39:I[290370,["/_next/static/immutable/chunks/0_ymm24_7ssm5.js","/_next/static/immutable/chunks/2v61o810dw1rf.js","/_next/static/immutable/chunks/0so_6hiupchv7.js","/_next/static/immutable/chunks/3b2h_bbz559px.js","/_next/static/immutable/chunks/2ol9wyej_s9h-.js","/_next/static/immutable/chunks/2luv2orik4flv.js","/_next/static/immutable/chunks/437mepgwkzb8b.js","/_next/static/immutable/chunks/3zi4-dnjqyzkc.js","/_next/static/immutable/chunks/1doaqstxtm3cr.js","/_next/static/immutable/chunks/1d__9ackz9giv.js","/_next/static/immutable/chunks/0xcz7j92bzv8_.js","/_next/static/immutable/chunks/0cxjiw1oj-yi4.js","/_next/static/immutable/chunks/1atq4abfx_cwi.js","/_next/static/immutable/chunks/42myhavf0mka7.js","/_next/static/immutable/chunks/3vuhqfn8nseb-.js","/_next/static/immutable/chunks/0x8hbtz4240sd.js","/_next/static/immutable/chunks/449p3g95ic_j-.js","/_next/static/immutable/chunks/20ip8817rbc4e.js","/_next/static/immutable/chunks/0rwu25h6frxyw.js","/_next/static/immutable/chunks/3yt0w6j-3807h.js"],"default"] 3a:I[218728,["/_next/static/immutable/chunks/0_ymm24_7ssm5.js","/_next/static/immutable/chunks/2v61o810dw1rf.js","/_next/static/immutable/chunks/0so_6hiupchv7.js","/_next/static/immutable/chunks/3b2h_bbz559px.js","/_next/static/immutable/chunks/2ol9wyej_s9h-.js","/_next/static/immutable/chunks/2luv2orik4flv.js","/_next/static/immutable/chunks/437mepgwkzb8b.js","/_next/static/immutable/chunks/3zi4-dnjqyzkc.js","/_next/static/immutable/chunks/1doaqstxtm3cr.js","/_next/static/immutable/chunks/1d__9ackz9giv.js","/_next/static/immutable/chunks/0xcz7j92bzv8_.js","/_next/static/immutable/chunks/0cxjiw1oj-yi4.js","/_next/static/immutable/chunks/1atq4abfx_cwi.js","/_next/static/immutable/chunks/42myhavf0mka7.js","/_next/static/immutable/chunks/3vuhqfn8nseb-.js","/_next/static/immutable/chunks/0x8hbtz4240sd.js","/_next/static/immutable/chunks/449p3g95ic_j-.js","/_next/static/immutable/chunks/20ip8817rbc4e.js","/_next/static/immutable/chunks/0rwu25h6frxyw.js","/_next/static/immutable/chunks/3yt0w6j-3807h.js"],"default"] 3b:T1214,// checker.cpp #include "testlib.h" #include using namespace std; int main(int argc, char *argv[]) { registerTestlibCmd(argc, argv); // Read number of test cases from input file int t; try { t = inf.readInt(); } catch (...) { quitf(_fail, "Checker error: cannot parse number of test cases from input"); } for (int tc = 1; tc <= t; ++tc) { int n; try { n = inf.readInt(); } catch (...) { quitf(_fail, "Checker error: cannot parse n for test case %d", tc); } vector a(n); long long totalXor = 0; long long minVal = LLONG_MAX, maxVal = LLONG_MIN; for (int i = 0; i < n; ++i) { try { a[i] = inf.readLong(); } catch (...) { quitf(_fail, "Checker error: cannot parse a[%d] for test case %d", i, tc); } totalXor ^= a[i]; minVal = min(minVal, a[i]); maxVal = max(maxVal, a[i]); } // Determine correct answer according to N&S condition bool possible; if (n == 1) possible = true; else possible = (totalXor != 0) && (minVal != maxVal); // Read participant verdict (from ouf) if (ouf.eof()) { quitf(_wa, "Wrong answer on test case %d: missing output", tc); } string verdict; try { verdict = ouf.readToken(); } catch (...) { quitf(_wa, "Wrong answer on test case %d: missing output", tc); } // normalize to lowercase for (char &c : verdict) c = char(tolower(c)); if (verdict == "no") { if (possible) { quitf(_wa, "Wrong answer on test case %d: answer is Yes but participant printed No", tc); } // correct: No and impossible continue; } if (verdict != "yes") { // Format problem or unexpected token quitf(_pe, "Wrong answer on test case %d: expected Yes or No, got \"%s\"", tc, verdict.c_str()); } // Participant said "Yes" if (!possible) { quitf(_wa, "Wrong answer on test case %d: answer is No but participant printed Yes", tc); } // If n == 1 and participant printed Yes, accept this test case (no operations needed) if (n == 1) { continue; } // Simulate operations: read n-1 pairs from participant output multiset current(a.begin(), a.end()); for (int op = 1; op <= n - 1; ++op) { if (ouf.eof()) { quitf(_wa, "Wrong answer on test case %d, operation %d: missing or invalid operation (expected two integers)", tc, op); } long long x, y; try { x = ouf.readLong(); } catch (...) { quitf(_wa, "Wrong answer on test case %d, operation %d: missing or invalid operation (expected two integers)", tc, op); } try { y = ouf.readLong(); } catch (...) { quitf(_wa, "Wrong answer on test case %d, operation %d: missing or invalid operation (expected two integers)", tc, op); } if (x == y) { quitf(_wa, "Wrong answer on test case %d, operation %d: x = y = %lld (would produce 0)", tc, op, x); } auto itX = current.find(x); if (itX == current.end()) { quitf(_wa, "Wrong answer on test case %d, operation %d: value %lld not found in current array", tc, op, x); } current.erase(itX); auto itY = current.find(y); if (itY == current.end()) { quitf(_wa, "Wrong answer on test case %d, operation %d: value %lld not found in current array", tc, op, y); } current.erase(itY); long long result = x ^ y; if (result == 0) { quitf(_wa, "Wrong answer on test case %d, operation %d: %lld XOR %lld = 0", tc, op, x, y); } current.insert(result); } if (current.size() != 1) { quitf(_fail, "Checker error on test case %d: array has %zu elements after operations (expected 1)", tc, current.size()); } // Optionally: allow trailing whitespace / newline. If you want to forbid extra tokens after all test cases, // you can check ouf.eof() after processing all test cases and issue _pe if not eof. } quitf(_ok, "ok"); return 0; } 3c:T564,At any moment, if the multiset is not all equal, then there exist two different values. Pick a value $v$ with maximum frequency, and pick some other value $u \ne v$. Perform one operation on the pair $(v,u)$, and insert $w = v \oplus u$. - Since $u \ne v$, we have $w \ne 0$, so we do not create $0$. Now we show the multiset after the operation is still **not** all equal. There are two cases: - **freq$(v) \ge 2$** After removing one copy of $v$, there is still at least one remaining copy of $v$. Also, $w \ne v$ (because $v \oplus u = v$ would imply $u=0$, impossible). So the multiset contains both $v$ and $w$ with $v \ne w$, hence it is not all equal. - **freq$(v) = 1$** Since $v$ has maximum frequency and it equals $1$, every value appears exactly once. So before the operation all numbers are distinct. After removing $v$ and $u$ and adding $w$, at least one element remains (unless $n=2$), and that remaining element is different from $w$: - if it were equal to $w$, then $w$ would appear twice, contradicting that all original values were distinct and only one new value was added while two distinct values were removed. So the multiset still has at least two different values, hence it is not all equal. Therefore, as long as the multiset is not all equal, we can pick two different elements, create a non-zero value, and keep the multiset not all equal.3d:T455,Consider extending a subsequence by appending $a_k$ after $a_j$ where $j \"26\"\n\n`\nIt is not possible.\n`\n\nIf we pair the indices like this, it works:\n\n(1,2) -> \"20\", (3,4) -> \"26\", (5,6) -> \"26\"\n","isSample":true}],"sampleCount":1,"inputFormat":"$$\nt \\\\\n\\left.\n\\begin{array}{l}\nn \\\\\ns\n\\end{array}\n\\right\\} \\times t\n$","outputFormat":"$$\n\\left.\n\\begin{array}{l}\n\\text{ans}\n\\end{array}\n\\right\\} \\times t\n$\n\nwhere $\\text{ans}$ is either `Yes` or `No`.","interactionFormat":"","timeLimit":2,"memoryLimit":256,"checker":{"type":"wcmp"},"solvedCount":52,"attemptedCount":56,"tags":["greedy","strings"],"learnPages":[{"title":"Page 1","content":"In every valid pair, the two characters must be one of `\"20\"`, `\"02\"`, `\"26\"`. \nSo **every pair contains exactly one `'2'`**.\n\nLet $c_0, c_2, c_6$ be the counts of characters `'0'`, `'2'`, `'6'` in $s$.\n\n- Pairs `\"20\"` and `\"02\"` use one `'2'` and one `'0'`.\n- Pair `\"26\"` uses one `'2'` and one `'6'`.\n\nSo all `'2'` characters must be matched either with a `'0'` or with a `'6'`, hence:\n$$\nc_2 = c_0 + c_6\n$$\nIf this is not true, the answer is immediately `No`."},{"title":"Page 2","content":"Every `'6'` must appear in a pair `\"26\"`, which means **it needs a `'2'` before it** (because for a chosen pair $(i,j)$ we must have $i

ARooks	500	52/53
BPairing like 2026	1000	52/56
CNatural Xor	1500	29/40
DWeird Arrays	2000	28/30
EMerge Three Vertices	2500	5/14
FConsecutive Subtree	3000	2/6
GZero Distance	3500	7/11