Let $ m$ and $ n$ be two positive integers. Let $ a_1$, $ a_2$, $ \ldots$, $ a_m$ be $ m$ different numbers from the set $ \{1, 2,\ldots, n\}$ such that for any two indices $ i$ and $ j$ with $ 1\leq i \leq j \leq m$ and $ a_i \plus{} a_j \leq n$, there exists an index $ k$ such that $ a_i \plus{} a_j \equal{} a_k$. Show that \[ \frac {a_1 \plus{} a_2 \plus{} ... \plus{} a_m}{m} \geq \frac {n \plus{} 1}{2}. \]

Let $p$ be a prime number and $a_1, a_2, \ldots, a_{(p+1)/2}$ different natural numbers less than or equal to $p.$ Prove that for each natural number $r$ less than or equal to $p$, there exist two numbers (perhaps equal) $a_i$ and $a_j$ such that \[p \equiv a_i a_j \pmod r.\]

Show that every rational number $r$ can be written as the sum of numbers in the form $\frac{a}{p^k}$ where $p$ is prime, $a$ is an integer and $k$ is natural.

A shipping company charges $.30l+.40w+.50h$ dollars to process a right rectangular prism-shaped box with dimensions $l,w,h$ in inches. The customers themselves are allowed to label the three dimensions of their box with $l,w,h$ for the purpose of calculating the processing fee. A customer finds that there are two different ways to label the dimensions of their box $B$ to get a fee of $\$8.10$, and two different ways to label $B$ to get a fee of $\$8.70$. None of the faces of $B$ are squares. Find the surface area of $B$, in square inches. [i]Proposed by Justin Hsieh[/i]

Prove that $n^2 + 8n + 15$ is not divisible by $n + 4$ for any positive integer $n$.

Find all positive integers $m,n$ such that $m^2+5n$ and $n^2+5m$ are perfect squares.

Determine all pairs $ (n,p)$ of positive integers, where $ p$ is prime, such that $ 3^p\minus{}np\equal{}n\plus{}p$.

Find all triples $(x,y,n)$ of positive integers such that \[ (x+y)(1+xy) = 2^{n} \]

Let $p$ be an odd prime. Show that for every integer $c$, there exists an integer $a$ such that $$a^{\frac{p+1}{2}} + (a+c)^{\frac{p+1}{2}} \equiv c\pmod p.$$

An [i]arithmetic sequence[/i] is an infinite sequence of the form $a_n=a_0+n\cdot d$ with $d\neq 0$. A [i]geometric sequence[/i] is an infinite sequence of the form $b_n=b_0 \cdot q^n$ where $q\neq 1,0,-1$. [list=a] [*] Does every arithmetic sequence of [b]integers[/b] have an infinite subsequence which is geometric? [*] Does every arithmetic sequence of [b]real numbers[/b] have an infinite subsequence which is geometric? [/list]

Let $(a_n), n = 0, 1, . . .,$ be a sequence of real numbers such that $a_0 = 0$ and \[a^3_{n+1} = \frac{1}{2} a^2_n -1, n= 0, 1,\cdots\] Prove that there exists a positive number $q, q < 1$, such that for all $n = 1, 2, \ldots ,$ \[|a_{n+1} - a_n| \leq q|a_n - a_{n-1}|,\] and give one such $q$ explicitly.

Let $n \ge 3$ be a positive integer. For every set $S$ with $n$ distinct positive integers, prove that there exists a bijection $f: \{1,2, \cdots n\} \rightarrow S$ which satisfies the following condition. For all $1 \le i < j < k \le n$, $f(j)^2 \neq f(i) \cdot f(k)$.

Let $S=\{1,2,3,\ldots,280\}$. Find the smallest integer $n$ such that each $n$-element subset of $S$ contains five numbers which are pairwise relatively prime.

From each set $ \{a_1,a_2,...,a_7\} \subset Z$ one can choose a number of elements whose sum is a multiple of $7$.

Find all prime numbers $p,q$ such that: $$p^4+p^3+p^2+p=q^2+q$$

[b]p1.[/b] A square is tiled by smaller squares as shown in the figure. Find the area of the black square in the middle if the perimeter of the square $ABCD$ is $14$ cm. [img]https://cdn.artofproblemsolving.com/attachments/1/1/0f80fc5f0505fa9752b5c9e1c646c49091b4ca.png[/img] [b]p2.[/b] If $a, b$, and $c$ are numbers so that $a + b + c = 0$ and $a^2 + b^2 + c^2 = 1$. Compute $a^4 + b^4 + c^4$. [b]p3.[/b] A given fraction $\frac{a}{b}$ ($a, b$ are positive integers, $a \ne b$) is transformed by the following rule: first, $1$ is added to both the numerator and the denominator, and then the numerator and the denominator of the new fraction are each divided by their greatest common divisor (in other words, the new fraction is put in simplest form). Then the same transformation is applied again and again. Show that after some number of steps the denominator and the numerator differ exactly by $1$. [b]p4.[/b] A goat uses horns to make the holes in a new $30\times 60$ cm large towel. Each time it makes two new holes. Show that after the goat repeats this $61$ times the towel will have at least two holes whose distance apart is less than $6$ cm. [b]p5.[/b] You are given $555$ weights weighing $1$ g, $2$ g, $3$ g, $...$ , $555$ g. Divide these weights into three groups whose total weights are equal. [b]p6.[/b] Draw on the regular $8\times 8$ chessboard a circle of the maximal possible radius that intersects only black squares (and does not cross white squares). Explain why no larger circle can satisfy the condition. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Suppose $n$ is a perfect square. Consider the set of all numbers which is the product of two numbers, not necessarily distinct, both of which are at least $n$. Express the $n-$th smallest number in this set in terms of $n$.

Find all positive integers $b$ for which there exists a positive integer $a$ with the following properties: - $a$ is not a divisor of $b$. - $a^a$ is a divisor of $b^b$

In some magic country, there are banknotes only of values $3$, $25$, $80$ hryvnyas. Businessman Victor ate in one restaurant of this country for $2024$ days in a row, and each day (except the first) he spent exactly $1$ hryvnya more than the day before (without any change). Could he have spent exactly $1000000$ banknotes? [i](Proposed by Oleksii Masalitin)[/i]

Find all ordered 6-tuples satisfy following system of modular equation: $ab+a'b' \equiv 1 $(mod 15) $bc+b'c' \equiv 1 $(mod 15) $ca+c'a' \equiv 1 $(mod 15) Given that $a,b,c,a',b',c' \epsilon (0;1;2;...;14)$

Let $ a$, $ b$, $ c$, $ d$, $ e$, $ f$ be positive integers and let $ S = a+b+c+d+e+f$. Suppose that the number $ S$ divides $ abc+def$ and $ ab+bc+ca-de-ef-df$. Prove that $ S$ is composite.

Let $[r,s]$ denote the least common multiple of positive integers $r$ and $s$. Find the number of ordered triples $(a,b,c)$ of positive integers for which $[a,b] = 1000$, $[b,c] = 2000$, and $[c,a] = 2000$

Prove that for all primes \( p \) such that \( p \equiv 3 \pmod{4} \) or \( p \equiv 5 \pmod{8} \), there exist integers \[ 1 \leq a_1 < a_2 < \cdots < a_{(p-1)/2} < p \] such that \[ \prod_{\substack{1 \leq i < j \leq (p-1)/2}} (a_i + a_j)^2 \equiv 1 \pmod{p}. \]

1. Find $N$, the smallest positive multiple of $45$ such that all of its digits are either $7$ or $0$. 2. Find $M$, the smallest positive multiple of $32$ such that all of its digits are either $6$ or $1$. 3. How many elements of the set $\{1,2,3,...,1441\}$ have a positive multiple such that all of its digits are either $5$ or $2$?

Let $n$ be a positive integer and $a_1, a_2, \dots , a_{2n}$ mutually distinct integers. Find all integers $x$ satisfying \[(x - a_1) \cdot (x - a_2) \cdots (x - a_{2n}) = (-1)^n(n!)^2.\]