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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 $1$.

The statement is affine in nature.

I can produce a complete, rigorous Kvant-style solution, but I need the text of problem M358 to proceed.

Let

Let the unknown triangle be $ABC$, and suppose that $H$ is the foot of the altitude from $A$ onto $BC$.

Let the given triangle have sides $a,b,c$ opposite angles $A,B,C$.

Consider first small cases to understand the tension between a European team dominating the European Championship yet performing worst in the World Championship.

Consider the operation on a small set of digits.

For two polygons the statement is immediate.

Let the tetrahedron have edge length $1$.

The statement asks for a classification.

Consider a small graph representing countries, where vertices are countries and edges connect neighboring countries.

Consider a single vertex where three hexagonal walls meet.

For $m=5$ the consecutive fractions

The problem involves a convex polyhedron intersected by three parallel planes $p_0$, $p_1$, $p_2$, with $p_1$ equidistant between the outer planes.

Let $m(N)$ denote the minimum possible number of distinct marked points.

Consider smaller analogues of the problem to understand its structure.

The condition

Let the rectangle have coordinates

Consider a hoop of radius $R$ placed over a fixed circle of radius $r < R$.

Consider a convex quadrilateral with vertices $A$, $B$, $C$, $D$ in order, and let $K$, $L$, $N$ be the midpoints of three of its sides.

Let $S$ denote the area of the polygon $A_1A_2\cdots A_n$.

Consider an acute-angled triangle with vertices $A$, $B$, and $C$ and corresponding angles $\alpha$, $\beta$, and $\gamma$, and sides $a = BC$, $b = AC$, $c = AB$.

Let the square have side length $a$.

Consider a trihedral angle, that is, three planes meeting at a common vertex, forming three plane angles $\alpha$, $\beta$, and $\gamma$ at the vertex.

Consider small convex polyhedra such as the tetrahedron, cube, and octahedron.

The problem asks for an orientation of all edges between $n$ points.

Let

The conditions place $H$ on the line $AB$ and $K$ on the line $BC$.

Consider three segments $AB$, $CD$, and $EF$ intersecting at a single point $O$, with $E$ on $AC$ and $F$ on $BD$.

Before I begin the full solution, I need to clarify: should I solve **all five subproblems (1–5) for point $C$**, or just a specific one from the list? Each has its own locus.

Let $H$ be the orthocenter of triangle $ABC$.

Consider triangle $ABC$ with isosceles triangles erected externally on each side.

I cannot write a solution to Kvant problem M355 because the actual problem statement is not present in your message.

Before proceeding, I need the precise textual statement of Kvant problem M353.

Write

Consider a small $n\times m$ chessboard, for instance $4\times 5$.

The numbers in the table are

Let the chosen pair be an unknown $2$ element subset of ${1,\dots,25}$.

Let $a_n$ be the sequence, with

Model the country by a connected graph.

The problem is a binary coding problem.

Consider first a $2 \times 2$ table:

For each of the given lines, orient it upward.

Represent the marked cell centers by a $0$-$1$ matrix.

Consider small polynomials such as $P(x) = x$, $P(x) = x+1$, or $P(x) = x^2$.

Consider small values of $k$ first.

Let the rotation about the circumcenter $O$ have angle $\varphi$, where $0<\varphi<180^\circ$.

Let the vertices of the convex $n$-gon be $P_1,P_2,\dots,P_n$ in cyclic order.

Consider small groups to understand how the structure of "liking" works.

Let the circle have radius $R$ and let a chord $AB$ be at a distance $h$ from the center $O$.

Consider a single pile with a small number of stones.

Consider a function $f:\mathbb{R}\to\mathbb{R}$.

The table may be taken to be the unit square $[0,1]\times[0,1]$.

I cannot write a solution to Kvant problem M319 because the actual problem statement is not present in your message.

Let

Consider the sum of squares of $k$ consecutive natural numbers beginning at $n$, expressed as

Consider the difference between a number and the product of its digits, denoted $N - P(N)$, where $N$ is a 9-digit number with digits $d_1, d_2, \dots, d_9$ in ${1,2,\dots,9}$ and $P(N) = d_1 d_2 \dot…

Consider an angle with vertex $O$ and denote its sides by rays $OA$ and $OB$.

Consider the growth process for small numbers.

For the first question, divisibility by $x^2+x+1$ suggests evaluating the polynomial at the nonreal cube roots of unity.

Consider first the case $n=2$, where the inequality takes the form $a_1\cos x + a_2\cos 2x \ge -1$ for all real $x$.

Let the removed corner be the unit square with vertices $(0,0)$, $(1,0)$, $(1,1)$, $(0,1)$.

The axioms resemble the algebraic properties of the bitwise exclusive-or operation.

Consider placing a small number of identical weights on the vertices of a $1 \times 1$ grid.

For $n=1$ the statement is immediate.

Consider a ruled sheet of paper with parallel lines spaced a fixed distance apart, and suppose a regular $n$-gon is drawn so that all vertices lie on these lines.

The problem involves four squares arranged on a plane with shared vertices, forming a chain: the second vertex of the first square coincides with a vertex of the second square, and so on, closing back…

Let the rows be numbered from top to bottom by $1,\dots,n$, and let $a_{ij}$ be the entry in row $i$, column $j$.

The inequality is homogeneous and symmetric in a suggestive way.

Consider a small example with numbers $1, 2, 3$.

Let the closed non-self-intersecting broken line have vertices

Model the congress by a simple graph.

Consider a sequence of natural numbers $a_1 < a_2 < a_3 < \dots$ such that every natural number $n$ can be represented uniquely as $a_j - a_i$ with $j > i$.

Consider small examples to understand the claim.

Consider small examples of convex polygons, starting with triangles and quadrilaterals, and examine what happens when each side is shifted outward by a fixed distance.

Consider a small table, for instance $2 \times 2$, with entries

Consider a triangle $ABC$ of area $1$ with midpoints $A_1$, $B_1$, and $C_1$ of the sides $BC$, $AC$, and $AB$ respectively.

Let the numbers on the cards be $a_1,\dots,a_n$, where each $a_i\in{\pm1}$.

Let $E$ be the number of segments whose endpoints have different colors.

Let the vectors be represented by points on the unit circle.

We seek the smallest positive value attained by the given differences.

Consider two circles of radii $R$ and $r$ that are externally tangent.