AoPS Academy
[/quote]
, do not answer with "
is a solution" only. Either you post any kind of proof or at least something unexpected (like "
is the smallest solution). Someone that does not see that is a solution of the above without your post is completely wrong here, this is an IMO-level forum.
be real numbers such that 
. Find the smallest possible value of 
.
+1 w
grid has 
cells painted such that the following condition holds:.
and it's circumcircle 
. With center 
, we construct a circle 
that intersects arc 
of circle , that doesn't contain 
, at point 
and arc 
, that doesn't contain 
, at point 
. Suppose that the intersection point 
of lines 
and 
lies on circle . Prove that line 
is perpendicular on line 
.
, and for all 
, define the sequence 
by the recurrence![\[
a_{n+1} = a_n^2 + 1
\]](http://latex.artofproblemsolving.com/3/1/1/3114595ded8c8961d611afaab9ad19e254973001.png)
Prove that there is no natural number 
such that![\[
\prod_{k=1}^{n} \left( a_k^2 + a_k + 1 \right)
\]](http://latex.artofproblemsolving.com/a/4/f/a4f88ee7e3d4e53e369319f74cd9b12be26c9681.png)
is a perfect square.
be distinct real numbers such that![\[
a + b + c + \frac{1}{abc} = \frac{19}{2}
\]](http://latex.artofproblemsolving.com/5/1/e/51eef77358db2b6c98f0fb0fb4e30db837a03642.png)
Find the maximum possible value of 
.
be a real numbers such that 
Prove that
Where 
2021 IMOC Problems
. What is the least possible positive integer value of 
such that there exists a non-negative integer 
for which the set 
is fragrant?
be real numbers such that![\[ a^2+x^2=b^2+y^2=c^2+z^2=(a+b)^2+(x+y)^2=(b+c)^2+(y+z)^2=(c+a)^2+(z+x)^2 \]](http://latex.artofproblemsolving.com/7/e/b/7ebfb2f663d390714310d3c43b9c68e1b101d632.png)
.
, let 
be the circumcentre, 
be the orthocentre, and 
be the centroid.
satisfy the following condition:
is constant.
. Find 
.
such that ![\[\left(g(m)+n\right)\left(g(n)+m\right)\]](http://latex.artofproblemsolving.com/5/a/c/5ac9fce315e8a524a3a01b345cfe76bc76056d68.png)
is a perfect square for all 
the set of all positive real numbers.
which have the property:![\[f(x)f(y)=2f(x+yf(x))\]](http://latex.artofproblemsolving.com/4/f/f/4ffbde35ff082585a188c1a4368d39338d1b07a4.png)
and 
.
be a point in the circumference such that 
is diameter of the circle and the point on the perpendicular of 
that lies on the circle. is a diameter we have 
and notice 
so 
is cyclic 
is tangent at we must have 
because they're looking at the same arc (small arc 
). So we have 
is isosceles with 
. 
.
.
Let 
.
)
.