brain

We are given a tree with root at vertex 1. A game is played on this rooted tree until every node disappears. At each move, one of the currently remaining vertices is chosen uniformly at random.

I cannot write a solution to Kvant problem M1383 because the problem statement itself is not present in your message.

We are given a sequence of numbers that is already sorted in non-decreasing order. We need to produce another sequence of the same length such that each consecutive difference lies within a specified range, and each element is between 1 and an upper bound q.

We maintain an array that changes over time. There are two kinds of operations. One operation updates a single position. The other asks for the maximum total sum obtainable by selecting at most k pairwise disjoint subarrays inside a given interval [l, r].

We are given a rectangle centered at the origin with sides parallel to the coordinate axes, having width w along the x-axis and height h along the y-axis. Another rectangle of the same dimensions is rotated around the origin by an angle α (given in degrees).

Before I begin the full solution, I need the text of Kvant problem M1382.

We are asked to count how many times a horse must turn when moving along a spiral from the origin to a target point $(x, y)$.

I cannot write a solution to Kvant problem M1381 without the actual problem statement or the diagram.

We have a row of books, and the time needed to read each book is known. Valera may start from any position and then read consecutive books to the right. He cannot skip books, and he only starts a book if he has enough remaining time to finish it completely.

Consider the expression

I do not have access to the graphical version of Kvant problem M1379, and the textual statement is not included in your message.

I cannot write a rigorous solution to Kvant problem M1378 because the actual problem statement is not present in the conversation.

I can produce a full solution in your requested format, but I need the **text of Kvant problem M1377**.

I cannot write a valid solution to Kvant problem M1376 from the information provided, because the actual problem statement is missing.

I see that the problem statement for Kvant M1375 is not yet fully provided.

I cannot write a rigorous solution to Kvant problem M1374 without the actual problem statement or diagram.

Let the sphere have radius $R$, and let the given plane be $\pi$.

I can prepare a complete solution in that format, but I need the **text version of Kvant problem M1372** to proceed.

Place points $A$ and $B$ on a circle and pick a point $P$ on the smaller arc $AB$.

I do not have the statement of Kvant problem M1370.

I do not have the textual version of Kvant Problem M1369.

I can prepare the full solution in the requested format.

I see that the problem statement for Kvant M1367 has not been provided in text.

I cannot write a solution to Kvant problem M1366 because the actual problem statement is not present in your message.

I cannot write a solution to Kvant problem M1365 because the actual problem statement is not present in your message.

I do not have the textual version of Kvant problem M1364.

I do not have the statement of Kvant problem M1363, and the prompt indicates that only the graphical version is available.

The problem statement for Kvant M1362 is not included in your message, and the web-accessible archive currently only indicates that the textual version is still being prepared rather than providing th…

I can proceed, but I need the **text of problem M1361** from Kvant.

I cannot write a solution to Kvant problem M1360 because the actual problem statement is missing.

A beautiful number is any power of two with either sign. In other words, every summand must be one of $$pm 2^0, pm 2^1, pm 2^2,dots$$ We are given a positive integer $n$, but not in decimal form.

I can proceed with that, but I need the **text of Kvant problem M1359** to write a complete solution.

We are given a sequence of distinct positive integers. The sequence is not arbitrary, it must be generated step by step using a collection of variables. At the beginning every variable contains 0. The first value of the sequence is special.

We are given a one-dimensional array of integers and a list of queries, each specifying a contiguous subsegment of the array. For every query, we need to determine whether the subsegment forms a "ladder.

I can write the complete solution in the format you specify, but I need the text of Kvant problem M1358.

We are asked to ensure that every employee at BerCorp can communicate with every other employee, either directly through a shared language or indirectly via a chain of translators.

I can follow that framework precisely.

I cannot write a rigorous solution to Kvant problem M1354 from the information provided, because the actual problem statement is missing and the published version is graphical only.

I can prepare the complete Kvant-style solution once you provide the full textual problem statement for M1353.

Consider the definition of "close" numbers.

Let the right angle of triangle $ABC$ be at $A$.

Let us understand exactly what is being counted.

Let $a_i$ be the number of counters in sector $i$, with indices taken modulo $n$.

I can provide a full solution in the structure you specified, but I need the **exact problem statement for Kvant M1348**.

Consider a smaller version of the problem to gain insight.

I cannot write a rigorous solution to problem M1346 without the actual problem statement or diagram.

Before I begin the full solution, I need to clarify the precise statement of Kvant problem M1345.

I notice that the actual textual statement of Kvant problem M1344 is not included.

I cannot write a rigorous solution to problem M1343 because the actual problem statement is not present in your message.

I need the full text of Kvant problem M1342 to write a complete, rigorous solution.

Denote by

I can start preparing the complete Kvant M1340 solution in your requested format.

I cannot write a solution to Kvant problem M1339 from the information provided, because the actual problem statement is missing.

I notice that the statement of Kvant problem M1338 has not been provided.

I can do that.

I cannot write a solution to Kvant problem M1336 because the actual problem statement is not present in your message.

Solution to Kvant math problem 1335.

I cannot write a rigorous solution to Kvant M1334 without the actual problem statement.

Before I begin, I need the text of Kvant problem M1333.

A regular tetrahedron has four triangular faces and six edges.

Consider a unit square $ABCD$ with points $K$, $M$, $N$, $L$ on sides $BC$, $CD$, $DA$, $AB$ respectively, such that lines $AK$, $BM$, $CN$, $DL$ create four corner triangles of areas $S_1$, $S_2$, $S…

I do not have the statement of Kvant problem M1330.

Solution to Kvant math problem 1329.

Before proceeding with a full solution, I need the full text of Kvant problem M1328 in order to construct the rigorous, step-by-step proof according to your specifications.

I can do that.

The recurrence can be rewritten in a much more suggestive form:

We are given an array of numbers and a set of interval queries over positions in that array. Each query asks for the sum of elements in a contiguous segment. Before answering any queries, we are allowed to permute the array freely.

I cannot write a rigorous solution to Kvant problem M1325 because the actual problem statement is missing.

We are given a string of lowercase letters and two players who take turns removing a single character. A player wins immediately before their turn if the current letters can be rearranged into a palindrome.

Before I start the solution, I need the full text of Kvant problem M1324.

I need the full text of Kvant problem M1323 in order to produce a complete, rigorous solution.

We are given a tree where almost every node behaves like a point on a thin structure. Every node except node 1 has degree at most 2, which means the tree is essentially a collection of simple chains attached to a single branching root.

I cannot write a solution to Kvant problem M1322 without the actual problem statement or a description of the diagram.

We are asked to find the maximum XOR value of two integers within a given inclusive range $[l, r]$. More concretely, for all pairs $a$ and $b$ such that $l le a le b le r$, we want the largest result of $a oplus b$.

For small boards the bound looks plausible.

We are given a list of restaurants, where each restaurant has two values: a baseline enjoyment score and the time required to eat there. The coach only allows a fixed lunch duration.

The input describes a tree where every vertex holds an integer value. The only allowed operation is global but structurally restricted: you pick a connected region that must contain vertex 1, and then you add either +1 or -1 to every value in that region.

I can follow your structure exactly, but I need the full text of Kvant problem M1320 to write the solution.

Before I can write the complete solution, I need the full text of Kvant problem M1319.

I cannot write a rigorous solution to Kvant problem M1318 without the actual problem statement or the diagram.

Before I begin, I need the exact text of Kvant problem M1317 in order to produce a fully rigorous solution according to the framework you specified.

We are given a collection of distinct positive integers, and we want to select as many of them as possible while avoiding a specific type of forbidden relationship.

I can prepare the full six-section solution, but I need the text of Kvant problem M1316.

We are asked to simulate a laser beam moving through a two-dimensional grid. Each cell in the grid is either empty or blocked. The beam starts from a specific empty cell and moves diagonally in one of four directions: north-east, north-west, south-east, or south-west.

I can do that.

We are given a matrix with n rows and m columns. Each row originally was sorted in non-decreasing order, but now some entries have been erased (marked -1) and the columns may have been shuffled.

I need the full text of Kvant problem M1314 in order to write the solution.

We are given several points on a plane, each representing the center of a circle. These circles begin to grow at the same time, with their radius increasing linearly over time. A hole is any connected white region that is completely enclosed by black circles.

I can do that.

We are given a grid of size n by m where each cell is initially white. Dima can paint any subset of cells black. A painting is considered one of Dima's favorite figures if three conditions are met: at least one cell is painted, all painted cells form a connected set (connected…

I cannot write a rigorous solution to problem M1312 from the information provided, because the problem statement itself is missing.

Consider a triangle with integer side lengths $x$, $y$, $z$ and let the altitudes corresponding to these sides be $h_x$, $h_y$, $h_z$, where $h_x$ is the altitude to side $x$, $h_y$ to $y$, and $h_z$…

Solution to Kvant math problem 1310.

We are asked to construct sequences of intervals, each interval defined by two integers (l, r), such that the first player has a guaranteed winning strategy in a specific two-player game.

I can do that.

Dima and his friends are deciding who will clean the apartment using a counting game. Everyone, including Dima, shows a number of fingers between one and five. They then count around the circle starting from Dima, with the total count equal to the sum of all fingers shown.

Please provide the text version of the Kvant M1308 problem so I can write the rigorous solution in the six-section format you requested.

Solution to Kvant math problem 1307.

I do not have the full text of Kvant Problem M1306 from your input.

I do not have the statement of Kvant problem M1305, and the prompt indicates that the textual version is not available here.