Andrew writes down all of the prime numbers less than $50$. How many times does he write the digit $2$?

Let $x,y,z,w$ be integers such that $2^x+2^y+2^z+2^w=24.375$. Find the value of $xyzw$.

Find the sum of all possible sums $a + b$ where $a$ and $b$ are nonnegative integers such that $4^a + 2^b + 5$ is a perfect square.

Prove that there exist infinitely many positive integers $ n$ such that $ n!$ is not divisible by $ n^2\plus{}1$.

Given are 100 different positive integers. We call a pair of numbers [i]good[/i] if the ratio of these numbers is either 2 or 3. What is the maximum number of good pairs that these 100 numbers can form? (A number can be used in several pairs.) [i]Proposed by Alexander S. Golovanov, Russia[/i]

Let $p, q$ be two different odd prime numbers and $n$ an integer such that $pq$ divides $n^{pq} + 1$. Prove that if $p^3q^3$ divides $n^{pq} + 1$ then either $p^2$ divides $n + 1$ or $q^2$ divides $n + 1$. Malik Talbi

Natural numbers $a, b, c, d$ are given such that $c>b$. Prove that if $a + b + c + d = ab-cd$, then $a + c$ is a composite number.

Let a, b, c, d, and e be positive integers. Are there any solutions to $a^2+b^3+c^5+d^7=e^{11}$?

Prove that there exist infinitely many integers $n$ such that $n^2+1$ is squarefree.

Prove that the product of five consecutive positive integers cannot be the square of an integer.

The sequence of polynomials $\left( P_{n}(X)\right)_{n\in Z_{>0}}$ is defined as follows: $P_{1}(X)=2X$ $P_{2}(X)=2(X^2+1)$ $P_{n+2}(X)=2X\cdot P_{n+1}(X)-(X^2-1)P_{n}(X)$, for all positive integers $n$. Find all $n$ for which $X^2+1\mid P_{n}(X)$

Find the smallest positive integer $n$ with the following property: there does not exist an arithmetic progression of $1999$ real numbers containing exactly $n$ integers.

Given a polynomial $P$ and a positive integer $k$, we denote the $k$-fold composition of $P$ by $P^{\circ k}$. A polynomial $P$ with real coefficients is called [b]perfect[/b] if for each integer $n$ there is a positive integer $k$ so that $P^{\circ k}(n)$ is an integer. Is it true that for each perfect polynomial $P$, there exists a positive $m$ so that for each integer $n$ there is $0<k\leq m$ for which $P^{\circ k}(n)$ is an integer?

Determine all natural numbers $m$ and $n$ such that \[ n \cdot (n + 1) = 3^m + s(n) + 1182, \] where $s(n)$ represents the sum of the digits of the natural number $n$.

Find the number of solutions to the given congruence$$x^{2}+y^{2}+z^{2} \equiv 2 a x y z \pmod p$$ where $p$ is an odd prime and $x,y,z \in \mathbb{Z}$. [i]Proposed by math_and_me[/i]

Prove that for all positive integers $n$ there are positive integers $a,b$ such that $$n\mid 4a^2+9b^2-1.$$

In $\triangle ABC$ with $AB=10$, $AC=13$, and $\measuredangle ABC = 30^\circ$, $M$ is the midpoint of $\overline{BC}$ and the circle with diameter $\overline{AM}$ meets $\overline{CB}$ and $\overline{CA}$ again at $D$ and $E$, respectively. The area of $\triangle DEM$ can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m, n$. Compute $100m + n$. [i]Based on a proposal by Matthew Babbitt[/i]

A student adds up rational fractions incorrectly: \[\frac{a}{b}+\frac{x}{y}=\frac{a+x}{b+y}\quad (\star) \] Despite that, he sometimes obtains correct results. For a given fraction $\frac{a}{b},a,b\in\mathbb{Z},b>0$, find all fractions $\frac{x}{y},x,y\in\mathbb{Z},y>0$ such that the result obtained by $(\star)$ is correct.

The Fibonacci numbers $f_n$ are defined for each natural number $n$ as follows: $f_0=f_1=1$ and for $n$ greater than or equal to $2$, by recurrence: $f_n=f_{n-1}+f_{n-2}$ Let $S=f_1+f_2+...+f_{2004}+f_{2005}$. Calculate the largest value of $N$, such that the Fibonacci number $f_N$ satisfies $f_N<S$

Given the number $\overline{a_1a_2\cdots a_n}$ such that $$\overline{a_n\cdots a_2a_1}\mid \overline{a_1a_2\cdots a_n}$$Then show $(\overline{a_1a_2\cdots a_n})(\overline{a_n\cdots a_2a_1})$ is a perfect square. [i]Proposed by Ezra Guerrero, Francisco Morazan[/i]

The sequence $a_1,a_2,a_3\ldots $, consisting of natural numbers, is defined by the rule: \[a_{n+1}=a_{n}+2t(n)\] for every natural number $n$, where $t(n)$ is the number of the different divisors of $n$ (including $1$ and $n$). Is it possible that two consecutive members of the sequence are squares of natural numbers?

[b]p1.[/b] A shape made by joining four identical regular hexagons side-to-side is called a hexo. Two hexos are considered the same if one can be rotated / reflected to match the other. Find the number of different hexos. [b]p2.[/b] The sequence $1, 2, 3, 3, 3, 4, 5, 5, 5, 5, 5, 6,... $ consists of numbers written in increasing order, where every even number $2n$ is written once, and every odd number $2n + 1$ is written $2n + 1$ times. What is the $2019^{th}$ term of this sequence? [b]p3.[/b] On planet EMCCarth, months can only have lengths of $35$, $36$, or $42$ days, and there is at least one month of each length. Victor knows that an EMCCarth year has $n$ days, but realizes that he cannot figure out how many months there are in an EMCCarth year. What is the least possible value of $n$? [b]p4.[/b] In triangle $ABC$, $AB = 5$ and $AC = 9$. If a circle centered at $A$ passing through $B$ intersects $BC$ again at $D$ and $CD = 7$, what is $BC$? [b]p5.[/b] How many nonempty subsets $S$ of the set $\{1, 2, 3,..., 11, 12\}$ are there such that the greatest common factor of all elements in $S$ is greater than $1$? [b]p6.[/b] Jasmine rolls a fair $6$-sided die, with faces labeled from $1$ to $6$, and a fair $20$-sided die, with faces labeled from $1$ to $20$. What is the probability that the product of these two rolls, added to the sum of these two rolls, is a multiple of $3$? [b]p7.[/b] Let $\{a_n\}$ be a sequence such that $a_n$ is either $2a_{n-1}$ or $a_{n-1} - 1$. Given that $a_1 = 1$ and $a_{12} = 120$, how many possible sequences $a_1$, $a_2$, $...$, $a_{12}$ are there? [b]p8.[/b] A tetrahedron has two opposite edges of length $2$ and the remaining edges have length $10$. What is the volume of this tetrahedron? [b]p9.[/b] In the garden of EMCCden, there is a tree planted at every lattice point $-10 \le x, y \le 10$ except the origin. We say that a tree is visible to an observer if the line between the tree and the observer does not intersect any other tree (assume that all trees have negligible thickness). What fraction of all the trees in the garden of EMCCden are visible to an observer standing at the origin? [b]p10.[/b] Point $P$ lies inside regular pentagon $\zeta$, which lies entirely within regular hexagon $\eta$. A point $Q$ on the boundary of pentagon $\zeta$ is called projective if there exists a point $R$ on the boundary of hexagon $\eta$ such that $P$, $Q$, $R$ are collinear and $2019 \cdot \overline{PQ} = \overline{QR}$. Given that no two sides of $\zeta$ and $\eta$ are parallel, what is the maximum possible number of projective points on $\zeta$? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $n>1$ be a positive integer. Show that the number of residues modulo $n^2$ of the elements of the set $\{ x^n + y^n : x,y \in \mathbb{N} \}$ is at most $\frac{n(n+1)}{2}$. [I]Proposed by N. Safaei (Iran)[/i]

We call a natural number venerable if the sum of all its divisors, including $1$, but not including the number itself, is $1$ less than this number. Find all the venerable numbers, some exact degree of which is also venerable.

Let $d(n)$ denote the number of positive divisors of $n$. For any given integer $a \geq 3$, define a sequence $\{a_i\}_{i=0}^\infty$ satisfying [list] [*] $a_{0}=a$, and [*] $a_{n+1}=a_{n}+(-1)^{n} d(a_{n})$ for each integer $n \geq 0$. [/list] For example, if $a=275$, the sequence would be \[275, \overline{281,279,285,277,279,273}.\] Prove that for each positive integer $k$ there exists a positive integer $N$ such that if such a sequence has period $2k$ and all terms of the sequence are greater than $N$ then all terms of the sequence have the same parity. [i]Proposed by Navid[/i]