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be an acute-angled triangle with 
, and let 
be its circumcentre and orthocentre respectively. Points 
lie on segments 
respectively, such that ![\[\angle ZOB=\angle YOC = 90^{\circ}.\]](http://latex.artofproblemsolving.com/8/f/e/8fe34cc0f04ff9a48f1a7eee89fc50d57272d936.png)
The perpendicular line from 
to line 
meets lines 
and 
at 
respectively. Let the tangents to the circumcircle of 
at points 
and 
meet at point 
. Prove that 
are concyclic.
marbles in 
colors such that there are exactly 
marbles of each color. She wants to arrange them on 
shelves, 
marbles on each shelf such that for any 
colors there is a shelf that has marbles of those colors.
be an inscribed circle of 
and touching 
, 
, 
at 
, 
, 
respectively. Let 
and 
be diameters of . Let 
and be the pole of 
and 
with respect to , respectively. 
cuts again at 
. 
cuts again at 
. The tangent at of cuts 
at 
. The tangent at of cuts 
at 
. Let 
and 
be midpoint of 
and 
, respectively.
, 
and perpendicular bisector of 
are concurrent.![$A \in \mathbb Q[x]$](http://latex.artofproblemsolving.com/d/1/5/d158f344cf30f8beb5f2199da53272bd8e304704.png)
with ![$\exists Q \in \mathbb Q[x], Q(A(x))= x^{2025!+2}+x^2+x+1$](http://latex.artofproblemsolving.com/1/1/9/119a7b617e09ba16bb266b44ab3c85acc529f88f.png)
and 
.
, prove there exists positive integer 
and positive real number 
, such that 
, for any 
, 
, 
holds.
be a acute triangle with 
. 
are the orthocenter and excenter. Let be a point on the same side of 
as 
, such that 
is equilateral. Let 
be a point on the same side of 
as , such that 
is equilateral. Let 
be a point on the same side of 
as , such that 
is equilateral. Let be the midpoint of 
. Prove that 
are collinear.
be real numbers satisfying 
and 
Prove that 
Let be real numbers satisfying 
and 
Prove that 
and a line 
in the plane.Let 
be reflections of 
across the line , respectively.Let 
be the midpoints of 
, respectively.Let 
be the reflections of across , respectively.Prove that the points lie on a line parallel to .
and 
are natural numbers such that 
is divisible by 
and 
is divisible by 
, then find the least possible value of 
.
) and you also need to be quite good at algebra. You should start with basic euclidean though and perhaps learn inversion first before trying bashing, you will build the fundamentals you need that way.
Prove that
Let 
Prove that
