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.
, 
, 
, the inequality![\[ \frac {1}{a^3 + b^3 + abc} + \frac {1}{b^3 + c^3 + abc} + \frac {1}{c^3 + a^3 + abc} \leq \frac {1}{abc}
\]](http://latex.artofproblemsolving.com/f/a/9/fa964e29de91f0e2f3cb32e335a52d4521bc38fd.png)
, , be positive real numbers such that 
. Prove that![\[ \frac {1}{a^{3}\left(b + c\right)} + \frac {1}{b^{3}\left(c + a\right)} + \frac {1}{c^{3}\left(a + b\right)}\geq \frac {3}{2}.
\]](http://latex.artofproblemsolving.com/b/2/9/b295b0b2a765e41a639657e13b724c60323fa208.png)
be a triangle. Let 
be the feet of the altitudes from 
respectively. Let 
be the projections of onto line 
. Show that 
.
be real numbers such that ![\[ \frac{1}{x^2-1} + \frac{1}{y^2-1} + \frac{1}{z^2-1} = 1. \]](http://latex.artofproblemsolving.com/8/f/5/8f5d49c9b44e3db3cd85fea7b6521e1dc02d4269.png)
Prove that ![\[ \frac{1}{x+1} + \frac{1}{y+1} + \frac{1}{z+1} \le 1. \]](http://latex.artofproblemsolving.com/c/0/4/c0438d6eba686d0a11fd4decb4afa9a14aa50bac.png)
are non-negative integers.
satisfying the following equation:
.
and 
be two sequences defined by:![\[
a_{n+1} = \frac{1 + a_n + a_n b_n}{b_n} \quad \text{and} \quad b_{n+1} = \frac{1 + b_n + a_n b_n}{a_n}
\]](http://latex.artofproblemsolving.com/a/a/0/aa0a5c5acedc990eb345fb50a8395db05307a4e7.png)
for all 
, with initial values 
and 
.![\[
a_{2024} < 5.
\]](http://latex.artofproblemsolving.com/f/c/a/fca6e6630f33429b5a7ea867d639a2e29ff881f2.png)
children, no two of the same height, lined up. Let us call a pair of different children 
good if between them there is no child whose height is greater than the height of one of 
and 
, but less than the height of the other. What is the greatest number of good pairs that could be formed? (Here, and 
are considered the same pair.)
where 
is in radians.
satisfy 
prove in equality.
be distinct reals such that 
Prove that 
be the altitude and 
be the intersection point of the altitudes of triangle 
. Point 
is symmetric to with respect to 
. The circumcircle of triangle 
intersects at points 
and 
. Prove that 
.
. is located on the diagonal 
, then the parallelogram is a rhombus. a rhombus whenever the circumcenter of the triangle is located on the diagonal ?
is called beautiful if the are at least 
integers 
such that in base 
we have that and 
end in the same digit. Find the sum of the two smallest beautiful numbers.
is called beautiful if the are at least integers such that in base we have that and end in the same digit. Find the sum of the two smallest beautiful numbers.
.
min
![\[ N^2 \equiv N \pmod{b} \]](http://latex.artofproblemsolving.com/a/c/4/ac4ee87facecbef82d166c38849d8add3b981029.png)
Rearranging, we obtain:![\[ N(N-1) \equiv 0 \pmod{b} \]](http://latex.artofproblemsolving.com/1/a/6/1a6a5cffdd4f21aacaa51f72d7100e6e0c4c72bd.png)
and 
are relatively prime, meaning they share no common factors. This follows directly from the Euclidean algorithm.
as its prime factorization:![\[ b = p_1^{e_1} p_2^{e_2} \dots p_k^{e_k} \]](http://latex.artofproblemsolving.com/0/9/9/099a60dfe024e610f64d2d922812e891d45c3892.png)
and exclude the remaining prime power factors 
.![\[ N \equiv 0 \pmod{a_1 a_2 \dots a_m} \]](http://latex.artofproblemsolving.com/f/7/c/f7c63c3f424decdeb95b8b60f69c1cc8808f0063.png)
![\[ N \equiv 1 \pmod{b_1 b_2 \dots b_n} \]](http://latex.artofproblemsolving.com/c/5/7/c57bf18091e8216d6e461222f91049c42c8e03be.png)
![\[ 2^{\text{number of distinct prime factors of } b} \]](http://latex.artofproblemsolving.com/3/0/7/30753fb35fcd010e056a65102f24ce572bed6087.png)
![\[ 2^x = 17 \]](http://latex.artofproblemsolving.com/5/e/8/5e8a8473383eaf71134d89a5cbd193b701a8ee25.png)
(as 
and 
is too small), we need 
,![\[ b_1 = 2 \times 3 \times 5 \times 7 \times 11 = 2310 \]](http://latex.artofproblemsolving.com/7/1/e/71e721575458606c673a9b7f7dd342260e43b42c.png)
![\[ b_2 = 2 \times 3 \times 5 \times 7 \times 13 = 2730 \]](http://latex.artofproblemsolving.com/f/0/5/f059046a885f2c60af9118b827da13bf92d33589.png)
![\[ 2310 + 2730 = 5040 \]](http://latex.artofproblemsolving.com/b/0/1/b012249c1119c95b517bb79fd8289d534f333584.png)
.