MathDB

Tattoo congress

Source: Russian TST 2016, Day 9 P1 (Group NG), P3 (Groups A & B)

4/19/2023

Several people came to the congress, each of whom has a certain number of tattoos on both hands. There are

n{}

types of tattoos, and each of the

n{}

types is found on the hands of at least

k{}

people. For which pairs

(n, k)

is it always possible for each participant to raise one of their hands so that all

n{}

types of tattoos are present on the raised hands?

combinatorics

Diophantine equation

Source:

3/26/2015

Find all

x, y, z\in\mathbb{Z}^+

such that

(x-y)(y-z)(z-x)=x+y+z

number theoryDiophantine equation

Inequality with polynomial

Source: Russian TST 2016, Day 8 P1 (Group NG), P2 (Groups A & B)

4/19/2023

For which even natural numbers

d{}

does there exists a constant

\lambda>0

such that any reduced polynomial

f(x)

of degree

d{}

with integer coefficients that does not have real roots satisfies the inequality

f(x) > \lambda

for all real numbers?

algebrapolynomialInequality

Marmelade war

Source: Russian TST 2016, Day 8 P1 (Groups A & B)

4/19/2023

There are 100 saucers in a circle. Two people take turns putting marmalade of various colors in empty saucers. The first person can choose one or three empty saucers and fill each of them with marmalade of arbitrary color. The second one can choose one empty saucer and fill it with marmalade of arbitrary color.
There should not be two adjacent saucers with marmalade of the same color. The game ends when all the saucers are filled. The loser is the last player to introduce a new color of marmalade into the game. Who has a winning strategy?

combinatoricsgame

Geo with complete quadrilateral

Source: Russian TST 2016, Day 11 P1 (Group NG)

4/19/2023

In the cyclic quadrilateral

ABCD

, the diagonal

BD

is divided in half by the diagonal

AC

. The points

E, F, G

and

H{}

are the midpoints of the sides

AB, BC, CD{}

and

DA

respectively. Let

P = AD \cap BC

and

Q = AB \cap CD{}

. The bisectors of the angles

APC

and

AQC

intersect the segments

EG

and

FH

at the points

X{}

and

Y{}

respectively. Prove that

XY \parallel BD

.

geometrycyclic quadrilateral

Easy inequality

Source: Russian TST 2016, Day 9 P1 (Groups A & B)

4/19/2023

The positive numbers

a, b, c

are such that

a^2<16bc, b^2<16ca

and

c^2<16ab

. Prove that

a^2+b^2+c^2<2(ab+bc+ca).

algebrainequalities

NT with divisibility

Source: Russian TST 2016, Day 10 P1 (Group NG)

4/19/2023

Find all natural

n{}

such that for every natural

a{}

that is mutually prime with

n{}

, the number

a^n - 1

is divisible by

2n^2

.

number theoryDivisibility

101 blue and 101 red points are selected on the plane, no 3 on straight line

Source: 2016 Kazakhstan MO grades XI P5

11/2/2020

101

blue and

101

red points are selected on the plane, and no three lie on one straight line. The sum of the pairwise distances between the red points is

1

(that is, the sum of the lengths of the segments with ends at red points), the sum of the pairwise distances between the blue ones is also

1

, and the sum of the lengths of the segments with the ends of different colors is

400

. Prove that you can draw a straight line separating everything red dots from all blue ones.

combinatoricsColoringpointscombinatorial geometry

Number theory with powers of two

Source: Russian TST 2016, Day 10 P1 (Group A), P2 (Group B)

4/19/2023

Let

a{}

and

b{}

be natural numbers greater than one. Let

n{}

be a natural number for which

a\mid 2^n-1

and

b\mid 2^n+1

. Prove that there is no natural

k{}

such that

a\mid 2^k+1

and

b\mid 2^k-1

.

number theoryDivisibility

Very similar to a Chinese TST problem

Source: Russian TST 2016, Day 11 P1 (Group A), P2 (Group B)

4/19/2023

A cyclic quadrilateral

ABCD

is given. Let

I{}

and

J{}

be the centers of circles inscribed in the triangles

ABC

and

ADC

. It turns out that the points

B, I, J, D

lie on the same circle. Prove that the quadrilateral

ABCD

is tangential.

geometrycyclic quadrilateraltangential quadrilateral

P1

Problems(13)