Show that $2013^2 +2013^2\times 2014^2 + 2014^2$ is a perfect sqare.

Compute $\sqrt[32]{259\times 23\times 11 +9}$.

Let $f(x)$ be a second degree function satisfying $f(-2)=0$ and $2x \lt f(x) \le\frac{x^2+4}{2}$. Find the value of $f(10)$.

Let $a$, $b$, $c$, $x$, $y$, and $z$ be positive numbers. Show that $$\sqrt{a^2+x^2} + \sqrt{b^2 + y^2}+\sqrt{c^2+z^2} \ge \sqrt{(a+b+c)^2 + (x+y+z)^2}$$

Let positive numbers $a$, $b$ and $c$ satisfy $a+b+c=8$. Find the minimal value of $\sqrt{a^2+1}+\sqrt{b^2+4}+\sqrt{c^2+9}$.

Solve $x^{x^{88}} - 88=0$.

Given a pointer pointing to the header of a linked list, how to detect whether the linked list has a loop?

An additional question: how can math knowledge help here?

Show that there exists a perfect sqaure whose leading $2024$ digits are all $1$.

For any given integer $N$, show that there exists a perfect square whose leading digits match $N$. For example, if $N=2024$, there must exist a perfect square whose leading four digits are $2024$.

Given an string as input, find the length of the longest palindrome it contains. A palindrome is a word which stays the same if reading backwards. Examples include madam, Hannah. 

Determine whether a given integer can be expressed as the sum of two perfect squares.

(Coin Change Problem) Given a set of $N$ possible denominations of coins, find the number of different combinations to pay for a desired sum.

You are given an integer array height of length $n$. There are $n$ vertical lines drawn such that the two endpoints of the $i$th line are $(i, 0)$ and ($i$, height[$i$]). Find two lines that together with the $x$-axis form a container, such that the container contains the most water. Return the maximum amount of water a container can store.

Input: height = $[1,8,6,2,5,4,8,3,7]$

Output: $49$

Explanation: The above vertical lines are represented by array $[1,8,6,2,5,4,8,3,7]$. In this case, the max area of water (blue section) the container can contain is $49$.

Design an algorithm that finds the number of ways in which you can traverse $N$ meters by doing jumps of $1$, $2$, $3$, $4$, or $5$ meter lengths.

On a $M\times N$ board, some cells are occupied. Find the size of the largest square of unoccupied cells. 

Given two strings s and t, return true if they are equal when both are typed into empty text editors. '#' means a backspace character. (Note that after backspacing an empty text, the text will continue empty.)

Example 1:

Input: s = "ab#c", t = "ad#c"

Output: true

Explanation: Both s and t become "ac".

Example 2:

Input: s = "ab##", t = "c#d#"

Output: true

Explanation: Both s and t become "".

Example 3:

Input: s = "a#c", t = "b"

Output: false

Explanation: s becomes "c" while t becomes "b".

You are given a string s consisting of lower case English letters. A duplicate removal consists of choosing two adjacent and equal letters and removing them. We repeatedly make duplicate removal on s until we no longer can. Return the final string after all such duplicate removals have been made.

Example:

Input : s = "abbaca"

Output :  "ca"

1. remove bb

2. remove aa

Let $T=\underbrace{333\cdots 3}_{3^{2024}}$. Find the largest power of $3$ that can divid $T$.

Lifting The Exponent (LTE) Let $v_p(n)$ be the largest power of a prime $p$ that divides a positive integer $n$, and $x$, $y$ be any two integers such that $p \not\mid x$ and $p \not\mid y$, then

• When $p$ is odd
  • If $p\mid x-y$, then $v_p(x^n-y^n) = v_p(x-y)+v_p(n)$
  • If $n$ is odd and $p\mid x+y$, then $v_p(x^n+y^n) = v_p(x+y)+vp(n)$
  • If $n$ is even and $p\mid x+y$, then $v_p(x^n+y^n) = 0$
• When $p=$
  • If $2\mid x-y$ and $n$ is even, then $v_2(x^n-y^n)=v_2(x-y)+v_2(x+y)+v_2(n)-1$
  • If $2\mid x-y$ and $n$ is odd, then $v_2(x^n-y^n)=v_2(x-y)$
• For all $p$
  • if $gcd(n,p)=1$, and $p\mid x-y$, then $v_p(x^n-y^n)=v_p(x-y)$
  • If $bcd(n,p)=1$, $p\mid x+y$ and n odd, then $v_p(x^n+y^n) = v_p(x+y)$

Given a string s, find the length of the longest substring without repeating characters.

Example 1:

Input: s = "abcabcbb"

Output: 3

Explanation: The answer is "abc", with the length of 3.

Example 2:

Input: s = "bbbbb"

Output: 1

Explanation: The answer is "b", with the length of 1.

Example 3:

Input: s = "pwwkew"

Output: 3

Explanation: The answer is "wke", with the length of 3.

Notice that the answer must be a substring, "pwke" is a subsequence and not a substring.

Given an $m \times n$ 2D binary grid grid which represents a map of '$1$'s (land) and '$0$'s (water), return the number of islands.

An island is surrounded by water and is formed by connecting adjacent lands horizontally or vertically. You may assume all four edges of the grid are all surrounded by water.

Example 1:

Input: grid = [

  ["1","1","1","1","0"],

  ["1","1","0","1","0"],

  ["1","1","0","0","0"],

  ["0","0","0","0","0"]

]

Output: 1

Example 2:

Input: grid = [

  ["1","1","0","0","0"],

  ["1","1","0","0","0"],

  ["0","0","1","0","0"],

  ["0","0","0","1","1"]

]

Output: 3

Constraints:

m == grid.length

n == grid[i].length

1 <= m, n <= 300

grid[i][j] is '0' or '1'.

You are given an m x n grid where each cell can have one of three values:

• $0$ representing an empty cell,
• $1$ representing a fresh orange, or
• $2$ representing a rotten orange.

Every minute, any fresh orange that is $4$-directionally adjacent to a rotten orange becomes rotten.

Return the minimum number of minutes that must elapse until no cell has a fresh orange. If this is impossible, return $-1$.

Example 1

Input: grid = $[[2,1,1],[1,1,0],[0,1,1]]$

Output: $4$

Example 2:

Input: grid = $[[2,1,1],[0,1,1],[1,0,1]]$

Output: $-1$

Explanation: The orange in the bottom left corner (row $2$, column $0$) is never rotten, because rotting only happens $4$-directionally.

Example 3:

Input: grid = $[[0,2]]$

Output: $0$

Explanation: Since there are already no fresh oranges at minute 0, the answer is just 0.

Constraints:

$m$ == grid.length

$n$ == grid[$i$].length

$1 \le m, n \le 10$

grid[$i$][$j$] is $0$, $1$, or $2$.

Simplify: $\left(\frac{1-\sqrt{5}}{2}\right)^{12}$.