brain

A, B, C, D, E are points on a circle O with radius equal to r.

Let a polynomial p(x) with integer coefficients take the value

Let X be an arbitrary nonempty set contained in the plane and

On a line a set of segments is given of total length less than

Let r and m (r \leqm) be natural numbers and Ak = 2k−1

Poldavia is a strange kingdom. Its currency unit is the bourbaki

Set

If 0 < k \leq1 and ai are positive real numbers, i = 1, 2, . . . , n,

(a) Find the equations of regular hyperbolas passing through

Let n \geq2 be an integer. Find the minimum k for which there

Two families of parallel lines are given in the plane, consisting

Show that if 994 integers are chosen from 1, 2, . . . , 1992 and

The circle x2 + y2 = r2 meets the coordinate axes at A =

A convex quadrilateral is inscribed in a circle of radius 1. Prove

A linear binomial l(z) = Az + B with complex coefficients A

Find all real numbers \lambda such that the equation

In a triangle ABC, the incircle touches the sides BC, CA, AB

A square ABCD is divided into (n −1)2 congruent squares,

Let a, b be coprime integers. Show that the equation ax2 +

Does there exist an integer such that its cube is equal to

The points S(i, j) with integer Cartesian coordinates 0 < i \leqn,

In the plane a point O and a sequence of points P1, P2, P3, . . .

Let the circles C1, C2, and C3 be orthogonal to the circle C

Determine the volume of the body obtained by cutting the

Let f(x) be a polynomial with integer coefficients. Prove that

The circle inscribed in the triangle A1A2A3 is tangent to

If T and T1 are two triangles with angles x, y, z and x1, y1, z1,

Construct the circle that is tangent to three given circles.

Let f : N oN be such that

The circles c1 and c2 are tangent at the point A. A straight

Given that n elements a1, a2, . . . , an are organized into n pairs

Given integers a1, . . . , a10, prove that there exists a nonzero

Let E be the set of all bijective mappings from R to R satisfying

Prove that n! cannot be the square of any natural number.

In a triangle P1P2P3 let PiQi be the altitude from Pi for

Let AA1, BB1, CC1 be the altitudes in an acute-angled triangle

There are n \geq3 job openings at a factory, ranked 1 to n in

Prove that for any natural number n, the number

Given any integer m > 1 prove that there exist infinitely

Show that there do not exist more than 27 half-lines (or rays)

Let squares be constructed on the sides BC, CA, AB of a trian-

Let p be a prime odd number. Is it possible to find p−1 natural

Prove the identity

Let f be a function from the real numbers to the real numbers

Denote by [x] the greatest integer not exceeding x. For all

If n1, n2, . . . , nk are natural numbers and n1+n2+\cdot \cdot \cdot+nk = n,

We wish to construct a matrix with 19 rows and 86 columns,

We are given three equal rectangles with the same center in

Find positive integers x1, x2, . . . , x29, at least one of which is

The sum of all the face angles about all of the vertices except

In a group of n people each one knows exactly three others. They

For each P inside the triangle ABC, let A(P), B(P), and

Prove that if x, y, z are real numbers such that x2+y2+z2 = 2,

Let P and Q be distinct points in the plane of a triangle ABC

Let ABCD be a tetrahedron such that AB \perpCD,

(a) Calculate x = (11+6

Let a, 0 < a < 1, be a real number and f a continuous function

The polynomial

The reviewers correctly identified that the previous proof failed at the planar lemma.

The reviewer identified a false claim in the previous proof:

The problem asks for the existence of a regular tetrahedron with one vertex on each of four given distinct parallel planes in $\mathbb{R}^3$.

The system involves five positive real numbers $(x_1, x_2, x_3, x_4, x_5)$ linked cyclically by inequalities of the form $(x_i^2 - x_{i+2}x_{i+4})(x_{i+1}^2 - x_{i+2}x_{i+4}) \le 0$, where indices are…

Consider small values of $m$ and $n$ to examine the expression

A cyclic quadrilateral is given.

Consider a set of ten distinct two-digit numbers, $S = {a_1, a_2, \dots, a_{10}}$, and examine the sums of all its non-empty subsets.

The reviewer identified only one critical flaw, namely the final deduction from

For small values of $m$, explicit examples suggest a graph-theoretic interpretation.

The path $XYZTX$ lies on the four faces adjacent cyclically around the tetrahedron.

The sequence begins

Let $P_1$ be a convex polyhedron with nine vertices $A_1,\dots,A_9$ in $\mathbb{R}^3$, and let $P_i = P_1 + (A_i - A_1)$ for $i=2,\dots,9$.

Define

The fraction

The problem asks us to count the number of bad pairs in an array. We are given a 0-indexed integer array nums, and a pair of indices (i, j) is considered bad if: - i < j - j - i !

The problem asks us to count how many strings of length n made of lowercase English letters can be rearranged to contain the substring "leet". In other words, a string is "good" if, after any permutation of its characters, "leet" appears as a contiguous sequence.

The problem describes a Martian with a row of eyes, each covered by a patch marked with an uppercase letter. The string of letters visible when all eyes are opened represents a sequence s of length n.

The problem requires computing the value of the last element in an array after a series of sequential cumulative sum operations over a fixed number of seconds. You start with an array a of length n where all elements are initialized to 1.

The problem asks us to maximize the total energy gained from a sequence of magicians arranged in a line, where each magician provides a certain energy value, which can be negative or positive.

The problem describes a painting laid out on a number line. Each segment of the painting is represented as a half-closed interval [start, end) and is painted with a unique color value.

This problem asks us to implement a throttling mechanism for a function. We are given a function fn and a delay interval t in milliseconds. We must return a new function, called a throttled function, that controls how often fn is allowed to execute.

This problem asks us to compute, for every possible subarray size, the best possible minimum value among all subarrays of that size. Given an array nums of length n, we must evaluate every window size from 1 to n. For each size k, we consider all contiguous subarrays of length k.

We have a rectangular grid with n rows and m columns. Vasya starts at position (xc, yc). Then he processes k movement vectors one by one. For a vector (dx, dy), he repeatedly moves: He keeps moving in that direction until the next move would leave the grid.

The problem asks us to transform a given string into a "fancy string". A fancy string is defined as a string that does not contain three consecutive identical characters anywhere in the string.

The problem asks us to count the number of special characters in a given string word. A character is defined as special if it appears in both lowercase and uppercase forms within the same string.

This problem asks us to determine the minimum total damage Bob will receive while fighting a group of enemies. Each enemy has two attributes: - damage[i], the amount of damage they inflict on Bob every second while alive. - health[i], the amount of health they start with.

The problem asks us to determine whether two strings s1 and s2 of equal length can be made identical using a specific type of swap operation.

The problem asks us to find two indices i and j in an integer array nums that satisfy two conditions simultaneously: the absolute difference between the indices must be at least indexDifference (abs(i - j) = indexDifference) and the absolute difference between the values at…

That is a very large, detailed reference document with multiple long sections, full walkthroughs, two language implementations, worked examples, test suites, and edge case analysis for LeetCode 2737.

The problem gives us two positive integers, n and k. We are allowed to perform one specific operation on n: choose any bit that is currently 1 in the binary representation of n and change it to 0.

We have two collections of objects. Markers are described by (color, diameter) and caps are also described by (color, diameter). A cap can be attached to a marker only if the diameters are equal.

We are given a triangular region of a square matrix called an n-degree staircase. In this staircase, each cell is accessible except for the area above the secondary diagonal, which runs from the top right to the bottom left.

We are asked to select soldiers from a battalion to participate in a beauty pageant over several days. Each soldier has a unique beauty value. On each day, we must send a group of soldiers whose combined beauty is unique compared to the other days.

In this problem, we are given a list of stock prices over different days. Each element in stockPrices is a pair: This represents a point on a 2D graph where: - The X-axis is the day - The Y-axis is the stock price on that day The line chart is formed by connecting consecutive…

That is a detailed, long-form solution guide request. Before I generate it, I need one missing detail: do you want the solution centered on the optimal DP + graph/component approach (grouping numbers by modulo k and solving independent chains with house-robber style DP), or…

We have n people placed on a line at strictly increasing coordinates. PMP starts at person s, which immediately counts as visited. Every later move must go either strictly left or strictly right, depending on the ticket used for that move.

The problem asks us to count n-digit integers that are "good" with respect to a given integer k. A "good" integer is one whose digits can be rearranged to form a k-palindromic integer.

The problem asks us to determine the minimum number of swap operations required to sort the values of a binary tree level by level in strictly increasing order. Specifically, at each level of the tree, we can only swap values of nodes that exist on that level.

In this problem, we are given a lowercase English string s, and we must split it into one or more contiguous substrings such that every substring is balanced. A substring is considered balanced when every distinct character inside it appears the same number of times.

The problem involves a set of people, each with an appointment on a unique day in the next n days. You do not know who is scheduled on which day, but you can query the organization in forms that list up to m names.

This problem asks us to determine the minimum number of coins required to acquire all fruits in a market where buying a fruit grants a special reward. You are given a 0-indexed array prices, where prices[i] represents the cost of purchasing the (i + 1)th fruit.

The problem asks us to count how many arrays of length n satisfy a divisibility condition while keeping every value within the range [1, maxValue]. An array arr is considered ideal if: 1. Every element is between 1 and maxValue. 2.