AoPS Academy
[/quote]
, do not answer with "
is a solution" only. Either you post any kind of proof or at least something unexpected (like "
is the smallest solution). Someone that does not see that is a solution of the above without your post is completely wrong here, this is an IMO-level forum.
greater than 10. Then, she multiplies all the digits of and obtains an odd number. Find all possible values of the units digit of .
and 
be prime numbers. Prove that if 
is a perfect square, then 
is also a perfect square.
cups labeled 
, where the 
-th cup has capacity liters. In total, there are liters of water distributed among these cups such that each cup contains an integer amount of water. In each step, we may transfer water from one cup to another. The process continues until either the source cup becomes empty or the destination cup becomes full.
Prove that from any configuration where each cup contains an integer amount of water, it is possible to reach a configuration in which each cup contains exactly 1 liter of water in at most 
steps.
Prove that in at most 
steps, one can go from any configuration with integer water amounts to any other configuration with the same property.
+1 w
satisfy 
. Prove that
When does the equality occur?
be the incircle of the triangle 
and with centre 
. Let 
be the circumcircle of the triangle 
. Circles and 
intersect at the point 
and 
. Let 
be the intersection of the common tangents of the circles 
and 
. Show that the circumcircle of the triangle 
is tangent to the circumcircle of the triangle .
, the point 
is the foot of the altitude from 
, and 
is a point on the segment 
. The lines through parallel to 
and 
meet 
at 
and 
, respectively. Points 
and 
lie on the circles 
and 
, respectively, such that 
and 
.
and are concyclic.
be vertices of regular pentagon inscribed in a circle whose radius is 
and center is at 
. Find all possible values of 
be an acute triangle and let 
be the midpoint of . A circle passing through 
and meets the sides and at points and 
respectively. Let 
be the point such that 
is a parallelogram. Suppose that lies on the circumcircle of . Determine all possible values of 
.
be a triangle with circumcircle 
. Let 
and 
respectively denote the midpoints of the arcs and that do not contain the third vertex. Let 
denote the midpoint of arc 
(the arc including ). Let be the incenter of . Let 
be the circle that is tangent to and internally tangent to at , and let 
be the circle that is tangent to and internally tangent to at . Show that the line 
, and the lines through the intersections of and , meet on .
be a circle with centre , and 
a convex quadrilateral such that each of the segments 
and 
is tangent to . Let be the circumcircle of the triangle 
. The extension of 
beyond meets at , and the extension of 
beyond 
meets at . The extensions of 
and 
beyond meet at and , respectively. Prove that ![\[A D+D T+T X+X A=C D+D Y+Y Z+Z C.\]](http://latex.artofproblemsolving.com/8/2/6/8261276053071cc6cc06dda824a883426bf1d4ac.png)
such that for every 
is satisfied? Explain the answer.
be a random point on the smaller arc of the circumcircle of square 
, and let 
be the intersection point of segments and 
. The feet of the tangents from point to the circumcircle of the triangle 
are and , where 
is the center of the square. Prove that points , , and lie on a single circle.
for which exist two non-congruent triangles, whose sides have integer lengths and the numerical value of the area of each triangle is .
,
for all 
