Eight consecutive positive integers are divided into 2 sets, such that the sum of the squares of the elements in the first set is equal to the sum of the squares of the elements in the second set. Prove that the sum of the lements in the first set is equal to the sum of the elements in the second one.

Find all the pairs of prime numbers $ (p,q)$ such that $ pq|5^p\plus{}5^q.$

Let \( n > 1 \) be a positive integer. List in increasing order all the irreducible fractions in the interval \([0, 1]\) that have a positive denominator less than or equal to \( n \): \[ \frac{0}{1} = \frac{p_0}{q_0} < \frac{p_1}{q_1} < \cdots < \frac{p_M}{q_M} = \frac{1}{1}. \] Determine, in function of \( n \), the smallest possible value of \( q_{i-1} + q_i + q_{i+1} \), for \( 0 < i < M \). For example, if \( n = 4 \), the enumeration is \[ \frac{0}{1} < \frac{1}{4} < \frac{1}{3} < \frac{1}{2} < \frac{2}{3} < \frac{3}{4} < \frac{1}{1}, \] where \( p_0 = 0, p_1 = 1, p_2 = 1, p_3 = 1, p_4 = 2, p_5 = 3, p_6 = 1, q_0 = 1, q_1 = 4, q_2 = 3, q_3 = 2, q_4 = 3, q_5 = 4, q_6 = 1 \), and the minimum is \( 1 + 4 + 3 = 3 + 2 + 3 = 3 + 4 + 1 = 8 \).

Let's call a ticket with a number from $000000$ to $999999$ [i]excellent [/i] if the difference between some two adjacent digits is $5$. Find the number of excellent tickets.

Let $n \ge 2$ be an integer, and let $A_n$ be the set \[A_n = \{2^n - 2^k\mid k \in \mathbb{Z},\, 0 \le k < n\}.\] Determine the largest positive integer that cannot be written as the sum of one or more (not necessarily distinct) elements of $A_n$ . [i]Proposed by Serbia[/i]

Let $a,b$ be positive integers.Prove that there are no positive integers on the interval $\bigg[\frac{b^2}{a^2+ab},\frac{b^2}{a^2+ab-1}\bigg)$.

$p$ is a polynomial with integer coefficients and for every natural $n$ we have $p(n)>n$. $x_k $ is a sequence that: $x_1=1, x_{i+1}=p(x_i)$ for every $N$ one of $x_i$ is divisible by $N.$ Prove that $p(x)=x+1$

Let $a_1,a_2,\ldots,a_n$ are different integer numbers in the range $[100,200]$ for which: $a_1+a_2+\ldots+a_n\ge11100$. Prove that it can be found at least number from the given in the representation of decimal system on which there are at least two equal digits. [i]L. Davidov[/i]

Find all functions $f:\mathbb{N} \rightarrow \mathbb{N}$, such that $f(a)+f(b)+ab \mid a^2f(a)+b^2f(b)+f(a)f(b)$ for all positive integers $a,b$.

Joe and Kathryn are both on the school math team, which practices every Wednesday after school until $4$ PM for competitions. The team was preparing for the $ 2005$ iTest when Joe realized how crazy he was for not asking Kathryn out – the way she worked those iTest problems, solving question after question, almost made him go insane sitting there that day. He never felt the same way when she worked on preparing for other competitions – they just aren’t the same. Kathryn always beat Joe at competitions, too. Joe admired her resolve and unwillingness to make herself look stupid, when so many other girls he knew at school tried to pretend they were stupid in order to attract guys. So as time ticked away and that afternoon’s Wednesday practice neared an end, Joe was determined to strike up a conversation with Kathryn and ask her out. He really wanted to impress her, so he thought he’d ask her a really hard history of math question that she didn’t know. Naturally, she’d want the answer, and be so impressed with Joe’s brilliance that she’d go out with him on Friday night. Great plan. Seriously. When Joe asked Kathryn after class, “Who was the mathematician that died in approximately $200$ B.C. that developed a method for calculating all prime numbers?” Kathryn gave the correct response. What name did she say?

Find all integers $n$ (not necessarily positive) such that $7n^3-3n^2-3n-1$ is a perfect cube.

Which of the following are true? $\textbf{(A)}~\text{For every }n\in\mathbb N,n^3-n\text{ is divisible by }6$ $\textbf{(B)}~\text{For every }n\in\mathbb N,n^7-n\text{ is divisible by }42$ $\textbf{(C)}~\text{Every perfect square is of the form }3m\text{ or }3m+1\text{ for some }n\in\mathbb N$ $\textbf{(D)}~\text{None of the above}$

Find the smallest nomial of this sequence that $a_1=1993^{1994^{1995}}$ and \[ a_{n+1}=\begin{cases}\frac{a_n}{2}&\text{if $n$ is even}\\a_n+7 &\text{if $n$ is odd.} \end{cases} \]

[b]p7.[/b] An ant starts at the point $(0, 0)$. It travels along the integer lattice, at each lattice point choosing the positive $x$ or $y$ direction with equal probability. If the ant reaches $(20, 23)$, what is the probability it did not pass through $(20, 20)$? [b]p8.[/b] Let $a_0 = 2023$ and $a_n$ be the sum of all divisors of $a_{n-1}$ for all $n \ge 1$. Compute the sum of the prime numbers that divide $a_3$. [b]p9.[/b] Five circles of radius one are stored in a box of base length five as in the following diagram. How far above the base of the box are the upper circles touching the sides of the box? [img]https://cdn.artofproblemsolving.com/attachments/7/c/c20b5fa21fbd8ce791358fd888ed78fcdb7646.png[/img] PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $n$ be a positive integer. From the set $A=\{1,2,3,...,n\}$ an element is eliminated. What's the smallest possible cardinal of $A$ and the eliminated element, since the arithmetic mean of left elements in $A$ is $\frac{439}{13}$.

For which $n$ is $n^4+6n^3+11n^2+3n+31$ a perfect square?

A [i]peacock [/i] is a ten-digit positive integer that uses each digit exactly once. Compute the number of peacocks that are exactly twice another peacock.

Sergei chooses two different natural numbers $a$ and $b$. He writes four numbers in a notebook: $a$, $a+2$, $b$ and $b+2$. He then writes all six pairwise products of the numbers of notebook on the blackboard. Let $S$ be the number of perfect squares on the blackboard. Find the maximum value of $S$. [i]S. Berlov[/i]

The sequence of natural numbers $1, 5, 6, 25, 26, 30, 31,...$ is made up of powers of $5$ with natural exponents or sums of powers of $5$ with different natural exponents, written in ascending order. Determine the term of the string written in position $167$.

Let $a_1,\ldots ,a_{10}$ be distinct positive integers, all at least $3$ and with sum $678$. Does there exist a positive integer $n$ such that the sum of the $20$ remainders of $n$ after division by $a_1,a_2,\ldots ,a_{10},2a_1,2a_2,\ldots ,2a_{10}$ is $2012$?

Let $ n\ge 2 $ be a natural number and $ A $ be a subset of $ \{1,2,\ldots ,n\} $ having the property that $ x+y $ belongs to $ A $ for any choosing of $ x,y $ such that $ x+y\le n. $ Prove that the arithmetic mean of the elements of $ A $ is at least $ \frac{n+1}{2} . $

[b]p4.[/b] For how many three-digit multiples of $11$ in the form $\underline{abc}$ does the quadratic $ax^2 + bx + c$ have real roots? [b]p5.[/b] William draws a triangle $\vartriangle ABC$ with $AB =\sqrt3$, $BC = 1$, and $AC = 2$ on a piece of paper and cuts out $\vartriangle ABC$. Let the angle bisector of $\angle ABC$ meet $AC$ at point $D$. He folds $\vartriangle ABD$ over $BD$. Denote the new location of point $A$ as $A'$. After William folds $\vartriangle A'CD$ over $CD$, what area of the resulting figure is covered by three layers of paper? [b]p6.[/b] Compute $(1)(2)(3) + (2)(3)(4) + ... + (18)(19)(20)$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Consider the following sequence of positive real numbers $\dots<a_{-2}<a_{-1}<a_0<a_1<a_2<\dots$ infinite in both directions. For each positive integer $k$ let $b_k$ be the least integer such that the ratio between the sum of $k$ consecutive terms and the greatest of these $k$ terms is less than or equal to $b_k$(This fact occurs for any sequence of $k$ consecutive numbers). Prove that the sequence $b_1,b_2,b_3,...$ coincides with the sequence $1,2,3,...$ or is eventually constant.

Find all positive integers $n$ such that $\phi(n)$ is the fourth power of some prime.

Using only prime numbers, a set is formed with the following conditions: Any one-digit prime number can be in the set. For a prime number with more than one digit to be in the set, the number that results from deleting only the first digit and also the number that results from deleting only the last digit must be in the set. Write, of the sets that meet these conditions, the one with the greatest number of elements. Justify why there cannot be one with more elements. Remember that the number $1$ is not prime.