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Let the numbers be $a,b,c>0$ with $abc=1$.

There are $99999-11111+1=88889$ cards.

For small values of $n$ the statement is easy to test.

Let the longest diagonal of a convex polygon have length $D$.

Represent the five numbers as five binary strings of length $n$, where the symbols are $1$ and $2$.

Represent each cell of the sheet by a vertex.

Let $s(n)$ denote the sum of the decimal digits of $n$.

Let the circle have center $O$ and radius $R$.

Let the common point of the angle bisector $AD$, the median $BM$, and the altitude $CH$ be $P$.

Let $AB=p$ and $AC=q$.

For the first sum,

Consider small values of $n$ to gain intuition.

Let the original seventeen-digit number be

Consider the equation $x^2 - mxy + y^2 = 1$ with $x, y \ge 0$ and integer $m>1$.

Let $A(R)$ denote the sum of the numbers in a rectangle $R$ whose sides follow the grid lines.

Let

Consider arranging seven points and seven lines such that each point lies on exactly three lines and each line contains exactly three points.

Consider small cases first.

The polyhedron has 19 faces and is circumscribed about a sphere of radius $10$.

The number

Each tile is an isosceles trapezoid with bases $3$ and $1$ and height $1$.

Each cut is made on a single existing piece and splits it into two pieces.

For $N=1$, a single circle of diameter $0$ centered at the point covers it, and the sum of diameters is $0<1$.

Let the maximum number of polygons met by a line be denoted by $k$.

Let each vertex be a point, and let its color at time $t$ be represented by a sign $s_v(t)\in{+1,-1}$.

Let the radius of each coin in the chain be $r$.

For the first part, the information-theoretic count is encouraging.

For small values of $n$ the statement is easy to test.

The numbering pattern is linear.

The condition on the denominators is much stronger than in the usual Egyptian fraction problem.

Let

Consider an angle formed by two rays meeting at a vertex $O$.

Let the circles have radii $r_1,r_2,\dots,r_n$.

Consider a single excited cell in an infinite linear chain at $t=0$.

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

Let the fork be at point $A$.

Consider small instances to develop intuition.

Consider a polynomial $p(x)$ with integer coefficients that takes the value $1$ at three distinct integers, say $a$, $b$, and $c$.

Consider a convex quadrilateral $ABCD$ with area $S$.

Consider first small values of $n$ to understand the structure of the sum of pairwise differences.

Let the rows be $R_1,\dots,R_n$, each a vector of length $n$.

The ruler has two fixed marks.

Consider the set of faces of a convex polyhedron.

Let $ABCD$ be a cyclic quadrilateral whose diagonals $AC$ and $BD$ intersect at $P$ and satisfy $AC \perp BD$.

The inequality is symmetric in $a$ and $b$.

A shuffle takes an initial segment of the deck and inserts it somewhere later, preserving the internal order of the removed block and of the remaining cards.

Let

Let $S_t$ be the set of sick Mites on day $t$.

Define

Let the circle have center $O$ and radius $R$.

The problem involves packing 18 tons of cargo into at least 35 containers, with seven spacecraft available, each capable of carrying 3 tons, and the assertion that any selection of 35 containers can b…

Let

For the first statement, computing small cases is instructive.

The quadrilateral is cut by two families of lines.

The operations are purely projective.

Compute the first few terms of the sequence $(a_n)$ directly from the recursive formula.

A cube is highly symmetric, so the number of axes of symmetry should be larger than in simpler polyhedra.

Consider the two Diophantine equations

Let the triangle be $ABC$.

For the numbers $1,2,\dots,30$, the total sum is

A triangular pyramid is a tetrahedron.

Let $s(n)$ denote the sum of the digits of the single number $n$.

The data of the problem are naturally encoded by a similarity.

Assume that the infinite digit string obtained by concatenating

The statement involves two circles.

For part 1 it is natural to reinterpret a nondecreasing tuple

For the planar statement, choose coordinates so that the two given perpendicular directions are the coordinate axes.

The polygon is rectilinear: every side lies on a grid line, hence every side is horizontal or vertical.

An isosceles trapezoid includes rectangles as a special case, since a rectangle has a pair of parallel sides and equal legs.

The recurrence

A reflection with respect to a point $A$ is the central symmetry $x\mapsto 2A-x$.

I cannot write a solution to Kvant problem M604 from the information provided, because the actual problem statement is missing.

The denominators suggest introducing

Let the three consecutive entries in row $n$ be

Let $H$ be the orthocenter of triangle $ABC$, let $M$ be the midpoint of $BC$, and let $D$ be the point on the circumcircle diametrically opposite $A$.

Let the circles intersect at points $A$ and $B$.

Let

Label the trees by the residues modulo $n$, arranged around the circle.

The structure of the election is a rooted tree.

Let the centers of the circles be the vertices $A,B,C,D$ of a convex quadrilateral, listed in cyclic order.

Consider a tetrahedron with vertices $A$, $B$, $C$, and $D$.

Consider the original game with 25 matches, where each player may take 1, 2, or 3 matches per turn, and the winner is the player whose total number of matches at the end is even.

Let

Consider a standard 12-hour analog clock with an hour hand and a minute hand.

Let $E$ be a set of $n$ elements and $S_1, S_2, \dots, S_m$ be the chosen subsets of $E$ (distinct from $E$) such that for any pair of elements of $E$, there is exactly one $S_i$ containing both.

Fix the segment $AB$ and let its length be $d$.

Each circle lies on the unit sphere.

For the square tiling, the centers of all squares form the standard square lattice $\mathbb Z^2$.

The statement is intentionally wrapped in game terminology, but the underlying process is a sequence of color merges. Each map cell has a panel color. Some cells also contain a symbol, and every symbol has its own color. We start by destroying one specific symbol.

Ta có một mảng số nguyên dương. Một phép giảm chọn ba phần tử liên tiếp sao cho phần tử giữa nhỏ hơn tổng hai phần tử hai bên. Khi đó bộ ba $$(c{i-1},ci,c{i+1})$$ được thay bằng một giá trị duy nhất $$x=c{i-1}-ci+c{i+1}.

We are given three characters, each standing on a different position along a one-dimensional half-line. Each position is an integer starting at 1, and each character has a movement range and a throwing range.

We are given a string made only of opening and closing parentheses. In one move, we are allowed to take a contiguous block, remove it, and then reinsert its characters anywhere in the remaining string, with full freedom to permute those removed characters and place each one…

Each array element can be modified independently. For a value $x$, we may choose any positive integer $b$, then replace $x$ by $x bmod b$. The goal is to maximize the MEX of the resulting array.

We are given a tree and a fixed set of vertices consisting of all leaves in the original tree. These leaves are determined once from the initial structure and do not change during the process.

Each test case gives a sequence of column heights. Think of column i as a vertical stack of unit blocks, occupying rows 1 up to ai. All blocks in a row are aligned across columns.

An array $a$ is fixed, and we append to it a permutation $p$ of ${1,2,dots,m}$ to form a longer array $bp$. For each such permutation, we count how many subarrays of $bp$ with even length are palindromes.

We are given several test cases, each consisting of an array. The task is to decide whether we can transform each array into a non-decreasing order using a specific operation: we may pick any subarray, compute its minimum and maximum values, and reverse it, but only if the sum…

A subarray product is divisible by $6$ if and only if the product contains at least one factor $2$ and at least one factor $3$. For each number, only its divisibility by $2$ and $3$ matters. The exact value is irrelevant.

We have a small assembly of n senators, each defined by a level and a loyalty score. Loyalty is a probability that a senator votes in favor of a proposal, given in 10% increments. If more than half of senators vote yes, the proposal passes.

We are given a sequence of positive integers. We repeatedly remove groups of elements until nothing remains. Each group must respect two constraints: if we look at the chosen indices in increasing order, the corresponding values must be nondecreasing, and the indices…