[/quote]
,
be a cyclic quadrilateral with circumcenter 
,
is not a diameter of its circumcircle. The lines 
and 
intersect at point 
,
lies between 
and 
lies between 
and 
is acute and let 
be its orthocenter. The points 
and 
on the lines 
and 
.
,
.
Prove that
Let 
Prove that
Let 
Prove that
Let 
Prove that
Let 
Prove that
.The solution to 
is only 
for some 
,![\[
\begin{cases}
a^n - 2b^n = 1, \\
b^n - 2c^n = 1.
\end{cases}
\]](http://latex.artofproblemsolving.com/3/f/5/3f51406329aed6e3a915205c0bce3965ef547527.png)
Prove that
Where 
Prove that
Let 
Prove that
Let 
Prove that
distinct positive integers and then writes in blue the 
sums of each two red numbers. At most how many of the blue numbers can be prime?
be a positive integer. Find the number of reduced fractions 
for which 
,
denotes the fractional part of 
.
,
,
be the centers of the circles circumscribing triangles ADE, BDF, CEF. Prove that the orthocenter of triangle 
is tangent to sides 
and 
at 
.
.
denote the set of positive integers. Find all functions 
such that for positive integers 
and 
![\[f(a^2 + b^2) = f(a)f(b) \text{ and } f(a^2) = f(a)^2.\]](http://latex.artofproblemsolving.com/c/1/f/c1f8ffe04cfcc45b498f3931a4796d5c56dc04d0.png)
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2 years have passed since I posted geometry problems from mock USAJMO and USAMO collected. Last days I started posting in HSO the Mock Contests of USAJMO level, not posted there or not existing already inside User Created Contests
So enjoy
USAJMO Mocks posted in separate threads, collected in post collections
USAMO Mocks, posted in separate threads, collected in post collections
AIME Mocks
ARML Mocks
HMMT Mocks
Mandelbrot Mocks
PUMaC Mocks
USAMTS Mocks
Different Type Mocks
USA Computational Geo Mocks (from AMC, AIME, HMMT, PUMaC Mocks)
Geometry from USA Computational Mocks (AIME + ARML + Other Computational + Different Type)
Geometry from AIME Mocks + 3D Geo from Mock AIMEs + Conics from Mock AIMEs
Geometry from AIME (official)
Collected among others inside here, among USA Mocks
after I finish with USAJMO Mocks, I shall start doing the same with USAMO Mocks (a few are here ) (more soon)
if you happen to come across other USAJMO mocks, please let me know to create their post collections, to include them in the list above
enjoy / start solving
So enjoy
USAJMO Mocks posted in separate threads, collected in post collections USAMO Mocks, posted in separate threads, collected in post collections AIME Mocks ARML Mocks HMMT Mocks Mandelbrot Mocks PUMaC Mocks USAMTS Mocks Different Type Mocks USA Computational Geo Mocks (from AMC, AIME, HMMT, PUMaC Mocks) Geometry from USA Computational Mocks (AIME + ARML + Other Computational + Different Type) Geometry from AIME Mocks + 3D Geo from Mock AIMEs + Conics from Mock AIMEs Geometry from AIME (official)Collected among others inside here, among USA Mocks
if you happen to come across other USAJMO mocks, please let me know to create their post collections, to include them in the list above
enjoy / start solving
This post has been edited 111 times. Last edited by parmenides51, Feb 6, 2024, 7:28 PM
