brain

Consider small examples of the problem to understand the dynamics.

We start with two numbers on the blackboard, (0, 1). Operation T replaces the top number by the sum of both numbers. Operation B does the same for the bottom number. After exactly n operations, we look at the number written by the last operation.

Consider a smaller cube, for instance $2\times2\times2$ or $4\times4\times4$, with a checkerboard coloring.

The formula describes a Boolean function built from constants 0 and 1, placeholders ?, and the operators AND, OR, and XOR. Each ? corresponds to a leaf of the plant. During an experiment, we choose which colony is placed on each leaf.

The problem is essentially about circular modular arithmetic and subset sums. We have a circular orbit of n stations, each separated by m miles. Captain Bitonix has a set of fuel tanks, each with a certain capacity.

We are given a set of snow drifts on a 2D grid with integer coordinates. Bajtek can move only in straight lines along the x or y axis, sliding from one drift until he hits another.

We are asked to simulate the process of distributing numbered footballs into a row of baskets according to a specific order. Each new ball must go into the basket that currently contains the fewest balls.

Consider an angle with vertex $A$ and two rays forming the sides.

The video lasts for c seconds. Watching one second of video consumes a units of data, while the internet connection downloads only b units per second. Since a b, starting immediately is impossible because data would be consumed faster than it arrives.

Begin by considering the simplest nontrivial cases.

The problem statement is missing.

We are asked to construct a special kind of directed forest where each vertex has at most one outgoing edge, and each edge carries a weight. Vertices are added one by one. When adding a vertex, we are optionally given a set of pairs consisting of an existing vertex and a number.

We are asked to analyze the polyline formed by summing several linear functions. Each function is of the form $yi(x) = ki cdot The input consists of $n$ lines, each providing the slope $ki$ for the positive $x$-side and a constant $bi$.

We are given the source code of a tiny language that contains only three kinds of statements: try, catch(type, message), and exactly one throw(type). Each try is paired with a later catch, forming a try-catch block.

The task is to sort a list of integers in non-descending order. You are given the size of the list, n, and then n integers, each on a separate line, with values ranging from 1 to 100.

We are asked to calculate the n-th hexagonal number. Hexagonal numbers represent a figurate number pattern where each number counts the total tiles forming a hexagon with successive layers. The formula for the n-th hexagonal number is given by $Hn = 2n^2 - n$.

The folding is performed only along the grid lines.

Let

The number is

Consider a parallelogram $ABCD$ with $AC > BD$ and a point $M$ on $AC$ such that $BCDM$ is cyclic.

Consider a small case to understand the structure.

Let a permutation of the deck be written as a sequence.

Consider an isosceles triangle $ACB$ with base $AB$ and vertex $C$.

The figure shows three allowed tile types.

Consider two trapezoids $ABCD$ and $A'B'C'D'$ such that the legs of each are equal to the bases of the other.

The input is a string that describes operations on a stack. Each digit means "push this number onto the stack". Each + or means "take the top two values from the stack, apply the operation, and push the result back".

We are given two positive integers and need to compute their least common multiple. The least common multiple, usually abbreviated as LCM, is the smallest positive integer that is divisible by both numbers without leaving a remainder.

Consider two positive numbers $x$ and $y$ such that $xy > x + y$.

The decimal expansion of $\dfrac1{3^{100}}$ is purely periodic after a finite preperiod, because $3^{100}$ is coprime to $10$.

For $n=2$, the conditions give $x_2=-x_1$ and $2x_1^2=1$, hence $x_1x_2=-\frac12\le-\frac12=-\frac1n$.

Begin by testing small values of $n$.

Consider a fixed chord $MN$ of a circle and a variable diameter $AB$ that does not intersect $MN$.

Let

Denote

Before I can write the full solution, I need the full text of Kvant problem M1273 in order to proceed.

Begin with small values of $n$ to gain intuition.

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

Let $a=AB$ and $c=AC$.

Consider triangle $ABC$ with an interior point $X$, and let $AX$, $BX$, $CX$ intersect the opposite sides at $A_1$, $B_1$, $C_1$ respectively.

Consider small values of $n$ to understand the behavior of the remainders $r_k$.

Consider small cases with integer points in a bounded region.

Consider a small portion of the infinite grid and attempt to construct a $2\times 2$ black square using only $3\times 3$ and $4\times 4$ flip operations.

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

Let the side lengths be $a,b,c$.

Consider small examples with three colors and a uniform degree graph.

Examine small integers $n>1$ to see when $\frac{2^n+1}{n^2}$ is an integer.

Before proceeding, I need the full textual statement of Kvant problem M1259.

I see that you have provided the framework for solving Kvant problem M1258, but I do not yet have the text of the problem itself.

Before I start the full solution, I need the precise statement of Kvant problem M1257.

The circles are equal and tangent to each other.

I cannot write a rigorous solution to Kvant problem M1255 because the actual problem statement is not present in your message.

Before I write the full solution, I need the precise text of Kvant problem M1254.

A complete solution cannot be written because the problem statement is not available.

I cannot write a solution to Kvant problem M1252 because the actual problem statement is not present.

I notice that the problem statement for Kvant M1251 is not fully given in text form.

The road system is a tournament: for every pair of cities exactly one directed edge joins them.

For the first statement, it is natural to place the original segment on the real line as $[0,L]$.

Consider tiling the plane with squares of side lengths $1, 2, 4, 8, \dots$ under the two constraints: using each size at most ten times or using each size once.

Let the arithmetic progression be

Begin by considering small cases, such as $n = 2$ or $n = 3$, to develop intuition.

Label the 30 senators as vertices of a graph where edges represent friendship.

For the quadratic game, the first player provides three numbers $a$, $b$, $c$, and the second player can assign them arbitrarily to the coefficients of $x^2$, $x$, and the constant term.

Consider a small regular polygon, for instance a regular hexagon or octagon, to visualize the positions of points $K$ on $AB$ and $N$ on $BC$.

For a small instance with piles $1,2,3$, one move cannot remove all stones, because the three pile sizes are different.

A broken line from $A$ to $C$ of length $2n$ along the grid lines must consist of exactly $n$ horizontal and $n$ vertical unit steps.

The problem statement is not actually present in your message.

I can follow your requested six-section format and rigorous style.

I cannot write a solution to Kvant problem M1237 because the actual problem statement is not included in your message, and I do not have access to the graphical version from the published issue.

I see the framework and instructions, but I do not yet have the actual text of Kvant problem M1236.

I cannot write a solution to Kvant problem M1235 because the actual problem statement is not included in your message.

I can start preparing the full Kvant-style solution, but I need the actual problem statement for M1234 to proceed.

We are given a directed system of roads connecting a small number of intersections. Each road has a capacity, meaning there is a maximum amount of traffic that can be assigned to it in total, regardless of direction.

We are given an $n times n$ grid that contains exactly $n-1$ ones, with all other cells being zeros. The grid itself is not fixed in place: we are allowed to reorder rows and columns arbitrarily by swapping any two rows or any two columns.

I can do that.

We are given a line of children represented as a string where each character is either a boy or a girl. The line evolves over time in discrete steps. During each second, every adjacent pair where a boy stands immediately before a girl swaps positions simultaneously.

We are given an array that changes over time through two kinds of operations. One operation overwrites a whole segment with a single value, effectively erasing previous information inside that interval.

I cannot write a solution to Kvant problem M1232 from the information provided, because the actual problem statement is missing.

We are given a connected undirected graph representing the roads of Bertown. Junctions are nodes, roads are weighted edges with positive lengths. The task is to choose a location for a new BerDonalds restaurant so that the maximum distance from it to any junction is minimized.

I notice the problem statement itself is not yet included.

We are given a row of stones, each painted one of three colors. The goal is to remove as few stones as possible so that after removals, no two adjacent stones share the same color.

We are given a sequence of balls arranged in a fixed order. Each ball carries two pieces of information, a color and a numeric value. From this sequence we are allowed to pick a subsequence while preserving the original order, or choose nothing at all.

Please provide the full textual statement of Kvant problem M1230.

We are given a strictly increasing list of positive integers. From this list, we want to choose a subsequence that is also strictly increasing, but with an additional constraint on adjacency: whenever two consecutive chosen numbers appear next to each other in the subsequence…

For the first expression, compute a few values:

We have a street with n positions where trees can be planted, numbered from west to east. Trees grow automatically by one meter per month, and we handle one query per month.

The expression is

We have two sequences of stones, each colored either red, green, or blue. One sequence belongs to Squirrel Liss, the other to Cat Vasya. Each animal starts on the first stone of their respective sequence.

Consider small tournaments first.

We are given a string consisting of the characters l and r. The characters describe how the squirrel moves when stones fall one by one. Stone i falls after stones 1...i-1 have already been placed.

We start with a number a. We want to extend it exactly n times. A single extension means appending one decimal digit to the right of the current number. The digit can be anything from 0 to 9, but after appending it, the resulting number must be divisible by b.

Error in message stream

Let the convex quadrilateral be $ABCD$.

Error in message stream

Let

The input is a long string consisting only of digits and the character -. Somewhere inside this string there may be many substrings that look like dates written exactly as dd-mm-yyyy. Our task is to find the valid date that appears most often as such a substring.

Consider a triangle $ABC$ with incenter $I$ and an arbitrary point $D$ on side $BC$.

We are given an 8×8 board where each cell is either white (W) or black (B). The goal is to determine whether this board can be transformed into a standard chessboard. A standard chessboard has alternating colors both horizontally and vertically.

Consider small cases first.

Let the given integers be considered modulo $m$.

We are given a 3×3 magic square where the three cells on the main diagonal have been erased. The missing cells are (0,0), (1,1), and (2,2), and appear as zeroes in the input. Every other value is known. A magic square has a single common sum S.