[Vijos] 1543 Extreme Value Problem (number theory + fib number)

Https://vijos.org/p/1543

A magic question ..

First, I forgot that N can be obtained for root, sad.

Then I typed the table and found that N and m are fib numbers ..

Strictly prove (the ghost knows why this is the right way, and how can it be the biggest if it can be replaced ?) :

(N ^ 2-mn-m ^ 2) ^ 2 = 1

(M ^ 2 + Mn-N ^ 2) ^ 2 = 1

(M (m + n)-N ^ 2) ^ 2 = 1

(M + n-n) (m + n)-N ^ 2) ^ 2 = 1

(M + n) ^ 2-N (m + n)-N ^ 2) ^ 2 = 1

Because (n' ^ 2-m' N'-m' ^ 2) ^ 2 = 1

Take n '= m + n, M' = n to enable the formula ..

Then there is violence.

#include <cstdio>#include <cstring>#include <cmath>#include <string>#include <iostream>#include <algorithm>#include <queue>using namespace std;typedef long long ll;#define rep(i, n) for(int i=0; i<(n); ++i)#define for1(i,a,n) for(int i=(a);i<=(n);++i)#define for2(i,a,n) for(int i=(a);i<(n);++i)#define for3(i,a,n) for(int i=(a);i>=(n);--i)#define for4(i,a,n) for(int i=(a);i>(n);--i)#define CC(i,a) memset(i,a,sizeof(i))#define read(a) a=getint()#define print(a) printf("%d", a)#define dbg(x) cout << (#x) << " = " << (x) << endl#define printarr2(a, b, c) for1(_, 1, b) { for1(__, 1, c) cout << a[_][__]; cout << endl; }#define printarr1(a, b) for1(_, 1, b) cout << a[_] << ‘\t‘; cout << endlinline const int getint() { int r=0, k=1; char c=getchar(); for(; c<‘0‘||c>‘9‘; c=getchar()) if(c==‘-‘) k=-1; for(; c>=‘0‘&&c<=‘9‘; c=getchar()) r=r*10+c-‘0‘; return k*r; }inline const ll max(const ll &a, const ll &b) { return a>b?a:b; }inline const int min(const int &a, const int &b) { return a<b?a:b; }ll k, a, b, ans;int main() {read(k);ll n=1, m=0;while(n+m<=k) {ll c=m;m=n;n+=c;}printf("%lld\n", m*m+n*n);return 0;}

Background

James's journey to mathematics 2.

Description

M and n are known as integers and meet the following two conditions:
① M, 'n ,..., K
② (N ^ 2-mn-m ^ 2) ^ 2 = 1
Compile a program to obtain a group of M and N that meet the preceding two conditions for the given K, and set the maximum value of M ^ 2 + N ^ 2. For example, if k = 1995, then M = 987, n = 1597, then M and N meet the conditions, and the M ^ 2 + N ^ 2 value can be maximized.

Format input format

Enter the value of K in only one row.

Output Format

Output only one line, m ^ 2 + N ^ 2 value.

Example 1 input 1 [copy]

1995

Sample output 1 [copy]

3524578

Restrictions

Each test point is 1 second.

Prompt

Source:
Shantou fxoi Group
Phoeagon
Thanx2 sivon
For tripley

vijos: 1543 Extreme Value Problem (number theory + fib number)