You are given three integers $$$n$$$, $$$l$$$, and $$$r$$$, count the number of arrays of size $$$n + 1$$$ that have the first element as the average of the whole array and every element of the array is between $$$l$$$ and $$$r$$$ inclusive. print the number of arrays modulo $$$10^9 + 7$$$.

constraints: $$$1 <= n <= 3000$$$, $$$1 <= l <= r <= 10^{18}$$$

samples: Input: $$$5$$$ $$$1$$$ $$$2$$$, Output: $$$20000$$$ Input: $$$4$$$ $$$2$$$ $$$7$$$, Output: $$$322$$$

This problem has DP as a tag in it, I was able to solve it without using DP but since I don't mind getting better at DP (xD) I would like to know how to solve it using DP. I would appreciate your help, thanks in advance.

Given an array $$$A$$$ of $$$N$$$ non-negative integers, and two different integers $$$x$$$ and $$$y$$$, could you permute the array $$$A$$$ into a new array $$$B$$$ in such a way that after creating the $$$XOR$$$ prefix of the array $$$B$$$, both $$$x$$$ and $$$y$$$ would appear in it?

the $$$XOR$$$ of an empty prefix equals $$$0$$$.

Input: $$$n$$$, $$$x$$$ and $$$y$$$ ($$$2 <= n <= 30$$$), ($$$0 <= x,y < 2^{10}, x \ne y$$$)

then we are given an array $$$A$$$ with length $$$n$$$ $$$a_1, a_2, ..., a_n$$$ ($$$0 <= a_i < 2^{10}$$$)

test cases:

Input: $$$n=5$$$, $$$x=1$$$, $$$y=7$$$, $$$A = [1, 2, 3, 4, 5]$$$

Output: YES

Input: $$$n=3$$$, $$$x=8$$$, $$$y=9$$$, $$$A = [1, 2, 3]$$$

Output: NO

This was a problem in a national contest, we couldn't solve it. I would appreciate your help, thanks in advance.

How to solve this problem using DP?

I'm so happy that today i finally reached pupil after 827 problems on codeforces & 2 years of competitive programming, it feels refreshing and gives you the motivation to continue.

I'm curious about something, is solving harder problems makes you able to solve the easier ones ?, let's say that i find problems with rating $$${X}$$$ are currently challenging for me, so if i just jump to problems with rating $$${Y > X}$$$ and get comfortable solving these (will be much more difficult but it's already difficult anyways) does it make me able to solve problems with rating $$${R < Y}$$$ ?

i recently solved this problem: https://mirror.codeforces.com/contest/1097/problem/B,

I know that it's an easy problem with a brute-force solution but, I'm curious how to solve it for $$${1 <= N <= 10^5}$$$, I'd appreciate your help, thanks in advance.

Work Smarter, Not Harder, can you give me insights on how does the "Smart" part apply in training to get better at CP, and/or does it even apply ? or it just falls to years of hard work ?, (can we accelerate the process ?)