[/quote]
,
such that![\[
2^a \cdot 3^b + 1 = 5^c.
\]](http://latex.artofproblemsolving.com/0/a/c/0ac1f22162eacf698847bceb33f9a337bd53c44b.png)
be an isosceles trapezoid with 
.
be the midpoint of 
.
and 
the circumcircles of the triangles 
and 
,
be the crossing point of the tangent to 
with the tangent to 
.
is tangent to 
,
denote the feet of the altitudes from 
,
.
and 
be the circles through 
,
other than 
such that![\[ f(x+f(y))=y+f(x+1),\]](http://latex.artofproblemsolving.com/b/a/3/ba35828951778c0d35f50b5b557796bd2cf9e338.png)
for all 
.
is defined by 
,
for 
.
of ordered pairs 
for which there is a finite set 
of positive integers such that 
,
.
,
,
with the following property: An ordered pair of nonnegative integers ![\[m < \alpha x+\beta y < M\]](http://latex.artofproblemsolving.com/a/9/b/a9bb35658173fdab821cc8180420e47d5437e0a2.png)
if and only if 
.
.
,![\[
f(xy)(f(x) - f(y)) = (x-y)f(x)f(y)
\]](http://latex.artofproblemsolving.com/0/9/b/09b8ed4e133ccf33b3f902fd9a9aa8e23c264341.png)
for all 
.
be 
positive integers such that the 
products![\[a_1 a_2 a_3 \cdots a_n, b_1 a_2 a_3 \cdots a_n, b_1 b_2 a_3 \cdots a_n, \dots, b_1 b_2 b_3 \cdots b_n\]](http://latex.artofproblemsolving.com/0/2/c/02cbff15c4d0118156a9d24be09360facb264d5a.png)
form a strictly increasing arithmetic progression in that order. Determine the smallest possible integer that could be the common difference of such an arithmetic progression.
and, for each 
,
be the unique integer in the range 
for which 
is divisible by 
the obtained sequence is 
.
eventually becomes constant.
.
and 
has center 
is tangent to the circumcircle of triangle 
satisfying
and 
such that 
is divisible by 
?
,
and 
and 
the reflections of 
be the intersection of 
,
and 
are perpendicular.
.
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 small and 
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)
be the number formed by the last 
.
.
.![\[ 999a + b = b^2 \leftrightarrow 999a = b(b-1)\]](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 
.
.
,
.
