MathDB

1

Scanning for The Coin

Seyed has 998 white coins, a red coin, and an unusual coin with one red side and one white side. He can not see the color of the coins instead he has a scanner which checks if all of the coin sides touching the scanner glass are white. Is there any algorithm to find the red coin by using the scanner at most 17 times?
Proposed by Seyed Reza Hosseini

6

1

Triangular Grid Polynomial with Sides in Grid

Consider a triangular grid of equilateral triangles with unit sides. Assume that

\mathcal{P}

is a non-self-intersecting polygon with perimeter 1399 and sides from the grid. Prove that

\mathcal{P}

has either an internal or an external 120-degree angle.
Proposed by Seyed Hessam Firouzi

5

1

Stairway to Heaven

Abolf is on the second step of a stairway to heaven in every step of this stairway except the first one which is the hell there is a devil who is either a human, an elf or a demon and tempts Abolf. The devil in the second step is Satan himself as one of three forms. Whenever an elf or a demon tries to tempt Abolf he resists and walks one step up but when a human tempts Abolf he is deceived and hence he walks one step down. However if Abolf is deceived by Satan for the first time he resists and does not fall down to hell but the second time he falls down to eternal hell. Every time a devil makes a temptation it changes its form from a human, an elf, a demon to an elf, a demon, a human respectively. Prove that Abolf passes each step after some time.
Proposed by Yaser Ahmadi Fouladi

4

1

Finding a Star of Equal Degrees

Given a graph with

99

vertices and degrees in

\{81,82,\dots,90\}

, prove that there exist

10

vertices of this graph with equal degrees and a common neighbour.
Proposed by Alireza Alipour

3

1

Crossing ٍٍChords

1399

points and some chords between them is given.

a)

In every step we can take two chords

RS,PQ

with a common point other than

P,Q,R,S

and erase exactly one of

RS,PQ

and draw

PS,PR,QS,QR

let

s

be the minimum of chords after some steps. Find the maximum of

s

over all initial positions.

b)

In every step we can take two chords

RS,PQ

with a common point other than

P,Q,R,S

and erase both of

RS,PQ

and draw

PS,PR,QS,QR

let

s

be the minimum of chords after some steps. Find the maximum of

s

over all initial positions.
Proposed by Afrouz Jabalameli, Abolfazl Asadi

2

1

Strategy for Higher Winning Probability

Morteza and Amir Reza play the following game. First each of them independently roll a dice

100

times in a row to construct a

100

-digit number with digits

1,2,3,4,5,6

then they simultaneously shout a number from

1

to

100

and write down the corresponding digit to the number other person shouted in their

100

digit number. If both of the players write down

6

they both win otherwise they both loose. Do they have a strategy with wining chance more than

\frac{1}{36}

?
Proposed by Morteza Saghafian

1

On the Scores of Soccer Without Draw

In a soccer league with

2020

teams every two team have played exactly once and no game have lead to a draw. The participating teams are ordered first by their points (3 points for a win, 1 point for a draw, 0 points for a loss) then by their goal difference (goals scored minus goals against) in a normal soccer table. Is it possible for the goal difference in such table to be strictly increasing from the top to the bottom?
Proposed by Abolfazl Asadi

2020 Iranian Combinatorics Olympiad

Subcontests

Scanning for The Coin

Triangular Grid Polynomial with Sides in Grid

Stairway to Heaven

Finding a Star of Equal Degrees

Crossing ٍٍChords

Strategy for Higher Winning Probability

On the Scores of Soccer Without Draw