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Possible values of determinant of 0-1 matrices
mathematics2004 4
N
3 hours ago
by loup blanc
Source: 2021 Simon Marais, A3
Let
be the set of all
matrices with at most two entries in each row equal to
and all other entries equal to
.
Determine the size of the set
.
Here
denotes the determinant of the matrix
.




Determine the size of the set

Here


4 replies
ISI UGB 2025
Entrepreneur 1
N
3 hours ago
by Knight2E4
Source: ISI UGB 2025
1.)
Suppose
is differentiable and
Show that for some 
3.)
Suppose
is differentiable with
If
then show that 
4.)
Let
be the unit circle in the complex plane. Let
be the map given by
We define
and
for
The smallest positive integer
such that
is called period of
Determine the total number of points
of period 
6.)
Let
denote the set of natural numbers, and let
be nine distinct tuples in
Show that there are
distinct elements in the set
whose product is a perfect cube.
8.)
Let
and let
be positive integers such that
Prove that
and determine when equality holds.
Suppose



3.)
Suppose
![$f:[0,1]\to\mathbb R$](http://latex.artofproblemsolving.com/e/a/0/ea02b990406104357c11912a8bf57bea7eeb519d.png)

![$|f'(x)|\le f(x)\;\forall\;x\in[0,1],$](http://latex.artofproblemsolving.com/b/9/d/b9d6088657cd85b37190b98abac1d94a18bb5916.png)

4.)
Let











6.)
Let





8.)
Let




1 reply
Recurrence trouble
SomeonecoolLovesMaths 3
N
3 hours ago
by Knight2E4
Let
be real numbers. Define
and
.
Prove that
and hence find the limit.



Prove that

3 replies
Trigo or Complex no.?
hzbrl 5
N
Today at 9:20 AM
by GreenKeeper
(a) Let
, where
. Verify by direct substitution that
satisfies the quadratic equation
and deduce that the value of
is
.
(b) Let
. Show that 
(c) If
, show that the value of
is
.
I could solve (a) and (b). Can anyone help me with the 3rd part please?






(b) Let


(c) If



I could solve (a) and (b). Can anyone help me with the 3rd part please?
5 replies
Quadruple Binomial Coefficient Sum
P162008 3
N
Today at 4:28 AM
by pineconee
Source: Self made by my Elder brother

3 replies
2023 Putnam A1
giginori 29
N
Yesterday at 10:52 PM
by kidsbian
For a positive integer
, let
. Find the smallest
such that
.




29 replies
A MATHEMATICA E BONITA
P162008 0
Yesterday at 7:54 PM
Source: Self made by my Elder brother
Let
where
and 
Now, consider the ratio
defined as

The summation function
is given by

Let
denotes the number of points of intersection between the curves

Define
as

The value of
is

And,
Number of subsets of
whose sum of elements is divisible by 
Finally, compute the value of



Now, consider the ratio


The summation function


Let


Define


The value of


And,



Finally, compute the value of

0 replies
Ultra-hyper saddle with logarithmic weight
randomperson1021 0
Yesterday at 5:22 PM
Fix integers
and
, a parameter
, and a real log-exponent
. For every real
define

Put

(1) Show that there exists a real constant
(independent of
and of
) such that
![$$
\lim_{x\to 1^{-}}
F_{a,\beta}^{(k,r)}(x)\,
e^{-\Lambda_{k,r,\lambda}\,(1-x)^{-\gamma}}
\;=\;
\begin{cases}
0, & a<c,\\[6pt]
\infty, & a>c.
\end{cases}
$$](//latex.artofproblemsolving.com/e/3/8/e38c6b43d513bcc697587fff610a846b3a1b0e0a.png)
(2) Determine this critical value
explicitly and verify that it coincides with the classical case
, namely
.
(3) Evaluate the finite, non-zero limit that occurs at the borderline
(your answer may depend on
but not on
).






Put

(1) Show that there exists a real constant



![$$
\lim_{x\to 1^{-}}
F_{a,\beta}^{(k,r)}(x)\,
e^{-\Lambda_{k,r,\lambda}\,(1-x)^{-\gamma}}
\;=\;
\begin{cases}
0, & a<c,\\[6pt]
\infty, & a>c.
\end{cases}
$$](http://latex.artofproblemsolving.com/e/3/8/e38c6b43d513bcc697587fff610a846b3a1b0e0a.png)
(2) Determine this critical value



(3) Evaluate the finite, non-zero limit that occurs at the borderline



0 replies
3rd AKhIMO for university students, P5
UzbekMathematician 1
N
Yesterday at 3:53 PM
by grupyorum
Source: AKhIMO 2025, P5
Show that for every positive integer
there exist nonnegative integers
and integers
such that




1 reply
Sum of three squares
perfect_radio 9
N
Yesterday at 1:36 PM
by RobertRogo
Source: RMO 2004, Grade 12, Problem 4
Let
be a field of characteristic
,
.
(a) Prove that
is the square of an element from 
(b) Prove that any element
from
can be written as the sum of three squares, each
, of elements from
.
(c) Can
be written in the same way?
Marian Andronache



(a) Prove that


(b) Prove that any element




(c) Can

Marian Andronache
9 replies
