brain

Let $\sigma(n)$ denote the sum of all positive divisors of $n$.

Let the position of the first turtle at time $t$ be $P(t)$ and the position of the second turtle be $Q(t)$.

Consider the Diophantine equation

Consider first the configuration of ten identical billiard balls arranged snugly in a triangular container.

Consider the situation with only two kangaroos first.

We are asked to minimize $(x+y)(x+z)$ under the constraint $xyz(x+y+z)=1$, with $x$, $y$, $z$ positive.

Let the three triangles be $T_W,T_R,T_G$, and let $M$ be a point lying in the interior of each of them.

There are eight teams, each playing once against every other team, so each team plays $7$ games.

A complete solution cannot be written from the information provided.

I cannot write a solution to Kvant problem M1154 because the actual problem statement is not present in your message.

I do not have the statement of Kvant problem M1152, and the prompt says that only the graphical version is currently available.

I cannot write a solution to Kvant problem M1151 from the information provided, because the actual problem statement is missing.

For $n=3$ the inequality becomes

Consider two rays $p$ and $q$ with vertices $P$ and $Q$, respectively.

Consider small values of $a$ and $n$ to understand the pattern.

The condition says that every closed path contains an even number of red edges.

Place the equilateral triangle $ABC$ in the plane with convenient coordinates.

Consider a circle with a point $P$ outside it and two tangents $PB$ and $PC$, forming an angle $\angle BPC > 90^\circ$.

Let

We are asked to evaluate a large sum over all triples of integers chosen independently from three ranges. Concretely, imagine three slots, where the first slot can be filled with any value from 1 to a, the second from 1 to b, and the third from 1 to c.

Consider first small circular arrangements of weights with integer masses and total mass divisible into parts.

We are asked to compute the expected total cost of a recursive deletion procedure applied to a connected graph with exactly as many edges as nodes.

Each click in the game independently becomes either a successful hit (O) or a miss (X). For a completed sequence, we split it into maximal consecutive blocks of O. If a block has length L, it contributes L² to the score. The total score is the sum of these squared block lengths.

Consider small tables first.

We are asked to help Vasya pick the best hours to read a textbook on a train ride. The train trip lasts for n hours, and each hour has a given light level between 0 and 100. Vasya wants to read for exactly k hours.

We are given a sequence of daily temperatures. We may change any temperature to any other value, and each modified position costs one change. The goal is to make the sequence follow a very specific pattern.

Consider a trapezoid $ABCD$ with $AB$ and $CD$ as the bases, $AB \parallel CD$, and a circle inscribed within it.

We are given two decks of cards, each with a specific order from top to bottom, and each card is either face up or face down. The first deck has n cards, the second deck has m cards.

We have n football players, numbered from 1 to n. Each practice consists of splitting all players into two non-empty teams.

We are given a fence made of n vertical boards, each with a specified height. Vasya has two paint colors, red and green, each with a limited total area he can paint. Every board must be painted exactly one color, and the total painted area of each color cannot exceed its limit.

We are asked to simulate a simplified version of the UEFA Champions League group stage draw. There are n teams, with n divisible by four, each assigned a unique rating. The goal is to divide the teams into groups of four, following a structured "basket" draw procedure.

Each intersection point is a crossing of two branches.

Vasya has a list of movies and a list of favorite actors. Each movie lists some of its cast, but some actor IDs may be missing (represented by 0).

We are asked to seat an even number of students, each either left-handed or right-handed, at desks that hold exactly two students. Each desk has a left and a right position.

We are asked to compute shortest paths in a family of recursively defined graphs called Doe graphs. Each graph is defined by an order n. The base cases are trivial: D(0) is a single vertex and D(1) is two vertices connected by one edge.

For a convex polyhedron whose faces are all squares, every face angle equals $90^\circ$.

We are asked to construct an undirected graph with exactly k triangles, where a triangle is a set of three vertices all connected pairwise.

The expression $n^2+n+3\sqrt n$ is not always an integer.

Codeforces 232E: Quick Tortoise

We have an array of plank heights. A fence piece is simply a contiguous segment. For a query segment $[l,r]$, we must count how many other segments of the same length match it. Matching has three requirements. The two segments must have equal length. They must be disjoint.

Consider first small polygons.

Testing small integer values for $A$, $M$, and $S$ helps to gain intuition about the inequality.

Let

Codeforces 232B: Table

Consider a right triangle $ABC$ with right angle at $A$ and altitude $AD$.

Three friends evaluate each contest problem independently. For every problem, we are given three values, each either 0 or 1. A value of 1 means that friend is confident they know how to solve the problem. A value of 0 means they are not confident.

Consider the sum

We are given a connected undirected graph that is guaranteed to be a vertex cactus. In a vertex cactus, every vertex belongs to at most one simple cycle. Cycles may touch the rest of the graph through articulation points, but two different cycles can never share a vertex.

We are asked to compute the sum of numbers visible on a box from a given viewpoint in three-dimensional space. The box is axis-aligned, meaning all edges run along the X, Y, and Z axes. Its minimal corner is at the origin, and the opposite corner is at coordinates $(x1, y1, z1)$.

We are asked to reconstruct an initial sequence of integers given the final result of a repeated transformation. Vasya has n cards, each containing an integer between 1 and l.

Valera has exactly four horseshoes. Each horseshoe has a color represented by an integer. For the party, he wants all four horseshoes to have different colors. If some of his horseshoes share the same color, he must buy replacement horseshoes of new colors.

We maintain an array of up to $10^5$ numbers. Two kinds of operations arrive online. The first operation changes a single array element. The second operation asks for a weighted sum on a segment.

We have a graph with n cities connected by m undirected roads. Each road either has asphalt (1) or does not (0). The king can pick a city and the workers will toggle the asphalt status on every road incident to that city: asphalted roads become non-asphalted and vice versa.

We are given an $n times m$ grid where each cell is either white (".") or black (""). Vasya claims to have painted a fractal on some sub-squares of this grid.

Let

We have two binary matrices. A shift (x, y) means that cell (i, j) of the first matrix is compared with cell (i + x, j + y) of the second matrix. For a fixed shift, the overlap factor is the sum of products of overlapping cells.

Consider the case $n=1$ first.

Consider first a simple convex polygon, such as a triangle or a square.

Codeforces 225C: Barcode

I cannot write a solution to Kvant problem M1129 because the actual problem statement is not present in the conversation.

Codeforces 225E: Unsolvable

Codeforces 225D: Snake

Consider first the case of a $2 \times 2$ chessboard with two pieces.

Codeforces 225A: Dice Tower

Codeforces 223C: Partial Sums

I notice that the problem statement itself is not yet provided.

Codeforces 223B: Two Strings

The statement resembles a converse of a familiar fact about equal angles subtending the same segment.

I cannot write a solution to Kvant problem M1125 from the information provided, because the actual problem statement is missing.

Consider a trapezoid $ABCD$ with bases $AB$ and $CD$, where $AB$ is the shorter base.

Label the cells by coordinates $(i,j)$, where $i,j\in\mathbb N$ and $i,j\ge 1$.

Let

Consider the sequence defined by $a_0 = 0$ and $a_n = P(a_{n-1})$ for $n \ge 1$, where $P(x)$ is a polynomial with integer coefficients and $P(x) > 0$ for $x \ge 0$.

Consider first small values of $k$.

Expanding the left-hand side gives

Let the sides of the given triangle $ABC$ be

Consider a rectangle drawn on a square grid where the unit squares are the cells.

Consider the first problem.

Consider a tetrahedron with vertices $A$, $B$, $C$, $D$ and let $a = AB$ and $b = CD$ be two skew edges.

Model the situation as a graph on $21$ vertices, the cities.

Starting with the numbers $1$ and $2$ on the board, the rule allows us to produce $ab + a + b$ whenever $a$ and $b$ are present.

Consider triangle $ABC$ with acute angles and its circumcircle $\Gamma$.

Consider the first few natural numbers and compute the greatest common divisors of all distinct pairs.

Let the vertices of an inscribed equilateral triangle be

Consider small cases first.

The inequality is homogeneous in the ratios of the sides.

Consider a convex hexagon $ABCDEF$.

We are given a connected planar graph drawn on the plane with no bridges, articulation points, loops, or multiple edges. Each vertex has explicit coordinates, and every edge is a straight line between two vertices that intersects no other edge except at its endpoints.

We are asked to find the shortest path a spider can take on a simple polygon from one vertex to another. The polygon can be concave, but it is guaranteed to have no self-intersections, and its vertices are given in counter-clockwise order.

The problem concerns unfolding a convex polyhedron along straight-line cuts so that its surface lies flat as a single polygon, with specified identifications of points on the boundary.

We are given a string consisting only of four bracket characters: (, ), [ and ]. The string is not necessarily balanced. Among all substrings of this string, we need to find one that forms a correct bracket sequence.

Let

Begin with the first part of the problem, which concerns tiling an infinite plane with $1\times 2$ dominoes after some non-overlapping dominoes are already placed.

We are given a sequence of integers on a blackboard and a position k. A shooshun can perform one operation that appends the k-th element of the current sequence to the end and removes the first element.

For $n=3$ it is natural to search among classical identities involving sums of three cubes.

Consider a finite set of logs lying on a straight riverbank, each forming an angle less than $45^\circ$ with the bank.

Consider small examples to gain intuition.

Consider the game for small values of $n$.

Consider small examples of isosceles triangles whose vertices have integer coordinates.

Let the circle have radius $R=\dfrac d2$.