Showing that is not a square
by Kyj9981, Apr 24, 2025, 10:27 AM
Find all 
such that 
is a perfect square.
such that 
is a perfect square.3 var inequalities
by sqing, Apr 23, 2025, 1:13 PM
Let 
and 
Prove that
Solution:
and 
Assume
then 
Equalityholds when
and 
Prove that
Solution:
and 
Assume
then 
Equalityholds when
This post has been edited 1 time. Last edited by sqing, 23 minutes ago
Equal angles with midpoint of $AH$
by Stuttgarden, Mar 31, 2025, 1:10 PM
Let 
be an acute triangle with circumcenter 
and orthocenter 
, satisfying 
. The tangent line at 
to the circumcicle of intersects 
in 
. Let 
be the midpoint of 
. Prove that 
.
be an acute triangle with circumcenter 
and orthocenter 
, satisfying 
. The tangent line at 
to the circumcicle of intersects 
in 
. Let 
be the midpoint of 
. Prove that 
.1:1 correspondance + Graph Theory
by jaydenkaka, Oct 24, 2024, 9:43 AM
Attachments:
This post has been edited 1 time. Last edited by jaydenkaka, Oct 24, 2024, 9:44 AM
ISL 2023 C2
by OronSH, Jul 17, 2024, 12:22 PM
Determine the maximal length 
of a sequence 
of positive integers satisfying both the following properties:
of a sequence 
of positive integers satisfying both the following properties:
- every term in the sequence is less than or equal to
, and - there does not exist a consecutive subsequence
(where 
) with a choice of signs 
for which ![\[s_ia_i+s_{i+1}a_{i+1}+\dots+s_ja_j=0.\]](//latex.artofproblemsolving.com/4/b/6/4b66fe235e9a56cc646dbd38aae72f8ddfeb77de.png)
,
(where 
)
for which ![\[s_ia_i+s_{i+1}a_{i+1}+\dots+s_ja_j=0.\]](http://latex.artofproblemsolving.com/4/b/6/4b66fe235e9a56cc646dbd38aae72f8ddfeb77de.png)
This post has been edited 1 time. Last edited by OronSH, Jul 17, 2024, 12:28 PM
2024 IMO P1
by EthanWYX2009, Jul 16, 2024, 1:13 PM
Determine all real numbers 
such that, for every positive integer 
the integer
is a multiple of 
(Note that 
denotes the greatest integer less than or equal to 
For example, 
and 
)
Proposed by Santiago Rodríguez, Colombia
such that, for every positive integer 
the integer
is a multiple of 
(Note that 
denotes the greatest integer less than or equal to 
For example, 
and 
)Proposed by Santiago Rodríguez, Colombia
This post has been edited 2 times. Last edited by EthanWYX2009, Jul 19, 2024, 5:33 AM
Reason: change to original text
Reason: change to original text
Mount Inequality erupts on a sequence :o
by GrantStar, Jul 9, 2023, 4:43 AM
Let 
be pairwise different positive real numbers such that
![\[a_n=\sqrt{(x_1+x_2+\dots+x_n)\left(\frac{1}{x_1}+\frac{1}{x_2}+\dots+\frac{1}{x_n}\right)}\]](//latex.artofproblemsolving.com/8/a/e/8ae0bd1f55eb2fb0b7209b62739a9b98f462ee56.png)
is an integer for every 
Prove that 
be pairwise different positive real numbers such that![\[a_n=\sqrt{(x_1+x_2+\dots+x_n)\left(\frac{1}{x_1}+\frac{1}{x_2}+\dots+\frac{1}{x_n}\right)}\]](http://latex.artofproblemsolving.com/8/a/e/8ae0bd1f55eb2fb0b7209b62739a9b98f462ee56.png)
is an integer for every 
Prove that 
This post has been edited 6 times. Last edited by GrantStar, Jul 9, 2023, 4:54 AM
Reason: Renaming source
Reason: Renaming source
Matrices :D :D :D
by utkarshgupta, Jul 4, 2016, 3:56 AM
Long time people 
This sure is nice and I wouldn't have been able to solve it without first solving the
version.
Problem (Romanian Mathematical Olympiad Grade XI) :
Let
be a real 
matrix, such that 
, for all 
. Prove that for all non-negative real numbers 
we have ![\[ \det(A+xI_n)\cdot \det(A+yI_n) \geq \det (A+\sqrt{xy}I_n)^2.\]](//latex.artofproblemsolving.com/f/c/c/fcc03110bf8f741c4be0a64df336378ca84802e4.png)
Solution :
Let
Obviously
is a polynomial.
So we have to show that
But the above result will follow directly once we establish that all the coefficients of are positive.
obviously.
I will call the kind of matrices asked in the question as ski matrix.
To show this I will use induction on the order of the matrix.
Let the result be true for all such
ski matrices.
Now consider any
such ski matrix and call it 
.
Then,
But obviously since 
is a sum of determinants of ski matrices of the order , all it's coefficients by the induction hypothesis are positive and hence so are it's integrals (except the constants which we have already established as zero).
Hence
has all it's coefficients positive and hence by Cauchy Schwartz inequality, we are done.
This sure is nice and I wouldn't have been able to solve it without first solving the
version.Problem (Romanian Mathematical Olympiad Grade XI) :
Let
be a real 
matrix, such that 
, for all 
. Prove that for all non-negative real numbers 
we have ![\[ \det(A+xI_n)\cdot \det(A+yI_n) \geq \det (A+\sqrt{xy}I_n)^2.\]](http://latex.artofproblemsolving.com/f/c/c/fcc03110bf8f741c4be0a64df336378ca84802e4.png)
Solution :
Let
Obviously
is a polynomial.So we have to show that
But the above result will follow directly once we establish that all the coefficients of
are positive.
obviously.I will call the kind of matrices asked in the question as ski matrix.
To show this I will use induction on the order of the matrix.
Let the result be true for all such
ski matrices.Now consider any
such ski matrix and call it 
.Then,
But obviously since 
is a sum of determinants of ski matrices of the order , all it's coefficients by the induction hypothesis are positive and hence so are it's integrals (except the constants which we have already established as zero).Hence
has all it's coefficients positive and hence by Cauchy Schwartz inequality, we are done.Domain of (a, b) satisfying inequality with fraction
by Kunihiko_Chikaya, Feb 26, 2014, 7:47 AM
For real constants 
, define a function 
Draw the domain of the points
such that the inequality :
![\[f(x) \leq f(x)^3-2f(x)^2+2\]](//latex.artofproblemsolving.com/1/2/c/12c4c4c850c1c2886f002dfcb0bead075e2ab679.png)
holds for all real numbers
.
, define a function 
Draw the domain of the points
such that the inequality :![\[f(x) \leq f(x)^3-2f(x)^2+2\]](http://latex.artofproblemsolving.com/1/2/c/12c4c4c850c1c2886f002dfcb0bead075e2ab679.png)
holds for all real numbers
.2013 China girls' Mathematical Olympiad problem 7
by s372102, Aug 13, 2013, 3:18 PM
As shown in the figure, 
and 
touches each other externally at a point , quadrilateral 
is inscribed in , and the lines 
, 
are tangent to at points 
and 
respectively. Line 
bisects 
and meets segment 
at 
. Line 
meets the arc 
(not passing through the point 
) at another point 
different from . Prove that is the circumcenter of 
.
and 
touches each other externally at a point , quadrilateral 
is inscribed in , and the lines 
, 
are tangent to at points 
and 
respectively. Line 
bisects 
and meets segment 
at 
. Line 
meets the arc 
(not passing through the point 
) at another point 
different from . Prove that is the circumcenter of 
.Attachments:
Max and min of Sum of d_k^2
by Kunihiko_Chikaya, Feb 27, 2012, 6:41 PM
Given points 
on the 
-plane.
Let
. Denote by 
the distance between 
and the line 
. Let 
.
Answer the following questions:
(1) Express in terms of 
.
(2) When
moves in the range of 
, express the maximum and minimum value of in terms of 
.
points 
on the 
-plane.Let
. Denote by 
the distance between 
and the line 
. Let 
.Answer the following questions:
(1) Express
in terms of 
.(2) When
moves in the range of 
, express the maximum and minimum value of in terms of 
.Stay insane,Coz it's your will, labour and pain,which takes you to the top of the mountain.
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