brain

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 $DA$.

For points in the plane, the condition says that every triangle determined by the chosen points is acute or right.

Computing the first few values of $m = n + s(n)$ quickly shows that many numbers can be represented in this form.

Before I begin writing the full Kvant-style solution, I need the text of problem M908.

Let the polygon be $P$, let its area be $S$, and let the radius of its inscribed circle be $r$.

Let the set be $A={a_1,a_2,\dots}$, with no divisibility relations between distinct elements.

Let the median be drawn from a vertex $A$ to the midpoint $M$ of the opposite side.

A convex polyhedron with six faces is a cube or a more general hexahedron.

Let

Let

Let the consecutive arcs of the circumcircle be

Let

The problem asks for integers $(x,y)$ such that $(x+y)^7 - x^7 - y^7$ is divisible by $7^7$, while $(x+y)xy$ is not divisible by $7$.

Consider the problem of choosing three points $A$, $B$, $C$ in the plane such that every point $P$ has at least one segment $PA$, $PB$, or $PC$ of irrational length.

The problem defines a binary operation $_$ on a set with three strong constraints: a generalized associativity condition $a_(b_c)=b_(c*a)$, left and right cancellation laws, and asks to prove commutat…

The border of width two around an $n\times n$ board is the set of squares obtained after embedding the board into an $(n+4)\times(n+4)$ board and removing the central $n\times n$ square.

The vectors described in the statement depend only on the face areas and outward unit normals.

Let the amounts of water be $a,b,c$, all positive integers.

Let $v(t)$ be the speed of the snail at time $t$.

Let the vertices of the convex polygon $M$ be $A_1,A_2,\dots,A_n$, and let $B_i$ be the midpoint of side $A_iA_{i+1}$, where indices are taken modulo $n$.

Consider small integer triples $(a,b,c)$ satisfying $a+b+c=0$.

For small values of $n$ the statement is easy to check directly.

Consider the circle inscribed in an angle with vertex $O$ and the two diametrically opposite points $A$ and $B$.

The black cells form a finite set.

The condition forbids the distance $d=0.

Represent each sign by a number in ${0,1}$, where $0$ denotes $+$ and $1$ denotes $-$.

Represent the cells by lattice points $(i,j)$, where $1\le i\le m$ and $1\le j\le n$, the coordinates being the centers of the cells.

For a triangle, suppose a line intersects two sides and cuts the triangle into two parts.

For each parallelogram $A_iB_iC_iD_i$, the diagonals bisect each other.

The basic fact about digit sums is that replacing a number by the sum of its digits does not change its residue modulo $9$.

Consider first the equilateral triangle case.

The problem involves two points $P$ and $Q$ inside triangle $ABC$ such that at vertices $A$ and $B$, the lines connecting the vertex to the points form equal angles with the corresponding angle bisect…

The system is

A triangle already satisfies the condition for each of its three sides, since the third vertex completes an equilateral triangle.

The angles form an arithmetic progression with common difference $4^\circ$:

Consider small initial colony sizes to understand the dynamics.

Let $x_n$ denote the length of the base of the $n$th trapezoid obtained in the process, with $x_0=AB=a$ and with the other base always equal to $b=CD$.

The inequality resembles the triangle inequality.

The table resembles a generalized Pascal triangle, where each entry is the sum of the three entries immediately above it.

Let the five positive numbers be $a$, $b$, $c$, $d$, $e$.

Let $V$, $E$, and $F$ denote the numbers of vertices, edges, and faces of the polyhedron.

Consider small sequences of $+1$ and $-1$ and compute the sum $x_1x_2 + x_2x_3 + \dots + x_{n-1}x_n + x_nx_1$.

Let the trapezoid have bases $AB$ and $CD$, with $AB>CD$, and let $E$ and $F$ be the midpoints of the legs.

The schedule repeats with period $\operatorname{lcm}(2,3,5)=30$.

The equation is

Let $ a connected set of cells.

Let

Let the three circles have common radius $r$, and let their common point be $P$.

Consider small rectangular boxes that can be tiled with $2 \times 2$ and $1 \times 4$ tiles.

Consider a cubic polynomial $x^3+ax^2+bx+c=0$ and suppose its roots form an arithmetic progression.

For a triangle the statement is trivial, since the three sides themselves already form a triangle containing the polygon.

Consider the square lattice $\mathbb{Z}^2$ with distinguished origin $O=(0,0)$.

Let the cars be arranged around the circle in their order along the road.

The statement as written contains a typographical error.

Consider first small examples.

For small values of $n$, the statement is easy to check directly.

Let the right triangle have legs of lengths $a$ and $b$, with hypotenuse $c = \sqrt{a^2 + b^2}$.

Consider the square $A_1 A_2 A_3 A_4$ with an arbitrary point $P$ inside it.

For a $1\times 1$ table the statement is trivial.

Let the two intersecting lines be $l_P$ and $l_Q$, meeting at a point $O$.

Let

Part a) suggests looking at projections.

Let the triangle have sides adjacent to angle $A$ equal to $10$ and $15$.

Consider small-degree polynomials to detect a pattern.

Represent the group by a simple graph.

Let the common value of the three face angles be $\alpha$:

Consider a $2 \times 2$ table filled with arbitrary numbers:

The problem concerns numbers whose squares end with the same digits as the number itself, sometimes called automorphic numbers.

Consider three consecutive integers $n-1$, $n$, $n+1$ and compute the sum of their squares.

For each line $l_i$, let $P_i$ denote the orthogonal projection of the plane onto $l_i$.

Let $R$ be the set of remaining integers, and let $A=R\setminus{1}$.

Let

Fix the player's marked set of $8$ squares.

Consider the examples given: $3^2 + 4^2 = 5^2$, $36^2 + 37^2 + 38^2 + 39^2 + 40^2 = 41^2 + 42^2 + 43^2 + 44^2$, and $55^2 + 56^2 + 57^2 + 58^2 + 59^2 + 60^2 = 61^2 + 62^2 + 63^2 + 64^2 + 65^2$.

Consider the pattern formed by concentric circles of radii $1,2,3,\dots$ and a fixed line $l$ through the center $O$, along with all tangents to the circles parallel to $l$.

The total number of numbers is $1025=2^{10}+1$.

The ring is the solid obtained from a sphere by drilling a cylindrical hole through its center.

For Part 1, denote by $P=AF\cap BG$, $Q=BG\cap CE$, $R=CE\cap AF$.

The numbers under consideration are exactly the positive integers whose decimal expansion consists only of zeros and ones.

Let $A$ and $B$ be the feet of the altitudes from $Q$ and $P$ onto the sides $PM$ and $QM$ respectively.

Consider small odd numbers to test the claim.

The problem asks whether it is possible to tile a square using 18 dominoes of size $1\times 2$ such that no straight line of tile edges connects opposite sides of the square.

Let

Consider small values of $n$ to see the pattern.

Consider the initial configuration of four ones and five zeros written around a circle.

The three given lines are the three internal angle bisectors of a triangle.

Let

Let each rectangle have side lengths $a\ge b$.

Let the five segment lengths be

Consider small cases of regular polygons to understand the combinatorial structure imposed by coloring.

We consider triangle $ABC$ with incenter $O$ and midpoint $M$ of side $BC$.

Let the numbers be $a,b,c>0$ with $abc=1$.

There are $99999-11111+1=88889$ cards.

For small values of $n$ the statement is easy to test.

Let the longest diagonal of a convex polygon have length $D$.