Find all polynomials $P$ with integer coefficients such that all the roots of $P^n(x)$ are integers. (here $P^n(x)$ means $P(P(...(P(x))...))$ where $P$ is repeated $n$ times)

Determine all positive integers $n$ such that $n$ equals the square of the sum of the digits of $n$.

Let $p \geq 5$ be a prime. (a) Show that exists a prime $q \neq p$ such that $q| (p-1)^{p}+1$ (b) Factoring in prime numbers $(p-1)^{p}+1 = \prod_{i=1}^{n}p_{i}^{a_{i}}$ show that: \[\sum_{i=1}^{n}p_{i}a_{i}\geq \frac{p^{2}}2 \]

Suppose that $a$ is an even positive integer and $A=a^{n}+a^{n-1}+\ldots +a+1,n\in \mathbb{N^{*}}$ is a perfect square.Prove that $8\mid a$.

Find all subsets $A$ of $\left\{ 1, 2, 3, 4, \ldots \right\}$, with $|A| \geq 2$, such that for all $x,y \in A, \, x \neq y$, we have that $\frac{x+y}{\gcd (x,y)}\in A$. [i]Dan Schwarz[/i]

[b]p1.[/b] Nikhil computes the sum of the first $10$ positive integers, starting from $1$. He then divides that sum by 5. What remainder does he get? [b]p2.[/b] In class, starting at $8:00$, Ava claps her hands once every $4$ minutes, while Ella claps her hands once every $6$ minutes. What is the smallest number of minutes after $8:00$ such that both Ava and Ella clap their hands at the same time? [b]p3.[/b] A triangle has side lengths $3$, $4$, and $5$. If all of the side lengths of the triangle are doubled, how many times larger is the area? [b]p4.[/b] There are $50$ students in a room. Every student is wearing either $0$, $1$, or $2$ shoes. An even number of the students are wearing exactly $1$ shoe. Of the remaining students, exactly half of them have $2$ shoes and half of them have $0$ shoes. How many shoes are worn in total by the $50$ students? [b]p5.[/b] What is the value of $-2 + 4 - 6 + 8 - ... + 8088$? [b]p6.[/b] Suppose Lauren has $2$ cats and $2$ dogs. If she chooses $2$ of the $4$ pets uniformly at random, what is the probability that the 2 chosen pets are either both cats or both dogs? [b]p7.[/b] Let triangle $\vartriangle ABC$ be equilateral with side length $6$. Points $E$ and $F$ lie on $BC$ such that $E$ is closer to $B$ than it is to $C$ and $F$ is closer to $C$ than it is to $B$. If $BE = EF = FC$, what is the area of triangle $\vartriangle AFE$? [b]p8.[/b] The two equations $x^2 + ax - 4 = 0$ and $x^2 - 4x + a = 0$ share exactly one common solution for $x$. Compute the value of $a$. [b]p9.[/b] At Shreymart, Shreyas sells apples at a price $c$. A customer who buys $n$ apples pays $nc$ dollars, rounded to the nearest integer, where we always round up if the cost ends in $.5$. For example, if the cost of the apples is $4.2$ dollars, a customer pays $4$ dollars. Similarly, if the cost of the apples is $4.5$ dollars, a customer pays $5$ dollars. If Justin buys $7$ apples for $3$ dollars and $4$ apples for $1$ dollar, how many dollars should he pay for $20$ apples? [b]p10.[/b] In triangle $\vartriangle ABC$, the angle trisector of $\angle BAC$ closer to $\overline{AC}$ than $\overline{AB}$ intersects $\overline{BC}$ at $D$. Given that triangle $\vartriangle ABD$ is equilateral with area $1$, compute the area of triangle $\vartriangle ABC$. [b]p11.[/b] Wanda lists out all the primes less than $100$ for which the last digit of that prime equals the last digit of that prime's square. For instance, $71$ is in Wanda's list because its square, $5041$, also has $1$ as its last digit. What is the product of the last digits of all the primes in Wanda's list? [b]p12.[/b] How many ways are there to arrange the letters of $SUSBUS$ such that $SUS$ appears as a contiguous substring? For example, $SUSBUS$ and $USSUSB$ are both valid arrangements, but $SUBSSU$ is not. [b]p13.[/b] Suppose that $x$ and $y$ are integers such that $x \ge 5$, $y \ge 3$, and $\sqrt{x - 5} +\sqrt{y - 3} = \sqrt{x + y}$. Compute the maximum possible value of $xy$. [b]p14.[/b] What is the largest integer $k$ divisible by $14$ such that $x^2-100x+k = 0$ has two distinct integer roots? [b]p15.[/b] What is the sum of the first $16$ positive integers whose digits consist of only $0$s and $1$s? [b]p16.[/b] Jonathan and Ajit are flipping two unfair coins. Jonathan's coin lands on heads with probability $\frac{1}{20}$ while Ajit's coin lands on heads with probability $\frac{1}{22}$ . Each year, they flip their coins at thesame time, independently of their previous flips. Compute the probability that Jonathan's coin lands on heads strictly before Ajit's coin does. [b]p17.[/b] A point is chosen uniformly at random in square $ABCD$. What is the probability that it is closer to one of the $4$ sides than to one of the $2$ diagonals? [b]p18.[/b] Two integers are coprime if they share no common positive factors other than $1$. For example, $3$ and $5$ are coprime because their only common factor is $1$. Compute the sum of all positive integers that are coprime to $198$ and less than $198$. [b]p19.[/b] Sumith lists out the positive integer factors of $12$ in a line, writing them out in increasing order as $1$, $2$, $3$, $4$, $6$, $12$. Luke, being the mischievious person he is, writes down a permutation of those factors and lists it right under Sumith's as $a_1$, $a_2$, $a_3$, $a_4$, $a_5$, $a_6$. Luke then calculates $$gcd(a_1, 2a_2, 3a_3, 4a_4, 6a_5, 12a_6).$$ Given that Luke's result is greater than $1$, how many possible permutations could he have written? [b]p20.[/b] Tetrahedron $ABCD$ is drawn such that $DA = DB = DC = 2$, $\angle ADB = \angle ADC = 90^o$, and $\angle BDC = 120^o$. Compute the radius of the sphere that passes through $A$, $B$, $C$, and $D$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Mr. Jones has eight children of different ages. On a family trip his oldest child, who is 9, spots a license plate with a 4-digit number in which each of two digits appears two times. "Look, daddy!" she exclaims. "That number is evenly divisible by the age of each of us kids!" "That's right," replies Mr. Jones, "and the last two digits just happen to be my age." Which of the following is not the age of one of Mr. Jones's children? $ \textbf{(A) } 4 \qquad \textbf{(B) } 5 \qquad \textbf{(C) } 6 \qquad \textbf{(D) } 7 \qquad \textbf{(E) } 8$

Let $a,b,c,d$ be the natural numbers such that $2^a+4^b+5^c=2014^d.$ Find all $(a,b,c,d).$

Let $n>1$ be a positive integer. Each cell of an $n\times n$ table contains an integer. Suppose that the following conditions are satisfied: [list=1] [*] Each number in the table is congruent to $1$ modulo $n$. [*] The sum of numbers in any row, as well as the sum of numbers in any column, is congruent to $n$ modulo $n^2$. [/list] Let $R_i$ be the product of the numbers in the $i^{\text{th}}$ row, and $C_j$ be the product of the number in the $j^{\text{th}}$ column. Prove that the sums $R_1+\hdots R_n$ and $C_1+\hdots C_n$ are congruent modulo $n^4$.

Find the number of ordered pairs $(x,y)$ of integers with $0 \le x < 2023$ and $0 \le y < 2023$ such that $y^3 \equiv x^2 \pmod{2023}.$

A palindrome is a word that doesn't matter if you read it from left to right or from right to left. Examples: OMO, lepel and parterretrap. How many palindromes can you make with the five letters $a, b, c, d$ and $e$ under the conditions: - each letter may appear no more than twice in each palindrome, - the length of each palindrome is at least $3$ letters. (Any possible combination of letters is considered a word.)

* Assume that the number of a tree’s leaves is a multiple of $15$. Neglecting the shade of the trunk and branches prove that one can rip off the tree $7/15$ of its leaves so that not less than $8/15$ of its shade remains.

Show that each number is of the form $$2^{5^{2^{5^{...}}}}+ 4^{5^{4^{5^{...}}}}$$ is divisible by $2008$, where the exponential towers can be any independent ones have height $\ge 3$.

Let $a\geq 2$ a fixed natural number, and let $a_{n}$ be the sequence $a_{n}=a^{a^{.^{.^{a}}}}$ (e.g $a_{1}=a$, $a_{2}=a^a$, etc.). Prove that $(a_{n+1}-a_{n})|(a_{n+2}-a_{n+1})$ for every natural number $n$.

[b]p1.[/b] I have $6$ envelopes full of money. The amounts (in dollars) in the $6$ envelopes are six consecutive integers. I give you one of the envelopes. The total amount in the remaining $5$ envelopes is $\$2018$. How much money did I give you? [b]p2. [/b]Two tangents $AB$ and $AC$ are drawn to a circle from an exterior point $A$. Let $D$ and $E$ be the midpoints of the line segments $AB$ and $AC$. Prove that the line DE does not intersect the circle. [b]p3.[/b] Let $n \ge 2$ be an integer. A subset $S$ of {0, 1, . . . , n − 2} is said to be closed whenever it satisfies all of the following properties: • $0 \in S$ • If $x \in S$ then $n - 2 - x \in S$ • If $x \in S$, $y \ge 0$, and $y + 1$ divides $x + 1$ then $y \in S$. Prove that $\{0, 1, . . . , n - 2\}$ is the only closed subset if and only if $n$ is prime. (Note: “$\in$” means “belongs to”.) [b]p4.[/b] Consider the $3 \times 3$ grid shown below $\begin{tabular}{|l|l|l|l|} \hline A & B & C \\ \hline D & E & F \\ \hline G & H & I \\ \hline \end{tabular}$ A knight move is a pair of elements $(s, t)$ from $\{A, B, C, D, E, F, G, H, I\}$ such that $s$ can be reached from $t$ by moving either two spaces horizontally and one space vertically, or by moving one space horizontally and two spaces vertically. (For example, $(B, I)$ is a knight move, but $(G, E)$ is not.) A knight path of length $n$ is a sequence $s_0$, $s_1$, $s_2$, $. . . $, $s_n$ drawn from the set $\{A, B, C, D, E, F, G, H, I\}$ (with repetitions allowed) such that each pair $(s_i , s_{i+1})$ is a knight move. Let $N$ be the total number of knight paths of length $2018$ that begin at $A$ and end at $A$. Let $M$ be the total number of knight paths of length $2018$ that begin at $A$ and end at $I$. Compute the value $(N- M)$, with proof. (Your answer must be in simplified form and may not involve any summations.) [b]p5.[/b] A strip is defined to be the region of the plane lying on or between two parallel lines. The width of the strip is the distance between the two lines. Consider a finite number of strips whose widths sum to a number $d < 1$, and let $D$ be a circular closed disk of diameter $1$. Prove or disprove: no matter how the strips are placed in the plane, they cannot entirely cover the disk $D$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Find all positive integers $a$, $b$, $c$, and $p$, where $p$ is a prime number, such that $73p^2 + 6 = 9a^2 + 17b^2 + 17c^2$.

Let $a$ and $b$ be positive integers, $a < b$, such that in the decimal expansion of the fraction $\dfrac{a}{b} $ the five digits $1,4,2,8,6$ appear somewhere, in that order and consecutively. Determine the lowest possible value $b$ can take .

Let be two real numbers $ a,b, $ that satisfy $ 3^a+13^b=17^a $ and $ 5^a+7^b=11^b. $ Show that $ a<b. $

Let $S$ be a nonempty set of positive integers such that, for any (not necessarily distinct) integers $a$ and $b$ in $S$, the number $ab+1$ is also in $S$. Show that the set of primes that do not divide any element of $S$ is finite. [i]Proposed by Carl Schildkraut[/i]

The function $f$ is defined on positive integers : if $n$ has prime factorization $p^{k_1}_{1} p^{k_2}_{2} ...p^{k_t}_{t}$ then $f(n) = (p_1-1)^{k_1+1}(p_2-1)^{k_2+1}...(p_t-1)^{k_t+1}$. If we keep using this function repeatedly, starting from any positive integer $n$, we will always get to $1$ after some number of steps. What is the smallest integer $n$ for which we need exactly $6$ steps to get to $1$?

Let $p>3$ be a prime and let $a_1,a_2,...,a_{\frac{p-1}{2}}$ be a permutation of $1,2,...,\frac{p-1}{2}$. For which $p$ is it always possible to determine the sequence $a_1,a_2,...,a_{\frac{p-1}{2}}$ if it for all $i,j\in\{1,2,...,\frac{p-1}{2}\}$ with $i\not=j$ the residue of $a_ia_j$ modulo $p$ is known?

Let $a_i$, $i=1,2,...,n$ be non-negative real numbers and $\sum_{i=1}^na_i =1$. Find $\max S=\sum_{i\mid j}a_i a_j $.

Let $p > 10$ be prime. Prove that there are positive integers $m$ and $n$ with $m + n < p$ exist for which $p$ is a divisor of $5^m7^n-1$.

Let $d$ be a positive integer and let $\varepsilon$ be any positive real. Prove that for all sufficiently large primes $p$ with $\gcd(p-1,d) \neq 1$, there exists an positive integer less than $p^r$ which is not a $d$th power modulo $p$, where $r$ is defined by \[ \log r = \varepsilon - \frac{1}{\gcd(d,p-1)}. \][i]Proposed by Shashwat Kishore[/i]

For positive integers $n$ and $k \geq 2$, define $E_k(n)$ as the greatest exponent $r$ such that $k^r$ divides $n!$. Prove that there are infinitely many $n$ such that $E_{10}(n) > E_9(n)$ and infinitely many $m$ such that $E_{10}(m) < E_9(m)$.