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An nxn Checkboard
MithsApprentice 26
N
by NicoN9
Source: USAMO 1999 Problem 1
Some checkers placed on an 
checkerboard satisfy the following conditions:
(a) every square that does not contain a checker shares a side with one that does;
(b) given any pair of squares that contain checkers, there is a sequence of squares containing checkers, starting and ending with the given squares, such that every two consecutive squares of the sequence share a side.
Prove that at least
checkers have been placed on the board.
checkerboard satisfy the following conditions:(a) every square that does not contain a checker shares a side with one that does;
(b) given any pair of squares that contain checkers, there is a sequence of squares containing checkers, starting and ending with the given squares, such that every two consecutive squares of the sequence share a side.
Prove that at least
checkers have been placed on the board.
26 replies
Is this FE solvable?
Mathdreams 4
N
by Mathdreams
Find all 
such that ![\[f(2x+y) + f(x+f(2y)) = f(x)f(y) - xy\]](//latex.artofproblemsolving.com/0/6/2/0623b7c328af6d088cfc5dd288afe3e17a94ff29.png)
for all reals 
and 
.
such that ![\[f(2x+y) + f(x+f(2y)) = f(x)f(y) - xy\]](http://latex.artofproblemsolving.com/0/6/2/0623b7c328af6d088cfc5dd288afe3e17a94ff29.png)
for all reals 
and 
.
4 replies
+1 w
Coaxial circles related to Gergon point
Headhunter 0
Source: I tried but can't find the source...
Hi, everyone.
In
, 
is the Gergon point and the incircle 
touch 
, 
, 
at 
, 
, 
respectively.
Let the circumcircles of
, 
, 
be 
, 
, 
respectively.
Reflect in 
and then we get the circle 
Reflect in 
and then the circle 
Reflect in 
and then the circle 
Prove that , , are coaxial.
In
, 
is the Gergon point and the incircle 
touch 
, 
, 
at 
, 
, 
respectively.Let the circumcircles of
, 
, 
be 
, 
, 
respectively.Reflect
in 
and then we get the circle 
Reflect
in 
and then the circle 
Reflect
in 
and then the circle 
Prove that
, , are coaxial.
0 replies
Equation with powers
a_507_bc 6
N
by EVKV
Source: Serbia JBMO TST 2024 P1
Find all non-negative integers 
and primes 
such that 
and primes 
such that 
6 replies
no numbers of the form 80...01 are squares
Marius_Avion_De_Vanatoare 2
N
by EVKV
Source: Moldova JTST 2024 P5
Prove that a number of the form 
(there is at least 1 zero) can't be a perfect square.
(there is at least 1 zero) can't be a perfect square.
2 replies
f((x XOR f(y)) + y) = (f(x) XOR y) + y
the_universe6626 3
N
by jasperE3
Source: Janson MO 5 P4
Find all functions 
such that
![\[f((x\oplus f(y))+y)=(f(x)\oplus y)+y\]](//latex.artofproblemsolving.com/d/3/5/d355f091797570bdc6c023eedb30e8ee9deac168.png)
Note: 
denotes the bitwise XOR operation. For example, 
.
(Proposed by ja.)
such that![\[f((x\oplus f(y))+y)=(f(x)\oplus y)+y\]](http://latex.artofproblemsolving.com/d/3/5/d355f091797570bdc6c023eedb30e8ee9deac168.png)
Note: 
denotes the bitwise XOR operation. For example, 
.(Proposed by ja.)
3 replies
2024 8's
Marius_Avion_De_Vanatoare 3
N
by EVKV
Source: Moldova JTST 2024 P2
Prove that the number 
is divisible by 2024.
is divisible by 2024.
3 replies
pretty well known
dotscom26 0
Let 
be a scalene triangle such that 
is its incircle. is tangent to at . A point (
) is located on .
Let
, 
, and 
be the incircles of the triangles 
, 
, and 
, respectively.
Show that the common tangent to and is also tangent to .
be a scalene triangle such that 
is its incircle. is tangent to at . A point (
) is located on .Let
, 
, and 
be the incircles of the triangles 
, 
, and 
, respectively.Show that the common tangent to
and is also tangent to .0 replies
and![\[ f(x+y)+f(x^2+f(y))=f(f(x))^2+f(x)+f(y)+y,\forall x,y\in\mathbb{R}\]](http://latex.artofproblemsolving.com/c/8/9/c895ae7fdf8d7f284ac9fc94cc077d6edad6cbf0.png)
Modular NT
oVlad 3
N
by EVKV
Source: Romania JBMO TST 2024 Day 1 P1
Find all the positive integers 
and 
such that 
is a prime number.
Cosmin Manea and Dragoș Petrică
and 
such that 
is a prime number.Cosmin Manea and Dragoș Petrică
3 replies
