2020 Iran RMM TST

Part of Iran RMM TST

Subcontests

(6)

1

Fantabulous Combo

There are n stations

1,2,...,n

in a broken road (like in Cars) in that order such that the distance between station

i

i+1

is one unit. The distance betwen two positions of cars is the minimum units needed to be fixed so that every car can go from its place in the first position to its place in the second (two cars can be in the same station in a position). Prove that for every

\alpha<1

n

100^n

positions such that the distance of every two position is at least

n\alpha

1

Circle in triangle

\omega

is strictly inside triangle

ABC

. The tangents from

A

\omega

BC

A_1,A_2

B_1,B_2,C_1,C_2

similarly. Prove that if five of six points

A_1,A_2,B_1,B_2,C_1,C_2

lie on a circle the sixth one lie on the circle too.

1

a number theory problem that makes you want to count

p>3

1

3

8

prove that the number triples

(a,b,c), p=a^2+bc, 0<b<c<\sqrt{p}

is odd.
Proposed by Navid Safaie

1

Find Poly

n>1

. Find all polynomials with complex coefficient and degree more than one such that

(p(x)-x)^2

p^n(x)-x

p^0(x)=x , p^i(x)=p(p^{i-1}(x))

)
Proposed by Navid Safaie

1

9x9 table

9\times 9

table is filled with zeroes.In every step we can either take a row add

1

to every cell and shift it one unit to right or take a column reduce every cell by

1

and shift it one cell down. Can the table with the top right

-1

and bottom left

+1

and all other cells zero be reached?

1

Trapezoid And Segment

ABCD

AD

BC

E, F

AB, CD

A_1, C_1

AD,BC

A_1, E, F, A

lie on a circle and so do

C_1, E, F, C

. Prove that lines

A_1C_1, BD, EF

are concurrent.