[/quote]
,
and 
such that 
is divisible by 
?
be a triangle with incenter 
,
and 
at 
and 
.
and 
the reflections of 
be the intersection of 
with 
,
be the midpoint of 
and 
are perpendicular.
be a convex quadrilateral with perpendicular diagonals intersecting at 
.
be three diameters of the circumcircle of an acute triangle 
be an arbitrary point in the interior of 
,
be the orthogonal projection of 
,
be the point such that 
is the midpoint of 
,
be the point such that 
,
be the point such that 
.
is similar to triangle 
such that for any 
,
there exists real numbers 
such that 
for at least 
indices 
.
and 
denotes the fractional part of 
.
be an acute-angled scalene triangle. Let 
and 
be points on the rays 
and 
,
.
and 
be points on the rays 
and 
,
.
and 
intersect at 
,
is parallel to 
be the circumcircle of a triangle 
the midpoints of the sides 
,
the midpoint of the arc 
.
and 
intersect the perpendicular bisectors of 
and 
intersect at 
.
.
is equal to the largest prime divisor of 
.
be the set of positive real numbers. Find all functions 
such that, for all positive real numbers 
,
and 
,
,
.
of positive integers that satisfy the equation: 
.
be the set of real numbers. Determine all functions 
such that ![\[ f(x^2 - y^2) = x f(x) - y f(y) \]](http://latex.artofproblemsolving.com/6/8/f/68f78e0df627f6ac04052433c01627bf16f0af8e.png)
for all pairs of real numbers 
and 
,
to be the total number of times the digit 
between 
and 
.
,
,
,
.
Y by
Let 
be a positive integer. A positive integer 
is called a -good number if it satisfies
the following two conditions:
1. has exactly 
digits in decimal representation (it cannot have leading zeros).
2. If the first digits and the last digits of are considered as two separate -digit
numbers (which may have leading zeros), the square of their sum is equal to.
For example,
is a 
-good number because
![\[(20 + 25)^2 = 2025.\]](//latex.artofproblemsolving.com/5/c/5/5c51bca485cea6c3750c48c09091eb5fd72b5192.png)
Find all 
-good numbers.
be a positive integer. A positive integer 
is called a -good number if it satisfiesthe following two conditions:
1.
has exactly 
digits in decimal representation (it cannot have leading zeros).2. If the first
digits and the last digits of are considered as two separate -digitnumbers (which may have leading zeros), the square of their sum is equal to
.For example,
is a 
-good number because![\[(20 + 25)^2 = 2025.\]](http://latex.artofproblemsolving.com/5/c/5/5c51bca485cea6c3750c48c09091eb5fd72b5192.png)
Find all 
-good numbers.This post has been edited 3 times. Last edited by CSJL, Mar 6, 2025, 8:27 AM
Reason: Format
Reason: Format
and 
we have 
where 
is a three digit number, we have 
we have 
so 
we have 
so 
and 
and 
but 
is too small and 
is too big so no solution
but 
is too big so no solution
and 
.
.![\[ (494 + 209)^2 = 703^2 = 700^2 + 6 \cdot 700 + 9 = 490000 + 4209 = 494209\]](http://latex.artofproblemsolving.com/a/9/f/a9fb989dd86d92262dc5c4a88a1f5c8e09ab7851.png)
.![\[ (998 + 1)^2 = 999^2 = (1000 - 1)^2 = 1000^2 - 2000 + 1 = 998001\]](http://latex.artofproblemsolving.com/3/f/7/3f7228ca02118f3c1f74e272976dad6cb243eda4.png)
.
.
.![\[ 999a + b = b^2 \leftrightarrow 999a = b(b-1)\]](http://latex.artofproblemsolving.com/c/2/b/c2b98de6e364c24ec68cdd95d1777fa9b8105fd7.png)
.
is positive, this implies 
.
.
divides one of 
.
,
.
,
,
.
and 
,
and 
,
and 
,
.
,
)![\[b = 703 \implies a = \frac{703 \cdot 702}{999} = 494 \implies y = 703 - 494 = 209,\]](http://latex.artofproblemsolving.com/6/8/8/6881af9915a964087da5dd31ef4bf667a2b520f8.png)
meaning the number is 
and 
,
and 
.
.
,
.
