Prove that $2005^2$ can be written in at least $4$ ways as the sum of 2 perfect (non-zero) squares.

A twin prime pair is a pair of primes $(p,q)$ such that $q = p + 2$. The Twin Prime Conjecture states that there are infinitely many twin prime pairs. What is the arithmetic mean of the two primes in the smallest twin prime pair? (1 is not a prime.) $\textbf{(A) }4$

For what positive integers $n$ are there positive rational, but not integer, numbers $a$ and $b$ such that both numbers $a + b$ and $a^n + b^n$ are integers?

Denote by $l(n)$ the largest prime divisor of $n$. Let $a_{n+1} = a_n + l(a_n)$ be a recursively defined sequence of integers with $a_1 = 2$. Determine all natural numbers $m$ such that there exists some $i \in \mathbb{N}$ with $a_i = m^2$. [i]Proposed by Nikola Velov, North Macedonia[/i]

Find all triples $m$; $n$; $l$ of natural numbers such that $m + n = gcd(m; n)^2$; $m + l = gcd(m; l)^2$; $n + l = gcd(n; l)^2$: [i]S. Tokarev[/i]

Find all positive integers $n$ and $k$ such that $(n+1)^n=2n^k+3n+1$.

Find all positive integers $ n $ such that there exists an integer $ m $ satisfying $$ \frac{1}{n}\sum\limits_{k=m}^{m+n-1}{k^2} $$ is a perfect square.

For every integer $ k \geq 2,$ prove that $ 2^{3k}$ divides the number \[ \binom{2^{k \plus{} 1}}{2^{k}} \minus{} \binom{2^{k}}{2^{k \minus{} 1}} \] but $ 2^{3k \plus{} 1}$ does not. [i]Author: Waldemar Pompe, Poland[/i]

We denote the number of positive divisors of a positive integer $m$ by $d(m)$ and the number of distinct prime divisors of $m$ by $\omega(m)$. Let $k$ be a positive integer. Prove that there exist infinitely many positive integers $n$ such that $\omega(n) = k$ and $d(n)$ does not divide $d(a^2+b^2)$ for any positive integers $a, b$ satisfying $a + b = n$.

Determine the least possible value of the natural number $n$ such that $n!$ ends in exactly $1987$ zeros. [hide="Note"]Note. Here (and generally in MathLinks) natural numbers supposed to be positive.[/hide]

A natural number $a$ is said [i]to be contained[/i] in the natural number $b$ if it is possible to obtain a by erasing some digits from $b$ (in their decimal representations). For example, $123$ is contained in $901523$, but not contained in $3412$. Does there exist an infinite set of natural numbers such that no number in the set is contained in any other number from the set?

[b]p1.[/b] Evaluate $1+3+5+··· +2019$. [b]p2.[/b] Evaluate $1^2 -2^2 +3^2 -4^2 +...· +99^2 -100^2$. [b]p3. [/b]Find the sum of all solutions to $|2018+|x -2018|| = 2018$. [b]p4.[/b] The angles in a triangle form a geometric series with common ratio $\frac12$ . Find the smallest angle in the triangle. [b]p5.[/b] Compute the number of ordered pairs $(a,b,c,d)$ of positive integers $1 \le a,b,c,d \le 6$ such that $ab +cd$ is a multiple of seven. [b]p6.[/b] How many ways are there to arrange three birch trees, four maple, and five oak trees in a row if trees of the same species are considered indistinguishable. [b]p7.[/b] How many ways are there for Mr. Paul to climb a flight of 9 stairs, taking steps of either two or three at a time? [b]p8.[/b] Find the largest natural number $x$ for which $x^x$ divides $17!$ [b]p9.[/b] How many positive integers less than or equal to $2018$ have an odd number of factors? [b]p10.[/b] Square $MAIL$ and equilateral triangle $LIT$ share side $IL$ and point $T$ is on the interior of the square. What is the measure of angle $LMT$? [b]p11.[/b] The product of all divisors of $2018^3$ can be written in the form $2^a \cdot 2018^b$ for positive integers $a$ and $b$. Find $a +b$. [b]p12.[/b] Find the sum all four digit palindromes. (A number is said to be palindromic if its digits read the same forwards and backwards. [b]p13.[/b] How ways are there for an ant to travel from point $(0,0)$ to $(5,5)$ in the coordinate plane if it may only move one unit in the positive x or y directions each step, and may not pass through the point $(1, 1)$ or $(4, 4)$? [b]p14.[/b] A certain square has area $6$. A triangle is constructed such that each vertex is a point on the perimeter of the square. What is the maximum possible area of the triangle? [b]p15.[/b] Find the value of ab if positive integers $a,b$ satisfy $9a^2 -12ab +2b^2 +36b = 162$. [b]p16.[/b] $\vartriangle ABC$ is an equilateral triangle with side length $3$. Point $D$ lies on the segment $BC$ such that $BD = 1$ and $E$ lies on $AC$ such that $AE = AD$. Compute the area of $\vartriangle ADE$. [b]p17[/b]. Let $A_1, A_2,..., A_{10}$ be $10$ points evenly spaced out on a line, in that order. Points $B_1$ and $B_2$ lie on opposite sides of the perpendicular bisector of $A_1A_{10}$ and are equidistant to $l$. Lines $B_1A_1,...,B_1A_{10}$ and $B_2A_1,...· ,B_2A_{10}$ are drawn. How many triangles of any size are present? [b]p18.[/b] Let $T_n = 1+2+3··· +n$ be the $n$th triangular number. Determine the value of the infinite sum $\sum_{k\ge 1} \frac{T_k}{2^k}$. [b]p19.[/b] An infinitely large bag of coins is such that for every $0.5 < p \le 1$, there is exactly one coin in the bag with probability $p$ of landing on heads and probability $1- p$ of landing on tails. There are no other coins besides these in the bag. A coin is pulled out of the bag at random and when flipped lands on heads. Find the probability that the coin lands on heads when flipped again. [b]p20.[/b] The sequence $\{x_n\}_{n\ge 1}$ satisfies $x1 = 1$ and $(4+ x_1 + x_2 +··· + x_n)(x_1 + x_2 +··· + x_{n+1}) = 1$ for all $n \ge 1$. Compute $\left \lfloor \frac{x_{2018}}{x_{2019}} \right \rfloor$. PS. You had better use hide for answers.

Consider the sets $M = \{0,1,2,, 2019\}$ and $$A=\left\{ x\in M\,\, | \frac{x^3-x}{24} \in N\right\} $$ a) How many elements does the set $A$ have? b) Determine the smallest natural number $n$, $n \ge 2$, which has the property that any $n$-element subset of the set $A $contains two distinct elements whose difference is divisible by $40$.

Jeff writes down the two-digit base-$10$ prime $\overline{ab_{10}}$. He realizes that if he misinterprets the number as the base $11$ number $\overline{ab_{11}}$ or the base $12$ number $\overline{ab_{12}}$, it is still a prime. What is the least possible value of Jeff’s number (in base $10$)? [i]Proposed byMuztaba Syed[/i]

Show that for each positive integer n, \[n!=\prod_{i=1}^n \; \text{lcm} \; \{1, 2, \ldots, \left\lfloor\frac{n}{i} \right\rfloor\}\] (Here lcm denotes the least common multiple, and $\lfloor x\rfloor$ denotes the greatest integer $\le x$.)

Show that for every integer $n\geq2$ there are two distinct powers of $n$ such that their sum is greater than $10^{2023}$ and their positive difference is divisible with $2023$.

[b]p1.[/b] The final Big $10$ standings for the $1996$ Women's Softball season were 1. Michigan 2. Minnesota З. Iowa 4. Indiana 5. Michigan State 6. Purdue 7. Northwestern 8. Ohio State 9. Penn State 10. Wisconsin (Illinois does not participate in Women's Softball.) When you compare the $1996$ final standings (above) to the final standings for the $1999$ season, you find that the following pairs of teams changed order relative to each other from $1996$ to $1999$ (there are no ties, and no other pairs changed places): (Iowa, Michigan State) (Indiana, Penn State) (Purdue, Wisconsin) (Iowa, Penn State) (Indiana, Wisconsin) (Northwestern, Penn State) (Indiana, Michigan State) (Michigan State, Penn State) (Northwestern, Wisconsin) (Indiana, Purdue) (Purdue, Northwestern) (Ohio State, Penn State) (Indiana, Northwestern) (Purdue, Penn State) (Ohio State, Penn State) (Indiana, Ohio State) Determine as much as you can about the final Big $10$ standings for the $1999$ Women's Softball season. If you cannot determine the standings, explain why you do not have enough information. You must justify your answer. [b]p2.[/b] a) Take as a given that any expression of the form $A \sin t + B \cos t$ ($A>0$) can be put in the form $C \sin (t + D)$, where $C>0$ and $-\pi /2 <D <\pi /2 $. Determine $C$ and $D$ in terms of $A$ and $B$. b) For the values of $C$ and $D$ found in part a), prove that $A \sin t + B \cos t = C \sin (t + D)$. c) Find the maximum value of $3 \sin t +2 \cos t$. [b]pЗ.[/b] А $6$-bу-$6$ checkerboard is completelу filled with $18$ dominoes (blocks of size $1$-bу-$2$). Prove that some horizontal or vertical line cuts the board in two parts but does not cut anу of the dominoes. [b]p4.[/b] a) The midpoints of the sides of a regular hexagon are the vertices of a new hexagon. What is the ratio of the area of the new hexagon to the area of the original hexagon? Justify your answer and simplify as much as possible. b) The midpoints of the sides of a regular $n$-gon ($n >2$) are the vertices of a new $n$-gon. What is the ratio of the area of the new $n$-gon to that of the old? Justify your answer and simplify as much as possible. [b]p5. [/b] You run a boarding house that has $90$ rooms. You have $100$ guests registered, but on any given night only $90$ of these guests actually stay in the boarding house. Each evening a different random set of $90$ guests will show up. You don't know which $90$ it will be, but they all arrive for dinner before you have to assign rooms for the night. You want to give out keys to your guests so that for any set of $90$ guests, you can assign each to a private room without any switching of keys. a) You could give every guest a key to every room. But this requires $9000$ keys. Find a way to hand out fewer than $9000$ keys so that each guest will have a key to a private room. b) What is the smallest number of keys necessary so that each guest will have a key to a private room? Describe how you would distribute these keys and assign the rooms. Prove that this number of keys is as small as possible. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $ S(n)$ be the sum of decimal digits of a natural number $ n$. Find the least value of $ S(m)$ if $ m$ is an integral multiple of $ 2003$.

Given an integer ${n>1}$, denote by $P_{n}$ the product of all positive integers $x$ less than $n$ and such that $n$ divides ${x^2-1}$. For each ${n>1}$, find the remainder of $P_{n}$ on division by $n$. [i]Proposed by John Murray, Ireland[/i]

Let $P_1, \ldots , P_s$ be arithmetic progressions of integers, the following conditions being satisfied: [b](i)[/b] each integer belongs to at least one of them; [b](ii)[/b] each progression contains a number which does not belong to other progressions. Denote by $n$ the least common multiple of the ratios of these progressions; let $n=p_1^{\alpha_1} \cdots p_k^{\alpha_k}$ its prime factorization. Prove that \[s \geq 1 + \sum^k_{i=1} \alpha_i (p_i - 1).\] [i]Proposed by Dierk Schleicher, Germany[/i]

How many three-digit numbers are there, which do not have a zero in their decimal representation and whose sum of digits is $7$?

Determine all natural $ n$ for which there exists natural $ m$ divisible by all natural numbers from 1 to $ n$ but not divisible by any of the numbers $ n \plus{} 1$, $ n \plus{} 2$, $ n \plus{} 3$.

Find all fractions which can be written simultaneously in the forms $\frac{7k- 5}{5k - 3}$ and $\frac{6l - 1}{4l - 3}$ , for some integers $k, l$.

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 \[ \frac {a_1 \plus{} a_2 \plus{} ... \plus{} a_m}{m} \geq \frac {n \plus{} 1}{2}. \]

Prove that: a) there are infinitely many natural numbers of the form 3p + 1, p is positive integer , which can be represented as the difference of 2 cubes of positive integers; b) there are infinitely many natural numbers of the form 5q + 1, q is positive integer , which can be represented as the difference of two cubes of positive integers.