Let $n\ge 3$ be an integer. Let $s(n)$ be the number of (ordered) pairs $(a;b)$ consisting of positive integers $a,b$ from the set $\{1,2,\dots ,n\}$ which satisfy $\gcd (a,b,n)=1$. Prove that $s(n)$ is divisible by $4$ for all $n\ge 3$. [i] (Vlad Robu) [/i]

Consider a $m\times n$ rectangular board consisting of $mn$ unit squares. Two of its unit squares are called [i]adjacent[/i] if they have a common edge, and a [i]path[/i] is a sequence of unit squares in which any two consecutive squares are adjacent. Two parths are called [i]non-intersecting[/i] if they don't share any common squares. Each unit square of the rectangular board can be colored black or white. We speak of a [i]coloring[/i] of the board if all its $mn$ unit squares are colored. Let $N$ be the number of colorings of the board such that there exists at least one black path from the left edge of the board to its right edge. Let $M$ be the number of colorings of the board for which there exist at least two non-intersecting black paths from the left edge of the board to its right edge. Prove that $N^{2}\geq M\cdot 2^{mn}$.

BdMO National 2018 Higher Secondary P3 Nazia rolls four fair six-sided dice. She doesn’t see the results. Her friend Faria tells her that the product of the numbers is $144$. Faria also says the sum of the dice, $S$ satisfies $14\leq S\leq 18$ . Nazia tells Faria that $S$ cannot be one of the numbers in the set {$14,15,16,17,18$} if the product is $144$. Which number in the range {$14,15,16,17,18$} is an impossible value for $S$ ?

Let $f(k)$ be the number of integers $n$ satisfying the following conditions: (i) $0\leq n < 10^k$ so $n$ has exactly $k$ digits (in decimal notation), with leading zeroes allowed; (ii) the digits of $n$ can be permuted in such a way that they yield an integer divisible by $11$. Prove that $f(2m) = 10f(2m-1)$ for every positive integer $m$. [i]Proposed by Dirk Laurie, South Africa[/i]

A confused cockroach is initially at vertex $A$ of a cube $ABCDEFGH$ with edges measuring $1$ meter, as shown in the figure. Every second, the cockroach moves $1$ meter, always choosing to go to one of the three adjacent vertices to its current position. For example, after $1$ second, the cockroach could stop at vertex $B$, $D$, or $E$. (a) In how many ways can the cockroach stop at vertex $G$ after $3$ seconds? (b) Is it possible for the cockroach to stop at vertex A after exactly $2023$ seconds? (c) In how many ways can the cockroach stop at A after exactly $2024$ seconds? Note: One way for the cockroach to stop at a vertex after a certain number of seconds differs from another way if, at some point, the cockroach is at different vertices in the trajectory. For example, there are $2$ ways for the cockroach to stop at $C$ after $2$ seconds: one of them passes through $A$, $B$, $C$, and the other through $A$, $D$, $C$. [img]https://cdn.discordapp.com/attachments/954427908359876608/1299721377124847616/Screenshot_2024-10-16_173123.png?ex=671e3b5b&is=671ce9db&hm=76962ee2949d8324c2f7022ef63f8b7d3c6fe3aabf4ecf526f44249439f204ac&[/img]

Prove that for every partition of $ \{ 1,2,3,4,5,6,7,8,9\} $ into two subsets, one of the subsets contains three numbers such that the sum of two of them is equal to the double of the third.

Let $n > 1$ be a positive integer. A 2-dimensional grid, infinite in all directions, is given. Each 1 by 1 square in a given $n$ by $n$ square has a counter on it. A [i]move[/i] consists of taking $n$ adjacent counters in a row or column and sliding them each by one space along that row or column. A [i]returning sequence[/i] is a finite sequence of moves such that all counters again fill the original $n$ by $n$ square at the end of the sequence. [list] [*] Assume that all counters are distinguishable except two, which are indistinguishable from each other. Prove that any distinguishable arrangement of counters in the $n$ by $n$ square can be reached by a returning sequence. [*] Assume all counters are distinguishable. Prove that there is no returning sequence that switches two counters and returns the rest to their original positions.[/list] [i]Mitchell Lee and Benjamin Gunby.[/i]

Given an $n \times n \times n$ grid of unit cubes, a cube is [i]good[/i] if it is a sub-cube of the grid and has side length at least two. If a good cube contains another good cube and their faces do not intersect, the first good cube is said to [i]properly[/i] contain the second. What is the size of the largest possible set of good cubes such that no cube in the set properly contains another cube in the set?

A number is called [i]trilegal[/i] if its digits belong to the set \(\{1, 2, 3\}\) and if it is divisible by \(99\). How many trilegal numbers with \(10\) digits are there?

The number of functions $g:\mathbb{R}^4\to\mathbb{R}$ such that, $\forall a,b,c,d,e,f\in\mathbb{R}$ : (i) $g(1,0,0,1)=1$ (ii) $g(ea,b,ec,d)=eg(a,b,c,d)$ (iii) $g(a+e, b, c+f, d)= g(a,b,c,d)+g(e,b,f,d)$ (iv) $g(a,b,c,d)+g(b,a,d,c)=0$ is : (A)$1$ (B)$0$ (C)$\text{infinitely many}$ (D)$\text{None of these}$ [Hide=Hint(given in question)] Think of matrices[/hide]

Let $A$ be a set of $N$ residues $\pmod{N^{2}}$. Prove that there exists a set $B$ of of $N$ residues $\pmod{N^{2}}$ such that $A + B = \{a+b|a \in A, b \in B\}$ contains at least half of all the residues $\pmod{N^{2}}$.

Let $ S \equal{} \{x_1, x_2, \ldots, x_{k \plus{} l}\}$ be a $ (k \plus{} l)$-element set of real numbers contained in the interval $ [0, 1]$; $ k$ and $ l$ are positive integers. A $ k$-element subset $ A\subset S$ is called [i]nice[/i] if \[ \left |\frac {1}{k}\sum_{x_i\in A} x_i \minus{} \frac {1}{l}\sum_{x_j\in S\setminus A} x_j\right |\le \frac {k \plus{} l}{2kl}\] Prove that the number of nice subsets is at least $ \dfrac{2}{k \plus{} l}\dbinom{k \plus{} l}{k}$. [i]Proposed by Andrey Badzyan, Russia[/i]

What is the number of $4$ digit natural numbers such that the sum of digits is even? (A)$4999$ (B)$5000$ (C)$5050$ (D)$4500$

Each square in a $3\times 3$ grid of squares is colored red, white, blue, or green so that every $2\times 2$ square contains one square of each color. One such coloring is shown on the right below. How many different colorings are possible?\\ [asy] size(8cm); pen grey1, grey2, grey3; grey1 = RGB(211, 211, 211); grey2 = RGB(173, 173, 173); grey3 = RGB(138, 138, 138); for(int i = 0; i < 4; ++i) { draw((i, 0)--(i, 3)); draw((0, i)--(3, i)); } filldraw((5, 3)--(6, 3)--(6, 2)--(5, 2)--cycle, grey1); label('B', (5.5, 2.5)); filldraw((6, 3)--(7, 3)--(7, 2)--(6, 2)--cycle, grey2); label('R', (6.5, 2.5)); filldraw((7, 3)--(8, 3)--(8, 2)--(7, 2)--cycle, grey1); label('B', (7.5, 2.5)); filldraw((5, 2)--(6, 2)--(6, 1)--(5, 1)--cycle, grey3); label('G', (5.5, 1.5)); filldraw((6, 2)--(7, 2)--(7, 1)--(6, 1)--cycle, white); filldraw((7, 2)--(8, 2)--(8, 1)--(7, 1)--cycle, grey3); label('G', (7.5, 1.5)); filldraw((5, 1)--(6, 1)--(6, 0)--(5, 0)--cycle, grey2); label('R', (5.5, 0.5)); filldraw((6, 1)--(7, 1)--(7, 0)--(6, 0)--cycle, grey1); label('B', (6.5, 0.5)); filldraw((7, 1)--(8, 1)--(8, 0)--(7, 0)--cycle, grey2); label('R', (7.5, 0.5)); [/asy] $\textbf{(A) }24\qquad\textbf{(B) }48\qquad\textbf{(C) }60\qquad\textbf{(D) }72\qquad\textbf{(E) }96$

Let $A=\{1,2,\ldots,n\}$. For a permutation $P=(P(1), P(2), \ldots, P(n))$ of the elements of $A$, let $P(1)$ denote the first element of $P$. Find the number of all such permutations $P$ so that for all $i,j \in A$: (a) if $i < j<P(1)$, then $j$ appears before $i$ in $P$; and (b) if $P(1)<i<j$, then $i$ appears before $j$ in $P$.

We have some identical paper squares which are black on one side of the sheet and white on the other side. We can join nine squares together to make a $3$ by $3$ sheet of squares by placing each of the nine squares either white side up or black side up. Two of these $3$ by $3$ sheets are distinguishable if neither can be made to look like the other by rotating the sheet or by turning it over. How many distinguishable $3$ by $3$ squares can we form?

In a permutation $(x_1, x_2, \dots , x_n)$ of the set $1, 2, \dots , n$ we call a pair $(x_i, x_j )$ discordant if $i < j$ and $x_i > x_j$. Let $d(n, k)$ be the number of such permutations with exactly $k$ discordant pairs. Find $d(n, 2)$ and $d(n, 3).$

Given a regular $n$-gon $A_{1}A_{2}...A_{n}$ (with $n\geq 3$) in a plane. How many triangles of the kind $A_{i}A_{j}A_{k}$ are obtuse ?

Let $a_1,a_2,\ldots$ be a sequence of integers with infinitely many positive and negative terms. Suppose that for every positive integer $n$ the numbers $a_1,a_2,\ldots,a_n$ leave $n$ different remainders upon division by $n$. Prove that every integer occurs exactly once in the sequence $a_1,a_2,\ldots$.

What is the maximal number of regions a circle can be divided in by segments joining $n$ points on the boundary of the circle ? [i]Posted already on the board I think...[/i]

In order to study a certain ancient language, some researchers formatted its discovered words into expressions formed by concatenating letters from an alphabet containing only two letters. Along the study, they noticed that any two distinct words whose formatted expressions have an equal number of letters, greater than $ 2, $ differ by at least three letters. Prove that if their observation holds indeed, then the number of formatted expressions that have $ n\ge 3 $ letters is at most $ \left[ \frac{2^n}{n+1} \right] . $

There is an unlimited supply of congruent equilateral triangles made of colored paper. Each triangle is a solid color with the same color on both sides of the paper. A large equilateral triangle is constructed from four of these paper triangles. Two large triangles are considered distinguishable if it is not possible to place one on the other, using translations, rotations, and/or reflections, so that their corresponding small triangles are of the same color. Given that there are six different colors of triangles from which to choose, how many distinguishable large equilateral triangles may be formed?

Find the sum of all even numbers greater than 100000, that u can make only with the digits 0,2,4,6,8,9 without any digit repeating in any number.

Let $n\geq 3$ be an integer. Find the number of ways in which one can place the numbers $1, 2, 3, \ldots, n^2$ in the $n^2$ squares of a $n \times n$ chesboard, one on each, such that the numbers in each row and in each column are in arithmetic progression.

Let $M, k>0$ integers. Let $X(M,k)$ the (infinite) set of all integers that can be factored as ${p_1}^{e_1} \cdot {p_2}^{e_2} \cdot \ldots \cdot {p_r}^{e_r}$ where each $p_i$ is not smaller than $M$ and also each $e_i$ is not smaller than $k$. Let $Z(M,k,n)$ the number of elements of $X(M,k)$ not bigger than $n$. Show that there are positive reals $c(M,k)$ and $\beta(M,k)$ such that $$\lim_{n \rightarrow \infty}{\frac{Z(M,k,n)}{n^{\beta(M,k)}}} = c(M,k)$$ and find $\beta(M,k)$