brain

We are asked to maintain two dynamically growing rooted trees with labeled edges, and after each operation, compute the number of "good combinations.

We are given a column of n tanks numbered 1 to n. Each tank i has a message receiving radius a[i]. During the exercise, exactly n messages must be sent from the front of the column to the back. After each message, the last tank in the message path moves to the front.

The octagon is the intersection of two congruent squares.

Let

We are given a line of tanks, each occupying a fixed position in a row from left to right. Each tank has a parameter that determines how far it can receive a message when it is the recipient.

Let

We are asked to simulate a training exercise where a line of tanks must pass messages from the front to the end, following specific rules. Initially, the tanks are numbered from 1 to n, and each tank i has a message receiving radius ai.

We are asked to schedule problems brought by multiple scientists for a special calculator. Each scientist provides a sequence of problems that must be solved in the given order.

Let $x$ be the smallest of $n$ consecutive natural numbers.

Consider small examples of social networks where each person has at least 10 friends.

Consider triangle $ABC$ with an altitude $CH$ and median $CK$.

Consider three lines in space, each pair of which is skew, and they are not all parallel to the same plane.

I begin by examining small natural numbers $a$ to see which of them satisfy the given conditions.

Consider small values of $n$ and $k$ to build intuition.

Consider small instances to gain intuition.

We are given a theater with n rows and m seats per row, forming an n × m grid. A line of k people is waiting to buy tickets. Each person has a preferred seat, represented as coordinates (x, y). When a person reaches the box office, they attempt to take their preferred seat.

We are asked to distribute a fixed scholarship budget among students based on their exam grades. Each student receives a mark of 3, 4, or 5, and students with the same mark must receive the same scholarship.

We are given a collection of procedure declarations and a collection of variables. Each procedure declaration consists of a name and a list of parameter types. A parameter type may be one of the concrete types int, string, or double, or it may be the wildcard type T.

We are given several drinks, and each drink contains some percentage of orange juice. Vasya mixes equal amounts of every drink into a single cocktail. The question is simple: after mixing them, what percentage of the final cocktail is orange juice?

The problem involves modeling bacterial growth in a test tube according to a discrete-time recurrence. Each bacterium splits into k bacteria every second, and an additional b bacteria are added due to abnormal effects.

The inequality is

We are asked to compute the minimum time for Qwerty's ship to intercept a moving planet in a 2D plane. The star Diatar is at the origin, Persephone orbits the star in a perfect circle of radius $R$ at constant linear speed $vp$, and Qwerty's ship starts at some arbitrary…

We are asked to count configurations of three lines through a point in space with prescribed pairwise angles, up to congruence.

Consider a square $ABCD$ and an arbitrary point $K$ inside it.

The tournament is a complete directed graph on $16$ vertices.

Construct triangle $ABC$ on paper and build the external squares $ABB_1A_2$, $BCB_1C_2$, $CAA_1C_2$.

Consider small repunit numbers of the form $R_n = 11\ldots1$ with $n$ ones.

Consider first a convex polygon in the plane with vertices $A_1, A_2, \dots, A_n$ and a point $O$ inside it.

Consider the definition of an exceptional set of $k$ numbers $a_1, a_2, \dots, a_k$, all strictly between 0 and 1.

The threshold $\sqrt{2/3}$ is suggestive because an equilateral triangle of side $a$ has altitude $\frac{\sqrt3}{2}a$, and when $a=\sqrt{2/3}$ the altitude equals $\frac1{\sqrt2}$.

The problem asks whether $x$ can be expressed using only addition, subtraction, and multiplication from given polynomials.

Place the square in coordinates:

Consider first a simplified scenario: a small $n\times n$ board, say $n=5$, with just a few hypothetical pieces each attacking a limited number of squares.

Suppose both players start with equal time and make alternating moves.

Let

The sequence $(x_n)$ begins with $x_1 = \frac12$ and satisfies the recurrence $x_{n+1} = x_n^2 + x_n$.

Consider a tournament of $8$ volleyball teams where each team plays every other team exactly once.

Unusual activity has been detected from your device.

For small values,

The statement asks for a dissection of an arbitrary triangle into four pieces such that the pieces can be rearranged into two triangles, each similar to the original triangle.

The sequence $(a_n)$ consists of distinct positive integers with the growth constraint $a_n < 100n$.

Let the side length of the square be $6$.

Consider first small examples.

Let

Consider first small numbers of participants.

Consider first the case of a rectangle inscribed in a triangle.

Write

The $25$ plots form the $5\times5$ grid graph.

Consider first small cases.

Position two parabolas in the plane with perpendicular axes.

For $n=1$, the partition is ${1}$ and ${2}$, hence

Consider first the case $k=2$, which corresponds to a regular decagon.

For the planar statement, the natural idea is to look at one fixed side of the square, say the left side.

The problem has two parts.

Let the angular speed be $\dfrac{360^\circ}{n}$ per second.

Consider the simplest nontrivial case $n=1$.

Interpret the $2n$ points as vertices of a graph $G$ with $2n$ vertices and $n^2+1$ edges.

Let the clans be represented by labels.

Consider triangle $ABC$ with an incircle touching sides $AB$, $BC$, and $CA$ at points $C_1$, $A_1$, and $B_1$ respectively.

Consider the equation $a^4 + b^4 + c^4 + d^4 = e^4$ modulo small primes to understand divisibility constraints.

Consider small values of $N$ to understand the dynamics of the seat-shifting process.

Consider a small blue region, for example, a disk of radius $r<1$.

Consider a unit cube in three-dimensional space with edges parallel to the axes.

Every gripper is located at a fixed point in space. Qwerty's ship is also fixed. A gripper can pull another gripper into the ship if two conditions hold simultaneously.

We are asked to fill an $n times n times n$ cube with numbers from 1 to $n^3$ in such a way that two conditions hold simultaneously. First, the numbers must form a "snake": each consecutive number must occupy a cube that is a face neighbor of the previous number.

We have two vertical walls, each represented by a string of length n. Position i on a wall is either safe (-) or blocked (X). The ninja starts at position 0 of the left wall. Every second he may move to one of three positions: 1. One cell upward on the same wall. 2.

We are given a connected network of cities, some of which contain portals. Each road between cities has a positive travel time, and Pavel starts in city 1.

We are asked to find the next string lexicographically larger than a given string s such that no substring of length d or more is a palindrome. A palindrome is a sequence that reads the same forwards and backwards.

We are given a lowercase string and may choose any non-empty subsequence of its characters while preserving their original order. Among all possible subsequences, we need the one that is lexicographically largest.

The problem involves a convex quadrilateral $ABCD$ with two given angles, $\angle A = \alpha$ and $\angle B = \beta$, and a special relation between its sides and area: the doubled area satisfies $2S…

For the first integral equality, the two integrals involve complementary functions: the tangent function on $[0,\pi/4]$ and the arctangent function on $[0,1]$.

Consider six-digit numbers from $000000$ to $999999$.

The inequality is cyclic rather than symmetric:

Consider triangle $ABC$ with circumcircle $\Gamma$.

Place quadrilateral $ABCD$ in the plane and select points $E$ on $AB$ and $F$ on $CD$.

Let the regular hexagon be $P_1P_2P_3P_4P_5P_6$, and let the points of the problem be chosen on its sides so that $A_i\in P_iP_{i+1}$, indices modulo $6$.

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