MathDB

King attacks marked cells

Source: Russian TST 2017, Day 5 P1 (Group NG)

4/1/2023

What is the largest number of cells that can be marked on a

100 \times 100

board in such a way that a chess king from any cell attacks no more than two marked ones? (The cell on which a king stands is also considered to be attacked by this king.)

boardcombinatoricschess king

Successful numbers

Source: Russian TST 2017, Day 6 P1 (Group NG)

4/1/2023

Let's call a number of the form

x^3+y^2

with natural

x, y

successful. Are there infinitely many natural

m

such that among the numbers from

m + 1

to

m + 2016^2

exactly 2017 are successful?

number theory

Nine point centers of four triangles

Source: Russian TST 2017, Day 6 P1 (Groups A & B)

4/1/2023

The diagonals of a convex quadrilateral divide it into four triangles. Prove that the nine point centers of these four triangles either lie on one straight line, or are the vertices of a parallelogram.

geometryNine point center

Sum of square roots is an integer

Source: Russian TST 2017, Day 7 P1 (Groups A & B)

4/1/2023

Prove that

\sqrt{a_1}+\sqrt{a_2}+\cdots+\sqrt{a_{119}}

is an integer, where

a_n=2-\frac{1}{n^2+\sqrt{n^4+1/4}}.

algebrasquare roots

Perfect Cubes

Source: 2013 USAJMO Problem 1

4/30/2013

Are there integers

a

and

b

such that

a^5b+3

and

ab^5+3

are both perfect cubes of integers?

modular arithmeticEulerLaTeXnumber theoryrelatively prime

P1

Problems(5)