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.
such that for all 
and 
,
such that 
or 
, there exist integers![\[
1 \leq a_1 < a_2 < \cdots < a_{(p-1)/2} < p
\]](http://latex.artofproblemsolving.com/6/8/e/68e3d73edec43b2cc480832adf23bbbb5f4c7bee.png)
such that![\[
\prod_{\substack{1 \leq i < j \leq (p-1)/2}} (a_i + a_j)^2 \equiv 1 \pmod{p}.
\]](http://latex.artofproblemsolving.com/2/9/e/29edf6ab8cee2a526e7753a140c5a291e987856a.png)
be distinct real numbers such that![\[
a + b + c + \frac{1}{abc} = \frac{19}{2}
\]](http://latex.artofproblemsolving.com/5/1/e/51eef77358db2b6c98f0fb0fb4e30db837a03642.png)
Find the maximum possible value of 
.
such that for all positive real numbers 
and 
, the following equation holds:![\[
\frac{x + f(y)}{x f(y)} = f\left( \frac{1}{y} + f\left( \frac{1}{x} \right) \right)
\]](http://latex.artofproblemsolving.com/7/d/f/7df25755bc33bef970fffb392fb5b6b789390881.png)
.The solution to 
is only 
for some 
, consider the two circles on 
plane: 
. Find the range of 
for which these circles have two intersection points, then find the equation of the line passing through these points.
on the perimeter of a circle with radius 1. Let the length of the cord 
. When two points move on the circle, find the maximum area of 
.
satisfies 
. Prove that 
.
be real numbers such that ![\[ab^2+bc^2+ca^2=6\sqrt{3}+ac^2+cb^2+ba^2.\]](http://latex.artofproblemsolving.com/3/d/0/3d015292c249a69a44d5f923092201d7a271b896.png)
Find the smallest possible value of 
.
be real numbers such that 
. Find the smallest possible value of 
.
into 
or 
can sometimes be quite complex. That's why I’ve written a program to assist with this process.
cards on a table, flipped face down. Madina knows that on each card a single number is written and that the numbers are different integers from 
to . In a move, Madina is allowed to choose any 
cards, and she is told a number that is written on one of the chosen cards, but not which specific card it is on. After several moves, Madina must determine the written numbers on as many cards as possible. What is the maximum number of cards Madina can ensure to determine?
is a sphere with a diameter AP and passing through the point F. This sphere is centered on the midline M, N of the triangle 
parallel to the side BC, where M, N are the midpoints of the sides AB, AC. If the point A lies on the line BC, then the points A, F and the locus of the vertices V of the right angles is a sphere centered on segment M, N, where M, N are the midpoints of the segments AB, AC. As the point P moves on the segment BC, the locus of the vertices V becomes a portion of the pencil of spheres centered on the segment MN and passing through the points A, F. If the point A does not lie on the line BC, the pencil of spheres is elliptic (the spheres intersect in a fixed circle with the diameter AF and perpendicular to the line BC). If the point A lies on the line BC, the pencil of spheres is parabolic (the spheres touch the plane perpendicular to the line BC at the point A).
be 
Therefore, given 
and 
all possible vertices of the right angle lie on the circle with diameter 
Thus, when all possible circles are drawn, it encloses a region as shown in the diagram below:![[asy]
draw(circle((3.500000,3.000000),4.609772));
draw(circle((3.550000,3.000000),4.571925));
draw(circle((3.600000,3.000000),4.534314));
draw(circle((3.650000,3.000000),4.496943));
draw(circle((3.700000,3.000000),4.459821));
draw(circle((3.750000,3.000000),4.422952));
draw(circle((3.800000,3.000000),4.386342));
draw(circle((3.850000,3.000000),4.350000));
draw(circle((3.900000,3.000000),4.313931));
draw(circle((3.950000,3.000000),4.278142));
draw(circle((4.000000,3.000000),4.242641));
draw(circle((4.050000,3.000000),4.207434));
draw(circle((4.100000,3.000000),4.172529));
draw(circle((4.150000,3.000000),4.137934));
draw(circle((4.200000,3.000000),4.103657));
draw(circle((4.250000,3.000000),4.069705));
draw(circle((4.300000,3.000000),4.036087));
draw(circle((4.350000,3.000000),4.002812));
draw(circle((4.400000,3.000000),3.969887));
draw(circle((4.450000,3.000000),3.937321));
draw(circle((4.500000,3.000000),3.905125));
draw(circle((4.550000,3.000000),3.873306));
draw(circle((4.600000,3.000000),3.841875));
draw(circle((4.650000,3.000000),3.810840));
draw(circle((4.700000,3.000000),3.780212));
draw(circle((4.750000,3.000000),3.750000));
draw(circle((4.800000,3.000000),3.720215));
draw(circle((4.850000,3.000000),3.690867));
draw(circle((4.900000,3.000000),3.661967));
draw(circle((4.950000,3.000000),3.633524));
draw(circle((5.000000,3.000000),3.605551));
draw(circle((5.050000,3.000000),3.578058));
draw(circle((5.100000,3.000000),3.551056));
draw(circle((5.150000,3.000000),3.524557));
draw(circle((5.200000,3.000000),3.498571));
draw(circle((5.250000,3.000000),3.473111));
draw(circle((5.300000,3.000000),3.448188));
draw(circle((5.350000,3.000000),3.423814));
draw(circle((5.400000,3.000000),3.400000));
draw(circle((5.450000,3.000000),3.376759));
draw(circle((5.500000,3.000000),3.354102));
draw(circle((5.550000,3.000000),3.332041));
draw(circle((5.600000,3.000000),3.310589));
draw(circle((5.650000,3.000000),3.289757));
draw(circle((5.700000,3.000000),3.269557));
draw(circle((5.750000,3.000000),3.250000));
draw(circle((5.800000,3.000000),3.231099));
draw(circle((5.850000,3.000000),3.212865));
draw(circle((5.900000,3.000000),3.195309));
draw(circle((5.950000,3.000000),3.178443));
draw(circle((6.000000,3.000000),3.162278));
draw(circle((6.050000,3.000000),3.146824));
draw(circle((6.100000,3.000000),3.132092));
draw(circle((6.150000,3.000000),3.118092));
draw(circle((6.200000,3.000000),3.104835));
draw(circle((6.250000,3.000000),3.092329));
draw(circle((6.300000,3.000000),3.080584));
draw(circle((6.350000,3.000000),3.069609));
draw(circle((6.400000,3.000000),3.059412));
draw(circle((6.450000,3.000000),3.050000));
draw(circle((6.500000,3.000000),3.041381));
draw(circle((6.550000,3.000000),3.033562));
draw(circle((6.600000,3.000000),3.026549));
draw(circle((6.650000,3.000000),3.020348));
draw(circle((6.700000,3.000000),3.014963));
draw(circle((6.750000,3.000000),3.010399));
draw(circle((6.800000,3.000000),3.006659));
draw(circle((6.850000,3.000000),3.003748));
draw(circle((6.900000,3.000000),3.001666));
draw(circle((6.950000,3.000000),3.000417));
draw(circle((7.000000,3.000000),3.000000));
draw(circle((7.050000,3.000000),3.000417));
draw(circle((7.100000,3.000000),3.001666));
draw(circle((7.150000,3.000000),3.003748));
draw(circle((7.200000,3.000000),3.006659));
draw(circle((7.250000,3.000000),3.010399));
draw(circle((7.300000,3.000000),3.014963));
draw(circle((7.350000,3.000000),3.020348));
draw(circle((7.400000,3.000000),3.026549));
draw(circle((7.450000,3.000000),3.033562));
draw(circle((7.500000,3.000000),3.041381));
draw(circle((7.550000,3.000000),3.050000));
draw(circle((7.600000,3.000000),3.059412));
draw(circle((7.650000,3.000000),3.069609));
draw(circle((7.700000,3.000000),3.080584));
draw(circle((7.750000,3.000000),3.092329));
draw(circle((7.800000,3.000000),3.104835));
draw(circle((7.850000,3.000000),3.118092));
draw(circle((7.900000,3.000000),3.132092));
draw(circle((7.950000,3.000000),3.146824));
draw(circle((8.000000,3.000000),3.162278));
draw(circle((8.050000,3.000000),3.178443));
draw(circle((8.100000,3.000000),3.195309));
draw(circle((8.150000,3.000000),3.212865));
draw(circle((8.200000,3.000000),3.231099));
draw(circle((8.250000,3.000000),3.250000));
draw(circle((8.300000,3.000000),3.269557));
draw(circle((8.350000,3.000000),3.289757));
draw(circle((8.400000,3.000000),3.310589));
draw(circle((8.450000,3.000000),3.332041));
draw(circle((8.500000,3.000000),3.354102));
dot((0,0),red);
dot((10,0),red);
dot((7,6),red);
label((0,0),"$B$",SW,red);
label((10,0),"$C$",SE,red);
label((7,6),"$A$",N,red);
draw((0,0)--(7,6)--(10,0)--cycle,red);
[/asy]](http://latex.artofproblemsolving.com/c/a/2/ca2b9e2b05e066faf4f85049352b24def792b887.png)
