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IMC 2009 Day 1 P2
joybangla 3
N
by lminsl
Let 
be real square matrices of the same order, and suppose 
is invertible. Prove that
![\[ (A-B)C=BA^{-1}\implies C(A-B)=A^{-1}B \]](//latex.artofproblemsolving.com/0/6/0/06007271017c2c48faa708366f1adc67db91f245.png)
be real square matrices of the same order, and suppose 
is invertible. Prove that![\[ (A-B)C=BA^{-1}\implies C(A-B)=A^{-1}B \]](http://latex.artofproblemsolving.com/0/6/0/06007271017c2c48faa708366f1adc67db91f245.png)
3 replies
Expand into a Fourier series
Tip_pay 2
N
by Mathzeus1024
Expand the function in a Fourier series on the interval 
2 replies
D1041 : A generalisation of Tchebychef's Inequality
Dattier 1
N
by Dattier
Source: les dattes à Dattier
Let ![$f,g \in C^1([0,1])$](//latex.artofproblemsolving.com/6/1/7/617c5be50035205b21074842d7266911aa1dc4d8.png)
.
Is it true that :
?
![$f,g \in C^1([0,1])$](http://latex.artofproblemsolving.com/6/1/7/617c5be50035205b21074842d7266911aa1dc4d8.png)
.Is it true that :
?
1 reply
Putnam 2013 A5
Kent Merryfield 10
N
by blackbluecar
For 
a list of 
real numbers 
is said to be area definite for 
if the inequality ![\[\sum_{1\le i<j<k\le m}a_{ijk}\cdot\text{Area}(\triangle A_iA_jA_k)\ge0\]](//latex.artofproblemsolving.com/e/7/7/e778445aa49547e4a227d16cc8685370a5d1a7d9.png)
holds for every choice of 
points 
in 
For example, the list of four numbers 
is area definite for 
Prove that if a list of numbers is area definite for 
then it is area definite for 
a list of 
real numbers 
is said to be area definite for 
if the inequality ![\[\sum_{1\le i<j<k\le m}a_{ijk}\cdot\text{Area}(\triangle A_iA_jA_k)\ge0\]](http://latex.artofproblemsolving.com/e/7/7/e778445aa49547e4a227d16cc8685370a5d1a7d9.png)
holds for every choice of 
points 
in 
For example, the list of four numbers 
is area definite for 
Prove that if a list of numbers is area definite for 
then it is area definite for 
10 replies
Reducing the exponents for good
RobertRogo 3
N
by RobertRogo
Source: The national Algebra contest (Romania), 2025, Problem 3/Abstract Algebra (a bit generalized)
Let be a ring with unity such that for every 
there exist 
such that 
. Prove that
a) If
then 
b) If there is an such that 
then the result from a) may no longer hold.
Authors: Laurențiu Panaitopol, Dorel Miheț, Mihai Opincariu, me, Filip Munteanu
be a ring with unity such that for every 
there exist 
such that 
. Prove thata) If
then 
b) If there is an
such that 
then the result from a) may no longer hold.Authors: Laurențiu Panaitopol, Dorel Miheț, Mihai Opincariu, me, Filip Munteanu
3 replies
If \(\prod_{i=1}^{n} (x + r_i) = \sum_{k=0}^{n} a_k x^k\), show that \[ \sum_{i=
Martin.s 1
N
by alexheinis
If 
, show that
![\[
\sum_{i=1}^{n} \tan^{-1} r_i = \tan^{-1} \frac{a_1 - a_3 + a_5 - \cdots}{a_0 - a_2 + a_4 - \cdots}
\]](//latex.artofproblemsolving.com/8/1/2/812c7c690b2237f45bc7453ff2b4160fb72c7b0d.png)
and
![\[
\sum_{i=1}^{n} \tanh^{-1} r_i = \tanh^{-1} \frac{a_1 + a_3 + a_5 + \cdots}{a_0 + a_2 + a_4 + \cdots}.
\]](//latex.artofproblemsolving.com/6/1/9/61962993a2c34e1c0ce961d7057127ae75876ac2.png)
, show that![\[
\sum_{i=1}^{n} \tan^{-1} r_i = \tan^{-1} \frac{a_1 - a_3 + a_5 - \cdots}{a_0 - a_2 + a_4 - \cdots}
\]](http://latex.artofproblemsolving.com/8/1/2/812c7c690b2237f45bc7453ff2b4160fb72c7b0d.png)
and![\[
\sum_{i=1}^{n} \tanh^{-1} r_i = \tanh^{-1} \frac{a_1 + a_3 + a_5 + \cdots}{a_0 + a_2 + a_4 + \cdots}.
\]](http://latex.artofproblemsolving.com/6/1/9/61962993a2c34e1c0ce961d7057127ae75876ac2.png)
1 reply
D1040 : A general and strange result
Dattier 1
N
by Dattier
Source: les dattes à Dattier
Let ![$f \in C([0,1];[0,1])$](//latex.artofproblemsolving.com/4/7/5/475518b7db0b787dd90b93f602c7b912c663fba3.png)
bijective, 
and ![$(a_k) \in [0,1]^\mathbb N$](//latex.artofproblemsolving.com/a/d/8/ad8a80d847c60ee8e916a1f9b08b7b03ec17c814.png)
with 
converge.
Is it true that
converge?
![$f \in C([0,1];[0,1])$](http://latex.artofproblemsolving.com/4/7/5/475518b7db0b787dd90b93f602c7b912c663fba3.png)
bijective, 
and ![$(a_k) \in [0,1]^\mathbb N$](http://latex.artofproblemsolving.com/a/d/8/ad8a80d847c60ee8e916a1f9b08b7b03ec17c814.png)
with 
converge.Is it true that
converge?
1 reply
functional equation in Z
Matheo_Lucas 2
N
by mrtheory
Find all functions 
such that
![\[
x f(2f(y) - x) + y^2 f(2x - f(y)) = \frac{f(x)^2}{x} + f(y f(y))
\]](//latex.artofproblemsolving.com/f/9/e/f9e1ccf18bc744433e70a733aa8a2424befc8497.png)
for all
with 
.
such that![\[
x f(2f(y) - x) + y^2 f(2x - f(y)) = \frac{f(x)^2}{x} + f(y f(y))
\]](http://latex.artofproblemsolving.com/f/9/e/f9e1ccf18bc744433e70a733aa8a2424befc8497.png)
for all
with 
.2 replies
Recurrence trouble
SomeonecoolLovesMaths 4
N
by Hello_Kitty
Let 
be real numbers. Define 
and 
.
Prove that
and hence find the limit.
be real numbers. Define 
and 
.Prove that
and hence find the limit.
4 replies
Functions
mclolikoi 3
N
by Mathzeus1024
Let us consider f as the following function : ![$ f(x)= \frac {x - \sqrt{2}}{ [x \sqrt{2} ] } $](//latex.artofproblemsolving.com/a/9/f/a9f640a2ba0f2c58a1a7085e843e82ae41c72951.png)
1- Find the definition domain
2-Prove that
is continous on 
3-Study the continuity of on 
4-Then draw the geometric representation of on ![$ ] \frac {1}{ \sqrt{2} } ; 2 \sqrt{2} [ $](//latex.artofproblemsolving.com/d/0/4/d04f852d2494e9cd545a1ff20462e7283595c9ce.png)
![$ f(x)= \frac {x - \sqrt{2}}{ [x \sqrt{2} ] } $](http://latex.artofproblemsolving.com/a/9/f/a9f640a2ba0f2c58a1a7085e843e82ae41c72951.png)
1- Find the definition domain
2-Prove that
is continous on 
3-Study the continuity of
on 
4-Then draw the geometric representation of
on ![$ ] \frac {1}{ \sqrt{2} } ; 2 \sqrt{2} [ $](http://latex.artofproblemsolving.com/d/0/4/d04f852d2494e9cd545a1ff20462e7283595c9ce.png)
3 replies
