Prove that if $ p$ is a prime number, then $ 7p+3^{p}-4$ is not a perfect square.

Let $a \in \mathbb{N}$ such that $a + n^2$ can be expressed as the sum of two squares for all $n \in \mathbb{N}$. Prove that $a$ is the square of a natural number.

Let $p$ be an odd prime and $a_1, a_2,...,a_p$ be integers. Prove that the following two conditions are equivalent: 1) There exists a polynomial $P(x)$ with degree $\leq \frac{p-1}{2}$ such that $P(i) \equiv a_i \pmod p$ for all $1 \leq i \leq p$ 2) For any natural $d \leq \frac{p-1}{2}$, $$ \sum_{i=1}^p (a_{i+d} - a_i )^2 \equiv 0 \pmod p$$ where indices are taken $\pmod p$

For positive integer $k>1$, let $f(k)$ be the number of ways of factoring $k$ into product of positive integers greater than $1$ (The order of factors are not countered, for example $f(12)=4$, as $12$ can be factored in these $4$ ways: $12,2\cdot 6,3\cdot 4, 2\cdot 2\cdot 3$. Prove: If $n$ is a positive integer greater than $1$, $p$ is a prime factor of $n$, then $f(n)\leq \frac{n}{p}$

Let $\mathbb{Z}^+$ be the set of positive integers. Determine all functions $f : \mathbb{Z}^+\to\mathbb{Z}^+$ such that $a^2+f(a)f(b)$ is divisible by $f(a)+b$ for all positive integers $a,b$.

The positive integers $a, b$ and $c$ satisfy that the three fractions $\frac{b}{a}$, $\frac{c + 100}{b}$ and $\frac{a + b + 169}{2c + 200}$ are all integers. Determine all possible values of $a$.

Find all distinct prime numbers $p,q,r,s$ such that $1-\frac{1}{p} - \frac{1}{q} -\frac{1}{r} - \frac{1}{s} =\frac{1}{pqrs}$

Suppose $a$ and $b$ are positive integers for which $8a^ab^b = 27a^bb^a$. Find $a^2 + b^2$.

Find all prime numbers $p$ and $q$ such that $2p^2q + 45pq^2$ is a perfect square.

Determine all pairs of integers $(x, y)$ such that $2xy^2 + x + y + 1 = x^2 + 2y^2 + xy$.

Consider all triples $(x,y,p)$ of positive integers, where $p$ is a prime number, such that $4x^2 + 8y^2 + (2x-3y)p-12xy = 0$. Which below number is a perfect square number for every such triple $(x,y, p)$? A. $4y + 1$ B. $2y + 1$ C. $8y + 1$ D. $5y - 3$ E. $8y - 1$

[b]p1.[/b] Fluffy the Dog is an extremely fluffy dog. Because of his extreme fluffiness, children always love petting Fluffy anywhere. Given that Fluffy likes being petted $1/4$ of the time, out of $120$ random people who each pet Fluffy once, what is the expected number of times Fluffy will enjoy being petted? [b]p2.[/b] Andy thinks of four numbers $27$, $81$, $36$, and $41$ and whispers the numbers to his classmate Cynthia. For each number she hears, Cynthia writes down every factor of that number on the whiteboard. What is the sum of all the different numbers that are on the whiteboard? (Don't include the same number in your sum more than once) [b]p3.[/b] Charles wants to increase the area his square garden in his backyard. He increases the length of his garden by $2$ and increases the width of his garden by $3$. If the new area of his garden is $182$, then what was the original area of his garden? [b]p4.[/b] Antonio is trying to arrange his flute ensemble into an array. However, when he arranges his players into rows of $6$, there are $2$ flute players left over. When he arranges his players into rows of $13$, there are $10$ flute players left over. What is the smallest possible number of flute players in his ensemble such that this number has three prime factors? [b]p5.[/b] On the AMC $9$ (Acton Math Competition $9$), $5$ points are given for a correct answer, $2$ points are given for a blank answer and $0$ points are given for an incorrect answer. How many possible scores are there on the AMC $9$, a $15$ problem contest? [b]p6.[/b] Charlie Puth produced three albums this year in the form of CD's. One CD was circular, the second CD was in the shape of a square, and the final one was in the shape of a regular hexagon. When his producer circumscribed a circle around each shape, he noticed that each time, the circumscribed circle had a radius of $10$. The total area occupied by $1$ of each of the different types of CDs can be expressed in the form $a + b\pi + c\sqrt{d}$ where $d$ is not divisible by the square of any prime. Find $a + b + c + d$. [b]p7.[/b] You are picking blueberries and strawberries to bring home. Each bushel of blueberries earns you $10$ dollars and each bushel of strawberries earns you $8$ dollars. However your cart can only fit $24$ bushels total and has a weight limit of $100$ lbs. If a bushel of blueberries weighs $8$ lbs and each bushel of strawberries weighs $6$ lbs, what is your maximum profit. (You can only pick an integer number of bushels) [b]p8.[/b] The number $$\sqrt{2218 + 144\sqrt{35} + 176\sqrt{55} + 198\sqrt{77}}$$ can be expressed in the form $a\sqrt5 + b\sqrt7 + c\sqrt{11}$ for positive integers $a, b, c$. Find $abc$. [b]p9.[/b] Let $(x, y)$ be a point such that no circle passes through the three points $(9,15)$, $(12, 20)$, $(x, y)$, and no circle passes through the points $(0, 17)$, $(16, 19)$, $(x, y)$. Given that $x - y = -\frac{p}{q}$ for relatively prime positive integers $p$, $q$, Find $p + q$. [b]p10.[/b] How many ways can Alfred, Betty, Catherine, David, Emily and Fred sit around a $6$ person table if no more than three consecutive people can be in alphabetical order (clockwise)? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Show that for any $n \not \equiv 0 \pmod{10}$ there exists a multiple of $n$ not containing the digit $0$ in its decimal expansion.

Prove that on the coordinate plane it is impossible to draw a closed broken line such that [i](i)[/i] the coordinates of each vertex are rational; [i](ii)[/i] the length each of its edges is 1; [i](iii)[/i] the line has an odd number of vertices.

For any $h = 2^{r}$ ($r$ is a non-negative integer), find all $k \in \mathbb{N}$ which satisfy the following condition: There exists an odd natural number $m > 1$ and $n \in \mathbb{N}$, such that $k \mid m^{h} - 1, m \mid n^{\frac{m^{h}-1}{k}} + 1$.

Let $n\geqslant 3$ be an integer. Prove that there exists a set $S$ of $2n$ positive integers satisfying the following property: For every $m=2,3,...,n$ the set $S$ can be partitioned into two subsets with equal sums of elements, with one of subsets of cardinality $m$.

Let $(x_n)$ be a sequence of positive integers defined by $x_1=2$ and $x_{n+1}=2x_n^3+x_n$ for all integers $n\ge1$. Determine the largest power of $5$ that divides $x_{2014}^2+1$.

A Pythagorean triple is a solution of the equation $x^2 + y^2 = z^2$ in positive integers such that $x < y$. Given any non-negative integer $n$ , show that some positive integer appears in precisely $n$ distinct Pythagorean triples.

Prove that from any set of seven natural numbers (not necessarily consecutive) one can choose three, the sum of which is divisible by three.

[b]p1.[/b] How many two-digit primes have a units digit of $3$? [b]p2.[/b] How many ways can you arrange the letters $A$, $R$, and $T$ such that it makes a three letter combination? Each letter is used once. [b]p3.[/b] Hanna and Kevin are running a $100$ meter race. If Hanna takes $20$ seconds to finish the race and Kevin runs $15$ meters per second faster than Hanna, by how many seconds does Kevin finish before Hanna? [b]p4.[/b] It takes an ant $3$ minutes to travel a $120^o$ arc of a circle with radius $2$. How long (in minutes) would it take the ant to travel the entirety of a circle with radius $2022$? [b]p5.[/b] Let $\vartriangle ABC$ be a triangle with angle bisector $AD$. Given $AB = 4$, $AD = 2\sqrt2$, $AC = 4$, find the area of $\vartriangle ABC$. [b]p6.[/b] What is the coefficient of $x^5y^2$ in the expansion of $(x + 2y + 4)^8$? [b]p7.[/b] Find the least positive integer $x$ such that $\sqrt{20475x}$ is an integer. [b]p8.[/b] What is the value of $k^2$ if $\frac{x^5 + 3x^4 + 10x^2 + 8x + k}{x^3 + 2x + 4}$ has a remainder of $2$? [b]p9.[/b] Let $ABCD$ be a square with side length $4$. Let $M$, $N$, and $P$ be the midpoints of $\overline{AB}$, $\overline{BC}$ and $\overline{CD}$, respectively. The area of the intersection between $\vartriangle DMN$ and $\vartriangle ANP$ can be written as $\frac{a}{b}$ , where $a$ and $b$ are relatively prime positive integers. Find $a + b$. [b]p10.[/b] Let $x$ be all the powers of two from $2^1$ to $2^{2023}$ concatenated, or attached, end to end ($x = 2481632...$). Let y be the product of all the powers of two from $2^1$ to $2^{2023}$ ($y = 2 \cdot 4 \cdot 8 \cdot 16 \cdot 32... $ ). Let 2a be the largest power of two that divides $x$ and $2^b$ be the largest power of two that divides $y$. Compute $\frac{b}{a}$ . [b]p11.[/b] Larry is making a s’more. He has to have one graham cracker on the top and one on the bottom, with eight layers in between. Each layer can made out of chocolate, more graham crackers, or marshmallows. If graham crackers cannot be placed next to each other, how many ways can he make this s’more? [b]p12.[/b] Let $ABC$ be a triangle with $AB = 3$, $BC = 4$, $AC = 5$. Circle $O$ is centered at $B$ and has radius $\frac{8\sqrt{3}}{5}$ . The area inside the triangle but not inside the circle can be written as $\frac{a-b\sqrt{c}-d\pi}{e}$ , where $gcd(a, b, d, e) =1$ and $c$ is squarefree. Find $a + b + c + d + e$. [b]p13.[/b] Let $F(x)$ be a quadratic polynomial. Given that $F(x^2 - x) = F (2F(x) - 1)$ for all $x$, the sum of all possible values of $F(2022)$ can be written as $\frac{a}{b}$ , where $a$ and $b$ are relatively prime positive integers. Find $a + b$. [b]p14.[/b] Find the sum of all positive integers $n$ such that $6\phi (n) = \phi (5n)+8$, where $\phi$ is Euler’s totient function. Note: Euler’s totient $(\phi)$ is a function where $\phi (n)$ is the number of positive integers less than and relatively prime to $n$. For example, $\phi (4) = 2$ since only $1$, $3$ are the numbers less than and relatively prime to $4$. [b]p15.[/b] Three numbers $x$, $y$, and $z$ are chosen at random from the interval $[0, 1]$. The probability that there exists an obtuse triangle with side lengths $x$, $y$, and $z$ can be written in the form $\frac{a\pi-b}{c}$ , where $a$, $b$, $c$ are positive integers with $gcd(a, b, c) = 1$. Find $a + b + c$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Berit has $11$ twenty kroner coins, $14$ ten kroner coins, and $12$ five kroner coins. An exchange machine can exchange three ten kroner coins into one twenty kroner coin and two five kroner coins, and the reverse. It can also exchange two twenty kroner coins into three ten kroner coins and two five kroner coins, and the reverse. (i) Can Berit get the same number of twenty kroner and ten kroner coins, but no five kroner coins? (ii) Can she get the same number each of twenty kroner, ten kroner, and five kroner coins?

Find all triples $(p,m,n)$ satisfying the equation $p^m-n^3=8$ where $p$ is a prime number and $m,n$ are nonnegative integers.

Determine all two-digit positive integers $n =\overline{ab}$ (in the decimal system) with the property that for all integers $x$ the difference $x^a - x^b$ is divisible by $n$.

We call a number [i]perfect[/i] if the sum of its positive integer divisors(including $1$ and $n$) equals $2n$. Determine all [i]perfect[/i] numbers $n$ for which $n-1$ and $n+1$ are prime numbers.

If $n \in \mathbb{N} n > 1$ prove that for every $n$ you can find $n$ consecutive natural numbers the product of which is divisible by all primes not exceeding $2n+1$, but is not divisible by any other primes.