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question:Let f be a function taking the nonnegative integers to the nonnegative integers, such that [2f(a^2 + b^2) = [f(a)]^2 + [f(b)]^2] for all nonnegative integers a and b. Let n be the number of possible values of f(50), and let s be the sum of the possible values of f(50). Find n times s.

answer:Setting a = 0 and b = 0, we get: [2f(0) = 2f(0)^2 implies f(0) = 0 text{ or } 1.] Setting a = 0 and b = 1, we get: [2f(1) = f(0)^2 + f(1)^2.] If f(0) = 0, then 2f(1) = f(1)^2 implies f(1) = 0 text{ or } 2. If f(0) = 1, then f(1)^2 - 2f(1) + 1 = (f(1) - 1)^2 = 0 implies f(1) = 1. We continue to build upon the results: [a = 5, b = 5: 2f(50) = 2f(5)^2 quad Rightarrow quad f(50) = f(5)^2.] Evaluating f(2), f(5) and f(25) for each case: - Case 1: f(0) = 0 and f(1) = 0. Then, f(2) = 0, f(5) = 0, and f(50) = 0. - Case 2: f(0) = 0 and f(1) = 2. Then, f(2) = 4, f(5) = 20, and f(50) = 400. - Case 3: f(0) = 1 and f(1) = 1. Then, f(2) = 1, f(5) = 1, and f(50) = 1. Thus, there are n = 3 different possible values of f(50), which are 0, 400, 1, and their sum s = 401. Therefore, n times s = 3 times 401 = boxed{1203}.

question:What are the unique substrings of length 2 in lexicographical order from the sentence 'The quick brown fox jumps over the lazy dog'?

answer:The query cannot be answered with the provided tools.

question:You walk into a room with a mysterious door made of dark wood, surrounded by ancient stones. Describe the door, and what the room is like. The door looms before you, its dark wood surface weathered to a soft, silvery gray. The grain of the wood is visible, like the lines on a wise elder's face, telling stories of years gone by. A heavy iron knocker in the shape of a snarling lion's head glares at you, as if daring you to reach out and grasp it. The door's surface is adorned with strange symbols, etched into the wood in a language you don't understand. They seem to shimmer and dance in the dim light of the room, like whispers from another world. The room itself is small and dimly lit, the air thick with the scent of old dust and forgotten memories. The walls are made of rough-hewn stone, the blocks fitting together with a precision that speaks to a long-forgotten craftsmanship. The floor is covered in a thin layer of dust, and cobwebs cling to the corners of the room like wispy bridal veils. A single torch flickers on the wall, casting eerie shadows on the stone. Now I need to describe what I see when I look through the keyhole. What do I see when I peer inside?

answer:As you bend down to peer through the keyhole, the iron lion's head seems to lean in closer, its snarling lips curled into a menacing grin. The air is stale and musty, and you can feel the weight of years of secrets and whispers emanating from the other side of the door. Through the narrow aperture, you see a sliver of a room that seems to be bathed in a soft, ethereal light. The air inside is misty, as if a fog has rolled in, obscuring the details of the space. You catch a glimpse of a long, winding staircase curving upward, its steps disappearing into the mist. The walls seem to be lined with a tattered, crimson-colored velvet, and you can make out the faint outline of a chandelier, its crystals dulled by years of grime. You sense movement, a flicker of shadowy figures darting just out of sight. The air is heavy with an otherworldly energy, and you can feel the weight of the unknown pressing against the door, as if something is waiting for you to turn the key and step inside. The symbols on the door seem to pulse with a subtle power, as if they're urging you to unlock the secrets that lie beyond.

question:Write what happens after the door creaks open.

answer:As the door creaks open, the sound echoes through the small room, disturbing the dust and cobwebs that cling to the stone walls. The iron lion's head on the door knocker seems to relax its snarl, as if it has finally relinquished its guard. The symbols on the door pulse with a soft, blue light, and the air around you seems to vibrate with anticipation. The door swings open, revealing the misty room you glimpsed through the keyhole. The air inside is cool and damp, and you can smell the scent of decay and forgotten memories. The mist swirls around your feet as you step across the threshold, and you can feel the weight of the room's secrets settling around you. As you enter, the chandelier above the staircase flickers to life, casting a warm, golden glow over the space. The crystals, now sparkling with a soft, inner light, refract the colors of the room, casting tiny rainbows on the walls. The tattered velvet drapes seem to rustle and sigh, as if they're releasing a collective breath of relief. The staircase, once shrouded in mist, now curves upward like a ghostly ribbon, inviting you to ascend into the unknown. The steps seem to whisper to you, their soft creaks and groans weaving a gentle melody that draws you deeper into the heart of the room. As you begin to climb, the room falls silent, except for the soft hum of the chandelier and the creaking of the stairs beneath your feet. You feel as though you're being drawn into a dream world, one that's been waiting for you all along. The air grows colder, the shadows deepening, and you sense that you're not alone in this place. Something is waiting for you, watching from the shadows, its eyes fixed on you as you ascend into the unknown.