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Consider a coat of area $1$ and five patches, each of area at least $\frac{1}{2}$.

Let the trapezoid have bases of lengths $b$ and $a$, with $a<b$.

Begin by examining the three-variable inequality

A strategy can be represented by a decision tree.

Consider small values of $n$ first.

Let the circle have center $O$ and radius $r$.

Let the angles of $T$ be $A,B,C$.

Let the bisectors of $\angle A$ and $\angle B$ meet at a point $P$.

Testing small values of $n$ shows that the divisibility condition $n^2+1 \mid n!$ is rarely satisfied for small integers, as $n^2+1$ grows faster than $n$.

Consider small examples of natural numbers and attempt to write them as sums of numbers whose reciprocals add to one.

Consider first the intersection of two cylinders of equal radius $r$ with axes perpendicular.

Let the optimal finishing time be $T$.

Consider the equation

Consider the case of a triangle first.

Let the roots of

The equation is

For problem c), the set of all solutions of

Consider three points on the plane.

For four points on a circle, label them by position vectors $a,b,c,d$ on a circle with center $O$, taken as the origin.

Let the rectangle be centered at the origin, with sides parallel to the coordinate axes.

Let the magic sum be $M$.

Consider first the case $n=4$.

Let us denote the initial right triangle as $A_0A_1A_2$, with right angle at $A_2$, and legs $|A_0A_2|=a$ and $|A_1A_2|=b$.

Consider a small social network where each person has exactly three friends.

Let the members of each country form a set of integers contained in ${1,2,\dots,1978}$.

Let

Let the given sphere have center $O$ and radius $R$.

Part 1 is a special case of Part 2.

Begin by examining small values to understand the recursive structure imposed by $g(n)=f(f(n))+1$.

For the first part, the condition means that every domino of the upper layer must cross the boundary between two dominoes of the lower layer.

The defining condition of a trigram is

Consider triangle $ABC$ with a point $P$ inside it, through which three lines are drawn, each parallel to one side of the triangle.

The statement concerns a regular frustum of a pyramid.

Let each cell be represented by a variable in $\mathbb F_2$, where $1$ means black and $0$ means white.

Consider two points $A$ and $B$ on a line and a motorist starting from $A$ and a cyclist starting from $B$, both moving toward each other at constant speeds $v_m$ and $v_c$.

For small values,

Consider an arithmetic progression $a$, $a+d$, $a+2d$, $\dots$, where $a$ and $d$ are natural numbers.

Denote

Consider the $8 \times 8$ chessboard with one chip on each square.

A homothety with ratio $k<0$ reverses directions through its center.

Represent the circle by the additive group $\mathbb R/\mathbb Z$, so that arc lengths are measured as fractions of the circumference.

Consider small $n \times n$ boards and simulate the game.

Label the convex quadrilateral $ABCD$ with consecutive sides $AB = a$, $BC = b$, $CD = c$, and $DA = d$.

The area of the orthogonal projection of a polyhedron onto a plane depends on the direction of projection.

For one segment the answer is trivial.

Let the convex quadrilateral be $ABCD$, and let its diagonals $AC$ and $BD$ intersect at $P$.

Consider the first few values of $a_n$.

Consider the sequence $x_n=(1+\sqrt{2}+\sqrt{3})^n$.

Let $W$ denote a winning position for the player to move and $L$ a losing position.

Consider the inequality

We seek an infinite bounded sequence $(x_n)$ such that every two distinct terms satisfy

Let the lake be the interior of a simple nonconvex polygon $P=A_1A_2\cdots A_n$.

The three machines modify cards in distinct ways.

Consider a square inscribed in the graph of $y = A \sin x$.

Starting with two points $A$ and $B$ at distance 1, reflecting one about the other generates points along the line $AB$.

Let

Consider a small round-robin tournament with $n$ teams.

Consider placing the digits $0,1,2$ in a small grid and examining rectangles of size $3 \times 4$.

Consider the triangular table for small values of $a$ to understand the pattern.

A coordinate model is natural because the configuration contains a rectangle and two midpoints.

Let the entry in row $i$, column $j$ be $a_{ij}$.

Equation (1), $2^x + 1 = 3^y$, suggests searching for powers of 2 that are one less than a power of 3.

For the one-dimensional problem, the state of the process is not the point $O_k$ itself but the set of material points lying in the interval of length $2r$ centered at $O_k$.

Let $F$ be the set of free squares.

The three segments $AA_1$, $BB_1$, $CC_1$ are parallel and not coplanar.

A power $3^k$ begins with the digit $9$ precisely when there exists an integer $m$ such that

Consider first a single square of area $1$.

Let the hexagon be $A B C D E F$ in cyclic order.

I cannot write a solution to Kvant problem M153 without the actual problem statement or the diagram.

I can produce the full Kvant-style solution structure for problem M154.

The statement concerns divisibility of numbers of the form $a^k+b^k$.

Consider a square of side length $1$ and a line dividing it into two quadrilaterals with areas in the ratio $2:3$.

Let the four circles have the same radius $r$.

Let

Let $P$ be the intersection of the tangents at $A$ and $C$.

Label the vertices of a regular $n$-gon by the residues modulo $n$.

Consider the first condition: the perimeters of the four triangles formed by three consecutive vertices of a quadrilateral are equal.

Consider small cases first.

Consider the first several terms of the sequence defined by taking the integer closest to the cumulative target $n\sqrt{2}$.

Let us compute the first terms.

Consider a cube with its twelve edges labeled by distinct numbers $1$ through $12$.

Consider small rectangles with integer sides.

Consider small positive integers $n$ and examine the condition that if $n$ is divisible by $p-1$ for some prime $p$, then $n$ must also be divisible by $p$.

The question asks whether there is a set $F$ which by itself cannot contain any semicircle of radius $1$, while two congruent copies of $F$ can together contain the whole unit circle.

Let us interpret the operations in reverse.

Let the altitude $BH$ be the $y$ axis, and let $H=(0,0)$.

Let the parallelogram have side vectors $\mathbf u=\overrightarrow{BA}$ and $\mathbf v=\overrightarrow{BC}$.

For $m=1$ and $n=2$,

The anaconda is an arbitrary polygonal line of total length $10$ contained in the unit square.

The stones have weights

Consider a quadrilateral with consecutive sides $a$, $b$, $c$, $d$ and area $S$.

Model the cellular shell as a polyhedral decomposition of a sphere.

The problem asks whether a sequence of allowed replacements can form a nontrivial cycle.

Consider small values of $n$ to understand the pattern of the product.

The equation is

Consider small values of $n$ and attempt to construct sequences of $+1$ and $-1$ satisfying the condition that for each $k=1,2,\ldots,n-1$, the sum of the $n$ pairwise products of numbers separated by…

Let the variable triangle have vertices $P\in AB$, $Q\in BC$, $R\in AC$.

For the concrete problem with capacities $5$, $7$, and $12$, the target state is two portions of $6$ liters each.

For a polynomial to be monotonically increasing on the whole real line, it is enough that its derivative be nonnegative everywhere.

Consider a cyclic quadrilateral $ABCD$ and extend opposite sides $AB$ and $CD$, $BC$ and $DA$.