2020 Turkey Team Selection Test

Part of Turkey Team Selection Test

Subcontests

(9)

1

Maximum value of festivities

66 dwarfs have a total of 111 hats. Each of the hats belongs to a dwarf and colored by 66 different colors. Festivities are organized where each of these dwarfs wears their own hat. There is no dwarf pair wearing the same colored hat in any of the festivities. For any two of the festivities, there exist a dwarf wearing a hat of a different color in these festivities. Find the maximum value of the number of festivities that can be organized.

1

Ordered list of students

There is at least one friend pair in a class of students with different names. Students in an ordered list of some of the students write the names of all their friends who are not currently written on the blackboard, in order. If each student on the list wrote at least one name on the board and the name of each student with at least one friend on the blackboard at the end of the process, call this list a

golden

list

. Prove that there exists a

golden

list

such that number of students in this list is even.

1

Number of equivalent's solutions

a,n

positive integers we show number of different integer 10-tuples

(x_1,x_2,...,x_{10})

(mod n)

x_1x_2...x_{10}=a (mod n)

f(a,n)

a,b

given positive integers , a) Prove that there exist a positive integer

c

such that for all

n\in \mathbb{Z^+}

\frac {f(a,cn)}{f(b,cn)}

is constant b) Find all

(a,b)

pairs such that minumum possible value of

c

c

satisfying condition in

(a)

1

Parallel chords

A_1,A_2,B_1,B_2,C_1,C_2

are points on a circle such that

A_1A_2 \parallel B_1B_2 \parallel C_1C_2

M

is a point on same circle

MA_1

B_2C_2

X

MB_1

A_2C_2

Y

MC_1

A_2B_2

Z

X , Y ,Z

1

Functional inequality

Z^+

be positive integers set.

f:\mathbb{Z^+}\to\mathbb{Z^+}

is a function and we show

f \circ f \circ ...\circ f

f_l

l\in \mathbb{Z^+}

f

l

times. Find all

f:\mathbb{Z^+}\to\mathbb{Z^+}

functions such that

(n-1)^{2020}< \prod _{l=1}^{2020} {f_l}(n)< n^{2020}+n^{2019}

n\in \mathbb{Z^+}

1

Tangential quadrilaterals

A_1A_2A_3A_4

is a tangential quadrilateral with perimeter

p_1

and sum of the diagonals

k_1

B_1B_2B_3B_4

is a tangential quadrilateral with perimeter

p_2

and sum of the diagonals

k_2

A_1A_2A_3A_4

B_1B_2B_3B_4

are congruent squares if

p_1^2+p_2^2=(k_1+k_2)^2

1

Minimum Value

x,y,z

be real numbers such that

0<x,y,z<1

. Find the minimum value of:

\frac {xyz(x+y+z)+(xy+yz+zx)(1-xyz)}{xyz\sqrt {1-xyz}}

1

p6 geometry

\triangle ABC

D

E

are respectively on

AB

AC

DE\parallel BC

P

is the intersection of

BE

CD

M

is the second intersection of

(APD)

(BCD)

N

is the second intersection of

(APE)

(BCE)

w

is the circle passing through

M

N

BC

. Prove that the lines tangent to

w

M

N

AP

1

Cubic equation

Find all pairs of

(a,b)

positive integers satisfying the equation:

\frac {a^3+b^3}{ab+4}=2020