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.
, let 
be a point on the segment 
and let 
be the intersection point of 
with the diagonal 
. The line perpendicular to the segment through intersects the side 
at point 
. Let 
be a point on the segment 
such that 
. Let 
be the intersection point of the diagonal 
and the segment 
.
and 
are concyclic.
+1 w
be the circumcenter and 
the orthocenter of an acute triangle 
Prove that the area of one of the triangles 
and 
is equal to the sum of the areas of the other two.
with any line through the centroid and I have no clue as to why that is true. Plz help
is a minuan pair if it satisfies the following two conditions:
is even.
are all the positive divisors of , ordered such that 
, then the set of all positive divisors of 
is precisely
.
have altitudes 
circumcenter and orthocenter . Let 
be a point on line 
. The points 
are on lines 
respectively such that 
and 
. Prove that 
is perpendicular to 
.
and 
meet at two distinct points 
and 
. Choose any line 
through , and let meet again at 
(other than ) and meet again at 
. On , let 
be the midpoint of the minor arc 
(i.e., the point on such that 
subtends the arc ), and on let 
be the midpoint of the minor arc 
. Prove that![\[
MN \parallel \text{(radical axis of } \omega_1, \omega_2).
\]](http://latex.artofproblemsolving.com/6/6/4/6647f0f7182bdf5096ec5f2956a272803082f4f7.png)
is on 
and 
is on .
.
is the circumcenter of 
, and 
is the circumcenter of 
intersects with at 
be a triangle with orthocentre . Let 
be the foot of altitudes of 
onto the opposite sides, respectively. Consider 
, the reflection of 
about line 
. Let line 
cut at distinct points 
, and let 
be the orthocenter of 
. Prove that points 
are concyclic.
be a continuous function such that 
show that 
is even.
grid, the number 1 is written. A move consists of choosing a positive integer 
and a cell, and then multiplying the numbers in that cell and its neighbors by . Is it possible, after a finite number of moves, for every cell on the grid to contain the number 
?
and 
be sequences of positive integers defined by the recurrence relations:![\[
a_{n+1} = n a_n + 1, \quad b_{n+1} = n b_n - 1
\]](http://latex.artofproblemsolving.com/e/8/f/e8fceef09debd22326dc85d36aa10a387650f797.png)
for all positive integers 
.
be positive integers, and define![\[ n = a + b + c + d + e + f. \]](http://latex.artofproblemsolving.com/9/d/d/9ddfecfa5800050bc15dcaa48d17d12e431fc358.png)
Suppose that divides both of the following expressions:![\[
abc + def \quad \text{and} \quad ab + bc + ca - de - ef - fd.
\]](http://latex.artofproblemsolving.com/c/5/8/c5817937f736a7b00c85e1f92b8e561ba26cf8b0.png)
Prove that is a composite number.
is called 
if there exists a permutation 
of the numbers 
, such that:
and 
have different parities for every 
;
the sum 
is a quadratic residue modulo for every 
.
compute the smallest possible value of ![\[\sum_{k=1}^{n}\left\lfloor\frac{a_k}{k}\right\rfloor\]](http://latex.artofproblemsolving.com/5/6/2/562c27c7779285b6439c4087f52933a41f29ee2e.png)
over all permutations 
of 
-
plane, given a parabola 
passing through 3 points 
and 
.
vary in the range of 
, draw the domain swept out by .
be a parabola in the 
plane that passes through 
, then we obtain the following set of equations:
(i);
(ii);
(iii).
(iv), and substitution of (iv) into (iii) produces 
(v). WLOG let us substitute (iv) & (v) into (ii) to obtain:
,
(vi). Solving (vi) for yields the quadratic:
(for real 
) (vii).![$t \in [-1,1]$](http://latex.artofproblemsolving.com/9/7/c/97ccc162ccb6a511d766bf5d9ea20f2ea4a9305f.png)
, then 
, or the domain of 
where:![$R_{1} = \{(x,y): x \in (-1,0], y \in [x,x^2+3x+1]\}$](http://latex.artofproblemsolving.com/0/a/0/0a06a5db27f9ff016027898c6f07cb4ca0e6e4bb.png)
and ![$R_{2} = \{(x,y): x \in (0,1), y \in [x,x^2-x+1]\}$](http://latex.artofproblemsolving.com/4/e/9/4e923c33a69f95704416f2856874c0616398af7f.png)
