brain

Let $A=\widehat A$, $B=\widehat B$, $C=\widehat C$.

Consider the equation

A plane section of a convex polyhedron changes combinatorially only when the plane passes through a vertex.

Consider triangle $ABC$ with bisectors $AK$ and $BM$ intersecting at $O$.

Consider the given odd natural numbers $a<b<c<d$ satisfying $ad=bc$, $a+d=2^k$, and $b+c=2^m$.

The condition that the circle with diameter $AB$ is tangent to the line $CD$ has a simple metric interpretation.

The complete graph on $n$ vertices is $K_n$.

Write

The problem concerns connecting 51 cities in a square-shaped country of side 1000 km with 11,000 km of highways.

Consider a circle $\Gamma_1$ with tangents $CA$ and $CB$ meeting at $C$, so $A$ and $B$ are points of tangency.

Label the cells on the boundary of the $n\times n$ square cyclically by

Working

The sequence begins as $1, 0, 1, 0, 1, 0$ and each subsequent term is defined as the last digit of the sum of the preceding six terms.

Consider a pyramid with apex $A$ and base $B_1B_2\dots B_n$.

Consider a smaller version of the problem first.

We begin by testing small integer values to see whether the equation $(5+3\sqrt{2})^m = (3+5\sqrt{2})^n$ admits any obvious solutions.

Let $O_1,O_2,O_3$ be the centers of the circles $C_1,C_2,C_3$.

Let the occupied rooms be represented by the multiset of integer positions of all pianists.

Let the boys' heights be $b_1,\dots,b_{17}$ and the girls' heights be $g_1,\dots,g_{17}$.

Consider first a right triangle.

Consider a small board, $n=3$.

Consider small values of $n$ to understand the behavior of the sums modulo $1$.

Consider the triangle $ABC$ with circumcircle $(O)$ and incircle $(Z)$.

Let

Let

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Before I begin, could you provide the full text of Kvant problem M854? I need the problem statement in order to construct the rigorous solution according to your specifications.

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The function $f_0(x) = ||x-1|-2||x|-3||$ involves nested absolute values.

Let the regular polygon have $n$ sides and circumradius $R$.

We consider the problem of forming a centrally symmetric polygon using two types of tiles: a “corner” formed by four $1\times1$ squares arranged in an L-shape, and $4\times1$ rectangles.

For the first representation, the coefficients are constrained by $0\le a_k\le k$.

Consider the first identity $\sin\alpha + \sin\beta + \sin\gamma$ under the constraint $\alpha + \beta + \gamma = 0$.

Consider a right triangle $ABC$ with right angle at $C$.

For a set of integers with no three-term arithmetic progression, the classical example is obtained by writing numbers in base $3$ and allowing only digits $0$ and $1$.

Consider the simpler case where two of the numbers are coprime.

Represent the previous encounters by a bipartite graph.

Compute the first few terms to detect patterns.

Consider first the case of dividing a square into smaller squares.

Consider small values of $m$ to detect a pattern.

The figure consists of a triangle subdivided into smaller regions, four of which are blue triangles of equal area.

The operation described on the board replaces one number with the sum of the other two numbers minus one.

I need the diagram from Figure 3 to solve the problem rigorously.

Before I begin the full solution, I need clarification: does the photograph provide any perspective cues such as vanishing points, the relative positions of the corners, or angles of the visible wall?

A parallelogram has two pairs of parallel sides.

Consider small $n$-gons to understand what a uniform set of blue vertices looks like.

Let $s(n)$ denote the sum of the decimal digits of $n$.

A natural number is representable as a sum of two squares if and only if every prime congruent to $3 \pmod 4$ appears with even exponent in its prime factorization.

Consider three segments $OA$, $OB$, and $OC$ of equal length $R$, with $B$ inside the angle $AOC$.

The quantities involving medians and altitudes suggest rewriting everything in terms of the sides.

Compute the first few terms of the series to detect a pattern.

Begin by examining small values of $k$.

Consider the planar case first.

We are given a small system of length-n arrays that evolves through a fixed number of global transformations. We start with an array a, and we repeatedly apply exactly u operations.

We are asked to reconstruct four binary strings consisting only of letters a and b, given the Hamming distances between every pair of strings. The Hamming distance between two strings counts the positions where the strings differ.

We are given a single number $f$, and we consider the infinite sequence generated by Fibonacci rules, but taken modulo 1013. The sequence starts with 0 and 1, and every next value is the sum of the previous two, reduced modulo 1013.

Place a right circular cylinder vertically with axis along the $z$-axis and center at the origin, so that $O=(0,0,0)$ is the midpoint of the axis.

We are given a permutation of integers from 1 to n. A permutation means each integer appears exactly once. The task is to count pairs of contiguous subarrays (segments) such that when we combine the two segments, the union of their elements forms a consecutive sequence of…

We are given a rectangular grid of size n × m representing a sheet of squared paper. Some squares are painted, forming a set A.

We are asked to compute the minimum and maximum bus fare collected given a number of grown-ups n and children m on a bus. Each grown-up pays one ruble for themselves, and each grown-up can take at most one child for free. Any additional children cost one ruble each.

We are given two circular “threat zones” on a plane. Each one is defined by a center point and a radius. The center is a city, and the radius describes how far the enemy ring extends from that city. We are allowed to place a single radar anywhere in the plane.

The data involve two right triangles erected externally on sides $AB$ and $BC$.

We are given a set of cities, but the twist is that the input does not describe the real road system directly. Instead, it describes the complement of it.

We are given a sequence of integers and a threshold value $k$. The task is to count how many contiguous subarrays contain some value that appears at least $k$ times inside that subarray.

Compute several small cases to see the pattern.

We are given a sequence of tokens that were spoken in order, each token being either int or pair. These tokens are supposed to form a valid nested type expression of a very specific grammar.

Consider first small values of $k$.

The task is to take a single positive integer and express it in base 2, meaning we rewrite it using only powers of two with coefficients 0 or 1. Instead of the usual decimal representation, we want the binary string that tells us which powers of two sum up to the number.

Consider a square $ABCD$ with a point $P$ inside it.

We are asked to analyze a string representing a program written in HQ9+, a toy language with four instructions: H, Q, 9, and +. Only the first three produce output when executed: H prints "Hello, World!

The problem asks us to generate a simple text pattern. We are given a single integer n, which represents the number of lines we need to print. The first line should contain a single asterisk, the second line two asterisks, and so on, until the n-th line contains n asterisks.

The task is to compute the sum of a number a and the reversed digits of another number b. Reversing a number means writing its digits in opposite order while ignoring any leading zeros that appear after reversal. For example, reversing 230 produces 32, and reversing 0 remains 0.

We are given a triangular structure of “weighing plates” arranged in rows. The top row has $n$ plates, the next has $n-1$, and so on until the last row which has a single plate. Each plate has a threshold weight.

We are asked to model the growth of a triangular plant over a number of years. Each plant triangle has an orientation: "upwards" or "downwards". The growth rules are deterministic: every year, each triangle produces four new triangles.

We are given a set of dwarves located at integer coordinates on a Cartesian plane, and optionally a set of subway stations also at integer coordinates.

We are asked to calculate the number of dwarves who can see the Great Mushroom King for multiple visits. Each visit is specified by three integers $k$, $l$, and $r$, which define a sequence of numbers $k cdot 2^l + 1, k cdot 2^l + 2, dots, k cdot 2^r + 1$.

Consider small values of $n$ and their divisor sums $\sigma(n)$.

We are asked to maximize a function defined on three non-negative real variables x, y, and z subject to a sum constraint. The function has the form f(x, y, z) = x^a · y^b · z^c, where a, b, c are non-negative integers.

We are asked to determine the type of divisibility rule that applies to a given number d in a given base b. The types are categorized as 2-type, 3-type, 11-type, 6-type, or 7-type depending on whether the divisibility can be checked using only the last few digits (2-type), the…

Let

The disk is a linear array of fixed-size cells, and each cell either stores a fragment of some file or is empty. Every file is already present on the disk, but its fragments may be scattered across arbitrary positions, while still appearing in the correct internal order for…

Each student initially owns a single notebook. There is a fixed hidden permutation p over students, meaning every student has exactly one “best friend” and every student is the best friend of exactly one other student.

We have a row of cubes, each painted in one of m colors, and we are allowed to remove up to k cubes to maximize the length of a consecutive segment of cubes all having the same color.

We are given two strings, s and t. The string s represents a set of letters we are allowed to rearrange freely, and the string t represents a benchmark name.

We are given a string consisting of uppercase and lowercase letters. Patrick wants to transform it into a "fancy" string, defined as having all uppercase letters on the left and all lowercase letters on the right. We can change the case of any letter at the cost of one action.

Petya's calculator allows addition, subtraction, increment by one, and reciprocal operations.

We are given a multiset of strings, each string representing a protein. From this collection, we must choose exactly $k$ strings and assign a score to the chosen subset. The score is defined over all unordered pairs of chosen strings.

We are given a collection of n protein strings, and we are asked to select exactly k of them such that the sum of the longest common prefixes (LCPs) between every pair in the selected subset is maximized. Each protein is a non-empty string of lowercase letters.

We are given a multiset of strings, each string representing a protein sequence. From these strings, we must choose exactly k of them.

We are given a square, monochrome image represented as an $n times n$ matrix, where each element is either 0 (white background) or 1 (black foreground). Within this image are several distinct geometric shapes, each either a square or a circle.

Consider first a small number of points on a circle, for example, four points dividing the circle into four equal arcs.

We are given a large binary grid representing a noisy black-and-white image. The white background contains several black connected regions, and each such region is guaranteed to be either a square or a circle, possibly rotated in any orientation.

We are given a large binary image represented as an $n times n$ grid, where each cell is either white or black. The black cells form several disconnected shapes, and each shape is guaranteed to be either a circle or a square, possibly rotated arbitrarily.

The task is to take a set of $n^2$ integers and arrange them into an $n times n$ matrix such that the sum of every row, every column, and both diagonals is equal. This sum is called the magic constant $s$.

We are asked to construct an $n times n$ magic square using a given multiset of $n^2$ integers. A magic square has the property that the sums of all rows, all columns, and both diagonals are equal.

Consider small values of $n$ and $k$ to identify a pattern.

We are asked to arrange a given set of $n^2$ integers into an $n times n$ matrix so that it forms a magic square. A magic square is defined as a matrix where the sum of every row, every column, and the two main diagonals is the same value $s$.

We are maintaining a hash table with a fixed number of slots, where each slot can hold at most one object. Each object has a unique identifier and a preferred starting position given by its hash value. When inserting an object, we first try its hash position.