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.
distinct colors, and I want to color dots in an 
array of dots, where 
. However, I am not allowed to color in one dot a specific color, and then color in a dot that is adjacent to the previously colored spot (diagonally, horizontally, vertically). What is the smallest that enables me to color in all of the dots successfully, and how many ways can I color that array in?
is to convert it into 
, given 
. To AIMEify a number ![$m\sqrt[k]{n}$](http://latex.artofproblemsolving.com/0/5/6/0560729683ba201b26cda879e9bd82316c72c2b9.png)
is to convert it into 
, where is power free (i.e. not divisible by the power of any prime). Define a function 
to AIMEify any square-root radical and any rational number such that it must always output a 3-digit number from 
to 
, inclusive.
such that there exists a real number 
such that 
, or prove none exist.
such that 
. Compute the value of the expression:![\[
P = \frac{a}{1 + a^2} + \frac{b}{1 + b^2} + \frac{c}{1 + c^2} - \frac{2}{(a + b)(b + c)(c + a)}.
\]](http://latex.artofproblemsolving.com/a/a/9/aa91c6bce9a71c55d5ae16070656079d45ac9017.png)
such that![\[
27(x + y + z)^3 = 240 + (3x + y - z)^3 + (3y + z - x)^3 + (3z + x - y)^3.
\]](http://latex.artofproblemsolving.com/f/4/f/f4fa142c59c216287674b7d824fd1f18a1ad257e.png)
Prove that 
.
be a prime number greater than 5. Prove that 
is divisible by 240.
such that![\[
(a + b\sqrt{2026})^2 = 2024 + 2025\sqrt{2026} \ ?
\]](http://latex.artofproblemsolving.com/8/8/6/886de7be16e4c852eb2d63ee8105b9081a2c4e1b.png)
such that![\[
2^x \cdot x^2 = 9y^2 + 6y + 88.
\]](http://latex.artofproblemsolving.com/9/d/0/9d02b030cfcdb4410b1c03c11a713514c5d45c85.png)
such that 
. Prove that![\[
\frac{a}{b^2 + 1} + \frac{b}{c^2 + 1} + \frac{c}{a^2 + 1} \geq \frac{3}{2}.
\]](http://latex.artofproblemsolving.com/e/3/0/e3085b895504ea669eb10359cc2d3063f10e1357.png)
. Prove that![\[
\frac{1 + bc + ca}{(1 + a + b)^2} + \frac{1 + ca + ab}{(1 + b + c)^2} + \frac{1 + ab + bc}{(1 + c + a)^2} \geq 1.
\]](http://latex.artofproblemsolving.com/9/a/4/9a402ff0ebf6800343d89195ec578c7307574cf1.png)
Problem 4
be acute with 
. Let the altitudes 
and 
intersect at 
. The line through 
perpendicular to 
intersects the extensions of 
and 
at points 
and 
, respectively. The line through 
perpendicular to intersects the extensions of and 
at points 
and 
, respectively. Let 
be the midpoint of .
and are similar.
is isosceles and that 
.
.
and 
, then writes down the following six numbers:![\[
ab,\quad a(a + 4),\quad ab(a + 4),\quad b(a + 4),\quad b(b + 4),\quad (a + 4)(b + 4).
\]](http://latex.artofproblemsolving.com/c/7/5/c759b446d441d55929a650bb9b80c3633163bd8b.png)
Among these six numbers, what is the maximum number of perfect squares that can appear? Explain why.
red line and 
blue lines in the plane in general position. Prove that there are at least![\[\frac{(n-1)(n-2)}{2}\]](http://latex.artofproblemsolving.com/d/f/4/df4d29a8f3b1d25d3a84e91a1085c82bf693141b.png)
regions with a monochromatic boundary. (An unbounded region is also considered a region, where its boundary is formed by some line segments and half-lines.)
, 
. The perpendicular bisector of the segment 
intersects the lines 
at the points 
respectively. Denote the mid-point of 
as 
. Suppose the circumcircle of 
intersects the line 
at points 
and 
, and 
are arranged in order on the line. Prove that 
.
be a triangle with circumcentre 
. The points 
and 
are interior points of the sides 
and 
respectively. Let 
and 
be the midpoints of the segments 
and 
. respectively, and let 
be the circle passing through and . Suppose that the line is tangent to the circle . Prove that 
and 
be positive integers. Show that if 
divides 
, then 
.
.Prove that there is a choice of plus or minus such that:
with the following property: for all positive divisors 
of , we have 
or 
is prime.
natural integer : 
?
and circle 
are externally tangent at point 
. From a point 
on circle , a tangent is drawn intersecting circle at points 
and 
. The line 
is extended to intersect circle at point 
. A point 
is taken on the arc 
of circle . The line 
is extended to intersect the angle bisector of 
at point 
. The circle passing through points 
and the circle passing through points 
intersect at another point 
. Prove that 
.
of positive integers with 
prime and ![\[ a^p=b!+p. \]](http://latex.artofproblemsolving.com/9/f/6/9f674dabcb364872cef4153198e2e362ae40a2fd.png)
can be defined as a sequence such that 
, and 
, such that 
. The function 
is such that 
. How many numbers within the interval 
for the function 
are perfect squares?
can be defined as a sequence such that , and , such that . The function is such that . How many numbers within the interval for the function are perfect squares? and 
are related?
is defined on the positive integers where 
for all positive integers . Also, the sequence is defined in two different ways that contradict each other if is allowed to be any positive integer. Maybe it is intended that we can only take 
?
for all positive integers such that , which is probably a crucial idea for whatever this problem is meant to be. there are such that and 
is a perfect square. Everything else is just confusing.
,
to be a perfect square. There are 
perfect squares between 
and 
inclusive, so our answer is