Qwen3-32B on A100 GPUs

def longest_palindrome(s: str) -> str: if not s: return "" # Transform: "abc" → "^#a#b#c#$" T = '^#' + '#'.join(s) + '#$' n = len(T) P = [0] * n C = R = 0 for i in range(1, n - 1): mirror = 2 * C - i if i < R: P[i] = min(R - i, P[mirror]) while T[i + P[i] + 1] == T[i - P[i] - 1]: P[i] += 1 if i + P[i] > R: C, R = i, i + P[i] max_len, center = max((P[i], i) for i in range(1, n - 1)) start = (center - max_len) // 2 return s[start:start + max_len]

Here's a production-ready URL shortener architecture: 1. Key Generation: Use a distributed ID generator (Snowflake-style) encoded as base62. A 7-char base62 key gives ~3.5 trillion unique URLs. Pre-generate keys in batches to avoid hot-spots. 2. Write Path: API → Kafka (durability) → Flink/Spark (dedup, validation) → PostgreSQL (sharded by key prefix, 16 shards). Each shard handles ~6M writes/day — well within Postgres capacity. 3. Read Path: Redis (LRU cache, 95% hit rate expected) → PostgreSQL fallback. Cache hot URLs (top 1% get ~80% of traffic). Use CDN for redirect latency < 10ms globally. 4. Scaling: Horizontal — add Kafka partitions and Postgres shards linearly. Key pre-generation decouples ID service from write volume. 5. Trade-offs: Eventual consistency on writes (Kafka lag ~100ms), strong consistency on reads (cache invalidation on update).

Proof by contradiction: Assume √2 is rational. Then √2 = a/b where a, b are coprime integers (b ≠ 0). Squaring both sides: 2 = a²/b² → a² = 2b². Therefore a² is even, which means a is even (if a were odd, a² would be odd). Write a = 2k for some integer k. Substituting: (2k)² = 2b² → 4k² = 2b² → b² = 2k². Therefore b² is even, which means b is even too. But we assumed a and b are coprime — they cannot both be even. Contradiction. ∴ √2 is irrational.