4 variables with quadrilateral sides 2
by mihaig, Apr 29, 2025, 8:47 PM
Let 
satisfying
Prove
satisfying
Prove
and 
.
?Number theory
by MuradSafarli, Apr 29, 2025, 7:39 PM
Prove that for any natural number 
:
![\[
1 \cdot 3 \cdot 5 \cdot \ldots \cdot (2n + 1) \mid (4n + 3)(4n + 5) \cdot \ldots \cdot (8n + 3).
\]](//latex.artofproblemsolving.com/3/1/a/31a0eac3c7ca298c8fa221083dc94e127e705784.png)
:![\[
1 \cdot 3 \cdot 5 \cdot \ldots \cdot (2n + 1) \mid (4n + 3)(4n + 5) \cdot \ldots \cdot (8n + 3).
\]](http://latex.artofproblemsolving.com/3/1/a/31a0eac3c7ca298c8fa221083dc94e127e705784.png)
This post has been edited 1 time. Last edited by MuradSafarli, 3 hours ago
Easy Combinatorics
by MuradSafarli, Apr 29, 2025, 6:40 PM
A student firstly wrote 
on the board. For each procces, the stutent deletes the number x and replaces it with either 
or 
or 
. Is this possible to make the number 
on the board?
on the board. For each procces, the stutent deletes the number x and replaces it with either 
or 
or 
. Is this possible to make the number 
on the board?prefix sum QRs
by optimusprime154, Apr 29, 2025, 5:23 PM
This post has been edited 3 times. Last edited by optimusprime154, 4 hours ago
functional equation interesting
by skellyrah, Apr 24, 2025, 8:32 PM
find all functions IR->IR such that 
This post has been edited 1 time. Last edited by skellyrah, Apr 25, 2025, 7:42 PM
Sum of products is n mod 2
by y-is-the-best-_, Sep 22, 2020, 11:35 PM
Let 
be different real numbers. Prove that
![\[\sum_{1 \leqslant i \leqslant n} \prod_{j \neq i} \frac{1-x_{i} x_{j}}{x_{i}-x_{j}}=\left\{\begin{array}{ll}
0, & \text { if } n \text { is even; } \\
1, & \text { if } n \text { is odd. }
\end{array}\right.\]](//latex.artofproblemsolving.com/6/5/2/6525fc6b451ff5a3c83ff2a0541b063a7cf065c3.png)
be different real numbers. Prove that![\[\sum_{1 \leqslant i \leqslant n} \prod_{j \neq i} \frac{1-x_{i} x_{j}}{x_{i}-x_{j}}=\left\{\begin{array}{ll}
0, & \text { if } n \text { is even; } \\
1, & \text { if } n \text { is odd. }
\end{array}\right.\]](http://latex.artofproblemsolving.com/6/5/2/6525fc6b451ff5a3c83ff2a0541b063a7cf065c3.png)
Functional equation on the set of reals
by abeker, Aug 25, 2017, 2:35 PM
Determine all functions 
satisfying
for all real numbers 
and 
.
satisfying
for all real numbers 
and 
.The number of integers
by Fang-jh, Apr 4, 2009, 10:20 AM
Prove that for any odd prime number 
the number of positive integer 
satisfying 
is less than or equal to 
where 
is a constant independent of 
the number of positive integer 
satisfying 
is less than or equal to 
where 
is a constant independent of 
This post has been edited 2 times. Last edited by Fang-jh, Apr 4, 2009, 2:23 PM
Perpendicularity
by April, Dec 28, 2008, 4:09 AM
Point 
lies inside triangle 
such that 
and 
. Point 
is the midpoint of segment 
. Point 
lies on segment 
with 
. Prove that 
.
lies inside triangle 
such that 
and 
. Point 
is the midpoint of segment 
. Point 
lies on segment 
with 
. Prove that 
.This post has been edited 1 time. Last edited by v_Enhance, Jan 25, 2016, 3:51 PM
Reason: \equal -> =
Reason: \equal -> =
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