brain

Let the two circles intersect at $A$ and another point $B$.

We are given two numbers: a positive integer $N$ and a prime $P$. The task is to compute a double sum over all ordered pairs $(i, j)$ where both indices run from $1$ to $N$.

We are given a rooted tree where every node starts with an initial positive integer value. Time evolves in discrete steps, and each node carries a value that changes every day according to a local rule involving its parent. At day zero, each node i has a value a[i].

We are given a small grid with obstacles and two special cells containing pieces A and B. Initially these two pieces occupy adjacent cells.

We are given an array of length $n$, and for every pair of indices $(l, r)$ with $l le r$, we define a value that depends on whether the greatest common divisor of the two endpoints equals a chosen parameter $k$.

We are given a number $N$ that is fully described by its prime factorization. Every prime $pi$ appears with the same exponent $m$, so $N = p1^m p2^m cdots pk^m$.

We start at position zero on an infinite number line and repeatedly roll a fair four-sided die, which produces one of the values 1, 2, 3, or 4 with equal probability. After each roll we move forward by the rolled amount.

We are simulating the final round of a four-team football group. Three matches are already known: China has already played two matches against Thailand, and now only the last round remains, consisting of China versus Korea and Thailand versus Singapore.

We are given an array of positive integers and we are allowed to overwrite elements arbitrarily. Each overwrite picks an index and assigns it any positive integer value. The goal is to transform the array into a geometric progression whose common ratio is a positive integer.

A direct attempt to force a contradiction from mixed colors on a single pentagon typically fails because cross edges can absorb local inconsistencies without producing a monochromatic triangle.

We are given a graph of towns connected by roads, and each town carries a population value. A “region” is simply any connected component formed using the roads that are currently usable.

We are given a binary string and a deterministic transformation that repeatedly shrinks it. Each transformation step replaces the string with a new one formed by taking XOR of adjacent characters.

The alternating sum is examined for small truncations to detect structure.

The previous argument attempted to manufacture a recurrence for Schur numbers by deleting minimal elements of color classes.

We are asked to distribute a fixed number of problems across a fixed number of days. Each day we choose how many problems to solve, and the sequence of daily counts must be non-decreasing.

We are given a calendar that is no longer the standard one year structure, but instead consists of a sequence of months, each with its own number of days. A date is represented as a pair of integers: a day inside a month and the month index.

We are given an $n times m$ grid where each cell has a non-negative weight. A random process picks two points in the grid, and those two points define a rectangle whose sides are parallel to the grid. The task is to compute the expected area of that rectangle.

We are given a sequence of integers and allowed to delete at most k of them. After deletions, the remaining numbers must form an arithmetic progression when read in their original order.

We are given a set of points in the plane, except one point is missing. There are already $n-1$ fixed points, and we are allowed to choose the coordinates of the final $n$-th point.

We are given a string, and we repeatedly perform an operation where we delete a contiguous segment. Each deletion removes a substring, and after deletions, the remaining characters collapse together as if the string is reindexed.

We are given a grid where movement is restricted to only two directions: right and down. Some cells contain obstacles, and normally stepping into an obstacle would block movement.

The task revolves around a multiset of values that are interpreted as exponents in a polynomial-like structure where addition is replaced by XOR.

We are looking at numbers formed using only two digits, 6 and 9. Any valid number is “nice” if every position is one of these two digits.

We are looking at all remainders produced by fixing a number $n$ and dividing it by every integer $i$ from $1$ to $n$. This gives a set of values of the form $n bmod i$.

We are given a collection of points on a 2D grid, where each point can be thought of as a cell with integer coordinates. For every point, we want to determine how far it is from the farthest other point, where distance is measured using Manhattan distance.

The expression involves a weighted sum of distinct positive integers $f(1),\dots,f(n)$ with weights $1/k^2$, which decrease strictly with $k$.

We are given a line, which we can think of as the x-axis. On this line there are several vertical people standing at distinct x-coordinates. Each person has a height, so you can think of each one as a vertical segment anchored on the line.

We are given a set of N items, where each item represents a T-shirt model. Two models are considered incompatible if they share either the same text color or the same background color.

We are given a weighted tree representing villages connected by roads. Every leaf village contains exactly one runner.

The configuration involves a triangle with $AB = AC$, a circle tangent to both equal sides at $P$ and $Q$, and also tangent internally to the circumcircle.

The sequence $f(1)<f(2)<f(3)<\cdots$ is strictly increasing and consists of positive integers, hence it defines an increasing subset $S\subset \mathbb{N}$.

The configuration consists of three mutually perpendicular segments $PA$, $PB$, $PC$ with common endpoint $P$, where $A,B,C$ lie on a fixed sphere centered at some point $O$.

The original failure comes entirely from an unverified claim about the multiplicative order of $978 \bmod 125$.

Testing small indices does not reveal any oscillatory behavior compatible with the condition $f(n+1) > f(f(n))$.

We are working with an $n times m$ grid that starts completely white. One operation chooses a single cell $(x,y)$ and a color among white, red, or blue, but the effect is not local: the whole row $x$ and the whole column $y$ are repainted in that color, overwriting anything…

We are given a set of players, each with a distinct height. We want to place all existing players in a single line and possibly add new players so that the final lineup forms a perfectly consecutive sequence of heights.

We are given an array of integers. Each operation modifies this array in a very specific way: we locate two special positions, one containing a maximum value and one containing a minimum value.

A high-rise building is described where apartments are numbered continuously starting from 1, beginning on the first floor, and every floor contains the same number of apartments.

We are given a very large positive integer written in decimal form. From this number we want to subtract another positive integer $b$, chosen by us, under two constraints. First, $b$ must be strictly smaller than the given number.

We are given a permutation of length $n$, and the task is to transform it into the identity permutation $[1,2,dots,n]$. The only allowed operation is reversing any subarray of length at least two. However, the transformation is constrained in two additional ways.

We are given a triangular arrangement of balls, where row 1 has 1 ball, row 2 has 2 balls, and so on up to row n. The balls are numbered consecutively row by row from top to bottom, and within each row from left to right.

We are given a single row of seats in an exam hall, represented as a string of length $n$. Each position is either occupied by a student or empty. Our task is to reorganize students by moving some of them from their current seats into empty ones.

We are given a connected undirected graph $G$ with up to $n$ vertices and $m$ edges, and inside this graph we are also given a spanning tree $T$ on the same set of vertices. Every edge of $T$ is guaranteed to exist in $G$, but $G$ may contain additional edges.

We are given several independent seasons of a football team. For each season, we know how many matches were won and lost, and we also know the total number of goals scored and conceded across all matches. No draws exist, so every match is strictly either a win or a loss.

The division condition is written as $a^2+b^2 = q(a+b) + r$ with $0 \le r < a+b$, and the constraint $q^2 + r = 1977$ forces $r = 1977 - q^2$.

We are given a sequence of operations applied to an unknown Latin square. A Latin square is an $N times N$ grid filled with numbers from $1$ to $N$, where each number appears exactly once in every row and in every column.

The expression is a trigonometric polynomial containing first and second harmonics.

The program is a directed acyclic graph of function calls. Each function has a base cost, and whenever it calls another function, that callee’s full execution cost is added immediately, and this effect propagates recursively through all further calls.

We are asked to reconstruct a permutation of the numbers from 1 to N given its position in a very specific ordering of all permutations. The ordering is not the usual lexicographic order.

We are given an undirected connected graph where every city has degree at least three and there is at most one road between any pair of cities.

The set $V_n$ consists of integers congruent to $1 \pmod n$, namely numbers of the form $1+kn$ with $k \ge 1$.

We are given a long continuous period defined by a start date and an end date, and inside this period we also receive a list of public holidays. Each holiday either repeats every year on a fixed month and day, or occurs only once in a specific year.

We are maintaining a live database of usernames under two operations: insertion and deletion. Each username is a short string, and every operation either tries to add it or remove it. When inserting a username, the system behaves like a reservation mechanism.

We are given an array and asked to reason about its inversion count under a restricted but flexible operation. The operation is a cyclic shift applied to the prefix of length k: we can take the first k elements, rotate them left any number of times, and append them back to the…

We are given an array of integers. For each value of a parameter $k$, we are allowed to repeatedly perform an operation that takes two positions $i$ and $j$ that are at least $k$ apart and copies the bitwise OR of $aj$ into $ai$.

We are given a set of points in four-dimensional Euclidean space. Each point has coordinates $(xi, yi, zi, wi)$. We are allowed to choose a single point $o = (ox, oy, oz, ow)$, and we measure its distance to every input point using standard Euclidean distance in 4D.

We are given a sequence of changes applied over time, where each element describes how the number of ants in an anthill changes after an observation. Positive values mean ants are added, negative values mean ants leave. We do not know the initial number of ants in the anthill.

The inequality system is local with two window lengths, $7$ and $11$, so every term participates in multiple overlapping constraints.

A square suggests a $90^\circ$ rotational symmetry about its center, and the repeated construction of equilateral triangles on each side introduces a $60^\circ$ rotational component locally.

The recurrence couples each term with the previous two terms in a nonlinear way, and the conclusion concerns the integer part of a logarithm, suggesting that the central structure is a hidden power of…

We are given a sequence of lectures, each lecture providing some amount of knowledge represented by an integer value. For every lecture, you make a decision in two layers: whether you attend it, and whether you apply a special “meditation” effect right before attending it.

We are given a collection of employees where each employee belongs to exactly one company, and each company is structured as a hierarchy rooted at a CEO.

We are given a collection of monsters, each described by two numbers. One value represents how much gold you need to have before you can defeat that monster, and the other represents how much gold you gain after defeating it.

We are given a sequence of mountain heights arranged in a line. A parameter $k$ defines when two neighboring mountains are considered “compatible”: if the height difference between adjacent positions exceeds $k$, then that adjacency is broken.

We are given an array of integers and we are asked to split it into several groups. The cost of a solution is the number of groups, and every element must belong to exactly one group.

We are given a very simple process that starts from a single object, called a part. Every time we press Enter, the number of parts increases by exactly one. After performing some number of presses, the system ends with exactly $n$ parts.

Each equation is a homogeneous linear relation in $q=2p$ integer variables with coefficients in ${-1,0,1}$.

We are given a sequence of points in the plane, and we look at the displacement vectors between consecutive points. Each such vector captures how we move from one point to the next.

A product of positive integers with fixed sum increases when the summands are replaced by a partition that favors factors close to the maximizer of the function $x \mapsto x^{1/x}$.

Let the box be a parallelepiped with side lengths $a,b,c>0$.

The polynomials are defined by iteration of $P_1(x)=x^2-2$, so that $P_n$ is the $n$-fold composition of $x^2-2$ with itself.

Let a convex quadrilateral $ABCD$ be given.

Condition (i) says that $P$ is a homogeneous polynomial of degree $n$.

The flawed argument tried to force a cyclotomic structure via Mann’s theorem.

We are simulating a traveler who moves back and forth between two stops, A and B, using two fixed timetables. One timetable lists departure times from A to B, and the other lists departure times from B to A.

Each establishment receives some number of visits over a fixed number of days. From this history we compute a frequency array $ci$, where each value represents how many times establishment $i$ was visited.

We are given a directed acyclic graph where two special vertices act as starting points. From each of these starting vertices, we must construct a path that follows directed edges forward.

We are modeling a training schedule that improves two independent skills over a number of days. Each skill has an initial level, a target level, and two ways to train it: a normal training mode that increases the skill slowly, and an intensive mode that boosts progress more…

We are given the number

We are given a sequence of positive integers. After removing duplicates, we are interested in building a directed structure over their positions in increasing index order.

The process described in the problem behaves like a two-dimensional counter that advances in a very regular pattern. We start from a current position identified by a pair of values, and each “press” moves this position forward.

There are several programming contests happening at the same time, and a set of participants must be distributed among them. Each contest has a single prize that only its winner receives.

Directed-angle accumulation across a multi-centered chain fails unless every angle is expressed as a single consistent invariant, because each construction triangle induces relations at two vertices r…

The statement concerns an arbitrary strictly increasing sequence of positive integers.

The expression to be minimized is

We are given a timeline of days, each day having a known “discomfort cost” if we choose to visit the museum on that day. Alongside this, there are multiple exhibitions, and each exhibition is active over a contiguous range of days.

The earlier approach fails because it invents a hierarchy of sets based on “sign changes” without defining a meaningful invariant that survives passage to finite differences.

We are given a single string representing a sequence of lyrics. We are allowed to perform exactly one compression operation.

We receive a sequence of snapshots from a game scoreboard, where each snapshot contains two integers representing the scores of two competing teams at that moment.

We are given a fixed itinerary consisting of n travel days, each occurring on a specific calendar day. For each of these days, there is a cost if we decide to buy a single ticket independently. These days are already sorted by time, and no two days share the same timestamp.

Each item has a price given with exactly two decimal places. You are allowed to split the set of items into groups, and each group can be paid for either in cash or by card. Card payment charges the exact sum of the items in the group.

We are given a factory modeled as a directed graph where each node represents a processing step. Every step independently fails with a given probability.

We are given an array of $n$ hidden integers, revealed only through queries. Each position corresponds to a canister of fuel, and querying index $i$ returns its value $ai$. The array is sorted in non-decreasing order.

We are given a log of orders where each order is a pair consisting of a menu item and a table number. Think of each order as an edge connecting a letter node (the dish) to a digit node (the table). The central computer knows all such edges.

We are given a frozen contest scoreboard with several teams already ordered from best to worst. Each team has a row describing its submission status per problem: accepted, rejected, pending, or no submission at all.

A rectangular box is placed in 3D space with its bottom face lying flat on a known horizontal plane. The height of the box is given, so the top face is just a vertical translation of the bottom face.

Direct computation of extremal configurations shows that the expression approaches $2$ when two opposite variables are small and the other two are large, and approaches $1$ when two adjacent variables…

Small cases were checked to understand what truly restricts the number of rectangles.

Let

A tree is given in parent representation, so every node except the root has a single parent and the edges are implicitly directed upward toward that root. We are asked to cover every tree edge exactly once using several directed segments.