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find all function F: IR->IR such that 
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Iranians playing with cards module a prime number.
Ryan-asadi 2
N
by AshAuktober
Source: Iranian Team Selection Test - P2
.........
2 replies
Coloring plane in black
Ryan-asadi 1
N
by AshAuktober
Source: Iran Team Selection Test - P3
..........
1 reply
An analytic sequence
Ryan-asadi 1
N
by AshAuktober
Source: Iran Team Selection Test - P1
..........
1 reply
Geometry
gggzul 6
N
by Captainscrubz
In trapezoid 
segments 
and 
are parallel. Angle bisectors of 
and 
meet at 
. Angle bisectors of 
and 
meet at 
. Prove that 
is cyclic
segments 
and 
are parallel. Angle bisectors of 
and 
meet at 
. Angle bisectors of 
and 
meet at 
. Prove that 
is cyclic
6 replies
Need help on this simple looking problem
TheGreatEuler 0
Show that 1+2+3+4....n divides 1^k+2^k+3^k....n^k when k is odd. Is this possible to prove without using congruence modulo or binomial coefficients?
0 replies
Geometry
Lukariman 5
N
by Lukariman
Given circle (O) and point P outside (O). From P draw tangents PA and PB to (O) with contact points A, B. On the opposite ray of ray BP, take point M. The circle circumscribing triangle APM intersects (O) at the second point D. Let H be the projection of B on AM. Prove that 
= 2∠AMP.
= 2∠AMP.
5 replies
inq , not two of them =0
win14 0
Let a,b,c be non negative real numbers such that no two of them are simultaneously equal to 0
0 replies
Number theory
MathsII-enjoy 5
N
by MathsII-enjoy
Prove that when 
| 
with 
are positive integers and 
is prime (
), we get: 
| 
with 
are positive integers and 
is prime (
), we get: 
5 replies
Number theory
Foxellar 0
It is known that for all positive integers 
,
![\[
1^2 + 2^2 + 3^2 + \ldots + k^2 = \frac{k(k + 1)(2k + 1)}{6}
\]](//latex.artofproblemsolving.com/b/5/d/b5d49e25bd8d8a211b72372ff967558e1a8365ce.png)
Find the smallest positive integer such that 
is divisible by 200.
,![\[
1^2 + 2^2 + 3^2 + \ldots + k^2 = \frac{k(k + 1)(2k + 1)}{6}
\]](http://latex.artofproblemsolving.com/b/5/d/b5d49e25bd8d8a211b72372ff967558e1a8365ce.png)
Find the smallest positive integer such that 
is divisible by 200.
0 replies
Combinatorics
P162008 4
N
by cazanova19921
Let 
Let ![$[n]$](//latex.artofproblemsolving.com/0/d/e/0deff9086fd7840ef23860f5bc924f1d4d2d2310.png)
denote the set of natural numbers less than or equal to 
Let![$f(m,n) = \sum_{(x_1,x_2,x_3, \cdots, x_m) \in [n]^{m}} \frac{x_1}{x_1 + x_2 + x_3 + \cdots + x_m} \binom{n}{x_1} \binom{n}{x_2} \binom{n}{x_3} \cdots \binom{n}{x_m} 2^{\left(\sum_{i=1}^{m} x_i\right)}$](//latex.artofproblemsolving.com/f/d/f/fdf7b2fcfb960e93d59f3f84cef4372262c69499.png)
Compute the sum of the digits of
Let ![$[n]$](http://latex.artofproblemsolving.com/0/d/e/0deff9086fd7840ef23860f5bc924f1d4d2d2310.png)
denote the set of natural numbers less than or equal to 
Let
![$f(m,n) = \sum_{(x_1,x_2,x_3, \cdots, x_m) \in [n]^{m}} \frac{x_1}{x_1 + x_2 + x_3 + \cdots + x_m} \binom{n}{x_1} \binom{n}{x_2} \binom{n}{x_3} \cdots \binom{n}{x_m} 2^{\left(\sum_{i=1}^{m} x_i\right)}$](http://latex.artofproblemsolving.com/f/d/f/fdf7b2fcfb960e93d59f3f84cef4372262c69499.png)
Compute the sum of the digits of
4 replies
interesting function equation (fe) in IR
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Source: mine
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or 
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Y by
skellyrah wrote:
find all function F: IR->IR such that
If IR means irrational numbers, and the problem is to find all
such that 
for all 
then setting 
gives a contradictionThis post has been edited 1 time. Last edited by jasperE3, Apr 23, 2025, 9:43 PM
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