MathDB

1

Kvant M2797 inequality

0 \leq a_1 \leq a_2 \leq ... \leq a_n

and

0 \leq b_1 \leq b_2 \leq ... \leq b_n

prove that

\left( \frac{a_1}{1 \cdot 2}+\frac{a_2}{2 \cdot 3}+...+\frac{a_n}{n(n+1)} \right) \times \left( \frac{b_1}{1 \cdot 2}+\frac{b_2}{2 \cdot 3}+...+\frac{b_n}{n(n+1)} \right) \leq \frac{a_1b_1}{1 \cdot 2}+\frac{a_2b_2}{2 \cdot 3}+...+\frac{a_nb_n}{n(n+1)}.

Proposed by A. Antropov

M2796

1

A lot of strips and cutting

Let's call a checkered polygon a strip, which can be traversed entirely, starting from some of its cells and then moving only in two directions - up or to the right. Several such strips can be inserted into each other by shifting by a vector

(-1.1)

. Prove that for any strip consisting of an even number of cells, there is such an odd

k

that if you combine

k

of the same strips by inserting them sequentially into each other, then the resulting polygon can be divided along the grid lines into two equal parts. Proposed by I. Markelov, S. Markelov

M2795

1

Rays from rational points aren't parallel

Is it possible to release a ray on a plane from each point with rational coordinates so that no two rays have a common point and at the same time, among the lines containing these rays, no two are parallel and do not coincide? Proposed by P. Kozhevnikov

M2794

1

Isogonal conjugate on (BIC)

The points

P

and

Q

lie inside the circle

\omega

. The perpendicular bisector to the segment

PQ

intersects

\omega

at points

A

and

D

. A circle centered on

D

passing through

P

and

Q

intersects

\omega

at points

B

and

C

. The segment

PQ

lies inside the triangle

ABC

. Prove that

\angle ACP = \angle BCQ

. Proposed by A. Zaslavsky

M2793

1

(GKH) and (ABC) are tangent

In acute triangle

ABC

(

AB<AC

) point

O

is center of its circumcircle

\Omega

. Let the tangent to

\Omega

drawn at point

A

intersect the line

BC

at point

D

. Let the line

DO

intersects the segments

AB

and

AC

at points

E

and

F

, respectively. Point

G

is constructed such that

AEGF

is a parallelogram. Let

K

and

H

be points of intersection of segment

BC

with segments

EG

and

FG

, respectively. Prove that the circle

(GKH)

touches the circle

\Omega

. Proposed by Dong Luu

M2792

1

The beetle crawls through the columns

There are

9

vertical columns in a row. In some places, horizontal sticks are inserted between adjacent columns, no two are at the same height. The beetle crawls from the bottom up; when he meets the wand, he crawls over it to the next column and continues to crawl up. It is known that if a beetle starts at the bottom of the first (leftmost) column, then it will end its journey on the ninth (rightmost) column. Is it always possible to remove one stick so that the beetle ends up at the top of the fifth column? (For example, if the sticks are arranged as in picture, the beetle will crawl along a solid line. If you remove the third one A stick in the path of the beetle, then it will crawl along the dotted line.) Proposed by G. Karavaev

M2791

1

Good cells in square N times N

A number is written in each cell of the

N \times N

square. Let's call cell

C

good if in one of the cells adjacent to

C

on the side, there is a number

1

more than in

C

, and in some other of the cells adjacent to

C

on the side, there is a number

3

more than in

C

. What is the largest possible number of good cells? Proposed by A. Chebotarev

M2790

1

Three of 9 vertics of nonagon form obtuse triangle

Prove that among the vertices of any convex nonagon, three can be found forming an obtuse triangle, none of whose sides coincide with the sides of the nonagon. Proposed by A. Yuran

M2788

1

Similar to All-Russian Regional

An equilateral triangle

\mathcal{T}{}

with side 111 is divided by straight lines parallel to its sides into equilateral triangles with side 1. The vertices of these small triangles, except the centre of

\mathcal{T}{}

are marked. Call a set of several marked points linear if[*]the marked points lie on a line

\ell

parallel to one of the sides of the triangle

\mathcal{T}

and; [*]if two marked points on

\ell

are in this set, every other marked point inbetween them is in the set. How many ways are there to split all the marked points into 111 linear sets?

M2785

1

Counting polylines

A finite set

S{}

of

n{}

points is given in the plane. No three points lie on the same line. The number of non-self-intersecting closed

n{}

-link polylines with vertices at these points will be denoted by

f(S).

Prove that
[*]

f(S)>0

for all sets

S{};

[*]

f(S)=1

if and only if all the points of

S{}

lie on the convex hull of

S{};

[*]if

f(S)>1

then

f(S)\geqslant n-1

, with equality if and only if one point of

S

lies inside the convex hull; [*]if exactly two points of

S{}

lie inside the convex hull, then

f(S)\geqslant\frac{(n-2)(n-3)}{2}.

Let

n\geqslant 3.

Denote by

F(n)

the largest possible value of the function

f(S)

over all admissible sets

S{}

of

n{}

points. Prove that

F(n)\geqslant3\cdot 2^{(n-8)/3}.

Proposed by E. Bakaev and D. Magzhanov

M2784

1

Unusual locus geometry

The bisectors

AD{}

and

BE{}

were drawn in the triangle

ABC{}

and they intersected at point

I{}.

Then everything was erased, leaving only the points

D{}

and

E{}.

Find the set of possible positions of the point

I{}.

Proposed by M. Didin

M2783

1

Sum of digits bounding

The sum of the digits of a natural number is

k{}.

What is the largest possible sum of digits for [*] the square of this number; [*]the fourth power of this number, given that

k\geqslant 4.

From the folklore

M2782

1

Graph theory

In a country, some cities are connected by two-way airlines, and one can get from any city to any other city in no more than

n{}

flights. Prove that all airlines can be distributed among

n{}

companies so that a route can be built between any two cities in which no more than two flights of each company would meet.
From the folklore

M2801

1

What have the Mayas ever done for us?

Yuri is looking at the great Mayan table. The table has

200

columns and

2^{200}

rows. Yuri knows that each cell of the table depicts the sun or the moon, and any two rows are different (i.e. differ in at least one column). Each cell of the table is covered with a sheet. The wind has blown aways exactly two sheets from each row. Could it happen that now Yuri can find out for at least

10000

rows what is depicted in each of them (in each of the columns)? Proposed by I. Bogdanov, K. Knop

M2800

1

A parallelogram and three equal segments

Let

ABCD

be a parallelogram. Let

M

be the midpoint of the arc

AC

containing

B

of the circumcircle of

ABC

. Let

E

be a point on segment

AD

and

F

a point on segment

CD

such that

ME=MD=MF

. Show that

BMEF

is cyclic. Proposed by A. Tereshin

M2798

1

A colour-blind grasshopper on the road

A straight road consists of green and red segments in alternating colours, the first and last segment being green. Suppose that the lengths of all segments are more than a centimeter and less than a meter, and that the length of each subsequent segment is larger than the previous one. A grasshopper wants to jump forward along the road along these segments, stepping on each green segment at least once an without stepping on any red segment (or the border between neighboring segments). Prove that the grasshopper can do this in such a way that among the lengths of his jumps no more than

8

different values occur. Proposed by T. Korotchenko

M2799

1

A polynomial guessing game with pairs

Let

n

be a positive integer. Ilya and Sasha both choose a pair of different polynomials of degree

n

with real coefficients. Lenya knows

n

, his goal is to find out whether Ilya and Sasha have the same pair of polynomials. Lenya selects a set of

k

real numbers

x_1<x_2<\dots<x_k

and reports these numbers. Then Ilya fills out a

2 \times k

table: For each

i=1,2,\dots,k

he writes a pair of numbers

P(x_i),Q(x_i)

(in any of the two possible orders) intwo the two cells of the

i

-th column, where

P

and

Q

are his polynomials. Sasha fills out a similar table. What is the minimal

k

such that Lenya can surely achieve the goal by looking at the tables? Proposed by L. Shatunov

M2781

1

Circumcenter and incenter lie on an Euler line

Let

A_1

be the midpoint of the smaller arc

BC

of the circumcircle of the acute-angled triangle

ABC.{}

The point

A_1

is reflected relative to the side

BC,

and then its image is reflected relative to the bisector of

\angle BAC{}

resulting in the point

A_2

. Similarly, the points

B_2

and

C_2

are constructed. Prove that the circumcenter and incenter of the triangle

ABC{}

lie on the Euler line of the triangle

A_2B_2C_2.

Proposed by A. Tereshin

M2780

1

Chromatic number inequality

Consider a natural number

n\geqslant 3

and a graph

G{}

with a chromatic number

\chi(G)=n

which has more than

n{}

vertices. Prove that there exist two vertex-disjoint subgraphs

G_1{}

and

G_2{}

of

G{}

such that

\chi(G_1)+\chi(G_2)\geqslant n+1.

Proposed by V. Dolnikov

M2779

1

A nice lcm problem

Prove that for any natural number

k{}

there is a natural number

n{}

such that

\mathrm{lcm}(1,2,\ldots,n)=\mathrm{lcm}(1,2,\ldots,n+k).

From the folklore

M2778

1

A parabola and a hyperbola

A parabola and a hyperbola are drawn on the coordinate plane. The graphs intersect at three points

A, B, C

and the axis of the parabola is the asymptote of the hyperbola. Prove that the intersection point of the medians of the triangle

ABC

lies on the axis of the parabola.
From the folklore

M2804

1

Weird geo with two equal circles and centroid

There are two equal circles of radius

1

placed inside the triangle

ABC

with side

BC = 6

. The circles are tangent to each other, one is inscribed in angle

B

, the other one is inscribed in angle

C

. (a) Prove that the centroid

M

of the triangle

ABC

does not lie inside any of the given circles. (b) Prove that if

M

lies on one of the circles, then the triangle

ABC

is isosceles.

M2802

1

Arithmetic means on a circle

The positive numbers

a_1, a_2, \ldots , a_{2024}

are placed on a circle clockwise in this order. Let

A_i

be the arithmetic mean of the number

a_i

and one or several following it clockwise. Prove that the largest of the numbers

A_1, A_2, \ldots , A_{2024}

is not less than the arithmetic mean of all numbers

a_1, a_2, \ldots , a_{2024}

.

M2805

1

Combinatorics with set and arithmetic means

Find the largest positive integer

n

, such that there exists a finite set

A

of

n

reals, such that for any two distinct elements of

A

, there exists another element from

A

, so that the arithmetic mean of two of these three elements equals the third one.

M2803

1

Easy permutation NT

Given is a permutation of

1, 2, \ldots, 2023, 2024

and two positive integers

a, b

, such that for any two adjacent numbers, at least one of the following conditions hold:
1) their sum is

a

; 2) the absolute value of their difference is

b

.
Find all possible values of

b

.

M2789

1

Game with fractions

Let

n>100

be a positive integer and originally the number

1

is written on the blackboard. Petya and Vasya play the following game: every minute Petya represents the number of the board as a sum of two distinct positive fractions with coprime nominator and denominator and Vasya chooses which one to delete. Show that Petya can play in such a manner, that after

n

moves, the denominator of the fraction left on the board is at most

2^n+50

, no matter how Vasya acts.

M2787

1

Innovative geo with areas and angles 18 degrees

Let

XY

be a segment, which is a diameter of a semi-circle. Let

Z

be a point on

XY

and 9 rays from

Z

are drawn that divide

\angle XZY=180^{\circ}

into

10

equal angles. These rays meet the semi-circle at

A_1, A_2, \ldots, A_9

in this order in the direction from

X

to

Y

. Prove that the sum of the areas of triangles

ZA_2A_3

and

ZA_7A_8

equals the area of the quadrilateral

A_2A_3A_7A_8

.

M2786

1

Coloring game

There are

100

white points on a circle. Asya and Borya play the following game: they alternate, starting with Asya, coloring a white point in green or blue. Asya wants to obtain as much as possible pairs of adjacent points of distinct colors, while Borya wants these pairs to be as less as possible. What is the maximal number of such pairs Asya can guarantee to obtain, no matter how Borya plays.

Kvant 2024

Subcontests

Kvant M2797 inequality

A lot of strips and cutting

Rays from rational points aren't parallel

Isogonal conjugate on (BIC)

(GKH) and (ABC) are tangent

The beetle crawls through the columns

Good cells in square N times N

Three of 9 vertics of nonagon form obtuse triangle

Similar to All-Russian Regional

Counting polylines

Unusual locus geometry

Sum of digits bounding

Graph theory

What have the Mayas ever done for us?

A parallelogram and three equal segments

A colour-blind grasshopper on the road

A polynomial guessing game with pairs

Circumcenter and incenter lie on an Euler line

Chromatic number inequality

A nice lcm problem

A parabola and a hyperbola

Weird geo with two equal circles and centroid

Arithmetic means on a circle

Combinatorics with set and arithmetic means

Easy permutation NT

Game with fractions

Innovative geo with areas and angles 18 degrees

Coloring game

Kvant 2024

Subcontests

Kvant M2797 inequality

A lot of strips and cutting

Rays from rational points aren&#039;t parallel

Isogonal conjugate on (BIC)

(GKH) and (ABC) are tangent

The beetle crawls through the columns

Good cells in square N times N

Three of 9 vertics of nonagon form obtuse triangle

Similar to All-Russian Regional

Counting polylines

Unusual locus geometry

Sum of digits bounding

Graph theory

What have the Mayas ever done for us?

A parallelogram and three equal segments

A colour-blind grasshopper on the road

A polynomial guessing game with pairs

Circumcenter and incenter lie on an Euler line

Chromatic number inequality

A nice lcm problem

A parabola and a hyperbola

Weird geo with two equal circles and centroid

Arithmetic means on a circle

Combinatorics with set and arithmetic means

Easy permutation NT

Game with fractions

Innovative geo with areas and angles 18 degrees

Coloring game

Rays from rational points aren't parallel