Problem p02851 (C++)

Buggy submission (status: Time Limit Exceeded):
#include <algorithm>
#include <cctype>
#include <cmath>
#include <cstdio>
#include <cstdlib>
#include <ctype.h>
#include <deque>
#include <iomanip>
#include <limits>
#include <map>
#include <queue>
#include <random>
#include <set>
#include <sstream>
#include <stack>
#include <stdio.h>
#include <string.h>
#include <unordered_map>
#include <utility>
#include <vector>
#define _USE_MATH_DEFINES
#include <iostream>
#include <math.h>
using namespace std;
typedef long long ll;
typedef pair<int, int> pii;
typedef pair<ll, ll> pll;
typedef pair<int, char> pic;
#define bit(x, v) ((ll)x << v)

const ll INF = 1000000007;
const int MAX = 210000;
const int MOD = 1000000007;

long long fac[MAX], finv[MAX], inv[MAX];
void COMinit() {
  fac[0] = fac[1] = 1;
  finv[0] = finv[1] = 1;
  inv[1] = 1;
  for (int i = 2; i < MAX; i++) {
    fac[i] = fac[i - 1] * i % MOD;
    inv[i] = MOD - inv[MOD % i] * (MOD / i) % MOD;
    finv[i] = finv[i - 1] * inv[i] % MOD;
  }
}

// 二項係数計算
long long COM(int n, int k) {
  if (n < k)
    return 0;
  if (n < 0 || k < 0)
    return 0;
  return fac[n] * (finv[k] * finv[n - k] % MOD) % MOD;
}

ll gcd(ll a, ll b) {
  if (b == 0)
    return a;
  return gcd(b, a % b);
}
int pr[100010];
int lank[100010];
void uini(int n) {
  for (size_t i = 0; i <= n; i++) {
    pr[i] = i;
  }
}

int parent(int x) {
  if (x == pr[x])
    return x;
  return pr[x] = parent(pr[x]);
}

int same(int x, int y) { return parent(x) == parent(y); }

bool unit(int x, int y) {
  int px = parent(x);
  int py = parent(y);

  if (px == py)
    return false;
  if (lank[px] < lank[py]) {
    pr[py] = px;
    lank[px] += lank[py] + 1;
  } else {
    pr[px] = py;
    lank[py] += lank[px] + 1;
  }
  return true;
}

ll bit[200010];
int max_n = 200000;
int pm = 0;
void add(int x) {

  while (max_n >= x) {
    bit[x]++;
    x += x & -x;
  }
}
void sub(int x) {
  while (max_n >= x) {
    bit[x]--;
    x += x & -x;
  }
}
ll merge(ll *a, int left, int mid, int right) {
  ll n1 = mid - left;
  ll n2 = right - mid;
  vector<int> L(n1 + 1);
  vector<int> R(n2 + 1);
  for (size_t i = 0; i < n1; i++) {
    L[i] = a[left + i];
  }
  for (size_t i = 0; i < n2; i++) {
    R[i] = a[mid + i];
  }

  L[n1] = INF;
  R[n2] = INF;
  ll i = 0;
  ll j = 0;
  ll r = 0;
  for (size_t k = left; k < right; k++) {
    if (L[i] <= R[j]) {
      a[k] = L[i];
      i++;
    } else {
      a[k] = R[j];
      r += n1 - i;
      j++;
    }
  }
  return r;
}
ll merge2(pair<int, char> *a, int left, int mid, int right) {
  ll n1 = mid - left;
  ll n2 = right - mid;
  vector<pair<int, char>> L(n1 + 1);
  vector<pair<int, char>> R(n2 + 1);
  for (size_t i = 0; i < n1; i++) {
    L[i] = a[left + i];
  }
  for (size_t i = 0; i < n2; i++) {
    R[i] = a[mid + i];
  }

  L[n1] = make_pair(INF, ' ');
  R[n2] = make_pair(INF, ' ');
  ll i = 0;
  ll j = 0;
  ll r = 0;
  for (size_t k = left; k < right; k++) {
    if (L[i].first <= R[j].first) {
      a[k] = L[i];
      i++;
    } else {
      a[k] = R[j];
      r += n1 - i;
      j++;
    }
  }
  return r;
}
ll mergeSort2(pair<int, char> *a, int left, int right) {
  ll res = 0;
  if (left + 1 < right) {
    int mid = (left + right) / 2;
    res = mergeSort2(a, left, mid);
    res += mergeSort2(a, mid, right);
    res += merge2(a, left, mid, right);
  }
  return res;
}
ll mergeSort(ll *a, int left, int right) {
  ll res = 0;
  if (left + 1 < right) {
    int mid = (left + right) / 2;
    res = mergeSort(a, left, mid);
    res += mergeSort(a, mid, right);
    res += merge(a, left, mid, right);
  }
  return res;
}
int partition(pair<int, char> *a, int p, int r) {
  pair<int, char> x = a[r];
  int i = p - 1;
  for (size_t j = p; j < r; j++) {
    if (a[j].first <= x.first) {
      i++;
      swap(a[i], a[j]);
    }
  }
  swap(a[i + 1], a[r]);
  return i + 1;
}
void quick(pair<int, char> *a, int p, int r) {
  if (p < r) {
    int q = partition(a, p, r);
    quick(a, p, q - 1);
    quick(a, q + 1, r);
  }
}

ll n;
int ci = 0;
ll P[1000010];
struct Node {
  int key;
  int priority;
  Node *parent, *left, *right;
  Node(int key, int priority);
  Node() {}
};
Node NIL;
Node::Node(int key, int priority) : key(key), priority(priority) {
  left = &NIL;
  right = &NIL;
}
Node *root = new Node();
void cenrec(Node *k) {
  if (k->key == NIL.key)
    return;
  cenrec(k->left);
  cout << " " << k->key;
  cenrec(k->right);
}
void fastrec(Node *k) {
  if (k->key == NIL.key)
    return;
  cout << " " << k->key;
  fastrec(k->left);
  fastrec(k->right);
}
void insert(Node *v) {
  Node *y = &NIL;
  Node *x = root;
  while (x->key != NIL.key) {
    y = x;
    if (v->key < x->key) {
      x = x->left;
    } else {
      x = x->right;
    }
  }
  v->parent = y;
  if (y->key == NIL.key) {
    root = v;
  } else if (v->key < y->key) {
    y->left = v;
  } else {
    y->right = v;
  }
}

Node *find(Node *k, ll v) {
  if (k->key == NIL.key)
    return &NIL;
  if (k->key == v)
    return k;
  if (v < k->key)
    return find(k->left, v);
  return find(k->right, v);
}
void delp12(Node *x) {
  if (x->key == NIL.key)
    return;
  Node *l = x->left;
  Node *r = x->right;
  Node *pr = x->parent;

  if (l->key == NIL.key && r->key == NIL.key) {
    if (pr->left == x) {
      pr->left = &NIL;
    } else
      pr->right = &NIL;
  } else if (l->key != NIL.key) {
    if (pr->left == x) {
      pr->left = l;
    } else
      pr->right = l;
    l->parent = pr;
  } else if (r->key != NIL.key) {
    if (pr->left == x) {
      pr->left = r;
    } else
      pr->right = r;
    r->parent = pr;
  }
}
Node *get_next(Node *k) {
  if (k->key == NIL.key)
    return &NIL;
  Node *res = get_next(k->left);
  if (res->key != NIL.key)
    return res;
  return k;
}
void del(Node *x) {

  if (x->key == NIL.key)
    return;
  Node *l = x->left;
  Node *r = x->right;
  Node *pr = x->parent;

  if (l->key != NIL.key && r->key != NIL.key) {
    Node *nex = get_next(r);
    x->key = nex->key;
    delp12(nex);
  } else {
    delp12(x);
  }
}
Node *rightRotate(Node *t) {
  Node *s = t->left;
  t->left = s->right;
  s->right = t;
  return s;
}
Node *leftRotate(Node *t) {
  Node *s = t->right;
  t->right = s->left;
  s->left = t;
  return s;
}
Node *_insert(Node *t, int key, int priority) {
  if (t->key == NIL.key) {
    return new Node(key, priority);
  }
  if (key == t->key) {
    return t;
  }

  if (key < t->key) {
    t->left = _insert(t->left, key, priority);
    if (t->priority < t->left->priority) {
      t = rightRotate(t);
    }
  } else {
    t->right = _insert(t->right, key, priority);
    if (t->priority < t->right->priority) {
      t = leftRotate(t);
    }
  }
  return t;
}
Node *delete1(Node *t, int key);
Node *_delete(Node *t, int key) {
  if (t->left->key == NIL.key && t->right->key == NIL.key) {
    return &NIL;
  } else if (t->left->key == NIL.key) {
    t = leftRotate(t);
  } else if (t->right->key == NIL.key) {
    t = rightRotate(t);
  } else {
    if (t->left->priority > t->right->priority) {
      t = rightRotate(t);
    } else
      t = leftRotate(t);
  }
  return delete1(t, key);
}
Node *delete1(Node *t, int key) {
  if (t->key == NIL.key) {
    return &NIL;
  }
  if (key < t->key) {
    t->left = delete1(t->left, key);
  } else if (key > t->key) {
    t->right = delete1(t->right, key);
  } else
    return _delete(t, key);
  return t;
}
int H;
int left(int i) { return i * 2 + 1; }
int right(int i) { return i * 2 + 2; }

ll heap[2000010];
void maxHeapify(int i) {
  int l = left(i);
  int r = right(i);
  int largest = 0;
  if (l < H && heap[l] > heap[i])
    largest = l;
  else
    largest = i;
  if (r < H && heap[r] > heap[largest])
    largest = r;

  if (largest != i) {
    swap(heap[i], heap[largest]);
    maxHeapify(largest);
  }
}
int pare(int i) { return (i - 1) / 2; }
void raise(int i) {
  int l = pare(i);
  if (l < 0)
    return;

  if (heap[l] < heap[i]) {
    swap(heap[i], heap[l]);
    raise(l);
  }
}
void minHeapify(int i) {
  int l = left(i);
  int r = right(i);
  int minimam = 0;
  if (l < H && heap[l] < heap[i])
    minimam = l;
  else
    minimam = i;
  if (r < H && heap[r] < heap[minimam])
    minimam = r;

  if (minimam != i) {
    swap(heap[i], heap[minimam]);
    minHeapify(minimam);
  }
}
void buildMaxHeap() {
  for (int i = H / 2; i >= 0; i--) {
    maxHeapify(i);
  }
}

int dx[] = {-1, 0, 1, 0};
int dy[] = {0, -1, 0, 1};
std::vector<int> find_all(const std::string str, const std::string subStr) {
  std::vector<int> result;

  int subStrSize = subStr.size();
  int pos = str.find(subStr);

  while (pos != std::string::npos) {
    result.push_back(pos);
    pos = str.find(subStr, pos + 1);
  }

  return result;
}
// ll memo[100010];
// ll next[100010];
// ll dm[100010];
// int f[100010];
// ll rec(int x) {
//
//	if (~memo[x]) return memo[x];
//	if (x == n) {
//		dm[n] = 1;
//		return 1;
//	}
//	ll *res = &memo[x];
//	*res = 0;
//	set<int> st;
//	st.insert(f[x]);
//	for (int i = x + 1; i <= n; i++)
//	{
//		if (~memo[i]) {
//			*res += memo[i] + 1;
//			*res %= INF;
//			break;
//		}
//
//		*res += rec(i);
//		*res %= INF;
//		if (st.find(f[i]) != st.end()) {break; }
//		st.insert(f[i]);
//	}
//
//	return *res;
// }
#define bit(x, v) ((ll)x << v)

class BIT {

  static const int MAX_N = 100010;

public:
  BIT() { memset(bit, 0, sizeof(bit)); }
  int bit[MAX_N + 1], n;
  int sum(int i) {
    int s = 0;
    while (i > 0) {
      s += bit[i];
      i -= i & -i;
    }
    return s;
  }

  void add(int i, int x) {
    while (i <= n) {
      bit[i] += x;
      i += i & -i;
    }
  }
  void clear() { memset(bit, 0, sizeof(bit)); }
  int a[MAX_N];
  void bable_swap_count() {
    ll ans = 0;
    for (size_t j = 0; j < n; j++) {
      ans += j - sum(a[j]);
      add(a[j], 1);
    }
    printf("%lld\n", ans);
  }

  int search(int s, int x) {

    ll half = (s + x) / 2;
    ll sh = sum(x);

    ll sl = sum(half);

    ll st = sum(s);
    if (sh - sl == 0) {
      return x;
    }
    if (sh - sl < x - half) {
      return search(half, x);
    }

    if (sl - st == 0) {
      return half;
    }

    if (sl - st < half - s) {
      return search(s, half);
    }
    return -1;
  }
};
vector<int> es[100010];
ll memo[2][100010];
ll rec(int x, int color, int frm) {

  if (~memo[color][x])
    return memo[color][x];
  ll res = 1;
  for (auto v : es[x]) {
    if (v == frm)
      continue;
    if (color == 1) {
      res *= rec(v, 0, x);
    } else {

      res *= rec(v, 0, x) + rec(v, 1, x);
    }
    res %= INF;
  }

  return memo[color][x] = res;
}
int ex[110];
vector<int> edg[110];
bool vis[110];
ll rec(int x) {
  vis[x] = true;
  ll r = 0;
  for (auto v : edg[x]) {
    if (vis[v])
      continue;
    r += rec(v);
  }

  if (r > 0) {
    r++;
  } else {
    r = ex[x];
  }
  return r;
}
int tb[110][110];
int visit[110];
bool check(int x, int fr) {
  visit[x] = true;
  for (size_t i = 1; i <= n; i++) {
    if (tb[x][i] == 1) {
      if (i == fr)
        continue;
      if (visit[i])
        return false;
      if (!check(i, x))
        return false;
    }
  }
  return true;
}
void solv() {
  ll k;
  cin >> n >> k;
  ll a[200010];
  ll s[200010];
  for (size_t i = 0; i < n; i++) {
    cin >> a[i];
  }

  s[0] = 0;
  for (size_t i = 1; i <= n; i++) {
    s[i] = s[i - 1] + a[i - 1];
  }

  map<ll, queue<int>> mp;
  mp[0].push(0);
  ll res = 0;
  for (ll j = 1; j <= n; j++) {
    ll v = (s[j] - j) % k;
    auto q = mp[v];
    while (!q.empty() && q.front() <= max(-1LL, j - k)) {
      q.pop();
    }
    res += q.size();
    q.push(j);
    mp[v] = q;
  }
  cout << res << endl;
}
int main() {
  // COMinit();
  solv();
  return 0;
}

Fixed submission:
#include <algorithm>
#include <cctype>
#include <cmath>
#include <cstdio>
#include <cstdlib>
#include <ctype.h>
#include <deque>
#include <iomanip>
#include <limits>
#include <map>
#include <queue>
#include <random>
#include <set>
#include <sstream>
#include <stack>
#include <stdio.h>
#include <string.h>
#include <unordered_map>
#include <utility>
#include <vector>
#define _USE_MATH_DEFINES
#include <iostream>
#include <math.h>
using namespace std;
typedef long long ll;
typedef pair<int, int> pii;
typedef pair<ll, ll> pll;
typedef pair<int, char> pic;
#define bit(x, v) ((ll)x << v)

const ll INF = 1000000007;
const int MAX = 210000;
const int MOD = 1000000007;

long long fac[MAX], finv[MAX], inv[MAX];
void COMinit() {
  fac[0] = fac[1] = 1;
  finv[0] = finv[1] = 1;
  inv[1] = 1;
  for (int i = 2; i < MAX; i++) {
    fac[i] = fac[i - 1] * i % MOD;
    inv[i] = MOD - inv[MOD % i] * (MOD / i) % MOD;
    finv[i] = finv[i - 1] * inv[i] % MOD;
  }
}

// 二項係数計算
long long COM(int n, int k) {
  if (n < k)
    return 0;
  if (n < 0 || k < 0)
    return 0;
  return fac[n] * (finv[k] * finv[n - k] % MOD) % MOD;
}

ll gcd(ll a, ll b) {
  if (b == 0)
    return a;
  return gcd(b, a % b);
}
int pr[100010];
int lank[100010];
void uini(int n) {
  for (size_t i = 0; i <= n; i++) {
    pr[i] = i;
  }
}

int parent(int x) {
  if (x == pr[x])
    return x;
  return pr[x] = parent(pr[x]);
}

int same(int x, int y) { return parent(x) == parent(y); }

bool unit(int x, int y) {
  int px = parent(x);
  int py = parent(y);

  if (px == py)
    return false;
  if (lank[px] < lank[py]) {
    pr[py] = px;
    lank[px] += lank[py] + 1;
  } else {
    pr[px] = py;
    lank[py] += lank[px] + 1;
  }
  return true;
}

ll bit[200010];
int max_n = 200000;
int pm = 0;
void add(int x) {

  while (max_n >= x) {
    bit[x]++;
    x += x & -x;
  }
}
void sub(int x) {
  while (max_n >= x) {
    bit[x]--;
    x += x & -x;
  }
}
ll merge(ll *a, int left, int mid, int right) {
  ll n1 = mid - left;
  ll n2 = right - mid;
  vector<int> L(n1 + 1);
  vector<int> R(n2 + 1);
  for (size_t i = 0; i < n1; i++) {
    L[i] = a[left + i];
  }
  for (size_t i = 0; i < n2; i++) {
    R[i] = a[mid + i];
  }

  L[n1] = INF;
  R[n2] = INF;
  ll i = 0;
  ll j = 0;
  ll r = 0;
  for (size_t k = left; k < right; k++) {
    if (L[i] <= R[j]) {
      a[k] = L[i];
      i++;
    } else {
      a[k] = R[j];
      r += n1 - i;
      j++;
    }
  }
  return r;
}
ll merge2(pair<int, char> *a, int left, int mid, int right) {
  ll n1 = mid - left;
  ll n2 = right - mid;
  vector<pair<int, char>> L(n1 + 1);
  vector<pair<int, char>> R(n2 + 1);
  for (size_t i = 0; i < n1; i++) {
    L[i] = a[left + i];
  }
  for (size_t i = 0; i < n2; i++) {
    R[i] = a[mid + i];
  }

  L[n1] = make_pair(INF, ' ');
  R[n2] = make_pair(INF, ' ');
  ll i = 0;
  ll j = 0;
  ll r = 0;
  for (size_t k = left; k < right; k++) {
    if (L[i].first <= R[j].first) {
      a[k] = L[i];
      i++;
    } else {
      a[k] = R[j];
      r += n1 - i;
      j++;
    }
  }
  return r;
}
ll mergeSort2(pair<int, char> *a, int left, int right) {
  ll res = 0;
  if (left + 1 < right) {
    int mid = (left + right) / 2;
    res = mergeSort2(a, left, mid);
    res += mergeSort2(a, mid, right);
    res += merge2(a, left, mid, right);
  }
  return res;
}
ll mergeSort(ll *a, int left, int right) {
  ll res = 0;
  if (left + 1 < right) {
    int mid = (left + right) / 2;
    res = mergeSort(a, left, mid);
    res += mergeSort(a, mid, right);
    res += merge(a, left, mid, right);
  }
  return res;
}
int partition(pair<int, char> *a, int p, int r) {
  pair<int, char> x = a[r];
  int i = p - 1;
  for (size_t j = p; j < r; j++) {
    if (a[j].first <= x.first) {
      i++;
      swap(a[i], a[j]);
    }
  }
  swap(a[i + 1], a[r]);
  return i + 1;
}
void quick(pair<int, char> *a, int p, int r) {
  if (p < r) {
    int q = partition(a, p, r);
    quick(a, p, q - 1);
    quick(a, q + 1, r);
  }
}

ll n;
int ci = 0;
ll P[1000010];
struct Node {
  int key;
  int priority;
  Node *parent, *left, *right;
  Node(int key, int priority);
  Node() {}
};
Node NIL;
Node::Node(int key, int priority) : key(key), priority(priority) {
  left = &NIL;
  right = &NIL;
}
Node *root = new Node();
void cenrec(Node *k) {
  if (k->key == NIL.key)
    return;
  cenrec(k->left);
  cout << " " << k->key;
  cenrec(k->right);
}
void fastrec(Node *k) {
  if (k->key == NIL.key)
    return;
  cout << " " << k->key;
  fastrec(k->left);
  fastrec(k->right);
}
void insert(Node *v) {
  Node *y = &NIL;
  Node *x = root;
  while (x->key != NIL.key) {
    y = x;
    if (v->key < x->key) {
      x = x->left;
    } else {
      x = x->right;
    }
  }
  v->parent = y;
  if (y->key == NIL.key) {
    root = v;
  } else if (v->key < y->key) {
    y->left = v;
  } else {
    y->right = v;
  }
}

Node *find(Node *k, ll v) {
  if (k->key == NIL.key)
    return &NIL;
  if (k->key == v)
    return k;
  if (v < k->key)
    return find(k->left, v);
  return find(k->right, v);
}
void delp12(Node *x) {
  if (x->key == NIL.key)
    return;
  Node *l = x->left;
  Node *r = x->right;
  Node *pr = x->parent;

  if (l->key == NIL.key && r->key == NIL.key) {
    if (pr->left == x) {
      pr->left = &NIL;
    } else
      pr->right = &NIL;
  } else if (l->key != NIL.key) {
    if (pr->left == x) {
      pr->left = l;
    } else
      pr->right = l;
    l->parent = pr;
  } else if (r->key != NIL.key) {
    if (pr->left == x) {
      pr->left = r;
    } else
      pr->right = r;
    r->parent = pr;
  }
}
Node *get_next(Node *k) {
  if (k->key == NIL.key)
    return &NIL;
  Node *res = get_next(k->left);
  if (res->key != NIL.key)
    return res;
  return k;
}
void del(Node *x) {

  if (x->key == NIL.key)
    return;
  Node *l = x->left;
  Node *r = x->right;
  Node *pr = x->parent;

  if (l->key != NIL.key && r->key != NIL.key) {
    Node *nex = get_next(r);
    x->key = nex->key;
    delp12(nex);
  } else {
    delp12(x);
  }
}
Node *rightRotate(Node *t) {
  Node *s = t->left;
  t->left = s->right;
  s->right = t;
  return s;
}
Node *leftRotate(Node *t) {
  Node *s = t->right;
  t->right = s->left;
  s->left = t;
  return s;
}
Node *_insert(Node *t, int key, int priority) {
  if (t->key == NIL.key) {
    return new Node(key, priority);
  }
  if (key == t->key) {
    return t;
  }

  if (key < t->key) {
    t->left = _insert(t->left, key, priority);
    if (t->priority < t->left->priority) {
      t = rightRotate(t);
    }
  } else {
    t->right = _insert(t->right, key, priority);
    if (t->priority < t->right->priority) {
      t = leftRotate(t);
    }
  }
  return t;
}
Node *delete1(Node *t, int key);
Node *_delete(Node *t, int key) {
  if (t->left->key == NIL.key && t->right->key == NIL.key) {
    return &NIL;
  } else if (t->left->key == NIL.key) {
    t = leftRotate(t);
  } else if (t->right->key == NIL.key) {
    t = rightRotate(t);
  } else {
    if (t->left->priority > t->right->priority) {
      t = rightRotate(t);
    } else
      t = leftRotate(t);
  }
  return delete1(t, key);
}
Node *delete1(Node *t, int key) {
  if (t->key == NIL.key) {
    return &NIL;
  }
  if (key < t->key) {
    t->left = delete1(t->left, key);
  } else if (key > t->key) {
    t->right = delete1(t->right, key);
  } else
    return _delete(t, key);
  return t;
}
int H;
int left(int i) { return i * 2 + 1; }
int right(int i) { return i * 2 + 2; }

ll heap[2000010];
void maxHeapify(int i) {
  int l = left(i);
  int r = right(i);
  int largest = 0;
  if (l < H && heap[l] > heap[i])
    largest = l;
  else
    largest = i;
  if (r < H && heap[r] > heap[largest])
    largest = r;

  if (largest != i) {
    swap(heap[i], heap[largest]);
    maxHeapify(largest);
  }
}
int pare(int i) { return (i - 1) / 2; }
void raise(int i) {
  int l = pare(i);
  if (l < 0)
    return;

  if (heap[l] < heap[i]) {
    swap(heap[i], heap[l]);
    raise(l);
  }
}
void minHeapify(int i) {
  int l = left(i);
  int r = right(i);
  int minimam = 0;
  if (l < H && heap[l] < heap[i])
    minimam = l;
  else
    minimam = i;
  if (r < H && heap[r] < heap[minimam])
    minimam = r;

  if (minimam != i) {
    swap(heap[i], heap[minimam]);
    minHeapify(minimam);
  }
}
void buildMaxHeap() {
  for (int i = H / 2; i >= 0; i--) {
    maxHeapify(i);
  }
}

int dx[] = {-1, 0, 1, 0};
int dy[] = {0, -1, 0, 1};
std::vector<int> find_all(const std::string str, const std::string subStr) {
  std::vector<int> result;

  int subStrSize = subStr.size();
  int pos = str.find(subStr);

  while (pos != std::string::npos) {
    result.push_back(pos);
    pos = str.find(subStr, pos + 1);
  }

  return result;
}
// ll memo[100010];
// ll next[100010];
// ll dm[100010];
// int f[100010];
// ll rec(int x) {
//
//	if (~memo[x]) return memo[x];
//	if (x == n) {
//		dm[n] = 1;
//		return 1;
//	}
//	ll *res = &memo[x];
//	*res = 0;
//	set<int> st;
//	st.insert(f[x]);
//	for (int i = x + 1; i <= n; i++)
//	{
//		if (~memo[i]) {
//			*res += memo[i] + 1;
//			*res %= INF;
//			break;
//		}
//
//		*res += rec(i);
//		*res %= INF;
//		if (st.find(f[i]) != st.end()) {break; }
//		st.insert(f[i]);
//	}
//
//	return *res;
// }
#define bit(x, v) ((ll)x << v)

class BIT {

  static const int MAX_N = 100010;

public:
  BIT() { memset(bit, 0, sizeof(bit)); }
  int bit[MAX_N + 1], n;
  int sum(int i) {
    int s = 0;
    while (i > 0) {
      s += bit[i];
      i -= i & -i;
    }
    return s;
  }

  void add(int i, int x) {
    while (i <= n) {
      bit[i] += x;
      i += i & -i;
    }
  }
  void clear() { memset(bit, 0, sizeof(bit)); }
  int a[MAX_N];
  void bable_swap_count() {
    ll ans = 0;
    for (size_t j = 0; j < n; j++) {
      ans += j - sum(a[j]);
      add(a[j], 1);
    }
    printf("%lld\n", ans);
  }

  int search(int s, int x) {

    ll half = (s + x) / 2;
    ll sh = sum(x);

    ll sl = sum(half);

    ll st = sum(s);
    if (sh - sl == 0) {
      return x;
    }
    if (sh - sl < x - half) {
      return search(half, x);
    }

    if (sl - st == 0) {
      return half;
    }

    if (sl - st < half - s) {
      return search(s, half);
    }
    return -1;
  }
};
vector<int> es[100010];
ll memo[2][100010];
ll rec(int x, int color, int frm) {

  if (~memo[color][x])
    return memo[color][x];
  ll res = 1;
  for (auto v : es[x]) {
    if (v == frm)
      continue;
    if (color == 1) {
      res *= rec(v, 0, x);
    } else {

      res *= rec(v, 0, x) + rec(v, 1, x);
    }
    res %= INF;
  }

  return memo[color][x] = res;
}
int ex[110];
vector<int> edg[110];
bool vis[110];
ll rec(int x) {
  vis[x] = true;
  ll r = 0;
  for (auto v : edg[x]) {
    if (vis[v])
      continue;
    r += rec(v);
  }

  if (r > 0) {
    r++;
  } else {
    r = ex[x];
  }
  return r;
}
int tb[110][110];
int visit[110];
bool check(int x, int fr) {
  visit[x] = true;
  for (size_t i = 1; i <= n; i++) {
    if (tb[x][i] == 1) {
      if (i == fr)
        continue;
      if (visit[i])
        return false;
      if (!check(i, x))
        return false;
    }
  }
  return true;
}
void solv() {
  ll k;
  cin >> n >> k;
  ll a[200010];
  ll s[200010];
  for (size_t i = 0; i < n; i++) {
    cin >> a[i];
  }

  s[0] = 0;
  for (size_t i = 1; i <= n; i++) {
    s[i] = s[i - 1] + a[i - 1];
  }

  map<ll, queue<int>> mp;
  mp[0].push(0);
  ll res = 0;
  for (ll j = 1; j <= n; j++) {
    ll v = (s[j] - j) % k;
    auto q = &mp[v];
    while (!q->empty() && q->front() <= max(-1LL, j - k)) {
      q->pop();
    }
    res += q->size();
    q->push(j);
  }
  cout << res << endl;
}
int main() {
  // COMinit();
  solv();
  return 0;
}