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.
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.
and 
such that
such that for every 
is satisfied? Explain the answer.
be an integer, and let 
be positive real numbers such that 
. Prove that![\[(1 + a_2)^2 (1 + a_3)^3 \dotsm (1 + a_n)^n > n^n.\]](http://latex.artofproblemsolving.com/7/7/f/77fb719e975225b59c64f1f3aa102c98a6c25f50.png)
colors, there exists a monochromatic regular hexagon.
is the smallest number such that if 
, for each coloring of the set 
with two colors there exists a monochromatic arithmetic progression of length 
. Prove that
.
points in plane. Prove that the number of isosceles triangles having their vertices among these points is 
. Find a configuration of points in plane such that the number of equilateral triangles with vertices among these points is 
.
satisfying 
and 
, in which 
denotes the sum of all positive divisors of , and 
denotes the largest prime divisor of .
be a polynomial with real coefficients.
. Prove that the polynomial 
cannot have 2024 real roots.
and 
. Prove that 
for all real numbers 
.
grid, one can find 
points such that no four of them are vertices of a square with sides parallel to lines of the grid. Imagine yourself as Erdos (!) and guess what is the best exponent instead of 
!
denote the set of positive integers. Find all functions 
satisfying the condition
for all 
be a scalene triangle, 
its medial triangle, and a point on the Thomson cubic (= 
) of . (Suppose that 
).
be the circumcevian triangle of wrt .
be the pedal triangle of wrt .
be the reflection in 
of 
. Define 
, 
cyclically.
be the reflection in 
of . Define 
, 
cyclically.
be the reflection in 
of . Define 
, 
cyclically.
, 
, 
and the orthocentroidal circle of are coaxal.
, the angle at vertex 
is 
. It is known that 
and the midpoint of the side . Denote by the midpoint of the broken line and by the intersection point of the lines and 
. Find the measure of the angle 
.
, the angle at vertex is . It is known that and the midpoint of the side . Denote by the midpoint of the broken line and by the intersection point of the lines and . Find the measure of the angle .
such that: 
and 
.
