MathDB

Two sequences are equal

Source: Russian TST 2019, Day 7 P1 (Group NG), P2 (Groups A & B)

3/27/2023

Let

a_0, a_1, \ldots , a_n

and

b_0, b_1, \ldots , b_n

be sequences of real numbers such that

a_0 = b_0 \geqslant 0

,

a_n = b_n > 0

and a_i=\sqrt{\frac{a_{i+1}+a_{i-1}}{2}}, b_i=\sqrt{\frac{b_{i+1}+b_{i-1}}{2}},for all

i=1,\ldots,n-1

. Prove that

a_1 = b_1

.

algebraSequences

Schedule of a school

Source: Russian TST 2019, Day 5 P1

3/22/2023

A school organizes optional lectures for 200 students. At least 10 students have signed up for each proposed lecture, and for any two students there is at most one lecture that both of them have signed up for. Prove that it is possible to hold all these lectures over 211 days so that no one has to attend two lectures in one day.

combinatoricsgraph theory

Inequality with divisors

Source: Russian TST 2019, Day 7 P1 (Groups A & B)

3/27/2023

A positive integer

n{}

is called discontinuous if for all its natural divisors

1 = d_0 < d_1 <\cdots<d_k

, written out in ascending order, there exists

1 \leqslant i \leqslant k

such that

d_i > d_{i-1}+\cdots+d_1+d_0+1

. Prove that there are infinitely many positive integers

n{}

such that

n,n+1,\ldots,n+2019

are all discontinuous.

number theoryDivisors

Pentagon geometry

Source: Russian TST 2019, Day 8 P1 (Groups A & B)

3/27/2023

A convex pentagon

APBCQ

is given such that

AB < AC

. The circle

\omega

centered at point

A{}

passes through

P{}

and

Q{}

and touches the segment

BC

at point

R{}

. Let the circle

\Gamma

centered at the point

O{}

be the circumcircle of the triangle

ABC

. It is known that

AO \perp P Q

and

\angle BQR = \angle CP R

. Prove that the tangents at points

P{}

and

Q{}

to the circle

\omega

intersect on

\Gamma

.

geometrypentagon

Isosceles triangle geo

Source: Russian TST 2019, Day 10 P1 (Group NG), P2 (Groups A & B)

3/27/2023

Point

M{}

is the middle of the side side

AB

of the isosceles triangle

ABC

. On the extension of the base

AC

, point

D{}

is marked such that

C{}

is between

A{}

and

D{}

, and point

E{}

is marked on the segment

BM

. The circumcircle of the triangle

CDE

intersects the segment

ME

a second time at point

F

. Prove that it is possible to make a triangle from the segments

AD, DE

and

AF

.

geometry

Graph problem

Source: Russian TST 2019, Day 9 P1 (Groups A & B)

3/27/2023

The shores of the Tvertsy River are two parallel straight lines. There are point-like villages on the shores in some order: 20 villages on the left shore and 15 villages on the right shore. We want to build a system of non-intersecting bridges, that is, segments connecting a couple of villages from different shores, so that from any village you can get to any other village only by bridges (you can't walk along the shore). In how many ways can such a bridge system be built?

graph theorycombinatorics

P1

Problems(6)