Determine all integers $ n\geq 2$ having the following property: for any integers $a_1,a_2,\ldots, a_n$ whose sum is not divisible by $n$, there exists an index $1 \leq i \leq n$ such that none of the numbers $$a_i,a_i+a_{i+1},\ldots,a_i+a_{i+1}+\ldots+a_{i+n-1}$$ is divisible by $n$. Here, we let $a_i=a_{i-n}$ when $i >n$. [i]Proposed by Warut Suksompong, Thailand[/i]

Show that a triangle whose side lengths are prime numbers cannot have integer area.

(a) Find all pairs $(x,k)$ of positive integers such that $3^k -1 = x^3$ . (b) Prove that if $n > 1$ is an integer, $n \ne 3$, then there are no pairs $(x,k)$ of positive integers such that $3^k -1 = x^n$.

An infinite arithmetic progression whose terms are positive integers contains the square of an integer and the cube of an integer. Show that it contains the sixth power of an integer.

A finite set $S$ of positive integers is given. Show that there is a positive integer $N$ dependent only on $S$, such that any $x_1, \dots, x_m \in S$ whose sum is a multiple of $N$, can be partitioned into groups each of whose sum is exactly $N$. (The numbers $x_1, \dots, x_m$ need not be distinct.)

Given $2021$ distinct positive integers non divisible by $2^{1010}$, show that it's always possible to choose $3$ of them $a$, $b$ and $c$, such that $|b^2-4ac|$ is not a perfect square.

For a prime $p$, a natural number $n$ and an integer $a$, we let $S_n(a,p)$ denote the exponent of $p$ in the prime factorisation of $a^{p^n} - 1$. For example, $S_1(4,3) = 2$ and $S_2(6,2) = 0$. Find all pairs $(n,p)$ such that $S_n(2013,p) = 100$.

In triangle $ ABC$, angle $ C$ is a right angle and $ CB > CA$. Point $ D$ is located on $ \overline{BC}$ so that angle $ CAD$ is twice angle $ DAB$. If $ AC/AD \equal{} 2/3$, then $ CD/BD \equal{} m/n$, where $ m$ and $ n$ are relatively prime positive integers. Find $ m \plus{} n$. $ \textbf{(A)}\ 10\qquad \textbf{(B)}\ 14\qquad \textbf{(C)}\ 18\qquad \textbf{(D)}\ 22\qquad \textbf{(E)}\ 26$

Find the number of integer $n$ from the set $\{2000,2001,...,2010\}$ such that $2^{2n} + 2^n + 5$ is divisible by $7$ (A): $0$, (B): $1$, (C): $2$, (D): $3$, (E) None of the above.

Find the number of three-digit palindromes that are divisible by $3$. Recall that a palindrome is a number that reads the same forward and backward like $727$ or $905509$.

Does there exists a subset of positive integers with infinite members such that for every two members $a,b$ of this set \[a^2-ab+b^2|(ab)^2\]

Show that for any $n > 5$ we can find positive integers $x_1, x_2, ... , x_n$ such that $\frac{1}{x_1} + \frac{1}{x_2} +... + \frac{1}{x_n} = \frac{1997}{1998}$. Show that in any such equation there must be two of the $n$ numbers with a common divisor ($> 1$).

Find three different positive integers whose sum is minimum than meet the condition that the sum of each pair of them is a perfect square.

$\text{(i)}$ Does there exist a positive integer $m > 2016^{2016}$ such that $\frac{2016^m-m^{2016}}{m+2016}$ is a positive integer? $\text{(ii)}$ Does there exist a positive integer $m > 2017^{2017}$ such that $\frac{2017^m-m^{2017}}{m+2017}$ is a positive integer? [i](Serbia MO 2016 P1)[/i]

Prove that if the integers $ a $ and $ b $ satisfy the equation $$ 2a^2 + a = 3b^2 + b,$$ then the numbers $ a - b $ and $ 2a + 2b + 1 $ are squares of integers.

Prove that it is not possible that numbers $(n+1)\cdot 2^n$ and $(n+3)\cdot 2^{n+2}$ are perfect squares, where $n$ is positive integer.

Given a positive integer $k$, find all polynomials $P$ of degree $k$ with integer coefficients such that for all positive integers $n$ where all of $P(n)$, $P(2024n)$, $P(2024^2n)$ are nonzero, we have $$\frac{\gcd(P(2024n), P(2024^2n))}{\gcd(P(n), P(2024n))}=2024^k.$$ [i]Allen Wang[/i]

Given is an arithmetic progression {$a_n$} of positive integers. Prove that there exist infinitely many $k$, such that $\omega (a_k)$ is even and $\omega (a_{k+1})$ is odd ($\omega (n)$ is the number of distinct prime factors of $n$). $\textit {Proposed by Viktor Simjanoski and Nikola Velov}$

Determine, with proof, whether or not there exist integers $a,b,c>2010$ satisfying the equation \[a^3+2b^3+4c^3=6abc+1.\]

$n$ and $d$ are positive integers such that $d$ divides $2n^2$. Prove that $n^2 + d$ cannot be a square.

Given a prime number $p$ and let $\overline{v_1},\overline{v_2},\dotsc ,\overline{v_n}$ be $n$ distinct vectors of length $p$ with integer coordinates in an $\mathbb{R}^3$ Cartesian coordinate system. Suppose that for any $1\leqslant j<k\leqslant n$, there exists an integer $0<\ell <p$ such that all three coordinates of $\overline{v_j} -\ell \cdot \overline{v_k} $ is divisible by $p$. Prove that $n\leqslant 6$.

Let $d$ denote the greatest common divisor of the natural numbers $a$ and $b$, and let $d'$ denote the greatest common divisor of the natural numbers $a'$ and $b'$. Prove that $dd'$ is the greatest common divisor of the four numbers $$ aa' , \ \ ab' , \ \ ba' , \ \ bb' .$$

[b]p1.[/b] What is the sum of the first $12$ positive integers? [b]p2.[/b] How many positive integers less than or equal to $100$ are multiples of both $2$ and $5$? [b]p3. [/b]Alex has a bag with $4$ white marbles and $4$ black marbles. She takes $2$ marbles from the bag without replacement. What is the probability that both marbles she took are black? Express your answer as a decimal or a fraction in lowest terms. [b]p4.[/b] How many $5$-digit numbers are there where each digit is either $1$ or $2$? [b]p5.[/b] An integer $a$ with $1\le a \le 10$ is randomly selected. What is the probability that $\frac{100}{a}$ is an integer? Express your answer as decimal or a fraction in lowest terms. [b]p6.[/b] Two distinct non-tangent circles are drawn so that they intersect each other. A third circle, distinct from the previous two, is drawn. Let $P$ be the number of points of intersection between any two circles. How many possible values of $P$ are there? [b]p7.[/b] Let $x, y, z$ be nonzero real numbers such that $x + y + z = xyz$. Compute $$\frac{1 + yz}{yz}+\frac{1 + xz}{xz}+\frac{1 + xy}{xy}.$$ [b]p8.[/b] How many positive integers less than $106$ are simultaneously perfect squares, cubes, and fourth powers? [b]p9.[/b] Let $C_1$ and $C_2$ be two circles centered at point $O$ of radii $1$ and $2$, respectively. Let $A$ be a point on $C_2$. We draw the two lines tangent to $C_1$ that pass through $A$, and label their other intersections with $C_2$ as $B$ and $C$. Let x be the length of minor arc $BC$, as shown. Compute $x$. [img]https://cdn.artofproblemsolving.com/attachments/7/5/915216d4b7eba0650d63b26715113e79daa176.png[/img] [b]p10.[/b] A circle of area $\pi$ is inscribed in an equilateral triangle. Find the area of the triangle. [b]p11.[/b] Julie runs a $2$ mile route every morning. She notices that if she jogs the route $2$ miles per hour faster than normal, then she will finish the route $5$ minutes faster. How fast (in miles per hour) does she normally jog? [b]p12.[/b] Let $ABCD$ be a square of side length $10$. Let $EFGH$ be a square of side length $15$ such that $E$ is the center of $ABCD$, $EF$ intersects $BC$ at $X$, and $EH$ intersects $CD$ at $Y$ (shown below). If $BX = 7$, what is the area of quadrilateral $EXCY$ ? [img]https://cdn.artofproblemsolving.com/attachments/d/b/2b2d6de789310036bc42d1e8bcf3931316c922.png[/img] [b]p13.[/b] How many solutions are there to the system of equations $$a^2 + b^2 = c^2$$ $$(a + 1)^2 + (b + 1)^2 = (c + 1)^2$$ if $a, b$, and $c$ are positive integers? [b]p14.[/b] A square of side length $ s$ is inscribed in a semicircle of radius $ r$ as shown. Compute $\frac{s}{r}$. [img]https://cdn.artofproblemsolving.com/attachments/5/f/22d7516efa240d00d6a9743a4dc204d23d190d.png[/img] [b]p15.[/b] $S$ is a collection of integers n with $1 \le n \le 50$ so that each integer in $S$ is composite and relatively prime to every other integer in $S$. What is the largest possible number of integers in $S$? [b]p16.[/b] Let $ABCD$ be a regular tetrahedron and let $W, X, Y, Z$ denote the centers of faces $ABC$, $BCD$, $CDA$, and $DAB$, respectively. What is the ratio of the volumes of tetrahedrons $WXYZ$ and $WAYZ$? Express your answer as a decimal or a fraction in lowest terms. [b]p17.[/b] Consider a random permutation $\{s_1, s_2, ... , s_8\}$ of $\{1, 1, 1, 1, -1, -1, -1, -1\}$. Let $S$ be the largest of the numbers $s_1$, $s_1 + s_2$, $s_1 + s_2 + s_3$, $...$ , $s_1 + s_2 + ... + s_8$. What is the probability that $S$ is exactly $3$? Express your answer as a decimal or a fraction in lowest terms. [b]p18.[/b] A positive integer is called [i]almost-kinda-semi-prime[/i] if it has a prime number of positive integer divisors. Given that there $are 168$ primes less than $1000$, how many almost-kinda-semi-prime numbers are there less than $1000$? [b]p19.[/b] Let $ABCD$ be a unit square and let $X, Y, Z$ be points on sides $AB$, $BC$, $CD$, respectively, such that $AX = BY = CZ$. If the area of triangle $XYZ$ is $\frac13$ , what is the maximum value of the ratio $XB/AX$? [img]https://cdn.artofproblemsolving.com/attachments/5/6/cf77e40f8e9bb03dea8e7e728b21e7fb899d3e.png[/img] [b]p20.[/b] Positive integers $a \le b \le c$ have the property that each of $a + b$, $b + c$, and $c + a$ are prime. If $a + b + c$ has exactly $4$ positive divisors, find the fourth smallest possible value of the product $c(c + b)(c + b + a)$. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Prove that there are infinitely many positive integer $n$ such that $n!$ is divisible by $n^3 -1$.

Find the sum of all integers $n$ that fulfill the equation \[2^n(6-n)=8n.\]