[/quote]
,
that satisfy the equation 
?
and 
Find the value of 
be a triangle with a circle 
with center 
tangent to sides 
at 
respectively. Suppose the circle with diameter 
intersects the circumcircle of 
is the reflection of 
over 
lie on 
are parallel to 
.
lie on the circumcircle of 
![\[f(xf(y) + x) = xy + f(x)\]](http://latex.artofproblemsolving.com/9/9/f/99f580ebc50846e6bc2c004667559922749a4dfa.png)
be a sequence of positive real numbers such that for every 
,![\[
a_n = \max_{1 \leq i \leq 2025} a_{n-i} - \min_{1 \leq i \leq 2025} a_{n-i}
\]](http://latex.artofproblemsolving.com/e/3/0/e30e9e533eb9a41cf847484e676ac4522edda665.png)
Prove that there exists a natural number 
such that for all 
,![\[
a_n < \frac{1}{1404}
\]](http://latex.artofproblemsolving.com/0/1/f/01f1a81eed2230332d51a15501ef2a6ad3a7ec82.png)
be an acute triangle with 
and let 
be its orthocenter. Let 
be a point on the perpendicular bisector of 
such that 
and 
are on different sides of 
,
a point on the perpendicular bisector of 
such that 
and 
a point on the perpendicular bisector of 
such that 
and 
lie on the opposite side of 
.
and 
square is filled with numbers 
.
squares is summed,and arranged in an increasing order. Is it possible to obtain the following sequences as a result of this operation?
lie on the perpendicular bisector of 
lie on a circle. Suppose 
are perpendicular to sides 
at points 
The tangent lines from points 
intersects at point 
Prove that: 
are parallel.
,
be four points in the plane, with 
and 
.![\[\frac{AB \cdot CD}{AC \cdot BD}, \]](http://latex.artofproblemsolving.com/f/3/c/f3cf9729b32b210faeea6e2d23da3582be2ae061.png)
and 
are orthogonal. (Intersecting circles are said to be orthogonal if at either common point their tangents are perpendicuar. Thus, proving that the circumcircles of the triangles 
such that for any positive real numbers 
and 
,
Proposed by Athanasios Kontogeorgis, Grecce, and Dorlir Ahmeti, Kosovo
Y by
Hello, I am going to be a 7th grader next year and I really want to qualify for USAJMO in 8th grade, so far I have these goals reached
1. AMC 10 Honor Roll A and B 2025
2. AMC 8 DHR and HR
3. AIME 3
This year on AIME something happened and I got a 3 on the AMC's I got a 105 on AMC 10 A and I got a 114 on AMC 10 B. I want to improve mostly on AIME but since the AMC 10 is coming up quicker what would you guys recommend for getting 110+ on both of the AMC 10's and getting a 6+ on AIME? So far I am only doing Alcumus and have no books so far.... Checking the table of contents on the books Alcumus provides the same topics. I was thinking to take WOOT 1 and AMC 10 Problem Series.
1. AMC 10 Honor Roll A and B 2025
2. AMC 8 DHR and HR
3. AIME 3
This year on AIME something happened and I got a 3
on the AMC's I got a 105 on AMC 10 A and I got a 114 on AMC 10 B. I want to improve mostly on AIME but since the AMC 10 is coming up quicker what would you guys recommend for getting 110+ on both of the AMC 10's and getting a 6+ on AIME? So far I am only doing Alcumus and have no books so far.... Checking the table of contents on the books Alcumus provides the same topics. I was thinking to take WOOT 1 and AMC 10 Problem Series.
