Let $S$ be a set of 6 integers taken from $\{1,2,\dots,12\}$ with the property that if $a$ and $b$ are elements of $S$ with $a<b$, then $b$ is not a multiple of $a$. What is the least possible value of an element in $S$? $\textbf{(A)}\ 2\qquad\textbf{(B)}\ 3\qquad\textbf{(C)}\ 4\qquad\textbf{(D)}\ 5\qquad\textbf{(E)}\ 7$

Dots are spaced one unit apart, horizontally and vertically. The number of square units enclosed by the polygon is [asy] for(int a=0; a<4; ++a) { for(int b=0; b<4; ++b) { dot((a,b)); } } draw((0,0)--(0,2)--(1,2)--(2,3)--(2,2)--(3,2)--(3,0)--(2,0)--(2,1)--(1,0)--cycle);[/asy] $ \text{(A)}\ 5\qquad\text{(B)}\ 6\qquad\text{(C)}\ 7\qquad\text{(D)}\ 8\qquad\text{(E)}\ 9 $

Circle I is circumscribed about a given square and circle II is inscribed in the given square. If $r$ is the ratio of the area of circle $I$ to that of circle $II$, then $r$ equals: $\text{(A)} \ \sqrt 2 \qquad \text{(B)} \ 2 \qquad \text{(C)} \ \sqrt 3 \qquad \text{(D)} \ 2\sqrt 2 \qquad \text{(E)} \ 2\sqrt 3$

Let $ n\geq 3 $ be an integer and let $ p\geq 2n-3 $ be a prime number. For a set $ M $ of $ n $ points in the plane, no 3 collinear, let $ f: M\to \{0,1,\ldots, p-1\} $ be a function such that (i) exactly one point of $ M $ maps to 0, (ii) if a circle $ \mathcal{C} $ passes through 3 distinct points of $ A,B,C\in M $ then $ \sum_{P\in M\cap \mathcal{C}} f(P) \equiv 0 \pmod p $. Prove that all the points in $ M $ lie on a circle.

Let $m\in N$ and $E(x,y,m)=(\frac{72}x)^m+(\frac{72}y)^m-x^m-y^m$, where $x$ and $y$ are positive divisors of 72. a) Prove that there exist infinitely many natural numbers $m$ so, that 2005 divides $E(3,12,m)$ and $E(9,6,m)$. b) Find the smallest positive integer number $m_0$ so, that 2005 divides $E(3,12,m_0)$ and $E(9,6,m_0)$.

Consider all the positive integers $N$ with the property that all of the divisors of $N$ can be written as $p-2$ for some prime number $p$. Then, there exists an integer $m$ such that $m$ is the maximum possible number of divisors of all numbers $N$ with such property. Find the sum of all possible values of $N$ such that $N$ has $m$ divisors. [i]Proposed by [b]FedeX333X[/b][/i]

The sequence $(x_n)$ is defined as follows: $$x_0=2,\, x_1=1,\, x_{n+2}=x_{n+1}+x_n$$ for every non-negative integer $n$. a. For each $n\geq 1$, prove that $x_n$ is a prime number only if $n$ is a prime number or $n$ has no odd prime divisors b. Find all non-negative pairs of integers $(m,n)$ such that $x_m|x_n$.

Let $ABC$ be a triangle with $AB = 16$, $AC = 10$, $BC = 18$. Let $D$ be a point on $AB$ such that $4AD = AB$ and let E be the foot of the angle bisector from $B$ onto $AC$. Let $P$ be the intersection of $CD$ and $BE$. Find the area of the quadrilateral $ADPE$.

Given $n \geq 3$ pairwise different prime numbers $p_1, p_2, ....,p_n$. Given, that for any $k \in \{ 1,2,....,n \}$ residue by division of $ \prod_{i \neq k} p_i$ by $p_k$ equals one number $r$. Prove, that $r \leq n-2 $.

Write the number \[ \frac 1{\sqrt{2}-\sqrt[3]{2}} \] as the sum of terms of the form $2^q,$ where $q$ is rational. (For example, $2^1 + 2^{-1/3} + 2^{8/5}$ is a sum of this form.) Prove that your sum equals $1/(\sqrt{2}-\sqrt[3]{2}).$

Let $n$ be a positive integer and let $a_k = \dfrac{1}{\binom{n}{k}}, b_k = 2^{k-n},\ (k=1..n)$. Show that $\sum_{k=1}^n \dfrac{a_k-b_k}{k} = 0$.

Let \( ABCDE \) be a convex pentagon whose vertices lie on a circle \( \Gamma \). The tangents to \( \Gamma \) at \( C \) and \( E \) intersect at \( X \), and the segments \( CE \) and \( AD \) intersect at \( Y \). Given that \( CE \) is perpendicular to \( BD \), that \( XY \) is parallel to \( BD \), that \( AY = BD \), and that \( \angle BAD = 30^\circ \), what is the measure of the angle \( \angle BDA \)? Proposed by João Gilberti Alves Tavares

A function $f : I \to \mathbb R$, defined on an interval $I$, is called concave if $f(\theta x + (1 - \theta)y) \geq \theta f(x) + (1 - \theta)f(y)$ for all $x, y \in I$ and $0 \leq \theta \leq 1$. Assume that the functions $f_1, \ldots , f_n$, having all nonnegative values, are concave. Prove that the function $(f_1f_2 \cdots f_n)^{1/n}$ is concave.

Find the sum : $C^{n}_{1}$ - $\frac{1}{3} \cdot C^{n}_{3}$ + $\frac{1}{9} \cdot C^{n}_{5}$ - $\frac{1}{27} \cdot C^{n}_{9}$ + ...

Let $m$ be a line in the plane and $M$ a point not in $m$. Find the locus of the focus of the parabolas with vertex $M$ that are tangent to $m$.

Does the set $\{1,2,3,...,3000\}$ contain a subset $ A$ consisting of 2000 numbers that $x\in A$ implies $2x \notin A$ ?!! :?:

Find all real numbers $x$ which satisfy $\frac{n}{3n+1}\leq x\leq \frac{4n+1}{2n-1}$, for all $n\in\mathbb{N}^*$. [i]Gheorghe Boroica[/i]

The numbers from 1 to $n^2$ are randomly arranged in the cells of a $n \times n$ square ($n \geq 2$). For any pair of numbers situated on the same row or on the same column the ratio of the greater number to the smaller number is calculated. Let us call the [b]characteristic[/b] of the arrangement the smallest of these $n^2\left(n-1\right)$ fractions. What is the highest possible value of the characteristic ?

A student is playing computer. Computer shows randomly 2002 positive numbers. Game's rules let do the following operations - to take 2 numbers from these, to double first one, to add the second one and to save the sum. - to take another 2 numbers from the remainder numbers, to double the first one, to add the second one, to multiply this sum with previous and to save the result. - to repeat this procedure, until all the 2002 numbers won't be used. Student wins the game if final product is maximum possible. Find the winning strategy and prove it.

Let $x_0, x_1, x_2, \ldots$ be the sequence defined by $x_i= 2^i$ if $0 \leq i \leq 2003$ $x_i=\sum_{j=1}^{2004} x_{i-j}$ if $i \geq 2004$ Find the greatest $k$ for which the sequence contains $k$ consecutive terms divisible by 2004.

[u]Round 6[/u] 16. If you roll four fair $6$-sided dice, what is the probability that at least three of them will show the same value. 17. In a tetrahedron with volume $1$, four congruent speres are placed each tangent to three walls and three other spheres. What is the radii of each of the spheres. 18. let $f(x)$ be twice the number of letters in $x$. What is the sum of the unique $x,y$ such that $x \ne y$ and $f(x)=y$ and $f(y)=x$. [u]Round 7[/u] [b]p19.[/b] How many $4$ digit numbers with distinct digits $ABCD$ with $A$ not equal to $0$ are divisible by $11$? [b]p20.[/b] How many ($2$-dimensional) faces does a $2014$-dimensional hypercube have? [b]p21.[/b] How many subsets of the numbers $1,2,3,4...2^{2014}$ have a sum of $2014$ mod $2^{2014}$? [u]Round 8[/u] [b]p22.[/b] Two diagonals of a dodecagon measure $1$ unit and $2$ units. What is the area of this dodecagon? [b]p23.[/b] Square $ABCD$ has point $X$ on AB and $Y$ on $BC$ such that angle $ADX = 15$ degrees and angle $CDY = 30$ degrees. what is the degree measure of angle $DXY$? [b]p24.[/b] A $4\times 4$ grid has the numbers $1$ through $16$ placed in it, $1$ per cell, such that no adjacent boxes have cells adding to a number divisible by $3$. In how many ways is this possible? [u]Round 9[/u] [b]p25.[/b] Let $B$ and $C$ be the answers to $26$ and $27$ respectively.If $S(x)$ is the sum of the digits in $x$, what is the unique integer $A$ such that $S(A), S(B), S(C) \subset A,B,C$. [b]p26.[/b] Let $A$ and $C$ be the answers to $25$ and $27$ respectively. What is the third angle of a triangle with two of its angles equal to $A$ and $C$ degrees. [b]p27.[/b] Let $A$ and $B$ be the answers to $25$ and $26$ respectively. How many ways are there to put $A$ people in a line, with exactly $B$ places where a girl and a boy are next to each other. [u]Round 10[/u] [b]p28.[/b] What is the sum of all the squares of the digits to answers to problems on the individual, team, and theme rounds of this years LMT? If the correct answer is $N$ and you submit $M$, you will recieve $\lfloor 15 - 10  \times \log (M - N) \rfloor $. [b]p29.[/b] How many primes have all distinct digits, like $2$ or $109$ for example, but not $101$. If the correct answer is $N$ and you submit $M$, you will recieve $\left\lfloor 15 \min \left( \frac{M}{N} , \frac{N}{M} \right)\right\rfloor $. [b]p30.[/b] For this problem, you can use any $10$ mathematical symbols that you want, to try to achieve the highest possible finite number. (So "Twenty-one", " $\frac{12}{100} +843$" and "$\sum^{10}_{i=0} i^2 +1$" are all valid submissions.) If your team has the $N$th highest number, you will recieve $\max (16 - N, 0)$. PS. You should use hide for answers. Rounds 1-5 have been posted [url=https://artofproblemsolving.com/community/c3h3156859p28695035]here[/url].. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Points $A_1, A_2, \ldots, A_{2022}$ are chosen on a plane so that no three of them are collinear. Consider all angles $A_iA_jA_k$ for distinct points $A_i, A_j, A_k$. What largest possible number of these angles can be equal to $90^\circ$? [i]Proposed by Anton Trygub[/i]

For any integer $n \ge2$, we define $ A_n$ to be the number of positive integers $ m$ with the following property: the distance from $n$ to the nearest multiple of $m$ is equal to the distance from $n^3$ to the nearest multiple of $ m$. Find all integers $n \ge 2 $ for which $ A_n$ is odd. (Note: The distance between two integers $ a$ and $b$ is defined as $|a -b|$.)

A rectangle $ D$ is partitioned in several ($ \ge2$) rectangles with sides parallel to those of $ D$. Given that any line parallel to one of the sides of $ D$, and having common points with the interior of $ D$, also has common interior points with the interior of at least one rectangle of the partition; prove that there is at least one rectangle of the partition having no common points with $ D$'s boundary. [i]Author: Kei Irie, Japan[/i]

Pål has more chickens than he can manage to keep track of. Therefore, he keeps an index card for each chicken. He keeps the cards in ten boxes, each of which has room for $2021$ cards. Unfortunately, Pål is quite disorganized, so he may lose some of his boxes. Therefore, he makes several copies of each card and distributes them among different boxes, so that even if he can only find seven boxes, no matter which seven, these seven boxes taken together will contain at least one card for each of his chickens. What is the largest number of chickens Pål can keep track of using this system?