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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.

The conditions mean that $KA \perp AB$, $KC \perp CD$, $HB \perp AB$, and $HD \perp CD$.

The quantity $T_k(n)$ is the $k$-th elementary symmetric polynomial in the numbers $1,2,\dots,n$:

Let the $n$th triple be $(a_n,b_n,c_n)$, with

Consider a rectangular parallelepiped with edges of length $a$, $b$, and $c$.

The graph described by Fig.

The problem asks for the maximal number of rooks or queens on an $8 \times 8$ chessboard such that each piece is attacked by at most one other piece.

Label the $n$ equal elementary arcs by the colors of the segments

Consider a simple case of a triangle circumscribed around a circle, where the inscribed circle is tangent to its sides at points $A', B', C'$.

The inequality can be written as

Consider first a simple case: a triangle circumscribed around a circle, with the incircle touching the sides at points $A'$, $B'$, and $C'$, forming the inscribed triangle.

Consider small cases of numbers of the form $0.

Consider a regular octagon with side length $a$ placed on a plane.

Represent friendship by a graph $G$ whose vertices are the knights, with an edge joining two friends.

Consider a cube $ABCDA'B'C'D'$ with an inscribed sphere, whose center coincides with the cube's center and whose radius is half the cube's edge length.

A $6 \times 6$ square contains $36$ unit squares.

Consider the task of placing $N$ points in the plane such that the distance between any two points $M_i$ and $M_j$ is a given number $r_{ij}$.

The desired inequality can be rewritten as

Denote the sides opposite $A_1,A_2,A_3$ by