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We are given a sequence of distinct integers in a fixed order. Every element must be assigned to exactly one of two subsequences. Inside each subsequence, the original order must be preserved. The goal is to make both subsequences arithmetic progressions.

We are given a weighted undirected graph whose capital is vertex 1. We must choose a set of roads that connects all cities. Since the chosen graph must be connected and have minimum total weight, the solution will always be a spanning tree whenever a solution exists.

The input is a valid XML-like text built from tags of the form <a and </a, where the tag name is a single lowercase letter. Tags can be nested and multiple XML fragments can appear one after another. The task is not to validate the XML. Validity is already guaranteed.

Consider small cases by simulating the procedure described.

We have n hobbits and want to create as many party days as possible. Each day corresponds to a guest list, which is some non-empty subset of the hobbits. The guest lists must satisfy two conditions.

Consider small examples of six natural numbers and examine the divisibility patterns.

We are asked about sequences of numbers (natural numbers or integers) such that every element in a certain target set (all naturals, all integers, or subsets thereof) can be represented uniquely as a…

Consider the problem for small values of $n$ to understand the geometric constraints.

A convex equiangular $n$-gon has exterior angle $2\pi/n$ at every vertex.

The coefficients are restricted to the set ${-1,0,1}$, and two neighboring coefficients cannot both be nonzero.

Let the decimal expansion of $x_k$ be

Consider an equilateral triangle $ABC$ with side length normalized to $1$ for convenience.

Consider small examples to understand how the set $A$ might grow.

Consider a spherical triangle with one side of length $120^\circ$.

For part 1, suppose every natural number appears exactly once.

I can proceed, but I need the text of problem M598 first.

The sequence $x_n=1+\frac12+\dots+\frac1n$ is the $n$-th harmonic number.

Label the vertices of the regular octagon cyclically by

Let

Consider a triangle $ABC$ with circumcircle $\Gamma$.

Consider first the expression $|\cos x| + |\cos 2x|$.

Let the given vectors be $v_1,\dots,v_n$.

The operation replaces two numbers $x,y$ by

The majority are chemists, and chemists are perfectly reliable.

Suppose such a family of lines exists.

Let the black sectors have angular lengths $\alpha_1,\dots,\alpha_k$, where each

Suppose, contrary to the statement, that none of the given numbers is prime.

Consider a triangle $ABC$ with sides $BC < AC < AB$.

Let $O$ be the center of the circumcircle of the isosceles triangle $ABC$, and let $M$ be the midpoint of $PQ$.

Consider the numbers $1978^m - 1$ and $1000^m - 1$ for small values of $m$.

Compute the first few values of $f$ for small natural numbers greater than $1$.

We are asked to consider sequences of $n$ distinct natural numbers $a_1 < a_2 < \dots < a_n < 2n$ with $n \ge 6$, and to find bounds for the minimum of their least common multiples and the maximum of…

The condition

Consider what it means for a number to be balanced.

Consider triangle $ABC$ with arbitrary points $A_1$ on $BC$, $B_1$ on $CA$, and $C_1$ on $AB$.

Each satellite moves along a circular orbit centered at $O$ with constant angular velocity.

Consider triangle $ABC$ and points $A_1$, $B_1$, $C_1$ on sides $BC$, $CA$, and $AB$, respectively, with cevians $AA_1$, $BB_1$, and $CC_1$ concurrent at $P$.

Let three consecutive terms be $a,ar,ar^2$, where all terms are integers.

The recurrence

The interval is

Consider a right triangle with legs $a$ and $b$ and hypotenuse $c$, where $c^2 = a^2 + b^2$.

Let

Consider a set of distinct natural numbers ${a_1, a_2, \dots, a_n}$ with the property that for any two elements $a_i$ and $a_j$, the sum $a_i + a_j$ is divisible by their difference $a_i - a_j$.

For the planar statement, the condition that no lattice points lie on the boundary except the vertices means that every side joins two lattice points with relatively prime coordinate differences.

We begin by examining the properties of perfect numbers modulo small integers.

Consider a cube of side length $1$ for simplicity.

We begin by examining the two sums for small values of $n$ to detect patterns.

Consider four points $A$, $B$, $C$, $D$ in the plane, and the scalar products $\overrightarrow{MA} \cdot \overrightarrow{MB}$ and $\overrightarrow{MC} \cdot \overrightarrow{MD}$ for a variable point $…

Consider first a smaller version of the problem.

Consider small examples to understand the problem concretely.

Consider a small number of weights, for instance $n=2$ or $n=3$, each with distinct masses $w_1<w_2<w_3$.

Consider the polynomial $x^{10}+a_9x^9+\dots+a_1x+1$ with all coefficients initially unspecified except for the leading and constant terms, which are $1$.

Let the vertices of the simple closed polygonal line be $A_1,A_2,\dots,A_n$ in cyclic order, and let $e_i=A_iA_{i+1}$, with indices taken modulo $n$.

Let the dwarfs act in order $1,2,\dots,7$ around the table.

Consider first the simplest nontrivial configuration of points, namely three points not lying on a line.

I cannot write a rigorous solution to Kvant problem M449 without the actual problem statement or the diagram.

I cannot write a solution to Kvant problem M447 because the actual problem statement is not present in your message.

I cannot write a solution to Kvant problem M445 from the information provided, because the actual problem statement is missing.

Before I begin writing the complete solution, I need the **full textual statement of Kvant problem M443**.

Let the vertices of the convex $2n$-gon be $A_1,A_2,\dots,A_{2n}$ in cyclic order.

For part 1, write

This problem comes from an April Fools contest where the statement intentionally hides the real task. We are given two non-negative integers. The required operation is: 1. Reverse the decimal representation of the second number. 2. Add the result to the first number. 3.

The input contains two numbers. The first number, a, determines the order of a recursively constructed polyline. The second number, b, is an index along that polyline. The picture in the statement is the key.

We are asked to partition all pairwise sums of two sets of ten numbers each into ten groups of ten, each with the same total.

We are given three small positive integers, a1, a2, and a3, each ranging from 1 to 20. The problem asks us to compute a single integer as output based on these three numbers.

We are asked to find a special number associated with a single integer input d, where d represents a divisor or parameter in a number-theoretic sequence.

This is one of Codeforces' classic "special" problems. Unlike ordinary algorithmic tasks, there is no meaningful input to process and no data structure or optimization challenge to solve. The contest provides access to a language called Secret through the custom test environment.

Consider the sum

This is one of the most unusual problems on Codeforces. The input is supposed to contain a single integer between 1 and 5. There are only five official test cases. The output must be a single integer between 1 and 3. The crucial detail is that there is no actual task to solve.

Consider the sum of the digits of perfect squares.

For the planar statement, the number $2$ strongly suggests a relation between the area of a convex figure and the area of a rectangle determined by two orthogonal widths.

The problem asks us to pick an integer $y$ for a given integer $x$ so that the minimum of $x$ and $y$ is as large as possible. We are given multiple test cases, each consisting of a single integer $x$ between -67 and 67.

The city has n real residents. Among them, exactly x are wizards, and all of those wizards will attend a demonstration. Nobody else will attend. The administration measures attendance as a percentage of the real city population, which remains n even if the wizards create clones.

Let

Suppose such an $N$ exists.

The left-hand side contains the three quantities

Let the cells of the infinite graph paper be indexed by integer coordinates $(x,y)$, where each cell corresponds to one pair of integers.

For small values of $n$,

Interpret the drawn segments as the edges of a graph whose vertices are the given points.

Consider the equation $f(x+a)-f(x)=0$ for a function $f$ continuous on $[0,1]$ with $f(0)=f(1)=0$.

Consider a triangle with sides $a$, $b$, $c$, and a point inside it through which three segments pass, each parallel to a side and all of equal length $x$.

Consider the sphere of radius $1$ centered at the origin in $\mathbb{R}^3$, and let the equatorial plane be the $xy$-plane.

Consider small examples to understand the structure of the problem.

The condition concerns preservation of adjacency.

A convex polyhedron has the property that every vertex has even degree.

Consider small values of $N$ to understand the structure of universal sequences.

I cannot write a rigorous solution to Kvant problem M398 from the information provided, because the problem statement itself is missing.

The side length of the equilateral triangle is much larger than the lower bound $1$ imposed on the sides of the desired triangles.

Let

I cannot write a solution to Kvant problem M391 from the information provided, because the actual problem statement is missing.

I cannot write a rigorous solution to Kvant problem M388 because the problem statement itself is not provided in the conversation.

I can proceed with that, but I need the text of Kvant Problem M386 to provide the complete solution.

Consider small examples to understand the problem.

I cannot write a solution to Kvant problem M381 because the actual problem statement is not present in your message.

I can proceed with this framework, but I need the text of Kvant problem M379 in order to produce a rigorous solution.

Let the black piece start at cell $1$ and the white piece at cell $N$.

Consider the triangle $ABC$ and the inequality $|AP| + |BP| + |CP| \ge |AC| + |BC|$ for an arbitrary point $P$ in the plane.

The circle $\gamma$ is centered at the orthocenter $H$ and lies inside the acute triangle $ABC$.

Choose coordinates so that the three cylinder axes are parallel to the coordinate axes.

Consider first a simple case of two numbers summing to $1$.