MathDB

2019 Saint Petersburg Grade 11 P7

Source: Saint Petersburg 2019

4/14/2019

Let

\omega

and

O

be respectively the circumcircle and the circumcenter of a triangle

ABC

. The line

AO

intersects

\omega

second time at

A'

.

M_B

and

M_C

are the midpoints of

AC

and

AB

, respectively. The lines

A'M_B

and

A'M_C

intersect

\omega

secondly at points

B'

and

C

, and also intersect

BC

at points

D_B

and

D_C

, respectively. The circumcircles of

CD_BB'

and

BD_CC'

intersect at points

P

and

Q

. Prove that

O

,

P

,

Q

are collinear.
(М. Германсков)
Thanks to the user Vlados021 for translating the problem.

geometry

10^{4038} points in 10^{2019} x 10^{2019} square

Source: St. Petersburg 2019 10.7

5/1/2019

In a square

10^{2019} \times 10^{2019}, 10^{4038}

points are marked. Prove that there is such a rectangle with sides parallel to the sides of a square whose area differs from the number of points located in it by at least

6

.

combinatorial geometrycombinatoricsrectangle

2019 plates in a circle with one cake in each, numbers from 1-16

Source: St. Petersburg 2019 9.7

5/2/2019

In a circle there are

2019

plates, on each lies one cake. Petya and Vasya are playing a game. In one move, Petya points at a cake and calls number from

1

to

16

, and Vasya moves the specified cake to the specified number of check clockwise or counterclockwise (Vasya chooses the direction each time). Petya wants at least some

k

pastries to accumulate on one of the plates and Vasya wants to stop him. What is the largest

k

Petya can succeed?

combinatoricsgameminimumgame strategy

7

Problems(3)