brain

The problem involves constructing a chord $MN$ of a circle with center $O$ seen from $A$ under a given angle $\alpha$, with additional geometric constraints.

The two inequalities are

Represent the configuration by numbers $a_1,\dots,a_n\in{0,1,2}$ arranged cyclically.

Consider a single fold of a convex polygon and a subsequent straight cut.

Testing small values helps build intuition about the inequality.

Let the inradius of triangle $AOB$ be $r_1$, of $BOC$ be $r_2$, of $COD$ be $r_3$, and of $DOA$ be $r_4$.

The condition is

Let $h_a,h_b,h_c$ be the altitudes of triangle $ABC$.

Consider the problem of reaching a target number from $0$ using only two operations: doubling the current number or adding $1$.

Consider the problem geometrically.

Let the two given circles be $\omega_1$ and $\omega_2$, intersecting at $A$ and $B$.

Consider small values of $n$ to understand the inequality.

The given equality resembles the identity for the sum of squares of the sides of a quadrilateral.

Compute a few values:

Let

For $n=3$ the problem asks for a single triangle whose three side lengths are irrational and whose area is a nonzero rational number.

Assume that a function $f:\mathbb N_0\to\mathbb N_0$ satisfies

Let $X(\sigma)$ denote the number of fixed points of a permutation $\sigma$ of an $n$ element set.

Consider the problem in terms of digit patterns.

Let $m=2n+1$.

Consider a hexagon $A_1A_2A_3A_4A_5A_6$ with a point $O$ from which all sides are seen under an angle of $60^\circ$.

The expression $989 \cdot 1001 \cdot 1007 + 320$ appears to involve three numbers spaced by six units: $989$, $1001$, $1007$.

Consider smaller versions of the game to understand the parity dynamics.

Let the tetrahedron have vertices $A,B,C,D$.

Consider a small number of families, say three or four, each in a distinct apartment.

Consider an angle $AOB$ with points $A$ on one side and $B$ on the other.

Let

Consider six points in the plane with all pairwise distances at most $1$.

We are asked to study vectors $(x;y)$ with non-negative integer coordinates and to decide when they can be written as sums of _generating vectors_, i.

Let the closed broken line have vertices $V_1,\dots,V_n$ and segments $e_i=V_iV_{i+1}$, where indices are taken modulo $n$.

Let the digits of the $n$-digit number $a$ be

The first part of the problem deals with a triangle $ABC$ with points $D$ on $AC$ and $E$ on $AB$, forming the intersecting lines $BD$ and $CE$ at $M$.

Interpret the cities and roads as a graph.

Consider two closed polygonal chains in the plane, each with an odd number of sides.

Let

Consider a finite subset of $\mathbb{Z}^2$ as a candidate for the marked points and examine what happens when we translate each by all vectors from the given finite set.

A move consists of writing a number that is not a divisor of any previously written number.

Consider small cases first.

Consider a circle with a small number of points to understand the behavior of arcs subtending at most $120^\circ$.

We have an alphabet of size m, consisting of the first m symbols from the sequence a..zA..Z. Some ordered pairs of symbols are forbidden. If pair (x, y) is forbidden, then symbol y cannot appear immediately after symbol x in the DNA string.

We are given two multisets of scores. Array a contains the scores obtained in the first tour, and array b contains the scores obtained in the second tour.

The fraction is not given as a single numerator and denominator. Instead, we receive two arrays. The product of all numbers in the first array is the numerator, and the product of all numbers in the second array is the denominator.

Consider four spheres in three-dimensional space.

We are given an array of positive integers and many range queries. For each query [l, r], we look only at the subarray between those positions and count how many values x satisfy a very specific condition: Inside that subarray, the value x appears exactly x times.

We are given an array that was originally sorted in non-decreasing order. At some point, either nothing happened or exactly one pair of elements may have been swapped.

We are given an array of positive integers a of length n. We need to count the number of pairs (l, r) with 1 ≤ l < r ≤ n such that if we take the element at position l and move it just before position r (shifting the elements in between right by one), the resulting array has…

We have all lattice points inside the rectangle $$0 le x le w,qquad 0 le y le h.$$ A valid answer is an ordered triple of points that forms a nondegenerate triangle whose area is a positive integer. The order matters, so every geometric triangle contributes up to $3!

We are given two permutations a and b of length n. Each permutation contains all integers from 1 to n exactly once. We are asked to compute, for every cyclic shift of b, a quantity called the distance.

We are given a stripe represented as a row of n cells, where each cell is painted one of k colors labeled with letters A through the k-th letter. The goal is to repaint as few cells as possible so that no two adjacent cells share the same color.

We are asked to choose a capital city in a tree-shaped country with one-way roads. Each city is a node, and each road is a directed edge. The goal is to orient all roads so that from the chosen capital, we can reach every other city by following the roads in their direction.

We have a linear parking lot with n spaces numbered from 1 to n. Cars arrive and depart over time. When a car arrives, we must assign it a spot that maximizes the distance to the nearest occupied space.

We start with a DNA string. Each mutation chooses a contiguous segment, keeps that segment in place, and inserts a transformed copy immediately after it.

The first few Fibonacci numbers with at least four digits are

Consider a convex $n$-gon with vertices labeled cyclically as $A_1, A_2, \dots, A_n$.

Consider a $3\times3$ cluster of pieces on an $8\times8$ chessboard.

Let the chosen points be

Consider a cylinder $\text{Ц}_1$ with radius $R_1$ and height $H_1$, and define its diameter-to-height ratio $k = \frac{2R_1}{H_1}$.

Consider the simplest nontrivial cases of the knight’s tour game.

Consider a small round-robin tournament with $n$ players.

Consider an acute-angled triangle $ABC$ with $\angle A = 60^\circ$.

Consider the geometry of the kingdom, which is a square of side $2$ km.

Let

Let the class contain $n$ students.

A regular pentagon is determined up to congruence by any three consecutive vertices.

For $n=1$ the three groups are ${1},{2},{3}$, and $3=1+2$.

Label the tetrahedron vertices as $A$, $B$, $C$, $D$.

The rectangle contains $mn$ cells.

Consider the equation $x^y - y^x = x + y$ with $x, y \in \mathbb{N}$.

Consider a pentagon and imagine cutting it into two smaller pentagons of equal area and shape.

Consider marking points on $[0,1]$ sequentially.

Consider small chocolate bars first.

Let the square have vertices $A,B,C,D$ in cyclic order.

Begin with small values of $n$ to detect a pattern.

Reflecting on the problem, the point $M$ is chosen on the line $\ell$ to minimize the sum $MA + MB$.

Consider simple convex polyhedra such as nested cubes, tetrahedra, or pyramids.

Begin by considering the configuration of two intersecting lines and points $D$ and $E$ on them.

The number $1987$ is prime, since it is not divisible by any prime not exceeding $\sqrt{1987}<45$.

Let the common measure of each arc be $x$.

Consider a convex quadrilateral $ABCD$ with extensions of opposite sides $AB$ and $CD$, and $AD$ and $BC$, intersecting at points $P$ and $Q$ respectively.

Consider two triangles with angles $\alpha, \beta, \gamma$ and $\alpha_1, \beta_1, \gamma_1$.

For small numbers of triangles the statement is false.

For numbers $1,2,\dots,2n$, suppose they are arranged in two rows and $n$ columns.

Consider how the mountaineer’s progress depends on the day’s starting point.

Consider a sphere of radius $1$ with a curve drawn on it, either open of length less than $\pi$ or closed of length less than $2\pi$.

Consider first a small example on a $3 \times 3$ portion of the grid.

Consider a regular $n$-gon $A_1 A_2 \dots A_n$ with center $O$.

Consider assigning integers to the vertices of a regular pentagon and performing the prescribed operation whenever a vertex carries a negative number.

For a polygon circumscribed about a circle of radius $r$, let the sides be $s_1,\dots,s_n$, with corresponding side lengths $\ell_1,\dots,\ell_n$.

The polynomial is

Consider small examples of pairwise coprime numbers, such as $a_1=2$, $a_2=3$, $a_3=5$.

Consider triangle $ABC$ with points $M$ on $AB$ and $N$ on $BC$.

Consider arrangements of circles in the plane where each circle touches several others.

For the first inequality,

Consider the sequence defined by $r_1=2$ and $r_{n+1}=r_1 r_2 \cdots r_n + 1$.

Let the parallelogram be represented by vectors.

Number the steps from $1$ at the bottom to $2n+1$ at the top.

We are given a shuffled deck of up to 52 cards arranged in a line, each represented by a value and a suit. Initially, every card forms its own pile.

We are given an undirected, connected graph representing cities and roads, where every road has the same travel time. Among all cities, city 1 and city n are special: we care about travel between these two endpoints.

The equality

We are given a rooted forest where each person has at most one parent. If we follow parent pointers upward, we eventually reach a root or fall off the structure. This defines a collection of trees. A “k-th ancestor” means applying the parent relation k times.