brain

We are given a sequence of platform heights generated by a modular arithmetic rule. The x-th platform sits at height (a · x) mod p, where a and p are coprime and we only consider the first n such positions.

We are given a directed graph representing a map a robot can traverse. The robot starts at a vertex s and must reach a target vertex t.

We are asked to build a sequence that appears as a subsequence of two given strings while avoiding a third string as a contiguous pattern.

We are given a list of strings, and the task is to find the largest possible subset of these strings such that all strings in the subset are suffixes of some single string. In other words, there should exist a string t that ends with every string in our chosen subset.

The rule replaces one chip by two chips placed one step north and one step east, provided those target cells are empty.

We are given a number $n le 10^5$. For every integer base $b ge 2$, we write $n$ in base $b$, but instead of usual digit symbols we represent each digit as a decimal number and concatenate them without separators.

We model the configuration as follows.

For the Fibonacci sequence $f_1=1$, $f_2=2$, $f_{k+1}=f_k+f_{k-1}$, the first terms are

Let $A=4^{5^6}+6^{5^4}$.

We are moving along a straight line from (0, 0) to (a, 0) at a constant speed v. Around us, there are n black cats, each starting at a point (xi, yi). Each cat can move in any direction at speed at most u.

Let $G$ be the graph whose vertices are parliamentarians and edges represent mutual enmity.

We are given a network of people, where every pair of people either knows each other in both directions or does not know each other at all.

Let the squares have side lengths $a_1, a_2, \dots, a_n$, so that

We are given a string representing Fridays the 13th across some period of time. Each character in the string can be "0" for a normal day, "1" for a particularly bad day, or "?" for a day the user cannot recall.

We are asked to construct a resistor network using only unit resistors to achieve a specific rational resistance $frac{a}{b}$.

Let

We are asked to simulate multiple read heads moving along a linear hard drive to read a set of target tracks as quickly as possible. Each head starts at a given track, can move left, right, or stay in place, and multiple heads can occupy the same track.

Consider small values of $n$.

We are asked to maximize the total amount of water Mike can pump across a network of stations in a sequence of days. Each station is a node in an undirected graph, and pipes between stations are edges with capacities.

The recurrence is

We are given a rooted tree where every node behaves like a reservoir that can either contain water or be empty. The root is fixed at node 1, and all edges are directed conceptually away from it.

The triangle is equilateral, so placing it in a coordinate system with one side horizontal allows algebraic control over all distances and angles.

We are given a sequence that describes how two wires, one called “plus” and the other “minus”, overlap as they run from the left side to the right side of a device. At each position along the path, exactly one of the wires is physically above the other.

Let the quadrilateral have diagonals intersecting at a point $O$.

The task is to determine how many spherical balloons of radius 1 can fit inside a cupboard shaped like a semicircle on top of a rectangle. The cupboard has a semicircular top with radius r and vertical walls of height h, forming a depth also equal to r.

The allowed operations on a fraction $\frac{a}{b}$ replace the integer pair $(a,b)$ by one of $(a-b,b)$, $(a+b,b)$, or $(b,a)$.

A note starts at one spy positioned at index s in a line of n spies. The note must eventually reach another spy at position f.

From the condition with $n=m=1$ we obtain

For three consecutive natural numbers $n, n+1, n+2$, the key structural feature is that any two of them are coprime.

We are working with a tree where every node represents a point in a connected acyclic graph. Initially only node 1 is colored red, while every other node starts blue. Over time, we perform two kinds of operations.

Let the two given lines be denoted $l_1$ and $l_2$.

We are given a 3×n grid representing a puzzle board. Each cell can be forbidden, free, or marked with a special circle. Dominoes, which are 1×2 or 2×1 tiles, must be placed to cover exactly two non-forbidden cells.

We are given a sequence of positive integers, all between 1 and 7, and the length of the sequence is divisible by three.

Work in the unit circle centered at $O$.

We are given an $n times n$ matrix initially filled with zeros. The problem asks us to handle two types of operations efficiently: one that computes the xor of all elements inside a submatrix, and another that xors a given value into every element of a submatrix.

We are given a set of boxes, each containing some number of candies. The objective is to redistribute candies such that exactly two boxes contain all the candies, and every other box is empty.

We are given a set of $n$ points in the 2D Cartesian plane, and we are asked to select four of these points to form a quadrilateral with the maximal area. The quadrilateral does not need to be convex, but it must have four vertices from the given points and cannot self-intersect.

We have several destinations located on a straight line at positions $a1,a2,dots,an$, all strictly positive and distinct. A route is simply a permutation of these destinations.

We are given a permutation of the numbers from 1 to n, but some elements have been replaced with -1. The permutation has no fixed points, meaning no element is in its original position.

given a permutation of the Problem Understanding We are given a permutation of numbers from 1 to n. the numbers from 1 to n. ImagineImagine running bubble sort on this permutation running ordinary bubble sort on this.

We are asked to count the number of bricks that get painted by both Iahub and Floyd in a specific range. The bricks are numbered with consecutive integers starting from 1.

The input is a string representing a sum where every number is either 1, 2, or 3, and the numbers are separated by plus signs. For example, the string 3+2+1+3 represents four summands. Xenia can only evaluate the expression if the numbers appear in non-decreasing order.

We have a circular city with n houses arranged clockwise along a ringroad. Each house has a unique number from 1 to n, and traffic flows only clockwise. Xenia starts at house 1 and has m tasks to complete in a specific order, each task located at a house number a[i].

We are given a row of n horses, initially numbered from 1 to n from left to right. Xenia performed at most three operations on the row.

We start with an array whose length is exactly $2^n$. The final value is produced by repeatedly combining adjacent elements. The first level uses bitwise OR, the next level uses bitwise XOR, then OR again, then XOR again, alternating until only one number remains.

Xenia has access to some subset of weights from 1 through 10 kilograms. For every weight type marked as available, she may use that weight any number of times. She places weights one at a time onto a balance scale.

We have an n × m grid. Every cell contains one shoe, and every shoe pair number appears exactly twice in the whole grid. A configuration is considered tidy when the two shoes belonging to every pair occupy adjacent cells. Adjacency is by a shared side, not a corner.

We have a dressing room represented as a rectangular grid of size n by m, where each cell contains a single shoe. Every shoe belongs to a pair, and each pair appears exactly twice in the grid.

The figure describes a standard cyclic filling of an $n\times n$ table with the numbers $1,2,\dots,n$ in such a way that each row is a cyclic shift of the previous one.

The 64 marked points are the centers of the squares of an $8\times 8$ grid, hence they can be identified as

Let $f_1(x)=x^2+p_1x+q_1$ and $f_2(x)=x^2+p_2x+q_2$.

Let

The given conditions state that the vectors $(x,y)$ and $(u,v)$ both lie on the unit circle and are orthogonal with respect to the standard dot product since $xu+yv=0$.

Let

We seek an arithmetic progression

Each number $x$ can be written uniquely as $x = \lfloor x \rfloor + t$, where $t \in (0,1)$.

We are given a source string s. We are interested only in substrings that appear inside s, and we count each distinct substring once, regardless of how many times it occurs. A substring is considered good if it satisfies every rule.

Let $ab=cd$.

The problem presents an image encoded as a 2D grid of integers where 0 represents the background and 1 represents a sun, including its rays. Each sun consists of a central ellipse (or circle) and a number of rays protruding from it.

We are given an array of integers of size n, and a sequence of m operations of three possible types. The first type replaces a single element in the array with a new value. The second type queries the sum of a range where each element is weighted by Fibonacci numbers.

Work is carried out in the residue field $\mathbb{F}_p$, where $p$ is odd, so $2$ is invertible.

We have a line of n students, each holding a distinct ball numbered from 1 to n. Students can swap their balls with each other, but each student has a limit on the total number of throws they can participate in.

We are given a pattern string describing a decimal code. Some positions already contain fixed digits. Some positions contain ?, meaning any digit may be placed there. Some positions contain letters from A to J. Each letter represents a digit, but with two constraints.

A number is powerful when every prime divisor appears with exponent at least $2$, so each such number has the form

Let the tetrahedron be $ABCD$, where $ABC$ is the base.

Let $A,B,C$ be a nondegenerate triangle with side lengths $BC=a$, $CA=b$, $AB=c$.

The first expression can be expanded into a difference of two homogeneous cyclic sums:

Introduce position vectors for the vertices $A,B,C,D$ in the plane.

The initial stack is the periodic sequence of length $4n$ with pattern $G,R,B,Y$ repeating every four cards, so every consecutive block of four cards contains all four colors exactly once.

We are given the complete graph on $n$ vertices with each edge oriented in exactly one direction.

Let $M$ be a convex polygon with area $S$.

Consider the first equation, $3^{x+1} + 100 = 7^{x-1}$.

For the first two equations, small natural numbers $x$ and $y$ can be tested directly.

The condition says that every point of the boundary of the square lies within distance $\tfrac12$ of the polygonal line $L$.

Model the airline network as a connected undirected graph.

Let the entries of the table be integers, and neighboring cells differ by at most $1$.

Let

The statement asks for a natural number with a specified property, not for all such numbers.

The equation is

Consider a convex quadrilateral $ABCD$ and construct squares externally on its sides $AB$, $BC$, $CD$, and $DA$, with centers $P$, $Q$, $R$, $T$ respectively.

Consider a square $ABCD$ and a parallelogram $PQRS$ that circumscribes it, with each vertex of the square touching a different side of the parallelogram.

The operation affects the two endpoints of an edge.

Consider an arbitrary acute-angled triangle $\triangle ABC$.

Consider a triangle $ABC$ and a point $M$ which is simultaneously the centroid, incenter, and orthocenter.

Computing the first few terms of the sequence $(x_i)$ modulo small integers may reveal patterns.

The statement concerns only the directions of the vectors, since scaling a nonzero vector does not change its angle with another vector.

Let

Consider small examples first.

Consider first the simplest case, $n=2$, with two circles inside a larger circle $\mathit\Gamma$.

Let

Let the cyclic quadrilateral be $ABCD$, and let its diagonals $AC$ and $BD$ intersect at $P$.

Introduce

Place coordinates on the board by identifying each square with the pair $(i,j)$, where $1\le i,j\le n$.

Let the lines through $O$ be $l_1,\dots,l_{1979}$.

Starting from $0$ and $1$, the first new number that can be obtained is $\frac12$, since the mean of $0$ and $1$ is $\frac12$.

Let

Let $BC=a$, and place the triangle in coordinates

We are given two arrays: one long array a of size up to 150,000, and a shorter array b of length up to 150,000 but used as a multiset that we may permute and temporarily “consume”. A threshold value h is also given.

We are given a point at the origin and another point $(x, y)$ that defines a rectangle aligned with the coordinate axes.

We are given a strictly increasing sequence of positive integers. The task is to choose a subset of these numbers such that the beauty of the subset is maximized.