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operator integral analysis
Hello_Kitty 1
N
by alexheinis
Let 
and an operator defined as 
for any continuous
.
- Find all
such 
- What if
now ?
and an operator defined as 
for any continuous
.- Find all
such 
- What if
now ?
1 reply
Limit of expression
enter16180 8
N
by YaoAOPS
Source: IMC 2025, Problem 10
For any positive integer 
, let 
be the number of pairs of integers 
such that the number 
is a perfect square. Prove that the limit
exists and find its value.
, let 
be the number of pairs of integers 
such that the number 
is a perfect square. Prove that the limit
exists and find its value.
8 replies
expected value of maximum of random process
enter16180 4
N
by Agsh2005
Source: IMC 2025, Problem 9
Let 
be a positive integer. Consider the following random process which produces sequence of distinct positive integers 
.
First,
is chosen randomly with 
for every positive integer 
. For 
. having chosen 
, arrange the remaining positive integers in increasing order as 
, and choose 
randomly with 
for every positive integer .
Let
. Show that
![$$
\mathbb{E}\left[Y_n\right]=\sum_{i=1}^n \frac{2^i}{2^i-1}
$$](//latex.artofproblemsolving.com/4/e/3/4e38d740482044146ce1f6be2601ef217d63661b.png)
where ![$\mathbb{E}\left[Y_n\right]$](//latex.artofproblemsolving.com/1/e/1/1e106e23a601c28219befc3d4f2c0f79114f6dda.png)
is the expected value of 
.
be a positive integer. Consider the following random process which produces sequence of distinct positive integers 
.First,
is chosen randomly with 
for every positive integer 
. For 
. having chosen 
, arrange the remaining positive integers in increasing order as 
, and choose 
randomly with 
for every positive integer .Let
. Show that![$$
\mathbb{E}\left[Y_n\right]=\sum_{i=1}^n \frac{2^i}{2^i-1}
$$](http://latex.artofproblemsolving.com/4/e/3/4e38d740482044146ce1f6be2601ef217d63661b.png)
where ![$\mathbb{E}\left[Y_n\right]$](http://latex.artofproblemsolving.com/1/e/1/1e106e23a601c28219befc3d4f2c0f79114f6dda.png)
is the expected value of 
.
4 replies
Fourier Series
EthanWYX2009 0
Source: 2025 Spring NSTE(2)-3
Let 
be real numbers. Define 
. Prove that:
![\[
\sum_{1 \leq i, j \leq n} 2^{\|x_i - x_j\|} \leq \sum_{1 \leq i, j \leq n} 2^{\|x_i - x_j + \frac{1}{2}\|}.
\]](//latex.artofproblemsolving.com/1/a/3/1a3c78a907f136fa4f8f9bead7306918775597eb.png)
Proposed by Site Mu
be real numbers. Define 
. Prove that:![\[
\sum_{1 \leq i, j \leq n} 2^{\|x_i - x_j\|} \leq \sum_{1 \leq i, j \leq n} 2^{\|x_i - x_j + \frac{1}{2}\|}.
\]](http://latex.artofproblemsolving.com/1/a/3/1a3c78a907f136fa4f8f9bead7306918775597eb.png)
Proposed by Site Mu
0 replies
Putnam 2016 A5
Kent Merryfield 10
N
by ransun
Suppose that 
is a finite group generated by the two elements 
and 
where the order of is odd. Show that every element of can be written in the form
![\[g^{m_1}h^{n_1}g^{m_2}h^{n_2}\cdots g^{m_r}h^{n_r}\]](//latex.artofproblemsolving.com/b/d/9/bd938445b37a8faa81cc97f80060f275d338a24b.png)
with 
and 
(Here 
is the number of elements of 
)
is a finite group generated by the two elements 
and 
where the order of is odd. Show that every element of can be written in the form![\[g^{m_1}h^{n_1}g^{m_2}h^{n_2}\cdots g^{m_r}h^{n_r}\]](http://latex.artofproblemsolving.com/b/d/9/bd938445b37a8faa81cc97f80060f275d338a24b.png)
with 
and 
(Here 
is the number of elements of 
)10 replies
Rotation of matrix and eignavalues
enter16180 2
N
by ZNatox
Source: IMC 2025, Problem 8
For an 
real matrix 
, denote by 
its counter-clockwise 
rotation.
(10 points) For example,
![$$
\left[\begin{array}{lll}
1 & 2 & 3 \\
4 & 5 & 6 \\
7 & 8 & 9
\end{array}\right]^R=\left[\begin{array}{lll}
3 & 6 & 9 \\
2 & 5 & 8 \\
1 & 4 & 7
\end{array}\right]
$$](//latex.artofproblemsolving.com/4/b/8/4b8267cc7835e97d7f8a0b03e9e487f14522c1c0.png)
Prove that if 
then for any eigenvalue 
of 
, we have 
or 
.
real matrix 
, denote by 
its counter-clockwise 
rotation.(10 points) For example,
![$$
\left[\begin{array}{lll}
1 & 2 & 3 \\
4 & 5 & 6 \\
7 & 8 & 9
\end{array}\right]^R=\left[\begin{array}{lll}
3 & 6 & 9 \\
2 & 5 & 8 \\
1 & 4 & 7
\end{array}\right]
$$](http://latex.artofproblemsolving.com/4/b/8/4b8267cc7835e97d7f8a0b03e9e487f14522c1c0.png)
Prove that if 
then for any eigenvalue 
of 
, we have 
or 
.
2 replies
Easy Limit problem
Fermat_Fanatic108 2
N
by Fermat_Fanatic108
Evaluate
![\[
\lim_{x \to 0^+} \left\{ \lim_{n \to \infty} \left( \frac{\left\lfloor 1^2 (\sin x)^x \right\rfloor + \left\lfloor 2^2 (\sin x)^x \right\rfloor + \cdots + \left\lfloor n^2 (\sin x)^x \right\rfloor}{n^3} \right) \right\},
\]](//latex.artofproblemsolving.com/5/b/f/5bf499eb70b5978e4b0e033e0706624ad341d6cc.png)
where 
denotes the floor function
![\[
\lim_{x \to 0^+} \left\{ \lim_{n \to \infty} \left( \frac{\left\lfloor 1^2 (\sin x)^x \right\rfloor + \left\lfloor 2^2 (\sin x)^x \right\rfloor + \cdots + \left\lfloor n^2 (\sin x)^x \right\rfloor}{n^3} \right) \right\},
\]](http://latex.artofproblemsolving.com/5/b/f/5bf499eb70b5978e4b0e033e0706624ad341d6cc.png)
where 
denotes the floor function2 replies
2024 Putnam A5
KevinYang2.71 10
N
by ray66
Consider the circle 
with radius 
and center at the origin 
, and a disk 
with radius 
and center at 
, where 
. Two points 
and 
are chosen independently and uniformly at random on . Which value(s) of 
minimize the probability that the chord 
intersects ?
with radius 
and center at the origin 
, and a disk 
with radius 
and center at 
, where 
. Two points 
and 
are chosen independently and uniformly at random on . Which value(s) of 
minimize the probability that the chord 
intersects ?
10 replies
2024 Putnam A2
KevinYang2.71 10
N
by ray66
For which real polynomials 
is there a real polynomial 
such that
![\[
p(p(x))-x=(p(x)-x)^2q(x)
\]](//latex.artofproblemsolving.com/7/e/2/7e291099736964fa228b0079072a8ceaf373da36.png)
for all real 
?
is there a real polynomial 
such that![\[
p(p(x))-x=(p(x)-x)^2q(x)
\]](http://latex.artofproblemsolving.com/7/e/2/7e291099736964fa228b0079072a8ceaf373da36.png)
for all real 
?
10 replies
