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.
to 
) until they have chosen seven such digits, and then consider the resulting seven-digit number by concatenating the digits in the order selected, with the seventh digit appearing last (i.e. 
). Alice wins if and only if the resulting number is the last seven decimal digits of some perfect seventh power. Please determine which player has the winning strategy.
containers placed along the track. Prove that there exists a position starting at which the car, beginning with an empty tank of gas, can complete a loop around the track without running out of gas. The tank of gas is assumed to be large enough.
such that there exists a unique integer 
such that 
with the following property:![\[ n!\mid a^n + 1
\]](http://latex.artofproblemsolving.com/e/2/3/e23d39aab685888993d355d8c93d4e6475f53487.png)
be reals such that 
Prove that
Let be reals such that 
Prove that
which are larger than 
and less than 
Prove that
Find 
.
Taking the square root of the second equation, we get
has a right angle at 
, with 
and 
. If 
and 
are points
and 
, respectively, such that 
, find the perimeter of quadrilateral
.
, 
, 
are positive integers such that 
. Find all possible values of 
.
to rearrange the terms and place the higher-degree terms at the front of the expression in order to have a more recognizable form.
![\begin{align*}
a^4+b^4+c^4-2a^2b^2-2a^2c^2-2b^2c^2 &=a^4-2a^2(b^2+c^2)+b^4-2b^2c^2+c^4 \\
&= a^4-2a^2(b^2+c^2)+(b^2+c^2)^2 - 4b^2c^2 \\
&= [a^2 - (b^2+c^2)]^2-(2bc)^2 \\
&= (a^2-b^2-c^2+2bc)(a^2-b^2-c^2-2bc)\\
&=[a^2-(b-c)^2][a^2-(b+c)^2] \\
&= (a+b-c)(a-b+c)(a+b+c)(a-b-c) = -63.
\end{align*}](http://latex.artofproblemsolving.com/0/c/4/0c430b7562ec7571fdf3e492939c354e0c3e7e20.png)
, we get
.
, 
, or 
, we get something of the form![\[
a^2(a+2b)(a-2b)=-63.
\]](http://latex.artofproblemsolving.com/1/0/0/1006c2fabc10f934c25b6e6ae822a59ce2dd2712.png)
and 
are the only square factors of 
, we have 
.
are 
and 
, including all their permutations.
are all distinct positive integers, there are no solutions. are 
.
. Three points 
are on the sides 
respectively such that 
. If 
, then find the length of 
?
have lengths 
and choose points 
above such that:
:
(i);
(ii);
(iii)
(via Wolfram Alpha). Hence:
.
and 
Prove that
.
,
is a right angle, using pythagorean theorem, we can get 
.
,
.
from point 
be the intersection of said altitude and 
.
both have a right angle and 
,
through AA Postulate. 
,
,
.
(CD = DE, as stated in the problem), 
,
(they were formed by an altitude, and are therefore both right angles), we can show that 
through HL Theorem. 
corresponds with 
,
.
,
and 
.
.
,such that 
P
Prove that
satisfying
Prove
