brain

For a fixed position of the cover, let $C$ be the convex cover and let $H$ be the hole.

The problem presents a geometric pattern generated on a coordinate plane and asks for the number of ways to sequentially paint this pattern with a given number of colors. Instead of thinking about coordinates, we can abstract the problem into combinatorial structures.

The expression

We are asked to count binary strings containing exactly n zeros and m ones that can be reduced to a single bit g after applying a sequence of specific "modifications." A modification takes the last two characters of a string and replaces them with a single new character.

Vasily has painted two rows of circles on the plane, each row containing m circles of the same radius R. The first row lies on the line y = 0 and the second on y = 2R.

We are dealing with a two-column grid of arbitrary height r, where each cell initially represents water. Two cities take turns reclaiming cells to turn them into land, but there is a restriction: reclaiming a cell blocks the three neighboring cells in the opposite column from…

A heptagon has $7$ vertices and $14$ diagonals.

We are given a single string made of lowercase letters. From this string, we are allowed to delete characters while keeping the remaining characters in order, and we are interested only in the resulting subsequences that are palindromes. The task is twofold.

Let $x=a^2,\; y=b^2,\; z=c^2,\; w=d^2$.

We are given a list of pies with positive prices and a store promotion: for each pie you pay full price for, you can take another pie that is strictly cheaper for free. The goal is to determine the minimum total amount you must spend to acquire all pies.

We are asked to relate two counting schemes over a sequence of randomly sized skyscrapers. Imagine a row of skyscrapers with random heights, where the height of each building follows a geometric distribution with probability $2^{-i}$ for height $i$.

We are given a collection of axis-aligned rectangles with integer coordinates, and we are allowed to pick any subset of them.

We are asked to help Piegirl buy sheets of stickers to construct a target string s. Each sheet contains exactly n stickers, and all stickers on a sheet are predetermined by the string we choose for that sheet.

Let the square be partitioned into a regular grid of $n \times n$ congruent squares, each of side length $1/n$.

We are asked to analyze a situation with coins of denominations that are powers of three: 1, 3, 9, 27, and so on. A buyer wants to pay an exact amount n but cannot do so because he lacks the right combination of coins.

We are given an $n times n$ grid with some blocked cells. Gerald is allowed to place chips only on the boundary cells, but corners are forbidden starting positions.

The expression

We are given a set of candidate points in the plane, and we must choose exactly three of them to serve as centers of three identical circles. All three circles must have the same radius, and they are allowed to touch but not overlap in their interiors.

We are given a rectangular grid of numbers with $n$ rows and $m$ columns. Each cell contains a non-negative integer. The "property" of the table is defined as the minimum value among the four corner cells.

A dissection of a convex polygon into regular polygons means that every piece is an equilateral triangle, a square, or a regular polygon of higher order, all glued edge-to-edge without overlap.

We are asked to construct a large collection of 8-digit strings, where each string is a “ticket”. Each ticket is considered valid if it is possible to insert arithmetic operations between its digits and fully parenthesize the resulting expression so that the final value…

We are given a list of numbers placed on a line, where each position represents a law and its value represents how “useful” or “valuable” it is. We must select exactly two contiguous blocks of fixed length k.

The sequence is defined by iteration of an integer polynomial $P$ satisfying $P(x)>x$ for all natural $x$.

We are given a collection of orders. From these, we must select exactly $p$ orders that will be enforced. Each chosen order has two effects: if the chairperson complies with it, it contributes some amount of “damage” measured by $ai$, and if she refuses, it causes…

We are given a string p, which acts as a container, and a target message s that we want to extract. To do this, we must construct a binary key q of length k.

We are asked to analyze a network of missile silos connected by underground passages, each guarded by a certain number of droids. The silos form a highly structured network: for any subset of silos of size k, there is exactly one silo connected directly to all of them.

The game can be thought of as a circle of n players taking turns performing one of two actions, denoted by 'a' for elbow and 'b' for nod. Vasya, at index 0, wants to maximize the number of times he can drink a glass of juice.

We are given a permutation of the numbers from 1 to n, but the permutation is not just data, it defines a fixed ordering of “beavers in a line”. Each number appears exactly once, but the position of each value changes over time due to swaps.

Two intersecting circles partition the plane into exactly three bounded regions: the common lens $R_0$, the two asymmetric caps $R_1$ and $R_2$ lying respectively in the first and second circle but ou…

We are given a linear sequence of trees, each carrying an integer value that represents how aesthetically pleasing that tree is. We are allowed to remove any subset of these trees, keeping the remaining ones in their original order.

The problem gives a directed graph where vertices represent locations and edges represent streets between them. Every street has two pieces of information: where it goes, and a fixed sequence of “visions”, which is just a list of vertices.

We are asked to model Neo's “deja vu” experiences as paths in a directed graph. The graph nodes represent shops, and directed edges represent streets. Each edge carries a sequence of visions, which is a list of shop indices that Neo sees when traveling that edge.

We are asked to simulate the movement of a vehicle, the Beaveractor, inside a square campus. The campus is a grid from (0,0) to (b,b).

A ticket is a length-$k$ word over the alphabet ${0,1,\dots,9}$.

We are asked to simulate the movement of a “Beaveractor” on a square campus of size b×b. The campus contains several arrows that act like teleportation instructions: whenever the Beaveractor reaches an arrow, it immediately changes its direction to match the arrow and…

We are asked to simulate the motion of a "Beaveractor" on a square campus of size $b times b$. The campus contains a set of directional arrows, each either horizontal or vertical, which force the Beaveractor to change its motion direction when it crosses them.

We are asked to reduce a positive integer to zero by repeatedly subtracting one of its digits. On each step, you must choose any digit that is present in the current number and subtract it. The goal is to minimize the number of subtraction operations needed to reach zero.

We begin with small values of $n$ to understand the structure.

We are given a positive integer n representing a "magic number" from the Julya calendar. The Smart Beaver can reduce n to zero by repeatedly subtracting one of its digits. For instance, if n = 24, the Beaver could subtract 2 or 4, producing 22 or 20 respectively.

We start with a non-negative integer n. In one operation, we look at the decimal representation of the current number, choose any digit that appears in it, and subtract that digit from the number.

We are given a permutation of n beavers, each with a unique ID from 1 to n. A high-tech machine can shave beavers in consecutive ID intervals, but only if the permutation contains all IDs in that interval in increasing order, not necessarily consecutively in positions.

We are given a linear sequence of trees, each with an integer esthetic appeal. The task is to remove some trees so that three conditions hold. First, the sum of the remaining trees' appeals is maximized.

We are given a small rectangular grid representing a cake. Each cell is either an ordinary cake cell (.) or contains a strawberry (S).

Let $S$ be a subset of ${a_1,\dots,a_n}$ and write $s(S)$ for its sum.

We have a graph with n cities and no roads initially. Some pairs of cities are forbidden, meaning we are not allowed to build a road directly between them. We must add the smallest possible number of roads so that two conditions hold.

We are given a rectangular grid representing the forest. Some cells are blocked by trees, one cell contains our starting position S, one cell contains the exit E, and some cells contain digits. A digit cell represents that many other breeders standing there initially.

For small primes the structure is very rigid.

The original graph is very restricted. Every vertex has degree at most two, which means every connected component is either a simple path or a simple cycle. We must build another graph on the same set of vertices.

The configuration imposes five independent parallelism relations between each side of a convex pentagon and a diagonal.

We are asked to design a placement of rocks on an n × n grid such that activating a single rock produces at least x sounds. Each rock has a fixed movement direction - up, down, left, or right - and rocks move until they hit either a wall or another rock.

Write $x=n+t$ with $n=[x]\in\mathbb{Z}$ and $t={x}\in[0,1)$.

We have an n × n grid. Some cells are usable (.), while some cells are forbidden (E). A spell may only be cast on a usable cell. When we cast a spell on cell (r, c), every cell in row r and every cell in column c becomes purified.

We need to construct an increasing sequence of n positive integers such that no later element is divisible by any earlier element. The input contains a single number n. We must output any sequence of length n satisfying two conditions.

Let $ABCD$ be a tetrahedron.

We are given a digit string a and an integer k. The actual plate is not a itself, but the string obtained by concatenating a with itself k times. From this long string, we may delete any subset of positions, except that we are not allowed to delete every digit.

We have n positive segment lengths. Their total sum is d, which is also the destination point on the number line. A route is simply an ordering of these lengths. While following a route, we keep a running prefix sum.

We are given a 2D grid with n rows and m columns. Each cell is either empty, where we can place a tower, or a hole, where no tower can be built.

We are given a binary array, every element is either 0 or 1. We must choose exactly one contiguous segment and flip every value inside it. A flip changes 0 to 1 and 1 to 0. After performing this single operation, we want the resulting array to contain as many ones as possible.

The object is a closed polygonal line drawn on the surface of a unit cube, with the condition that every face of the cube contains at least one entire segment of the polygonal line.

We are asked to simulate a process of turning cells in a cylindrical grid from sea into land, one by one, while ensuring that a sea path connecting the top row to the bottom row always exists.

The transformation replaces each entry in a row by the frequency of that value in the same row.

We are given the exact number of football games that must be played, and we need to find every possible initial number of teams that produces exactly that many games under a specific tournament format.

Let the circle have center $O$ and radius $R$.

Let $A,B,C$ be the angles of $\triangle ABC$.

Let the positions of the three pedestrians at time $t$ be represented by vectors $A(t), B(t), C(t)$ in the plane.

An infinite decimal expansion determines an infinite sequence of digits, hence an infinite word over the alphabet ${0,1,\dots,9}$.

Assume such a configuration exists and consider the finite set of triangles.

Place the square in a coordinate system with algebraic convenience so that perpendicularity can be tested by a dot product condition.

Let $M_0$ and $M_1$ be convex polygons.

Each circle contributes boundary pieces only where it is the lowest among the $N$ radii in some direction, since the intersection of disks can be described as the set of points satisfying $d(x,O_i)\le…

Each edge of the convex polyhedron is oriented, so the 1-skeleton becomes an orientation of a connected planar graph embedded on the sphere.

The outer parallelogram $P_1$ admits an affine normalization to a unit square without changing incidence relations such as “lying on a side” and “being parallel to fixed directions.

An $n$-digit number is a sequence of digits $d_1d_2\ldots d_n$ where $d_1 \in {1,\dots,9}$ and $d_i \in {0,\dots,9}$ for $i \ge 2$.

Let $O = AC \cap BD$ in the trapezoid $ABCD$ with $AB \parallel CD$.

We are given a finite or otherwise fixed collection of forbidden words over the alphabet ${a,b,c}$, each forbidden word having length at least $2$, and all forbidden words having pairwise distinct len…

Let $\gamma_n = \angle C_{n+1} C_n O$.

Let the triangle have vertices $A_1,A_2,A_3$.

Let the total weight be $S$, and suppose the $N$ weights are partitioned into $K$ piles each of sum $T$, so $S = KT$.

Consider small convex polygons whose diagonals are defined as segments joining non-adjacent vertices.

The configuration involves a convex hexagon with side lengths bounded below or above and three “long” diagonals connecting every second vertex.

For small values, direct checking clarifies the constraint.

The game is played on the graph of an $n\times n$ chessboard, where vertices are squares and edges correspond to standard knight moves $(\pm2,\pm1)$ and $(\pm1,\pm2)$.

Consider the circle through three consecutive vertices $A_{i-1},A_i,A_{i+1}$.

Let the centers of the spheres be $O_1$ and $O_2$, with radii $R_1$ and $R_2$.

Let the given points be $O$, $I$, and $I_a$, where $O$ is the circumcenter, $I$ the incenter, and $I_a$ one of the excenters of triangle $ABC$.

The condition says no color appears more than $\frac{n}{2}$ times.

Let the triangle be $ABC$ with circumcenter $O$.

For a triple of points $A,B,C$, the condition that the triangle is obtuse means that one of the three angles exceeds $90^\circ$, equivalently one of the three opposite-side inequalities of the form

Let the parallelogram be mapped by an affine transformation to the unit square, since affine maps preserve parallelism, ratios of areas, and the condition of a point lying on a segment.

Let the four points be $A,B,C,D$ in space, not lying in one plane.

We interpret the situation as a simple undirected graph on $N$ vertices, where each vertex represents a person and each edge represents a mutual acquaintance.

Let $f(x)=ax^{2}+bx+c$ and assume the equation $f(x)=x$ has no real roots.

Let the digits of the infinite sequence be $a_1,a_2,a_3,\dots$, where each $a_i \in {0,1,\dots,9}$.

Let $ABCD$ be a cyclic quadrilateral with diagonals $AC$ and $BD$ intersecting at $P$.

Let $A$ and $B$ be fixed, and let $l$ be a fixed line through $A$ not containing $B$.

The expression is a finite alternating sum of simple fractions with shifts in the denominator.

The wire must be bent into the full frame of a cube of side $10$, which is the 1-skeleton of a cube graph with $8$ vertices and $12$ edges, each of length $10$.

Let $A$ be the vertex of the angle whose bisector contains $P$.

A regular hexagon of side length $1$ provides three natural directions of equal unit segments forming angles of $60^\circ$.

Each row contains $n$ numbers arranged increasingly, so the $k$-th column consists of the $k$-th smallest element in each row.