Problem p03422 (C++)

Buggy submission (status: Time Limit Exceeded):
#include <algorithm>
#include <bitset>
#include <cassert>
#include <complex>
#include <cstdio>
#include <cstring>
#include <functional>
#include <iostream>
#include <map>
#include <queue>
#include <set>
#include <stack>
#include <string>
#include <unordered_map>
#include <vector>
using namespace std;

typedef long long ll;
typedef unsigned long long ull;
typedef pair<ll, ll> P;
typedef pair<ll, P> Q;
typedef complex<double> C;
#define cx real()
#define cy imag()
const ll INF = 1LL << 60;
const double DINF = 1e30;
const ll mod = 1000000007;
const ll dx[4] = {1, 0, -1, 0};
const ll dy[4] = {0, -1, 0, 1};
const C I = C(0, 1);
const double EPS = 1e-10;
const ll NCK_MAX = 510000;

ll gcd(ll a, ll b) {
  if (b == 0)
    return a;
  return gcd(b, a % b);
}

ll extgcd(ll a, ll b, ll &x, ll &y) {
  if (b == 0) {
    x = 1, y = 0;
    return a;
  }
  ll q = a / b, g = extgcd(b, a - q * b, x, y);
  ll z = x - q * y;
  x = y;
  y = z;
  return g;
}

ll invmod(ll a, ll m) { // a^-1 mod m
  ll x, y;
  extgcd(a, m, x, y);
  x %= m;
  if (x < 0)
    x += m;
  return x;
}

ll *fac, *finv, *inv;

void nCk_init(ll mod) {
  fac = new ll[NCK_MAX];
  finv = new ll[NCK_MAX];
  inv = new ll[NCK_MAX];
  fac[0] = fac[1] = 1;
  finv[0] = finv[1] = 1;
  inv[1] = 1;
  for (ll i = 2; i < NCK_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;
  }
}

ll nCk(ll n, ll k, ll mod) {
  if (fac == NULL)
    nCk_init(mod);
  if (n < k)
    return 0;
  if (n < 0 || k < 0)
    return 0;
  return fac[n] * (finv[k] * finv[n - k] % mod) % mod;
}

template <typename T> class Zip {
  vector<T> d;
  bool flag;
  void init() {
    sort(d.begin(), d.end());
    d.erase(unique(d.begin(), d.end()), d.end());
    flag = false;
  }

public:
  Zip() { flag = false; }
  void add(T x) {
    d.push_back(x);
    flag = true;
  }
  ll getNum(T x) {
    if (flag)
      init();
    return lower_bound(d.begin(), d.end(), x) - d.begin();
  }
  ll size() {
    if (flag)
      init();
    return (ll)d.size();
  }
};

class UnionFind {
  vector<ll> par, rank; // par > 0: number, par < 0: -par
public:
  UnionFind(ll n) : par(n, 1), rank(n, 0) {}
  ll getSize(ll x) { return par[find(x)]; }
  ll find(ll x) {
    if (par[x] > 0)
      return x;
    return -(par[x] = -find(-par[x]));
  }
  void merge(ll x, ll y) {
    x = find(x);
    y = find(y);
    if (x == y)
      return;
    if (rank[x] < rank[y]) {
      par[y] += par[x];
      par[x] = -y;
    } else {
      par[x] += par[y];
      par[y] = -x;
      if (rank[x] == rank[y])
        rank[x]++;
    }
  }
  bool isSame(ll x, ll y) { return find(x) == find(y); }
};

template <typename T> class SegmentTree {
  ll n;
  vector<T> node;
  function<T(T, T)> fun, fun2;
  bool customChange;
  T outValue, initValue;

public:
  void init(ll num, function<T(T, T)> resultFunction, T init, T out,
            function<T(T, T)> changeFunction = NULL) {
    // changeFunction: (input, beforevalue) => newvalue
    fun = resultFunction;
    fun2 = changeFunction;
    customChange = changeFunction != NULL;
    n = 1;
    while (n < num)
      n *= 2;
    node.resize(2 * n - 1);
    fill(node.begin(), node.end(), init);
    outValue = out;
    initValue = init;
  }
  void valueChange(ll num, T value) {
    num += n - 1;
    if (customChange)
      node[num] = fun2(value, node[num]);
    else
      node[num] = value;
    while (num > 0)
      num = (num - 1) / 2,
      node[num] = fun(node[num * 2 + 1], node[num * 2 + 2]);
  }
  T rangeQuery(ll a, ll b, ll l = 0, ll r = -1, ll k = 0) { // [a, b)
    if (r == -1)
      r = n;
    if (a <= l && r <= b)
      return node[k];
    if (b <= l || r <= a)
      return outValue;
    ll mid = (l + r) / 2;
    return fun(rangeQuery(a, b, l, mid, 2 * k + 1),
               rangeQuery(a, b, mid, r, 2 * k + 2));
  }
};

template <typename T> class Graph {
  struct edge {
    ll to;
    T cost;
  };
  struct edge_data {
    ll from, to;
    T cost;
  };

  ll v;
  vector<vector<edge>> e, re;
  vector<edge_data> ed;
  vector<bool> used;
  vector<ll> vs, cmp;
  bool isDirected, isMinasEdge;

public:
  Graph(ll _v, bool _isDirected = true, ll range_add = 0) {
    // range_add 0:no / 1:in / 2:out / 3:in+out
    //_v++;
    v = _v, isDirected = _isDirected;
    isMinasEdge = false;
    e.resize(v), re.resize(v);
  }
  void add_edge(ll s, ll t, T cost = 1) {
    e[s].push_back((edge){t, cost});
    if (!isDirected)
      e[t].push_back((edge){s, cost});
    else
      re[t].push_back((edge){s, cost});
    ed.push_back((edge_data){s, t, cost});
    if (cost < 0)
      isMinasEdge = true;
  }
  vector<T> dijkstra(ll s) {
    vector<T> d(v, INF);
    d[s] = 0;
    auto edge_cmp = [](const edge &a, const edge &b) {
      return a.cost > b.cost;
    };
    priority_queue<edge, vector<edge>, decltype(edge_cmp)> pq(edge_cmp);
    pq.push((edge){s, 0});
    while (!pq.empty()) {
      edge temp = pq.top();
      pq.pop();
      if (d[temp.to] < temp.cost)
        continue;
      for (const edge &next : e[temp.to]) {
        T cost = temp.cost + next.cost;
        if (d[next.to] > cost) {
          d[next.to] = cost;
          pq.push((edge){next.to, cost});
        }
      }
    }
    return d;
  }
  vector<T> bellmanford(ll s) {
    vector<T> d(v, INF);
    d[s] = 0;
    for (ll i = 0; i < v; i++) {
      for (const edge_data &temp : ed) {
        if (d[temp.from] != INF && d[temp.to] > d[temp.from] + temp.cost)
          d[temp.to] = d[temp.from] + temp.cost;
        if (!isDirected && d[temp.to] != INF &&
            d[temp.from] > d[temp.to] + temp.cost)
          d[temp.from] = d[temp.to] + temp.cost;
      }
    }
    for (ll i = 0; i < v; i++) {
      for (const edge_data &temp : ed) {
        if (d[temp.from] != INF && d[temp.to] > d[temp.from] + temp.cost)
          d[temp.to] = -INF;
        if (!isDirected && d[temp.to] != INF &&
            d[temp.from] > d[temp.to] + temp.cost)
          d[temp.from] = -INF;
      }
    }
    return d;
  }
  vector<T> shortest_path(ll s) {
    if (isMinasEdge)
      return bellmanford(s);
    else
      return dijkstra(s);
  }
  T kruskal() {
    // if (isDirected)
    UnionFind uf(v);
    auto edge_data_cmp = [](const edge_data &a, const edge_data &b) {
      return a.cost < b.cost;
    };
    sort(ed.begin(), ed.end(), edge_data_cmp);
    T ans = 0;
    for (const edge_data &temp : ed) {
      if (uf.isSame(temp.from, temp.to))
        continue;
      uf.merge(temp.from, temp.to);
      ans += temp.cost;
    }
    return ans;
  }
  void scc_dfs(ll s) {
    used[s] = true;
    for (const edge &i : e[s])
      if (!used[i.to])
        scc_dfs(i.to);
    vs.push_back(s);
  }
  void scc_rdfs(ll s, ll k) {
    used[s] = true;
    cmp[s] = k;
    for (const edge &i : re[s])
      if (!used[i.to])
        scc_rdfs(i.to, k);
  }
  vector<ll> scc() {
    used.resize(v);
    fill(used.begin(), used.end(), false);
    cmp.resize(v);
    vs.clear();
    for (ll i = 0; i < v; i++)
      if (!used[i])
        scc_dfs(i);
    used.resize(v);
    fill(used.begin(), used.end(), false);
    ll k = 0;
    for (ll i = vs.size() - 1; i >= 0; i--)
      if (!used[vs[i]])
        scc_rdfs(vs[i], k++);
    return cmp;
  }
};

class RollingHash {
  int base;
  vector<__int128> hash;
  vector<__int128> pw;
  const __int128 hashmod = (1ull << 61) - 1;

public:
  RollingHash(string s, int base = 10007)
      : base(base), hash(s.length() + 1, 0), pw(s.length() + 1, 1) {
    for (int i = 0; i < (int)s.length(); i++) {
      hash[i + 1] = (hash[i] * base + s[i]) % hashmod;
      pw[i + 1] = pw[i] * base % hashmod;
    }
  }
  ll get(ll a, ll b) { // [a, b)
    __int128 tmp = hashmod + hash[b] - hash[a] * pw[b - a] % hashmod;
    if (tmp >= hashmod)
      tmp -= hashmod;
    return (ll)tmp;
  }
};

ll n, a, k, ans;

ll get(ll a, ll k) {
  if (a < k)
    return 0;
  if (a % k == 0)
    return a / k;
  return get(a - a / k - 1, k);
}

int main() {
  scanf("%lld", &n);
  for (ll i = 0; i < n; i++) {
    scanf("%lld%lld", &a, &k);
    ans ^= get(a, k);
  }
  printf("%s\n", ans == 0 ? "Aoki" : "Takahashi");
}


Fixed submission:
#include <algorithm>
#include <bitset>
#include <cassert>
#include <complex>
#include <cstdio>
#include <cstring>
#include <functional>
#include <iostream>
#include <map>
#include <queue>
#include <set>
#include <stack>
#include <string>
#include <unordered_map>
#include <vector>
using namespace std;

typedef long long ll;
typedef unsigned long long ull;
typedef pair<ll, ll> P;
typedef pair<ll, P> Q;
typedef complex<double> C;
#define cx real()
#define cy imag()
const ll INF = 1LL << 60;
const double DINF = 1e30;
const ll mod = 1000000007;
const ll dx[4] = {1, 0, -1, 0};
const ll dy[4] = {0, -1, 0, 1};
const C I = C(0, 1);
const double EPS = 1e-10;
const ll NCK_MAX = 510000;

ll gcd(ll a, ll b) {
  if (b == 0)
    return a;
  return gcd(b, a % b);
}

ll extgcd(ll a, ll b, ll &x, ll &y) {
  if (b == 0) {
    x = 1, y = 0;
    return a;
  }
  ll q = a / b, g = extgcd(b, a - q * b, x, y);
  ll z = x - q * y;
  x = y;
  y = z;
  return g;
}

ll invmod(ll a, ll m) { // a^-1 mod m
  ll x, y;
  extgcd(a, m, x, y);
  x %= m;
  if (x < 0)
    x += m;
  return x;
}

ll *fac, *finv, *inv;

void nCk_init(ll mod) {
  fac = new ll[NCK_MAX];
  finv = new ll[NCK_MAX];
  inv = new ll[NCK_MAX];
  fac[0] = fac[1] = 1;
  finv[0] = finv[1] = 1;
  inv[1] = 1;
  for (ll i = 2; i < NCK_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;
  }
}

ll nCk(ll n, ll k, ll mod) {
  if (fac == NULL)
    nCk_init(mod);
  if (n < k)
    return 0;
  if (n < 0 || k < 0)
    return 0;
  return fac[n] * (finv[k] * finv[n - k] % mod) % mod;
}

template <typename T> class Zip {
  vector<T> d;
  bool flag;
  void init() {
    sort(d.begin(), d.end());
    d.erase(unique(d.begin(), d.end()), d.end());
    flag = false;
  }

public:
  Zip() { flag = false; }
  void add(T x) {
    d.push_back(x);
    flag = true;
  }
  ll getNum(T x) {
    if (flag)
      init();
    return lower_bound(d.begin(), d.end(), x) - d.begin();
  }
  ll size() {
    if (flag)
      init();
    return (ll)d.size();
  }
};

class UnionFind {
  vector<ll> par, rank; // par > 0: number, par < 0: -par
public:
  UnionFind(ll n) : par(n, 1), rank(n, 0) {}
  ll getSize(ll x) { return par[find(x)]; }
  ll find(ll x) {
    if (par[x] > 0)
      return x;
    return -(par[x] = -find(-par[x]));
  }
  void merge(ll x, ll y) {
    x = find(x);
    y = find(y);
    if (x == y)
      return;
    if (rank[x] < rank[y]) {
      par[y] += par[x];
      par[x] = -y;
    } else {
      par[x] += par[y];
      par[y] = -x;
      if (rank[x] == rank[y])
        rank[x]++;
    }
  }
  bool isSame(ll x, ll y) { return find(x) == find(y); }
};

template <typename T> class SegmentTree {
  ll n;
  vector<T> node;
  function<T(T, T)> fun, fun2;
  bool customChange;
  T outValue, initValue;

public:
  void init(ll num, function<T(T, T)> resultFunction, T init, T out,
            function<T(T, T)> changeFunction = NULL) {
    // changeFunction: (input, beforevalue) => newvalue
    fun = resultFunction;
    fun2 = changeFunction;
    customChange = changeFunction != NULL;
    n = 1;
    while (n < num)
      n *= 2;
    node.resize(2 * n - 1);
    fill(node.begin(), node.end(), init);
    outValue = out;
    initValue = init;
  }
  void valueChange(ll num, T value) {
    num += n - 1;
    if (customChange)
      node[num] = fun2(value, node[num]);
    else
      node[num] = value;
    while (num > 0)
      num = (num - 1) / 2,
      node[num] = fun(node[num * 2 + 1], node[num * 2 + 2]);
  }
  T rangeQuery(ll a, ll b, ll l = 0, ll r = -1, ll k = 0) { // [a, b)
    if (r == -1)
      r = n;
    if (a <= l && r <= b)
      return node[k];
    if (b <= l || r <= a)
      return outValue;
    ll mid = (l + r) / 2;
    return fun(rangeQuery(a, b, l, mid, 2 * k + 1),
               rangeQuery(a, b, mid, r, 2 * k + 2));
  }
};

template <typename T> class Graph {
  struct edge {
    ll to;
    T cost;
  };
  struct edge_data {
    ll from, to;
    T cost;
  };

  ll v;
  vector<vector<edge>> e, re;
  vector<edge_data> ed;
  vector<bool> used;
  vector<ll> vs, cmp;
  bool isDirected, isMinasEdge;

public:
  Graph(ll _v, bool _isDirected = true, ll range_add = 0) {
    // range_add 0:no / 1:in / 2:out / 3:in+out
    //_v++;
    v = _v, isDirected = _isDirected;
    isMinasEdge = false;
    e.resize(v), re.resize(v);
  }
  void add_edge(ll s, ll t, T cost = 1) {
    e[s].push_back((edge){t, cost});
    if (!isDirected)
      e[t].push_back((edge){s, cost});
    else
      re[t].push_back((edge){s, cost});
    ed.push_back((edge_data){s, t, cost});
    if (cost < 0)
      isMinasEdge = true;
  }
  vector<T> dijkstra(ll s) {
    vector<T> d(v, INF);
    d[s] = 0;
    auto edge_cmp = [](const edge &a, const edge &b) {
      return a.cost > b.cost;
    };
    priority_queue<edge, vector<edge>, decltype(edge_cmp)> pq(edge_cmp);
    pq.push((edge){s, 0});
    while (!pq.empty()) {
      edge temp = pq.top();
      pq.pop();
      if (d[temp.to] < temp.cost)
        continue;
      for (const edge &next : e[temp.to]) {
        T cost = temp.cost + next.cost;
        if (d[next.to] > cost) {
          d[next.to] = cost;
          pq.push((edge){next.to, cost});
        }
      }
    }
    return d;
  }
  vector<T> bellmanford(ll s) {
    vector<T> d(v, INF);
    d[s] = 0;
    for (ll i = 0; i < v; i++) {
      for (const edge_data &temp : ed) {
        if (d[temp.from] != INF && d[temp.to] > d[temp.from] + temp.cost)
          d[temp.to] = d[temp.from] + temp.cost;
        if (!isDirected && d[temp.to] != INF &&
            d[temp.from] > d[temp.to] + temp.cost)
          d[temp.from] = d[temp.to] + temp.cost;
      }
    }
    for (ll i = 0; i < v; i++) {
      for (const edge_data &temp : ed) {
        if (d[temp.from] != INF && d[temp.to] > d[temp.from] + temp.cost)
          d[temp.to] = -INF;
        if (!isDirected && d[temp.to] != INF &&
            d[temp.from] > d[temp.to] + temp.cost)
          d[temp.from] = -INF;
      }
    }
    return d;
  }
  vector<T> shortest_path(ll s) {
    if (isMinasEdge)
      return bellmanford(s);
    else
      return dijkstra(s);
  }
  T kruskal() {
    // if (isDirected)
    UnionFind uf(v);
    auto edge_data_cmp = [](const edge_data &a, const edge_data &b) {
      return a.cost < b.cost;
    };
    sort(ed.begin(), ed.end(), edge_data_cmp);
    T ans = 0;
    for (const edge_data &temp : ed) {
      if (uf.isSame(temp.from, temp.to))
        continue;
      uf.merge(temp.from, temp.to);
      ans += temp.cost;
    }
    return ans;
  }
  void scc_dfs(ll s) {
    used[s] = true;
    for (const edge &i : e[s])
      if (!used[i.to])
        scc_dfs(i.to);
    vs.push_back(s);
  }
  void scc_rdfs(ll s, ll k) {
    used[s] = true;
    cmp[s] = k;
    for (const edge &i : re[s])
      if (!used[i.to])
        scc_rdfs(i.to, k);
  }
  vector<ll> scc() {
    used.resize(v);
    fill(used.begin(), used.end(), false);
    cmp.resize(v);
    vs.clear();
    for (ll i = 0; i < v; i++)
      if (!used[i])
        scc_dfs(i);
    used.resize(v);
    fill(used.begin(), used.end(), false);
    ll k = 0;
    for (ll i = vs.size() - 1; i >= 0; i--)
      if (!used[vs[i]])
        scc_rdfs(vs[i], k++);
    return cmp;
  }
};

class RollingHash {
  int base;
  vector<__int128> hash;
  vector<__int128> pw;
  const __int128 hashmod = (1ull << 61) - 1;

public:
  RollingHash(string s, int base = 10007)
      : base(base), hash(s.length() + 1, 0), pw(s.length() + 1, 1) {
    for (int i = 0; i < (int)s.length(); i++) {
      hash[i + 1] = (hash[i] * base + s[i]) % hashmod;
      pw[i + 1] = pw[i] * base % hashmod;
    }
  }
  ll get(ll a, ll b) { // [a, b)
    __int128 tmp = hashmod + hash[b] - hash[a] * pw[b - a] % hashmod;
    if (tmp >= hashmod)
      tmp -= hashmod;
    return (ll)tmp;
  }
};

ll n, a, k, ans;

ll get(ll a, ll k) {
  if (a < k)
    return 0;
  if (a % k == 0)
    return a / k;
  ll minas = a / k + 1;
  ll diff = a % k;
  ll mul = (diff + minas - 1) / minas;
  return get(a - mul * minas, k);
}

int main() {
  scanf("%lld", &n);
  for (ll i = 0; i < n; i++) {
    scanf("%lld%lld", &a, &k);
    ans ^= get(a, k);
  }
  printf("%s\n", ans == 0 ? "Aoki" : "Takahashi");
}