle of an isosceles trapezoid

ABCD

, in which

AD

is parallel to

BC

. Let

X

be the reflection point of

D

with respect to

BC

. Point

Q

is on the arc

BC

of

\Omega

that does not contain

A

. Let

P

be the intersection of

DQ

and

BC

. A point

E

satisfies that

EQ

is parallel to

PX

, and

EQ

bisects

\angle BEC

. Prove that

EQ

also bisects

\angle AEP

.
Proposed by Li4.

Lines connecting centroids

Let

ABC

be a triangle. Let

ABC_1, BCA_1, CAB_1

be three equilateral triangles that do not overlap with

ABC

. Let

P

be the intersection of the circumcircles of triangle

ABC_1

and

CAB_1

. Let

Q

be the point on the circumcircle of triangle

CAB_1

so that

PQ

is parallel to

BA_1

. Let

R

be the point on the circumcircle of triangle

ABC_1

so that

PR

is parallel to

CA_1

.
Show that the line connecting the centroid of triangle

ABC

and the centroid of triangle

PQR

is parallel to

BC

.
Proposed by usjl

A

2

Standard FE

Let

f:\mathbb{N}\to\mathbb{R}_{>0}

be a given increasing function that takes positive values. For any pair

(m,n)

of positive integers, we call it disobedient if

f(mn)\neq f(m)f(n)

. For any positive integer

m

, we call it ultra-disobedient if for any nonnegative integer

N

, there are always infinitely many positive integers

n

satisfying that

(m,n), (m,n+1),\ldots,(m,n+N)

are all disobedient pairs.
Show that if there exists some disobedient pair, then there exists some ultra-disobedient positive integer.
Proposed by usjl

Polynomials with same degree

Given some monic polynomials

P_1, \ldots, P_n

with real coefficients, for any real number

y

, let

S_y

be the set of real number

x

such that

y = P_i(x)

for some

i = 1, 2, ..., n

. If the sets

S_{y_1}, S_{y_2}

have the same size for any two real numbers

y_1, y_2

, show that

P_1, \ldots, P_n

have the same degree.
Proposed by usjl

2023 Taiwan TST Round 1

Subcontests

Divisible FE

Cycle of Permutations

Sum of Decimal Parts

An identity on the rationals

Maximal Bipartite Graph without Perfect Matchings

A line that bisects two angles

Lines connecting centroids

Standard FE

Polynomials with same degree