Combi Geo
by Adywastaken, May 10, 2025, 3:58 PM
A regular polygon with 
vertices is given. To each vertex, a natural number from the set 
is assigned. Prove that there are 
vertices 
such that if the numbers 
are assigned to them respectively, then 
and 
is a parallelogram.
vertices is given. To each vertex, a natural number from the set 
is assigned. Prove that there are 
vertices 
such that if the numbers 
are assigned to them respectively, then 
and 
is a parallelogram.Inequality, inequality, inequality...
by Assassino9931, May 10, 2025, 9:38 AM
Let 
be real numbers such that ![\[ab^2+bc^2+ca^2=6\sqrt{3}+ac^2+cb^2+ba^2.\]](//latex.artofproblemsolving.com/3/d/0/3d015292c249a69a44d5f923092201d7a271b896.png)
Find the smallest possible value of 
.
Binh Luan and Nhan Xet, Vietnam
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 
.Binh Luan and Nhan Xet, Vietnam
This post has been edited 1 time. Last edited by Assassino9931, Today at 9:51 AM
where 
for some non-negative 
such that
for every 
.Divisibility..
by Sadigly, May 9, 2025, 7:37 AM
Find all consecutive even numbers, such that the square of their product is divisible by the sum of their squares.
consecutive even numbers, such that the square of their product is divisible by the sum of their squares.This post has been edited 3 times. Last edited by Sadigly, Yesterday at 4:34 PM
Surjective number theoretic functional equation
by snap7822, May 1, 2025, 12:18 PM
Let 
be a function satisfying the following conditions:
.
Proposed by snap7822
be a function satisfying the following conditions:
- For all
, if 
and 
, then 
; 
is surjective.
,
and 
,
;
is surjective.
.Proposed by snap7822
LOTS of recurrence!
by SatisfiedMagma, May 14, 2023, 1:33 PM
There is a rectangular plot of size 
. This has to be covered by three types of tiles - red, blue and black. The red tiles are of size 
, the blue tiles are of size and the black tiles are of size 
. Let 
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.
![\[ \binom{m}{r} = \begin{cases}
\dfrac{m!}{r!(m-r)!}, &\text{ if $0 \le r \le m$,} \\
0, &\text{ otherwise}
\end{cases}\]](//latex.artofproblemsolving.com/b/8/d/b8d653661e421db1fce7d93837e7bce31337bb30.png)
for integers 
.
. This has to be covered by three types of tiles - red, blue and black. The red tiles are of size 
, the blue tiles are of size and the black tiles are of size 
. Let 
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.- Prove that
for all 
.
- Prove that
for all 
.
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 
.This post has been edited 1 time. Last edited by SatisfiedMagma, May 14, 2023, 1:33 PM
Many Equal Sides
by mathisreal, Nov 10, 2022, 9:57 PM
Let 
be a triangle with 
and 
. Let 
and 
be the midpoints of 
and 
respectively. The point 
is inside of 
such that 
is equilateral. Let 
and 
. Prove that 
.
be a triangle with 
and 
. Let 
and 
be the midpoints of 
and 
respectively. The point 
is inside of 
such that 
is equilateral. Let 
and 
. Prove that 
.Vectors in a tilted square
by mathwizard888, Jul 19, 2017, 4:35 PM
Find all positive integers 
such that the following statement holds: Suppose real numbers 
, 
, 
, 
, 
, 
, , 
satisfy 
for all 
. Then there exists 
, 
, , 
, each of which is either 
or 
, such that
![\[ \left| \sum_{i=1}^n \varepsilon_i a_i \right| + \left| \sum_{i=1}^n \varepsilon_i b_i \right| \le 1. \]](//latex.artofproblemsolving.com/6/5/8/658ddc5e8626e7d5ef29809c8c3ac641e3a87757.png)
such that the following statement holds: Suppose real numbers 
, 
, 
, 
, 
, 
, , 
satisfy 
for all 
. Then there exists 
, 
, , 
, each of which is either 
or 
, such that![\[ \left| \sum_{i=1}^n \varepsilon_i a_i \right| + \left| \sum_{i=1}^n \varepsilon_i b_i \right| \le 1. \]](http://latex.artofproblemsolving.com/6/5/8/658ddc5e8626e7d5ef29809c8c3ac641e3a87757.png)
Equation of integers
by jgnr, Jun 2, 2008, 6:30 PM
For an arbitrary positive integer 
, define 
as the product of the digits of (in decimal). Find all positive integers such that 
.
, define 
as the product of the digits of (in decimal). Find all positive integers such that 
.
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