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Find the largest integer not exceeding $1992

Let L denote the set of all lattice points of the plane (points

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

Let a, b, c be integers different from zero. It is known that the

Find all triples (x, y, z) of integers such that

Let r1, . . . , rn be the radii of n spheres. Call S1, S2, . . . , Sn the

A circle of radius 1 rolls around a circle of radius

The n points P1, P2, . . . , Pn are placed inside or on the bound-

Let E = {1, 2, . . ., 16} and let M be the collection of all

Let a regular 7-gon A0A1A2A3A4A5A6 be inscribed in a circle.

Find all square numbers S1 and S2 such that S1 −S2 = 1989.

In a plane a circle with radius R and center w and a line 
ambda

For a finite set E of cardinality n \geq3, let f(n) denote the

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

Is it possible to cover a rectangle of dimensions m imes n with

Three disks of diameter d are touching a sphere at their centers.

Prove that the equation 4x+6x = 9x has no rational solutions.

All the irreducible positive rational numbers such that the prod-

Find the triples of positive integers x, y, z satisfying

Denote by an the greatest number that is not divisible by 3

Let S be a set of positive integers n1, n2, . . . , n6 and let n(f)

(a) Let ABC be a triangle with AB = 12 and AC = 16. Suppose M is the

Let S =

The sequence (an) of real numbers is defined as follows:

Given positive integers k, m, n with km \leqn and nonnegative

Let f(x) = xm + a1xm−1 + \cdot \cdot \cdot + am−1x + am and g(x) =

A smooth solid consists of a right circular cylinder of height

Let the polynomials

Let f : R oR be a continuous function. Suppose that the

Find the relations among the angles of the triangle ABC whose

Given f(x) \leqx for all real x and

If a, b, c, d are real numbers such that a2 + b2 + c2 + d2 \leq1,

From point P on arc BC of the circumcircle about triangle

For a point O inside a triangle ABC, denote by A1, B1, C1

A family of sets A1, A2, . . . , An has the following properties:

A cylindrical container has height 6 cm and radius 4 cm. It

Prove that a < b implies that a3 −3a \leqb3 −3b + 4. When

Given a graph with n vertices and a positive integer m that is

A fixed point A inside a circle is given. Consider all chords

Let C1, C2 be circles of radius 1/2 tangent to each other and

All edges and all diagonals of regular hexagon A1A2A3A4A5A6

Find all integers x, y, z that satisfy

Prove that there exist 78 lines in the plane such that they have

Prove that the system of equations

Given n real numbers a1 \leqa2 \leq\cdot \cdot \cdot \leqan, define

Given a triangle ABC, let R be the radius of its circumcir-

Prove that the volume V and the lateral area S of a right circular

A circle of radius ho is tangent to the sides AB and AC of the

Find the maximum value of

(a) Prove that (a1 +a2 +\cdot \cdot \cdot+ak)2 \leqk(a2

Let a and b be coprime integers, greater than or equal to 1.

Let b \geq2 be a positive integer.

Prove that the equation

Let an =

On the sides AB and AC of triangle ABC two points K and

Determine all functions f : R oR satisfying the following two

In the triangle ABC, let B1 be on AC, E on AB, G on BC,

Consider the set S of all the different odd positive integers

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 split the array nums into one or more contiguous non-empty subarrays such that the total cost is minimized. For every subarray, we define a special quantity called its importance value.

This problem describes a simulation of robots moving on a one-dimensional line, each with a given starting position, health, and direction. Every robot moves at the same speed, and collisions occur when two robots meet at the same position.

The problem asks us to identify the longest square streak in an array of integers nums. A square streak is a subsequence of at least length 2 where, after sorting, every element is the square of the previous element. In other words, if the sorted subsequence is [x1, x2, ...

We are given a fixed string consisting of lowercase letters, and many independent queries. Each query specifies a contiguous segment of this string, and for that segment we must count how many substrings are palindromes.

That is a long, structured technical guide. To make sure I target the correct problem and avoid wasting your time, can you confirm the exact LeetCode problem title for 2762?

This problem asks us to find the last day we can cross a grid from the top row to the bottom row, walking only on land. The grid is initially all land (0), and each day, specific cells are flooded with water (1) according to the cells array.

The problem asks us to find the largest integer num such that the accumulated price of all numbers from 1 to num does not exceed a given threshold k.

This problem gives us an undirected graph with n nodes and a list of edges. A connected trio is a group of exactly three distinct nodes where every pair of nodes has an edge between them. In graph theory terms, this is simply a triangle.

Vasya scribbled an Internet address in his notebook, but he was in a hurry and omitted all punctuation characters like :, /, and ..

We are working with a rooted infinite tree where every node always has exactly $k$ outgoing edges to children. Each of those $k$ edges has a fixed weight: the first is 1, the second is 2, and so on up to $k$.

The problem asks us to find the minimum possible sum of a "mountain triplet" within a given array of integers. A mountain triplet is defined as three indices (i, j, k) such that i < j < k, nums[i] < nums[j], and nums[k] < nums[j].

The problem provides two binary strings, s and target, of equal length n. You are allowed to perform a specific bitwise operation on s any number of times, which involves picking two distinct indices i and j and updating s[i] to s[i] OR s[j] and s[j] to s[i] XOR s[j].

Here’s a complete, detailed solution guide for LeetCode 2443 - Sum of Number and Its Reverse, fully following your formatting rules.

The problem presents a scenario in which Alice has an original array arr of length n consisting of positive integers. She chooses a positive integer k and generates two new arrays: lower and higher.

The problem is asking us to compute a special sum over a given array nums by repeatedly concatenating the first and last elements of the array until it is empty.

We are asked to model a journey along a straight line of cities connected by one-way roads, where a car travels at a constant speed of one kilometer per hour and consumes one liter of fuel per kilometer.

The problem asks us to design a data structure that supports two operations efficiently: 1. Assign a number to a given index. 2. Find the smallest index currently assigned to a given number.

We have a mushroom-growing contest with two phases separated by a break. Each participant has two speeds, and the problem is that we do not know the order they will use them. During the first phase of length t1, mushrooms grow at one speed.

The problem asks us to compute the number of same-end substrings within specified subranges of a given string s. A substring is same-end if its first and last character are identical. We are given multiple queries in the form [li, ri], each representing a substring s[li..ri].

The problem requires us to find the number of arithmetic triplets in a strictly increasing array of integers. An arithmetic triplet (i, j, k) satisfies the conditions i < j < k, nums[j] - nums[i] == diff, and nums[k] - nums[j] == diff.

The infinite string in this problem is built recursively. Start with: In general: The alphabet contains 36 symbols: The full string after 30 steps has length: $ For k = 30, the length is about 10^9, which matches the input bounds.

The problem is asking us to analyze call records from a Calls table and determine the peak calling hour for each city. Each row in the table contains a callerid, recipientid, a timestamp (calltime), and the city where the call originated.

We are given a valid parenthesis sequence. Every opening bracket has a unique matching closing bracket, and the pairs are properly nested. We want to assign colors to brackets under two rules.

The problem asks for the k-th smallest product that can be formed by multiplying one element from the sorted array nums1 with one element from the sorted array nums2. Both arrays may contain negative numbers, zeros, and positive numbers.

We are asked to determine the earliest square where the opposition can hold a demonstration given the interference of the city administration. There are n squares arranged by increasing distance from the city center, with square 1 being the most central.

The problem asks us to identify all valid chemical bonds that can form between elements in a given table. The Elements table contains three columns: symbol, type, and electrons. The type column is an enumeration of 'Metal', 'Nonmetal', and 'Noble'.

The problem gives us a linked list and asks us to place its values into an m x n matrix in clockwise spiral order. We begin filling from the top-left corner of the matrix, which is position (0, 0). The traversal direction follows the standard spiral pattern: 1.