MathDB

1

Yet another guinea pig

110

guinea pigs for each of the

110

species, arranging as a

110\times 110

array. Find the maximum integer

n

such that, no matter how the guinea pigs align, we can always find a column or a row of

110

guinea pigs containing at least

n

different species.

N

1

Not eventually odd and not eventually even

For each positive integer

n

, define

V_n=\lfloor 2^n\sqrt{2020}\rfloor+\lfloor 2^n\sqrt{2021}\rfloor

. Prove that, in the sequence

V_1,V_2,\ldots,

there are infinitely many odd integers, as well as infinitely many even integers.
Remark.

\lfloor x\rfloor

is the largest integer that does not exceed the real number

x

.

6

1

Rotating Red Points on Circles

Let

n

be a positive integer and

N=n^{2021}

. There are

2021

concentric circles centered at

O

, and

N

equally-spaced rays are emitted from point

O

. Among the

2021N

intersections of the circles and the rays, some are painted red while the others remain unpainted.
It is known that, no matter how one intersection point from each circle is chosen, there is an angle

\theta

such that after a rotation of

\theta

with respect to

O

, all chosen points are moved to red points. Prove that the minimum number of red points is

2021n^{2020}

.
[I]Proposed by usjl.

4

1

Maximum of Minimum Nonzero Sum

Let

n

be a positive integer. For each

4n

-tuple of nonnegative real numbers

a_1,\ldots,a_{2n}

,

b_1,\ldots,b_{2n}

that satisfy

\sum_{i=1}^{2n}a_i=\sum_{j=1}^{2n}b_j=n

, define the sets

A:=\left\{\sum_{j=1}^{2n}\frac{a_ib_j}{a_ib_j+1}:i\in\{1,\ldots,2n\} \textup{ s.t. }\sum_{j=1}^{2n}\frac{a_ib_j}{a_ib_j+1}\neq 0\right\},

B:=\left\{\sum_{i=1}^{2n}\frac{a_ib_j}{a_ib_j+1}:j\in\{1,\ldots,2n\} \textup{ s.t. }\sum_{i=1}^{2n}\frac{a_ib_j}{a_ib_j+1}\neq 0\right\}.

Let

m

be the minimum element of

A\cup B

. Determine the maximum value of

m

among those derived from all such

4n

-tuples

a_1,\ldots,a_{2n},b_1,\ldots,b_{2n}

.
[I]Proposed by usjl.

3

1

Easy pudding equation in Taiwan TST

Find all triples

(x, y, z)

of positive integers such that

x^2 + 4^y = 5^z.

Proposed by Li4 and ltf0501

C

1

Simple Grid Dividing Game

Let

n

and

k

be positive integers satisfying

k\leq2n^2

. Lee and Sunny play a game with a

2n\times2n

grid paper. First, Lee writes a non-negative real number no greater than

1

in each of the cells, so that the sum of all numbers on the paper is

k

. Then, Sunny divides the paper into few pieces such that each piece is constructed by several complete and connected cells, and the sum of all numbers on each piece is at most

1

. There are no restrictions on the shape of each piece. (Cells are connected if they share a common edge.) Let

M

be the number of pieces. Lee wants to maximize

M

, while Sunny wants to minimize

M

. Find the value of

M

when Lee and Sunny both play optimally.

G

1

Yet Another Beautiful OI Geometry Problem

Let

ABC

be a triangle with incenter

I

and circumcircle

\Omega

. A point

X

on

\Omega

which is different from

A

satisfies

AI=XI

. The incircle touches

AC

and

AB

at

E, F

, respectively. Let

M_a, M_b, M_c

be the midpoints of sides

BC, CA, AB

, respectively. Let

T

be the intersection of the lines

M_bF

and

M_cE

. Suppose that

AT

intersects

\Omega

again at a point

S

.
Prove that

X, M_a, S, T

are concyclic.
Proposed by ltf0501 and Li4

2021 Taiwan TST Round 1

Subcontests

Yet another guinea pig

Not eventually odd and not eventually even

Rotating Red Points on Circles

Maximum of Minimum Nonzero Sum

Easy pudding equation in Taiwan TST

Simple Grid Dividing Game

Yet Another Beautiful OI Geometry Problem