Let $a, b, c, d, e$ $(a < b < c < d < e)$be positive integers. FInd the greatest possible value of the expression $\frac{1}{[a,b]} + \frac{1}{[b,c]} + \frac{1}{[c,d]} + \frac{1}{[d,e]}$, where $[x,y]$ denotes the least common multiple of $x{}$ and $y{}$.

Joe the teacher is bad at rounding. Because of this, he has come up with his own way to round grades, where a [i]grade[/i] is a nonnegative decimal number with finitely many digits after the decimal point. Given a grade with digits $a_1a_2 \dots a_m.b_1b_2 \dots b_n$, Joe first rounds the number to the nearest $10^{-n+1}$th place. He then repeats the procedure on the new number, rounding to the nearest $10^{-n+2}$th, then rounding the result to the nearest $10^{-n+3}$th, and so on, until he obtains an integer. For example, he rounds the number $2014.456$ via $2014.456 \to 2014.46 \to 2014.5 \to 2015$. There exists a rational number $M$ such that a grade $x$ gets rounded to at least $90$ if and only if $x \ge M$. If $M = \tfrac pq$ for relatively prime integers $p$ and $q$, compute $p+q$. [i]Proposed by Yang Liu[/i]

One of two wizards, named Steven, was told the sum of three positive integers and the other, named Peter, their product. “If I knew”, said Steven, “that your number is greater than mine, I could find the integers”. “But my number is less than yours,” replied Peter, “and the integers are $X$, $Y$ and $Z$”. Find these numbers. (L Borisov)

A [i]trio [/i] is a set of three distinct integers such that two of the numbers are divisors or multiples of the third. Which [i]trio [/i] contained in $\{1, 2, ... , 2002\}$ has the largest possible sum? Find all [i]trios [/i] with the maximum sum.

For which real numbers $p$ does the equation $x^{2}+px+3p=0$ have integer solutions ?

For a set $S$ we denote its cardinality by $|S|$. Let $e_1,e_2,\ldots,e_k$ be non-negative integers. Let $A_k$ (respectively $B_k$) be the set of all $k$-tuples $(f_1,f_2,\ldots,f_k)$ of integers such that $0\leq f_i\leq e_i$ for all $i$ and $\sum_{i=1}^k f_i$ is even (respectively odd). Show that $|A_k|-|B_k|=0 \textrm{ or } 1$.

Given that $p$ and $p^4 + 34$ are both prime numbers, compute $p$.

Let \(m\) be a positive integer. Find, in terms of \(m\), all polynomials \(P(x)\) with integer coefficients such that for every integer \(n\), there exists an integer \(k\) such that \(P(k)=n^m\). [i]Proposed by Raymond Feng[/i]

Marko chose two prime numbers $a$ and $b$ with the same number of digits and wrote them down one after another, thus obtaining a number $c$. When he decreased $c$ by the product of $a$ and $b$, he got the result $154$. Determine the number $c$.

Define a sequence $\{a_n\}_{n\ge 1}$ such that $a_1=1,a_2=2$ and $a_{n+1}$ is the smallest positive integer $m$ such that $m$ hasn't yet occurred in the sequence and also $\text{gcd}(m,a_n)\neq 1$. Show all positive integers occur in the sequence.

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 $.

[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].

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 $E(9,6,m_0)$.

Compute the sum of the two smallest positive integers $b$ with the following property: there are at least ten integers $0 \le n < b$ such that $n^2$ and $n$ end in the same digit in base $b$.

Let $P$ be the set of all prime numbers. Let $A$ be some subset of $P$ that has at least two elements. Let's say that for every positive integer $n$ the following statement holds: If we take $n$ different elements $p_1,p_2...p_n \in A$, every prime number that divides $p_1 p_2 \cdots p_n-1$ is also an element of $A$. Prove, that $A$ contains all prime numbers.

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 $|a -b|$.)

For how many three-element sets of positive integers $\{a,b,c\}$ is it true that $a \times b \times c = 2310$? $\textbf{(A)}\ 32 \qquad \textbf{(B)}\ 36 \qquad \textbf{(C)}\ 40 \qquad \textbf{(D)}\ 43 \qquad \textbf{(E)}\ 45$

Show that $240$ divides all numbers of the form $p^4 - q^4$, where p and q are prime numbers strictly greater than $5$. Show also that $240$ is the greatest common divisor of all numbers of the form $p^4 - q^4$, with $p$ and $q$ prime numbers strictly greater than $5$.

Determine all real numbers $x$, such that $x^n+x^{-n}$ is an integer for all integers $n$.

Let $f$ be a non-constant function from the set of positive integers into the set of positive integer, such that $a-b$ divides $f(a)-f(b)$ for all distinct positive integers $a$, $b$. Prove that there exist infinitely many primes $p$ such that $p$ divides $f(c)$ for some positive integer $c$. [i]Proposed by Juhan Aru, Estonia[/i]

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.

Let $p_n$ be the $n^{\mbox{th}}$ prime counting from the smallest prime $2$ in increasing order. For example, $p_1=2, p_2=3, p_3 =5, \cdots$ (a) For a given $n \ge 10$, let $r$ be the smallest integer satisfying \[2\le r \le n-2, \quad n-r+1 < p_r\] and define $N_s=(sp_1p_2\cdots p_{r-1})-1$ for $s=1,2,\ldots, p_r$. Prove that there exists $j, 1\le j \le p_r$, such that none of $p_1,p_2,\cdots, p_n$ divides $N_j$. (b) Using the result of (a), find all positive integers $m$ for which \[p_{m+1}^2 < p_1p_2\cdots p_m\]

Find all functions $f:\mathbb{Z}_{\ge 1}\to \mathbb{R}_{>0}$ such that for all positive integers $n$ the following relation holds: $$\sum_{d|n} f(d)^3=\left (\sum_{d|n} f(d) \right )^2,$$ where both sums are taken over the positive divisors of $n$. [i] (Vlad Robu) [/i]

Find all triples $(a,b,c)$ such that $a,b,c$ are integers in the set $\{2000,2001,\ldots,3000\}$ satisfying $a^2+b^2=c^2$ and $\text{gcd}(a,b,c)=1$.

Do there exist positive integers $m,n$, such that $m^{20}+11^n$ is a square number?