Problem:
Score : 600 points

Problem Statement

Takahashi has N balls with positive integers written on them. The integer written on the i-th ball is A_i.
He would like to form some number of pairs such that the sum of the integers written on each pair of balls is a power of 2.
Note that a ball cannot belong to multiple pairs.
Find the maximum possible number of pairs that can be formed.

Here, a positive integer is said to be a power of 2 when it can be written as 2^t using some non-negative integer t.

Constraints

1 \leq N \leq 2\times 10^5

1 \leq A_i \leq 10^9

A_i is an integer.

Input

Input is given from Standard Input in the following format:

N
A_1 A_2 ... A_N

Output

Print the maximum possible number of pairs such that the sum of the integers written on each pair of balls is a power of 2.

Sample Input 1

3
1 2 3

Sample Output 1

1

We can form one pair whose sum of the written numbers is 4 by pairing the first and third balls.
Note that we cannot pair the second ball with itself.

Sample Input 2

5
3 11 14 5 13

Sample Output 2

2

Solution in C++14 (GCC 5.4.1) (judge verdict: Wrong Answer)

#include <bits/stdc++.h>
#include <boost/multiprecision/cpp_int.hpp>
using namespace std;
//#define int long long
using bll = boost::multiprecision::cpp_int;
using ll = long long;
//constexpr int INF = 1e9;//INT_MAX=(1<<31)-1=2147483647
constexpr ll INF = (ll)1e18;//(1LL<<63)-1=9223372036854775807
constexpr ll MOD = (ll)1e9 + 7;
constexpr double EPS = 1e-9;
constexpr int dx[4]={1,0,-1,0};
constexpr int dy[4]={0,1,0,-1};
#define p(var) std::cout<<var<<std::endl
#define PI (acos(-1))
#define rep(i, n)   for(ll i=0, i##_length=(n); i< i##_length; ++i)
#define repeq(i, n) for(ll i=1, i##_length=(n); i<=i##_length; ++i)
#define all(v) (v).begin(), (v).end()
#define uniq(v) (v).erase(unique((v).begin(), (v).end()), (v).end());
template<typename T> inline void pv(vector<T> v) { for(ll i=0, N=v.size(); i<N; i++) cout<< v[i] << (i==N-1 ? '\n' : ' '); }
template<typename T> inline T gcd(T a, T b) { return b ? gcd(b,a%b) : a; }
template<typename T> inline T lcm(T a, T b) { return a / gcd(a,  b) * b; }
template<typename T1, typename T2> inline T1 power(T1 x, T2 n){ return n ? power(x*x%MOD,n/2)*(n%2?x:1)%MOD : 1; }
template<typename T1, typename T2> inline bool chmax(T1 &a, T2 b) { return a < b && (a = b, true); }
template<typename T1, typename T2> inline bool chmin(T1 &a, T2 b) { return a > b && (a = b, true); }
template<typename T> class dvector : public std::vector<T> {
public:
	dvector() : std::vector<T>() {}
	explicit dvector(size_t n, const T& value = T()) : std::vector<T>(n,value) {}
	dvector(const std::vector<T>& v) : std::vector<T>(v) {}
	dvector(const std::initializer_list<T> il) : std::vector<T>(il) {}
	dvector(const typename std::vector<T>::iterator first, const typename std::vector<T>::iterator last) : std::vector<T>(first, last) {}
	dvector(const std::string::iterator first, const std::string::iterator last) : std::vector<T>(first, last) {}
	T& operator[](size_t n) { return this->at(n); }
	const T& operator[](size_t n) const { return this->at(n); }
};
class dbool{
public:
	dbool() : boolvalue(false) {}
	dbool(bool b) : boolvalue(b) {}
	dbool(const dbool &b) : boolvalue(b.boolvalue) {}
	operator bool&() { return boolvalue; }
	operator const bool&() const { return boolvalue; }
private:
	bool boolvalue;
};
template<typename T1, typename T2> ostream& operator<<(ostream& s, pair<T1, T2>& p) {return s << "(" << p.first << ", " << p.second << ")";}
template<typename T> ostream& operator<<(ostream& s, dvector<T>& v) {
	for (int i = 0, len = v.size(); i < len; ++i){ s << v[i]; if (i < len - 1) s << "\t"; } return s; }
template<typename T> ostream& operator<<(ostream& s, dvector< dvector<T> >& vv) {
	for (int i = 0, len = vv.size(); i < len; ++i){ s << vv[i] << endl; } return s; }
template<typename T1, typename T2> ostream& operator<<(ostream& s, map<T1, T2>& m) {
	s << "{" << endl; for (auto itr = m.begin(); itr != m.end(); ++itr){ s << "\t" << (*itr).first << " : " << (*itr).second << endl; } s << "}" << endl; return s; }
template<typename T> ostream& operator<<(ostream& s, set<T>& se) {
	s << "{ "; for (auto itr = se.begin(); itr != se.end(); ++itr){ s << (*itr) << "\t"; } s << "}" << endl; return s; }
template<typename T> ostream& operator<<(ostream& s, multiset<T>& se) {
	s << "{ "; for (auto itr = se.begin(); itr != se.end(); ++itr){ s << (*itr) << "\t"; } s << "}" << endl; return s; }
#ifdef LOCAL_DEV
	#define debug(var) std::cout<<#var" = "<<var<<std::endl
#else
	#define debug(var)
#endif
#ifdef LOCAL_TEST
	#define vector dvector
	#define bool dbool
#endif

/*-----8<-----8<-----*/

/*
Dinic(V):= 頂点数 V で初期化する。
add_edge(from, to, cap):= 頂点 from から to に容量 cap の辺を張る。
max_flow(s, t):= 頂点 s から t に最大流を流し、その流量を返す。
*/
template< typename flow_t >
class Dinic {
public:
	const flow_t INF;

	class edge {
	public:
		int to;
		flow_t cap;
		int rev;
		bool isrev;
		int idx;
	};

	vector< vector< edge > > graph;
	vector< int > min_cost, iter;

	Dinic(int V) : INF(numeric_limits< flow_t >::max()), graph(V) {}

	void add_edge(int from, int to, flow_t cap, int idx = -1) {
		graph[from].emplace_back((edge) {to, cap, (int) graph[to].size(), false, idx});
		graph[to].emplace_back((edge) {from, 0, (int) graph[from].size() - 1, true, idx});
	}

	bool bfs(int s, int t) {
		min_cost.assign(graph.size(), -1);
		queue< int > que;
		min_cost[s] = 0;
		que.push(s);
		while(!que.empty() && min_cost[t] == -1) {
			int p = que.front();
			que.pop();
			for(auto &e : graph[p]) {
				if(e.cap > 0 && min_cost[e.to] == -1) {
					min_cost[e.to] = min_cost[p] + 1;
					que.push(e.to);
				}
			}
		}
		return min_cost[t] != -1;
	}

	flow_t dfs(int idx, const int t, flow_t flow) {
		if(idx == t) return flow;
		for(int &i = iter[idx], size = graph[idx].size(); i < size; i++) {
			edge &e = graph[idx][i];
			if(e.cap > 0 && min_cost[idx] < min_cost[e.to]) {
				flow_t d = dfs(e.to, t, min(flow, e.cap));
				if(d > 0) {
					e.cap -= d;
					graph[e.to][e.rev].cap += d;
					return d;
				}
			}
		}
		return 0;
	}

	flow_t max_flow(int s, int t) {
		flow_t flow = 0;
		while(bfs(s, t)) {
			iter.assign(graph.size(), 0);
			flow_t f = 0;
			while((f = dfs(s, t, INF)) > 0) flow += f;
		}
		return flow;
	}

	void output() {
		for(int i = 0, size = graph.size(); i < size; i++) {
			for(auto &e : graph[i]) {
				if(e.isrev) continue;
				auto &rev_e = graph[e.to][e.rev];
				cout << i << "->" << e.to << " (flow: " << rev_e.cap << "/" << e.cap + rev_e.cap << ")" << endl;
			}
		}
	}
};



signed main() {
	ll N;
	cin>>N;
	vector<ll> a(N,0);
	map<ll,ll> index;
	rep(i,N){
		cin>>a[i];
		index[a[i]]=i+1;
	}


	ll maxval=*max_element(all(a));
	ll t=1;
	vector<ll> two;
	while(t<maxval){
		t*=2;
		two.push_back(t);
	}

	Dinic<ll> g(2*N+2);
	repeq(i,N){
		g.add_edge(0, i, 1);
		g.add_edge(N+i, 2*N+1, 1);
	}
	rep(i,N){
		ll from=i+1;
		rep(j,two.size()){
			ll t=two[j]-a[i];
			ll to=index[t];
			if(t>0 && to>0 && from>to){
				g.add_edge(from, N+to, 1);			
			}
		}
	}
	ll ans=g.max_flow(0, 2*N+1);
	//g.output();
	p(ans);
	return 0;
}