[/quote]
,
,
as the product of the digits of 
.
consecutive even numbers, such that the square of their product is divisible by the sum of their squares.
be a function satisfying the following conditions:
,
and 
,
;
is surjective.
.
be a triangle with 
and 
.
and 
be the midpoints of 
and 
respectively. The point 
is inside of 
such that 
is equilateral. Let 
and 
.
.
.
,
.
denote the number of ways this can be done. For example, clearly 
because we can have either a red or a blue tile. Also 
since we could have tiled the plot as: two red tiles, two blue tiles, a red tile on the left and a blue tile on the right, a blue tile on the left and a red tile on the right, or a single black tile.
for all 
.
for all 
.![\[ \binom{m}{r} = \begin{cases}
\dfrac{m!}{r!(m-r)!}, &\text{ if $0 \le r \le m$,} \\
0, &\text{ otherwise}
\end{cases}\]](http://latex.artofproblemsolving.com/b/8/d/b8d653661e421db1fce7d93837e7bce31337bb30.png)
for integers 
.
can be written in the form
where 
for some non-negative 
such that
for every 
.
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 
.
,
,
,
,
,
,
satisfy 
for all 
.
,
,
,
or 
,![\[ \left| \sum_{i=1}^n \varepsilon_i a_i \right| + \left| \sum_{i=1}^n \varepsilon_i b_i \right| \le 1. \]](http://latex.artofproblemsolving.com/6/5/8/658ddc5e8626e7d5ef29809c8c3ac641e3a87757.png)
vertices is given. To each vertex, a natural number from the set 
is assigned. Prove that there are 
such that if the numbers 
are assigned to them respectively, then 
and 
is a parallelogram.
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this is my 100th post! i cant believe it. 
P.S. can everyone who reads this give it an upvote? thx!
P.S. can everyone who reads this give it an upvote? thx!
This post has been edited 1 time. Last edited by JohannIsBach, Mar 10, 2025, 4:08 PM
more posts to get to 1000!
more 2 go
