brain

We are given a binary number as a string, representing an integer in base 2, and we are allowed to delete exactly one digit. The goal is to choose which digit to remove so that the resulting binary number, when interpreted as a decimal integer, is as large as possible.

We are given an array a. For every position i, we must choose an integer bi such that 1 ≤ bi ≤ ai. A sequence b is called good when its least common multiple is equal to its maximum element.

Consider a triangle $ABC$ with sides $AB$ and $AC$ given.

The recurrence is

Begin by testing the inequality for small values of $n$ to develop intuition.

We are given a rooted tree with n nodes, where node 1 is the root. Each node contains a list of numbers that starts empty. We then perform m operations.

We are given a permutation of integers from 1 to n and a sequence of m swap operations. Normally, these operations would sort the array into ascending order, but the swap program is broken: each move either swaps the indicated positions or does nothing, each with probability 1/2.

Codeforces 258B: Little Elephant and Elections

Codeforces 257B: Playing Cubes

Compute the sum for small values of $n$ to gain intuition.

Consider an acute-angled triangle $ABC$ with angle bisector $AD$ from vertex $A$.

We are given an array of integers where each element is at least as large as the previous one and at most double the previous one. We are asked to assign either a plus or minus sign to each element to form a sum that lies between zero and the first element of the array.

To understand $k(n)$, consider $n=15$, where the problem gives an example: three triples $(1,6,8)$, $(2,4,9)$, $(3,5,7)$, all sums equal to $15$, and all nine numbers distinct.

Consider small tables first.

Consider a convex hexagon that can be subdivided into $N$ parallelograms of equal area.

We have a single elevator in a building with floors 1..m. Each person appears at a known time t, starts on floor s, and wants to go to floor f.

Consider first partitioning the integers into arithmetic progressions with positive integer differences.

A plane intersects a tetrahedron along a triangle and a sphere along a circle.

The position of the game is completely determined by the current number of matches in the pile and the set of numbers already written on the sheet.

We are asked to determine the smallest possible angle, with its vertex at the origin, that can enclose all given mannequins on a plane. Each mannequin has coordinates $(xi, yi)$, and no mannequin is located at the origin itself.

I do not yet have the full textual version of Kvant problem M1209.

We have k wall sockets available in the apartment. There are m electrical devices that eventually need power. We also own n power strips, where the i-th strip provides a[i] sockets. A power strip is not free to use.

I need the textual statement of Kvant problem M1208 in order to produce a rigorous solution according to your strict rules.

I see the problem statement itself is not fully provided yet.

I see that the problem statement for Kvant M1206 is not yet provided.

We are given an interval of integers from a to b. We want to choose a length l such that every contiguous segment of length l inside this interval contains at least k prime numbers. Among all lengths that satisfy this condition, we need the smallest one.

I cannot write a solution to Kvant problem M1205 because the actual problem statement is not present in the conversation.

Consider three circles centered at points $A$, $B$, and $C$, each expanding at the same constant rate.

Let the side of the large square be $1000$ m.

We are given a triangular table of numbers. Each row has fewer or equal cells than the row above it, forming a structure like a Young tableau. Every cell contains a distinct integer between 1 and the total number of cells.

Let the two rays from $A$ be $r_1$ and $r_2$.

We are given a target string t and a set of n source strings s1, s2, ..., sn. The goal is to construct t by repeatedly taking single characters from the source strings.

Let the numbers of voters for parties $A,B,C$ in a district be $a,b,c$, expressed as fractions of the district electorate.

We are given a tree with $n$ nodes, and our goal is to decompose it into another tree, called a T-decomposition. Each node in this decomposition is a non-empty subset of the original nodes, and we must satisfy three conditions.

We are asked to determine how many cash registers Valera needs in his fast-food cafe so that every visitor can be served immediately, assuming each visitor arrives at a specific time during a single day and each service takes less than a minute.

Consider a small number of arcs on a circle.

We are asked to pick three positive integers not greater than n such that their least common multiple is maximized. The integers do not have to be distinct, so we could repeat numbers if it helps achieve a larger LCM.

We are given one large text string s, which is fixed for the entire input. Then we receive many query strings x. For a query x, we are not interested only in x itself. Any cyclic rotation of x is considered equivalent.

Consider the quartic polynomial $P(x) = ax^4 + bx^3 + cx^2 + dx + e$ and the quadratic polynomial $Q(x) = ax^2 + (c-b)x + (e-d)$.

The problem asks for the largest set of 10-digit binary words such that no two words can be obtained from each other by reversing a contiguous subsequence of even sum.

Let

Denote the transformation applied to a chosen pair $(a,b)$ by

Consider the given inequality

Let the rectangle have vertices

Let

The polyhedron has all edges of equal length and every edge is tangent to a sphere.

Let $S_A$ denote the central symmetry with center $A$.

For the first part, reinterpret the table as a bipartite graph.

For $n=1$, a single line divides the plane into two regions.

I cannot write a solution to Kvant problem M1188 from the information provided because the actual problem statement is missing.

Let the required order be $a_1,a_2,\dots,a_{m-1}$.

Solution to Kvant math problem 1186.

I do not yet have the full text of Kvant problem M1185.

I do not have access to the published graphical version of Kvant M1184, and you mentioned the textual version is being prepared.

I cannot write a solution to Kvant problem M1183 because the actual problem statement is not present in your message.

Consider small instances of the chessboard and examine the parity of black squares occupied by pieces when each row and each column contains exactly one piece.

Consider two spheres intersecting along a circle.

Consider the first recurrence, $a_{n+1} = \frac{n}{n+1}(a_n+1)$ with $a_1 = 0$.

The inequality involves the three classical quantities

Consider the inequality for small values of $n$ to understand its structure.

The expression to be proved resembles the area formula for a quadrilateral written in terms of the lengths of two diagonals of some auxiliary figure.

Consider the problem for small values of $n$ first.

Compute the first few values of the sequence to look for patterns.

Let the three lines through the interior point $P$ meet the sides of triangle $ABC$ in such a way that they cut off three corner triangles of areas $S_1,S_2,S_3$.

Let

Let

A triangulation of a convex $n$-gon contains exactly $n-3$ diagonals.

Consider a rectangle $ABCD$ with sides $AB = CD = a$ and $BC = AD = b$.

Model the country by a graph with $1989$ vertices and $4000$ edges.

Let $p(i)$ denote the position of $i$ in the permutation.

The inequality involves the side lengths $a$, $b$, $c$ of a triangle and three numbers $p$, $q$, $r$ summing to zero.

Consider a square of side length $n$ on a standard graph paper with $1\times1$ cells.

Let $\sigma(n)$ denote the sum of all positive divisors of $n$.

Let the position of the first turtle at time $t$ be $P(t)$ and the position of the second turtle be $Q(t)$.

Consider the Diophantine equation

Consider first the configuration of ten identical billiard balls arranged snugly in a triangular container.

Consider the situation with only two kangaroos first.

We are asked to minimize $(x+y)(x+z)$ under the constraint $xyz(x+y+z)=1$, with $x$, $y$, $z$ positive.

Let the three triangles be $T_W,T_R,T_G$, and let $M$ be a point lying in the interior of each of them.

There are eight teams, each playing once against every other team, so each team plays $7$ games.

A complete solution cannot be written from the information provided.

I cannot write a solution to Kvant problem M1154 because the actual problem statement is not present in your message.

I do not have the statement of Kvant problem M1152, and the prompt says that only the graphical version is currently available.

I cannot write a solution to Kvant problem M1151 from the information provided, because the actual problem statement is missing.

For $n=3$ the inequality becomes

Consider two rays $p$ and $q$ with vertices $P$ and $Q$, respectively.

Consider small values of $a$ and $n$ to understand the pattern.

The condition says that every closed path contains an even number of red edges.

Place the equilateral triangle $ABC$ in the plane with convenient coordinates.

Consider a circle with a point $P$ outside it and two tangents $PB$ and $PC$, forming an angle $\angle BPC > 90^\circ$.

Let

We are asked to evaluate a large sum over all triples of integers chosen independently from three ranges. Concretely, imagine three slots, where the first slot can be filled with any value from 1 to a, the second from 1 to b, and the third from 1 to c.

Consider first small circular arrangements of weights with integer masses and total mass divisible into parts.

We are asked to compute the expected total cost of a recursive deletion procedure applied to a connected graph with exactly as many edges as nodes.

Each click in the game independently becomes either a successful hit (O) or a miss (X). For a completed sequence, we split it into maximal consecutive blocks of O. If a block has length L, it contributes L² to the score. The total score is the sum of these squared block lengths.

Consider small tables first.

We are asked to help Vasya pick the best hours to read a textbook on a train ride. The train trip lasts for n hours, and each hour has a given light level between 0 and 100. Vasya wants to read for exactly k hours.

We are given a sequence of daily temperatures. We may change any temperature to any other value, and each modified position costs one change. The goal is to make the sequence follow a very specific pattern.

Consider a trapezoid $ABCD$ with $AB$ and $CD$ as the bases, $AB \parallel CD$, and a circle inscribed within it.

We are given two decks of cards, each with a specific order from top to bottom, and each card is either face up or face down. The first deck has n cards, the second deck has m cards.