problem_id stringlengths 5 5 | question stringlengths 425 1.72k | answer int64 0 98.7k | correct_count int64 3 16 |
|---|---|---|---|
10549 | In a specialized logistics warehouse, the floor is mapped as a grid of storage bays denoted by coordinates $(x, y)$, where both $x$ and $y$ are integers ranging from $-4$ to $4$ inclusive. This grid defines a region $\Gamma$ containing all possible bay locations.
A logistics manager needs to place a set of automated r... | 31 | 3 |
11105 | In a specialized logistics network, an $n \times n$ grid of servers is established, where $n = 2025$. Each server is assigned a non-zero power output value. The network architecture is governed by a strict "Balance Constraint": the power output of any individual server must be exactly $k$ times smaller than the total c... | 6,069 | 3 |
11358 | A specialized deep-sea research station utilizes a security console with four vertical pressure valves, each displaying a discrete integer reading from 0 to 9. The readings are currently calibrated to a baseline state of (0, 0, 0, 0).
To reach a specific target pressure configuration $(a_1, a_2, a_3, a_4)$, an operat... | 1,202 | 4 |
11478 | In a specialized digital archive, data packets are stored as sequences of non-zero integer security codes. A master sequence is defined as a string of $n$ codes, denoted as $(a_1, a_2, a_3, \ldots, a_n)$, where $n \ge 3$. Each individual code $a_i$ is a single-digit integer from the set $\{1, 2, 3, 4, 5, 6, 7, 8, 9\}$.... | 6 | 3 |
11650 | A specialized logistics hub manages a sequence of 10 unique shipping containers, labeled with identification numbers from 1 to 10. A quality control officer is testing different "loading sequences," which are permutations $(a_1, a_2, \dots, a_{10})$ of these 10 containers.
A loading sequence is classified as "Balanced... | 768 | 4 |
11721 | In a remote archipelago, there are several circular configurations of islands. Each configuration forms a perfect regular $n$-gon of distinct islands, where $n$ is an odd integer between $3$ and $100$ inclusive. On each island in a configuration, a unique radio tower is assigned a transmission frequency level from the ... | 2,499 | 3 |
12291 | In a specialized logistics hub, four automated couriers named Fred, Gerd, Hans, and Ingo are assigned to exactly one of four loading docks: $6a$, $6b$, $7a$, and $7b$.
A technical audit is conducted involving only Fred and the two couriers assigned to the 7th-grade docks ($7a$ and $7b$). During this audit, Hans repor... | 31,432 | 3 |
12406 | A high-tech agricultural research facility consists of a massive plot of land organized into a $5 \times 5$ grid of 25 experimental cultivation zones. The lead scientist is testing five different types of organic fertilizers, labeled with potency ratings of 1, 2, 3, 4, and 5.
To prevent cross-contamination and ensure... | 17 | 5 |
12512 | In a futuristic data-processing facility, an automated system generates "Resource Integrity Codes" (RICs) based on a signal frequency $x$, where $x$ can be any positive real-valued measurement. For a given frequency $x$, the resulting RIC value is calculated by summing two distinct processing components:
1. The first... | 590 | 3 |
12572 | A specialized environmental research station monitors four distinct bacterial colonies, labeled Alpha, Beta, Gamma, and Delta. The initial populations of these colonies are represented by the positive constants $a$, $b$, $c$, and $d$, respectively. After a specific duration $x$, each population evolves according to a g... | 0 | 5 |
12614 | In a specialized digital logistics hub, a system uses a growth constant $\lambda$, defined as the positive solution to the quadratic stability equation $t^2 - 1998t - 1 = 0$. This constant determines the scaling factor for data packets processed through a recursive algorithm.
The algorithm generates a sequence of inte... | 1,000 | 3 |
12648 | In the high-stakes world of digital logistics, two competing algorithms, Max (the "Requester") and Min (the "Assigner"), are determining the final output of a specialized subtraction engine. The engine consists of two four-digit memory registers, represented as:
$$[A_1][A_2][A_3][A_4] - [B_1][B_2][B_3][B_4]$$
There are... | 4,000 | 3 |
12701 | A high-security telecommunications network is built using $2n$ long-distance fiber-optic cables (where $n \geq 2$), arranged such that every single cable crosses every other cable exactly once, and no three cables meet at the same junction. Each resulting junction point must be fitted with a signal processor, which can... | 174 | 3 |
12728 | In a remote industrial logistics hub, there are exactly 38 distinct shipping containers, labeled with the identification integers $\{1, 2, 3, \dots, 38\}$. To determine the priority of these containers, 2017 logistics managers are asked to submit a complete priority manifest. Each manifest is a strict ranking of all 38... | 1,442 | 6 |
12854 | A specialized logistics firm manages a network of $n$ distinct storage hubs, labeled $H_1, H_2, \dots, H_n$, where $n$ is an integer such that $n \ge 2$. Each hub $H_i$ contains a non-empty collection of unique digital security keys.
The firm adheres to a strict "divergence protocol" for its data integrity: for any tw... | 113 | 3 |
12936 | In the coastal logistics hub of Veridia, a shipping manager handles daily consignments of cargo crates. Each individual crate has a weight that is a positive integer number of tons, with a strict safety regulation that no single crate can weigh more than 10 tons.
The manager’s goal is to determine the maximum total we... | 133 | 5 |
12939 | In a sprawling logistics complex, a grid-based storage floor is divided into a layout of $2m$ rows and $2n$ columns of individual square cargo zones. To secure the facility, structural support pillars must be installed at the intersections of the grid lines (the corners of the square zones).
The safety regulation stat... | 420 | 7 |
13033 | A high-security research facility is divided into a $3 \times 3$ grid of nine specialized containment vaults. To balance the facility's power grid, nine unique energy cores—with power ratings of exactly $1, 2, 3, 4, 5, 6, 7, 8,$ and $9$ units—must be distributed such that each vault contains exactly one core.
The faci... | 180 | 4 |
13055 | In a specialized logistics warehouse, a technician is organizing a series of crates labeled with strictly increasing identification numbers, where each ID must be a positive integer. The technician follows a strict protocol for the quality control checks:
1. Every crate whose position in the sequence is $(3k-2)$—speci... | 590 | 4 |
13160 | In a remote sector of the ocean, three research buoys—$A$, $B$, and $C$—form a triangular perimeter. The distance between $A$ and $B$ is $3\sqrt{30} - \sqrt{10}$ kilometers, the distance between $B$ and $C$ is exactly $12$ kilometers, and the distance between $C$ and $A$ is $3\sqrt{30} + \sqrt{10}$ kilometers. Two smal... | 11,271 | 3 |
13192 | In a highly secure automated manufacturing facility, a control system is governed by a specialized efficiency function $f(x) = \sum_{i=0}^{n} a_i x^i$. The performance coefficients $a_i$ for this system are strictly required to be integers.
The facility’s output is calibrated based on the energy levels of two specific... | 22 | 3 |
13194 | A specialized logistics software is used to verify the integrity of "Batch Codes" ($n$), which are positive integers. The software applies a security rule involving two variable operational parameters, $a$ and $b$, which can be any positive integers.
The rule states that for a Batch Code $n$ to be considered "Stable,"... | 744 | 3 |
13241 | In the year 2400, a logistics officer is auditing a series of interplanetary cargo containers, each assigned a unique identification serial number $n$. These serial numbers are natural numbers ranging from 100 to 400, inclusive.
The officer utilizes two different digital scanning protocols to verify the integrity of t... | 5,215 | 4 |
13249 | A specialized logistics company uses three-digit identification codes for its shipping containers. The company only uses "Type-A" containers, which are defined by a specific symmetric security sequence: the first digit must be 1, the third digit must be 1, and the middle digit $a$ can be any integer from 0 to 9.
A tec... | 55 | 4 |
13274 | In the competitive world of architectural design, a prestigious guild uses a numerical ranking system for its members, assigning each architect a unique identification number from the set $\{1, 2, 3, \ldots, 2014, 2015\}$.
Four specific architects—let’s call them Architect $A$, Architect $B$, Architect $C$, and Archi... | 464 | 3 |
13283 | An ancient maritime navigation society uses a specific "Safety Coefficient" to evaluate the stability of twin-hull vessels. For two hulls with integer lengths $x$ and $y$ (measured in nautical units, where $x$ and $y$ are positive integers), the coefficient is calculated by dividing the sum of the squares of their leng... | 16 | 7 |
13294 | A boutique landscaping firm is designing a circular meditation garden with six distinct stone pedestals placed at the vertices of a perfect regular hexagon. The designer has six unique ornamental shrubs, each assigned a growth-tier rating from 1 to 6.
The shrubs must be planted such that exactly three of them act as ... | 8 | 3 |
13298 | In a specialized logistics hub, a cargo manager is processing exactly 1,200 standard shipping containers. Within this fleet, there are a series of "fractional crates" identified by the codes $n/1000$, where $n$ represents every positive integer less than 1,000.
The manager’s protocol requires three distinct steps for ... | 863 | 5 |
13315 | In a remote industrial complex, an automated circular conveyor system operates with exactly 6 distinct workstations, numbered 1 through 6. The system processes batches of raw material using a simple "pulse" command. Each time a pulse is sent, the material moves from its current workstation to a specific next workstatio... | 306 | 3 |
13346 | In a specialized digital archive, data packets are organized into sequential clusters. For any group of $n$ unique data packets, the total number of possible internal configurations is given by the factorial $n!$. The system's storage efficiency is measured by the "trailing zero count" of these configurations.
A techn... | 44 | 5 |
13355 | A specialized global architecture firm is evaluating the structural integrity of a series of large-scale circular pavilions. Each pavilion is identified by an ID number $n$, representing the total number of load-bearing columns supporting it. The firm is specifically auditing "complex" pavilions—those where $n$ is a co... | 22 | 3 |
13372 | In a specialized circular particle accelerator facility known as Omega, a complex web of laser beams is used to stabilize plasma. There are a finite number of laser beams, each acting as a straight-line chord spanning the circular chamber. The safety protocols of the facility require a high level of redundancy: every s... | 128 | 4 |
13395 | A digital architect is designing a series of modular storage systems. For a system of size $n$, there are $n+1$ available slots labeled $\{0, 1, 2, \dots, n\}$. The architect needs to determine the number of valid "Symmetric Trios," which are configurations of three components $(a_1, a_2, a_3)$ placed in these slots su... | 375 | 4 |
13407 | A specialized automated bio-reactor system processes a chemical compound through discrete stages. The concentration of the compound at the start of the process, denoted as $a_0$, is determined by the reactor’s model number $M$ (where $M$ is a positive integer) according to the formula $a_0 = \frac{2M+1}{2}$.
The syste... | 5,049 | 5 |
13412 | A specialized digital architect is designing a signal encryption sequence using a growth factor $b$, where $b = \frac{1}{2} (-1 + 3\sqrt{5})$. To generate a specific signal pulse, the architect uses a polynomial formula representing a sum of weighted energy states:
\[ S = a_{2014}b^{2014} + a_{2013}b^{2013} + \dots + a... | 15 | 8 |
13413 | An industrial chemical processor is testing a new stabilizer compound for a filtration system. The efficiency of the system is represented by a non-constant performance rating $f(t)$, where $t \in (0,1]$ represents the concentration level of a specific catalyst in the solution.
The engineering team discovers that when... | 0 | 4 |
13419 | In a remote mountain range, a research station is designed as a perfect equilateral triangle. Along the perimeter and interior of this triangular plot, there are exactly 10 sensor nodes arranged in a grid of rows: the top row has 1 node, the second row has 2 nodes, the third row has 3 nodes, and the base row has 4 node... | 60 | 3 |
13446 | In a specialized laboratory, a digital signal generator continuously broadcasts a repeating sequence of data packets: **2, 0, 1, 5, 2, 0, 1, 5, 2, 0, 1, 5...**
An engineer named Elena is tasked with creating a specific security code for a server. To generate the code, she must select exactly **two separate continuous ... | 2,520 | 6 |
13447 | A high-security tech conglomerate is designing a circular network of data servers to handle encrypted transactions. Each server in the ring is allocated a specific number of "Type-S" security tokens and "Type-E" encryption keys.
Due to strict interference protocols, the system architecture must ensure that for any tw... | 8 | 5 |
13462 | In a remote archipelago, there are $n$ mysterious lighthouses arranged in a perfect circle. Each lighthouse is either in "Positive Phase" (represented by the value $1$) or "Negative Phase" (represented by the value $-1$). A researcher needs to determine the overall parity of the system—specifically, the product of the ... | 62 | 3 |
13482 | A specialized logistics company is tasked with organizing a sequence of cargo containers selected from a pool of 1,967 uniquely numbered crates, labeled 1 through 1967. The operations manager must arrange a selection of these crates in a single file line according to two strict weight-class protocols based on their ide... | 821 | 4 |
13483 | In a remote digital archives facility, a sequence of data packets is processed over a series of 124 distinct time intervals, labeled $n = 1, 2, 3, \dots, 124$. For each interval $n$, the system calculates a "Cumulative Signal Density" value, $H_n$, which is the sum of the reciprocal transmission rates from the first in... | 1,240 | 3 |
13491 | In a specialized digital archive, a sequence of data packets $a_1, a_2, a_3, \ldots, a_n$ is classified as "Optimized" if it strictly follows two protocols. First, the size of each packet must be a positive integer, and they must be arranged in strictly increasing order such that $a_1 < a_2 < a_3 < \cdots < a_n$. Secon... | 3,204 | 7 |
13493 | In a specialized logistics network, a "Core Hub" has a power capacity defined by $p^c$, where $p$ is a prime number and $c$ is a positive integer. This hub is powered by two auxiliary generators with outputs $a^p$ and $b^p$, where $a$ and $b$ are positive integers. For the network to remain stable, the combined output ... | 170 | 5 |
13509 | In a sprawling logistics network, a central hub processes shipments identified by positive integer codes ($1, 2, 3, \dots$). The facility employs exactly $2008$ different shipping lines (colors) to categorize these shipments. Every positive integer code must be assigned to exactly one shipping line.
A set of codes $S$... | 4,014 | 5 |
13510 | In the futuristic city of Numeria, two high-speed magnetic transit tracks are laid out on a coordinate grid. The "North-South Line" follows the parabolic path defined by the equation $y = x^2 - k$, where $k$ is a positive integer representing the track's depth below the main boulevard. The "East-West Line" follows a si... | 285 | 6 |
13512 | In the competitive world of artisanal coffee roasting, there are 6 unique origin labels that a collector can find, one inside each bag of "Mystery Roast" beans. Every bag contains exactly one label, and each of the 6 origins is distributed with equal probability.
A collector named Julian decides to start buying bags o... | 191 | 6 |
13548 | In a specialized logistics network, a series of cargo containers must be assigned unique positive integer identification codes. To prevent electromagnetic interference between any two containers during transit, the "Signal Gap" between their codes must meet a specific safety standard.
The safety regulation states that... | 18 | 5 |
13551 | In a specialized logistics hub, two security access codes, represented by four-digit integers $m$ and $n$, are programmed such that $n$ is exactly the reverse of $m$.
When these two codes are multiplied together, the resulting product $mn$ forms an eight-digit serial number used to calibrate a high-precision laser. T... | 11,561 | 5 |
13554 | A specialized logistics company is managing a 10-day supply chain schedule. Each day $i$ (from $1$ to $10$), a specific number of cargo containers, $a_i$, are processed through a port.
The scheduling follows a strict "Supply-Constraint Rule":
- On the first and second days, exactly 1 container is processed ($a_1 = 1... | 238 | 6 |
13566 | In a specialized biological research facility, a group of scientists identifies "Stable Colony Growth Patterns" involving three separate cultures of bacteria, with populations denoted as $a$, $b$, and $c$. For a set of cultures to be considered a *Stable Growth Set*, the following conditions must be met:
1. The populat... | 28 | 4 |
13571 | In a specialized maritime logistics zone, two circular cargo hubs, Hub B and Hub C, are being constructed along a straight 6-kilometer supply line designated as Segment BC. Hub B is positioned such that its boundary is tangent to the supply line exactly at the western terminal, Point B. Hub C is positioned such that it... | 180 | 4 |
13608 | In a remote $6 \times 6$ grid of solar-powered sensors, each unit is currently in one of two states: "Active" or "Inactive." Maintenance drones can be deployed to perform a "Phase-Shift" operation. A Phase-Shift consists of selecting any three adjacent sensors in a single row or a single column and reversing their stat... | 0 | 8 |
13627 | In a specialized cyber-security facility, an encryption protocol uses a security key $n$, where $n$ is a whole number greater than 6. For any given key $n$, the system generates a sequence of "clearance codes" $1 = a_1 < a_2 < \dots < a_k < n$, which consist of every positive integer less than $n$ that shares no common... | 374 | 4 |
13649 | In the city of Arithmos, a massive 8x8 square plaza is being paved with decorative stone tiles. The plaza is laid out in a grid with 8 rows and 8 columns of tiles.
The construction follows a strict cultural tradition:
1. Every tile located in the very first northernmost row and every tile in the very first westernmost... | 2,508 | 3 |
13666 | A digital architect is designing a series of security codes for a mainframe. A security level $n$ (where $n \ge 1$ is an integer) is classified as "Stable" if there exists at least one "Validator Key" associated with it.
A Validator Key must satisfy two specific criteria:
1. It must be a positive integer that is a mu... | 19 | 4 |
13676 | A specialized signal processing unit is designed to generate a complex waveform \( f(x) \) by synthesizing a series of \( n \) individual oscillators. Each oscillator \( i \) produces a sub-signal \( f_i(a_i x) \), where \( a_i \) is a specific frequency tuning constant and \( f_i \) is either a pure sine wave (\(\sin\... | 2,047 | 3 |
13689 | A regional logistics company manages a fleet of $n$ delivery drones, where $n \geq 5$ is a positive integer. Each drone is assigned a unique tail number from the set $\{1, 2, \dots, n\}$.
The company’s two supervisors, Anya and Becky, are organizing a maintenance schedule. To begin, Anya arranges all $n$ drones in a ... | 1,248 | 4 |
13697 | In a specialized logistics network, every package is identified by a positive integer ID, $n$. For each ID, we define two efficiency metrics:
1. **Storage Index** $\sigma(n)$: The total number of distinct positive divisors of $n$.
2. **Core Factor** $\operatorname{rad} n$: The product of the distinct prime divisors o... | 164 | 7 |
13699 | A specialized digital security vault uses a 5-digit access code composed of five distinct non-zero digits (selected from the set $\{1, 2, 3, 4, 5, 6, 7, 8, 9\}$).
The vault’s internal logic system is programmed with a unique "Divisibility Protocol":
1. The 5-digit code must be perfectly divisible by every single one... | 14,728 | 5 |
13732 | In the city of Planaria, three corporate plazas—named **DIAL**, **FOR**, and **FRIEND**—are constructed as perfectly regular polygons. A surveyor notes that the distance between the two specific corner markers labeled $I$ and $D$ is exactly $1$ unit.
An architect is commissioned to design a triangular park, denoted as... | 33 | 3 |
13737 | A grand international architecture competition requires participants to design a square city grid of size $n \times n$ city blocks. In each block, the architect must install a solar spire with a height chosen from the set $\{1, 2, \dots, n\}$ meters. The zoning laws for this competition are strict: for every single row... | 110 | 6 |
13764 | In the futuristic city of Neoterra, a digital architect is designing a recursive security protocol using encrypted data strings within a specialized 11-bit processor environment (represented by the field $\mathbb{F}_{11}$). The architect defines a sequence of data-structure polynomials, $f_0, f_1, f_2, \dots$, where th... | 301 | 3 |
13780 | In the competitive world of high-performance server cooling, a system’s "Efficiency Rating" is a function $f$ that maps an integer workload level $x \in \mathbb{Z}$ to an integer power consumption value $f(x) \in \mathbb{Z}$.
A cooling system is classified as "Ultra-Resilient" if it satisfies a specific safety redund... | 498 | 4 |
13792 | A specialized chemical refinery uses two reactors to process a compound. The efficiency of the first reactor is determined by the ratio $E_1 = \frac{a}{a + \lambda b}$ and the efficiency of the second reactor is $E_2 = \frac{b}{b + \lambda a}$, where $a$ and $b$ represent the positive flow rates of two catalyst types a... | 6 | 5 |
13818 | In a futuristic data-storage facility, an encryption sequence is classified as "Corrupted" if it contains the digit 1 followed immediately by the digit 3 (the substring "13"). If this specific sequence does not appear, the data is classified as "Secure."
A cybersecurity engineer is investigating a strange pattern in t... | 1,287 | 6 |
13819 | In a specialized thermodynamics laboratory, three distinct pressure readings, measured in kilopascals, are recorded as $a$, $b$, and $c$. These three values are the real roots of the characteristic system equation $x^3 - 3x^2 - 7x - 1 = 0$.
An engineer is calculating the "Resonance Coefficient" ($\Omega$) of the syste... | 1,913 | 5 |
13825 | In a remote sector of the galaxy, two logistics commanders, Commander Gerd and Commander Bernd, are competing for a supply contract using six automated delivery drones, each assigned a power frequency level from 1 to 6. At the start of the simulation, the fleet is split evenly: 3 drones are assigned to Gerd and 3 to Be... | 9 | 5 |
13837 | In a futuristic digital city, an architect is designing a high-security encryption gate. To unlock the gate, two security codes, $m$ and $n$, must be selected from a secure server containing integers ranging from $0$ to $2023$, inclusive.
The gate’s processing core operates on a cycle of $2024$ units. For the codes to... | 1,012 | 4 |
13846 | In a futuristic circular space station, there are $n$ docking ports arranged at the vertices of a perfectly regular $n$-gon. A security protocol requires that exactly five of these ports must be activated with red warning lights.
The station's central computer manages a set of "Symmetry Protocols." Each protocol corr... | 14 | 3 |
13861 | A remote-controlled survey drone is deployed from a base station (Point $A$) to map a triangular plot of land bounded by two landmarks, $B$ and $C$. The drone’s internal navigation system records that the angle between the paths to the two landmarks, $\angle BAC$, has a sine value of $4/5$, and this angle is known to b... | 34 | 4 |
13871 | A logistics company, "Metro-Link," operates a warehouse with a floor layout consisting of $m$ rows and $n$ columns of storage bays. The total number of bays, $mn$, is always an even number. A contractor, Jetze, is tasked with installing modular "Dual-Units" (dominoes). Each Dual-Unit is a single physical piece that mus... | 11 | 4 |
13887 | In the city of Metros, a power grid is managed by a sequence of $2n$ substations, where $n$ is a positive integer. Each substation $i$ (from $1$ to $2n$) produces a specific amount of energy $x_i$, measured as a real number of gigawatts.
A power grid configuration is classified as "Stable" if it satisfies the followin... | 2 | 3 |
13895 | In a futuristic solar farm, a grid of $5 \times 5$ identical solar panels is installed to capture energy. The grid consists of exactly 25 panels arranged in five rows and five columns. To monitor the health of the system, a set of specialized sensors must be installed on top of certain panels.
The maintenance protoco... | 8 | 5 |
13909 | In a specialized security facility, every digital access code is exactly four digits long, where each digit is limited to the values 1, 2, or 3. However, due to a hardware restriction on the keypad, no valid "Primary Code" is allowed to end with the digit 3. All four-digit sequences meeting these criteria (using only 1... | 9 | 3 |
13916 | A specialized cybersecurity firm classifies certain encryption keys, represented by odd prime numbers $p$, as "stable" based on the behavior of a recursive signal processing algorithm.
A prime $p$ is considered "stable" if an analyst can generate an infinite sequence of positive integers $a_0, a_1, a_2, \dots$ such t... | 33 | 3 |
13920 | In a futuristic data-management facility, a master encryption key is defined by the security constant $n = 2^{8} \cdot 3^{9} \cdot 5^{10} \cdot 7^{11}$. The facility generates "Echo Codes" for various digital assets. For any asset identified by a positive integer ID $k$, the system calculates a value $f(k)$, which repr... | 1,410 | 3 |
13936 | In a specialized digital archive, a compression algorithm processes a sequence of data packets labeled with positive integers $n \in \{1, 2, 3, \dots\}$. Each packet $n$ is assigned a priority value $f(n)$ determined by the following protocol:
1. The priority of the first packet is $f(1) = 1$, and the priority of the... | 92 | 4 |
13942 | In the city of Metros, two automated resource distributors, X-Unit and Y-Unit, operate under strict protocols based on integer commands $a$ and $b$. The output levels of these units, represented by the real numbers $x$ and $y$ respectively, are governed by the following regulatory system:
1. The integer floor of X-Uni... | 12 | 16 |
13949 | In the coastal city of Gridhaven, a new 8x8 solar array has been constructed, consisting of 64 individual panels arranged in a perfect square grid. Due to a recent atmospheric event, several panels have been supercharged with "Green Energy."
The city engineers have discovered that a specialized drone, known as a "B-Ty... | 41 | 5 |
13953 | A high-security data vault uses a rotating encryption system based on a cycle of $n = 2020$ possible states, indexed from $0$ to $2019$. A state index $k$ is considered "accessible" if there exists a set of three security keys—represented by integers $x, y,$ and $z$—that satisfy two specific synchronization protocols:
... | 101 | 5 |
13955 | In the competitive world of high-speed logistics, two rival dispatchers are managing a cargo load that is being filled in rounds. The game begins with the first dispatcher selecting an initial weight $n$, which must be a positive integer less than 10 units.
Following this, the dispatchers take turns increasing the we... | 6 | 6 |
13957 | A specialized logistics hub manages a fleet of "Core Units," which are identified by square-free serial numbers $k$. For any given system capacity $n$, a resource function $f_n(x)$ calculates how many of these Core Units can operate under a load factor $x$, defined as the number of square-free integers $k$ such that $k... | 60 | 4 |
13966 | In a remote digital archiving facility, a automated processing system generates a sequence of data packets, $(a_n)$, where $n$ represents the chronological index of each packet. The system initializes the first packet with a value of $a_1 = 0$.
For all subsequent steps where $n \geq 1$, the value of the next packet $a... | 17 | 3 |
13981 | In the high-tech semiconductor facility known as Sector 3, a specialized chemical enhancement process is governed by a precise growth formula. Engineers are tracking a reaction involving a proprietary compound with a stability constant exactly equal to $(3+\sqrt{6})$.
To create a high-density processor layer, this com... | 37 | 6 |
13985 | A network security engineer is designing a $4 \times 4$ authentication matrix for a high-security server room. The matrix must be populated using 16 unique security tokens. Each token is defined by two distinct attributes: one of four possible encryption protocols and one of four possible hardware manufacturers. There ... | 64 | 8 |
13986 | In the competitive world of aerospace engineering, three different fuel additives—Alpha, Beta, and Gamma—are being tested for their efficiency in a prototype combustion chamber. The chemical behavior of these additives is governed by a characteristic cubic system defined by the equation $a_{3} x^{3} - x^{2} + a_{1} x -... | 92 | 6 |
13987 | In a futuristic data-storage grid, a scientist is configuring a $6 \times 6$ array of memory cells. Each of the 36 cells can be set to one of two states: "Dormant" (represented by 0) or "Active" (represented by 1).
To prevent system overheating, the grid must adhere to strict stability protocols regarding sequences o... | 15 | 6 |
13991 | A specialized square solar farm is bounded by four perimeter fences: the Northern Boundary ($AB$), the Eastern Boundary ($BC$), the Southern Boundary ($CD$), and the Western Boundary ($DA$).
To optimize energy capture, four laser-guided sensors are positioned along the boundaries. A technician starts at the Northwest... | 2 | 6 |
13992 | A team of six specialized logistics drones is carrying out a cargo transport mission. At the start of the mission, the drones are carrying $1, 2, 3, 4, 5,$ and $6$ metric tons of fuel, respectively.
By the end of the mission, some fuel has been redistributed among the drones to optimize the return flight, but the tot... | 34 | 7 |
13994 | A specialized deep-space research station tracks cosmic radiation pulses over a sequence of $n$ consecutive days, labeled $k = 1, 2, \dots, n$. For each day $k$, the station records the "Cumulative Resonance" of the signal, which is mathematically defined as the factorial of the day number, $k!$.
The station’s compute... | 229 | 12 |
14002 | A high-security data vault is protected by a sequence of six digital locks, represented by the values $(a, b, c, d, e, f)$. Each lock must be assigned a positive integer code, and the total sum of these six codes must equal exactly $3p$, where $p$ is a prime number greater than 3.
To synchronize the vault's encryption... | 15 | 11 |
14003 | In a remote high-tech server farm, a technician is tasked with installing cooling modules on an $8 \times 8$ grid of processors. Each processor is identified by its row index $i$ and column index $j$, where $1 \le i, j \le 8$.
The technician has 21 specialized cooling plates, each designed to cover a $1 \times 3$ (or... | 36 | 16 |
14005 | In a circular radar-monitored zone $\omega$, two stationary beacons, $B$ and $C$, are positioned on the perimeter. A mobile surveillance drone $A$ orbits along the perimeter such that the angle formed between the signals from $A$ to $B$ and $C$ is always $60^\circ$.
A central relay station $D$ is located at the exact... | 1 | 3 |
14008 | In a specialized research facility, a group of $N$ distinct security sensors is arranged in a circular formation. Each sensor is assigned a unique identification code, which must be a positive integer no greater than 1000.
To maintain the integrity of the network, the system architecture requires a specific parity che... | 664 | 8 |
14019 | A high-security digital archive contains 2012 unique data canisters, labeled sequentially from 1 to 2012. To breach the archive’s encryption, a hacker needs to gain access to a specific subset of these canisters, which we will call the "Key Set."
The archive’s security protocol is triggered if the Key Set is large eno... | 1,008 | 5 |
14020 | In a remote digital architecture firm, a designer is tasked with creating "Safety Tilesets" for various rectangular server rooms. Each room is paved with a grid of $m \times n$ tiles. To meet security protocols, every tile must be set to one of two states: **Encrypted (E)** or **Open (O)**.
A configuration is consider... | 48 | 3 |
14021 | In a specialized laboratory, an experimental sequence is determined by a sequence of five distinct non-zero pressure settings, each represented by a single-digit integer (1 through 9). These settings are arranged in five consecutive slots to form a five-digit control code.
A technician is tasked with optimizing the se... | 85,230 | 3 |
14030 | In a futuristic data-management facility, a supercomputer is processing a massive archive containing a specific number of encrypted files, denoted by the integer $V = 2010^{2010}$.
The system security protocol requires identifying a specific subset of "Active Keys." A number is considered an Active Key if it is a pos... | 503 | 7 |
14036 | In the high-stakes world of quantum encryption, a security architect is designing "Validation Protocols," which are specific types of mathematical sequences represented by polynomials $f$ with integer coefficients.
To ensure the integrity of a global server network, a protocol is deemed "Stable" only if it satisfies ... | 188 | 4 |
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