[/quote]
,
such that the following equation holds:![\[
7^a + 1 = 2b^2
\]](http://latex.artofproblemsolving.com/e/0/8/e08d3fc7feefd7b4acc4a0f9c7e8bcfa328b5791.png)
have the property that 
is a perfect square for infinitely many positive integers 
?
and 
respectively be the incentre, circumcentre and circumcircle of triangle 
.
are chosen on 
,
is the foot of the altitude from 
to 
.
are similarly defined, prove that lines 
and 
concurr on 
.
and let 
and 
be tangency points of the incircle of 
,
and 
respectively. Let the circle with centre 
passing through 
and 
.
,
and 
,
.
and 
be feet of perpendiculars from 
and 
respectively. Prove that 
,
and 
are concurrent.
that is not isosceles at 
,
be its incircle, which is tangent to 
at 
,
and 
intersect the line passing through 
at 
and 
,
and perpendicular to 
intersect 
and 
at 
and 
,
are collinear and that 
are concurrent.
positive integer numbers such that the least common multiple (LCM) of all these numbers is not a perfect square. Mykhailo consecutively hides one of these numbers and writes down the LCM of the remaining 
numbers that are not hidden. What is the maximum number of the 
satisfying: 
.
min
is a prime number and 
are positive integers. Prove that there exist infinitely 
are divisible by 
if 
if 
primes, and let 
be the sum of the divisors of ![\[\left|\sum_{d|n}\frac{\mu(d)\sigma(d)}{d}\right| \geq \frac{1}{n}, \]](http://latex.artofproblemsolving.com/8/e/b/8eb88de7cd7ea8e2853dbc80cae24bdc6b77f9be.png)
be a sequance of natural integers. Proof that, any number in the sequance can be written only as 
for any 
and not necessarily distinct primes 
for 
.
are same in some case.
![\[2^{xy}f(xy-1)+2^{x+y+1}f(x)f(y)=4xy-2,\forall x,y\in\mathbb{R}\]](http://latex.artofproblemsolving.com/d/1/d/d1de7196fd3ba60732c14fa2b156ec880b4829d0.png)
has no integer solutions
play a game with a stack of 
coins from the stack. The player who takes the last coin wins. Prove that there are infinitely many integers 
Y by Adventure10, Mango247
In an infinite grid of regular triangles, Niels and Henrik are playing a game they made up.
Every other time, Niels picks a triangle and writes
in it, and every other time, Henrik picks a triangle where he writes a 
. If one of the players gets four in a row in some direction (see figure), he wins the game.
Determine whether one of the players can force a victory.
Every other time, Niels picks a triangle and writes
in it, and every other time, Henrik picks a triangle where he writes a 
. If one of the players gets four in a row in some direction (see figure), he wins the game.Determine whether one of the players can force a victory.
This post has been edited 1 time. Last edited by parmenides51, Sep 3, 2019, 9:33 AM
regular triangles, and observe that every triangle contributes to exactly 
edge in a hexagon. The strategy for player Henrik is to always play in the triangle sharing an edge with the triangle placed by Niels. This implies that Niels will never get multiple in a row across 
hexagons, so he can never get 
in a row. This is a strategy for Henrik to always draw.
