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is inscribed in a circle 
. Inscribed in a triangle circle touchs the sides 
in a point 
. 
— the circle inscribed in a segment 
circle of , and passing through a point . Let points 
and 
— the centers of circles and an extra inscribed circle (touching side ) respectively. Prove, that lines 
and 
are parallel.
be cyclic quadrilateral. Let 
and 
intersect at 
, and let 
and 
intersect at 
. Let 
and are points on and such that 
. Let 
and 
be the intersections of 
with the diagonals of . Prove that circumcircles of triangles 
and 
are tangent at a fixed point.
be an acute triangle. Points 
, and 
lie on a line in this order and satisfy 
. Let and be the midpoints of 
and 
, respectively. Suppose triangle 
is acute, and let 
be its orthocentre. Points and lie on lines 
and 
, respectively, such that 
and are concyclic and pairwise different, and 
and are concyclic and pairwise different. Prove that 
and are concyclic.
with diameter 
is given. The point lies in the interior of the circle, but not on . The line 
intersects in 
and . The tangent to at intersects the line through perpendicular to , at . The point 
lies on , and is such that 
is tangent to and 
.
, and are collinear.
colours (so that adjacent vertices have different colours).
edges can be removed from the graph so that it remains connected.
be the incenter and the circumcenter of triangle , and 
be the circumcenters of triangle 
. Let 
be the midpoints of segments 
. Prove that the circumcenter of triangle 
, , is the midpoint of segment 
.
Each entry of an 
matrix equals either 
or 
Let 
denote the sum of eight products of entries in each row. Also, let 
denote the sum of eight products of entries in each column. Find the maximum possible value of 
In other words, find![$$ {\rm max}\ \left[ \sum_{i=1}^8\ \prod_{j=1}^8\ a_{ij} -
\sum_{j=1}^8\ \prod_{i=1}^8\ a_{ij} \right]
$$](http://latex.artofproblemsolving.com/2/d/e/2de7fd40366073445d289356d0f4e66b542cb60c.png)
Same question, but for a 
matrix.
Show that there exists a constant 
such that for all positive integer 
![\[\sum_{k\le n^{\varepsilon}}(n\text{ mod } k)>cn^{2\varepsilon}.\]](http://latex.artofproblemsolving.com/f/e/9/fe933e886e28c2ecf1fa880d0e650ac7a2998aaa.png)
Proposed by Cheng Jiang
without further conclusions.
is possible for a function - this does not mean that 
.
or the "cancellation" of is called injectivity. You have to prove the function is injective first before cancellation.
from the equation 
in any way - so 
is wrong for that function as well. (thus is not injective)
without further conclusions. is possible for a function - this does not mean that . or the "cancellation" of is called injectivity. You have to prove the function is injective first before cancellation.
,we can divide by 
on both sides to get 
(or simply avoid pointwise trap).
.Let 
for 
.
gives 
.Take some 
such that 
.Then we have 
or 
.But 
gives that is even which means 
or 
, contradiction. Thus we are done.
be the assertion of the problem.
value we get that 
or 
(Just 
works).
Prove that
such that the equality 
holds for all pairs of real numbers 
.
is an obvious solution. Now, let 
.
or 
.
Adding these two together we get:
