2020 Turkey EGMO TST

Part of Turkey EGMO TST

Subcontests

(6)

1

A geometry about concyclic points

A, B, C, D, E

\Gamma

cycle clockwise.

[AE \cap [CD = \{M\}

[AB \cap [DC = \{N\}

. The line parallels to

EC

and passes through

M

intersects with the line parallels to

BC

and passes through

N

K

. Similarly, the line parallels to

ED

and passes through

M

intersects with the line parallels to

BD

and passes through

N

L

. Show that the lines

LD

KC

\Gamma

1

A combi with painting chess table

Every square of a

2020 \times 2020

chess table is painted in red or white. For every two columns and two rows, at least two of the intersection squares satisfies that they are in the same column or row and they are painted in the same color. Find the least value of number of columns and rows that are completely painted in one color.

1

Max of (xy+1)(yz+1)(zx+1)

x,y,z

are positive real numbers such that:

xyz+x+y+z=6

xyz+2xy+yz+zx+z=10

Find the maximum value of:

(xy+1)(yz+1)(zx+1)

1

Labelled boxes

33!

empty boxes labeled from

1

33!

. In every move, we find the empty box with the smallest label, then we transfer

1

ball from every box with a smaller label and we add an additional

1

ball to that box. Find the smallest labeled non-empty box and the number of the balls in it after

33!

1

An equation of a function about prime divisors

p(m)

is the number of distinct prime divisors of a positive integer

m>1

f(m)

\bigg \lfloor \frac{p(m)+1}{2}\bigg \rfloor

th smallest prime divisor of

m

. Find all positive integers

n

satisfying the equation:

f(n^2+2) + f(n^2+5) = 2n-4

1

Proving AH is tangent to MNH

H

is the orthocenter of a non-isosceles acute triangle

\triangle ABC

M

is the midpoint of

BC

BB_1, CC_1

are two altitudes of

\triangle ABC

N

is the midpoint of

B_1C_1

AH

is tangent to the circumcircle of

\triangle MNH