Find all positive integers $r$ with the property that there exists positive prime numbers $p$ and $q$ so that $$p^2 + pq + q^2 = r^2 .$$

Determine the largest positive integer $n$ that divides $p^6 - 1$ for all primes $p > 7$.

Find all polynomials with real coefficients $P$ such that \[ P(x)P(2x^2-1)=P(x^2)P(2x-1)\] for every $x\in\mathbb{R}$.

A positive integer $x$ satisfies the following: \[\{\frac{x}{3}\}+\{\frac{x}{5}\}+\{\frac{x}{7}\}+\{\frac{x}{11}\}=\frac{248}{165}\] Find all possible values of \[\{\frac{2x}{3}\}+\{\frac{2x}{5}\}+\{\frac{2x}{7}\}+\{\frac{2x}{11}\}\] where $\{y\}$ denotes the fractional part of $y$.

Solve the following equation in $\mathbb{Z}$: \[3^{2a + 1}b^2 + 1 = 2^c\]

Prove that for each positive integer $n$ there are at most two pairs $(a, b)$ of positive integers with following two properties: (i) $a^2 + b = n$, (ii) $a+b$ is a power of two, i.e. there is an integer $k \ge 0$ such that $a+b = 2^k$.

The PMO Magician has a special party game. There are $n$ chairs, labelled $1$ to $n$. There are $n$ sheets of paper, labelled $1$ to $n$. [list] [*] On each chair, she attaches exactly one sheet whose number does not match the number on the chair. [*] She then asks $n$ party guests to sit on the chairs so that each chair has exactly one occupant. [*] Whenever she claps her hands, each guest looks at the number on the sheet attached to their current chair, and moves to the chair labelled with that number. [/list] Show that if $1 < m \leq n$, where $m$ is not a prime power, it is always possible for the PMO Magician to choose which sheet to attach to each chair so that everyone returns to their original seats after exactly $m$ claps.

Let $a$ be the smallest and $A$ the largest of $n$ distinct positive integers. Prove that the least common multiple of these numbers is greater than or equal to $n a$ and that the greatest common divisor is less than or equal to $\frac{A}{n}$

Find all pair of positive integers $(m,n)$ such that $$mn(m^2+6mn+n^2)$$is a perfect square. [i]Proposed by Li4 and Untro368[/i]

The sequence of integers $a_1,a_2,a_3 ,.. $such that $a_1 = 1$, $a_2 = 2$ and for every natural $n \ge 1$ $$a_{n+2}=\begin{cases} 2001a_{n+1} - 1999a_n , \text{\,\,if\,\,the\,\,product\,\,} a_{n+1}a_n \text{is\,\,an\,\,even\,\,number} /\\ a_{n+1}-a_n , \text{\,\,if\,\,the\,\,product\,\,} a_{n+1}a_n \text{is\,\,an\,\,odd\,\,number} \end{cases}$$ Is there such a natural $m$ that $a_m= 2000$?

Find all integer $b$ such that $[x^2]-2012x+b=0$ has odd number of roots.

Are there infinitely many positive integers that [b]can’t[/b] be presented as a sum of no more than fifteen fourth degrees of positive integers. (For example 15 isn’t such number as it can be presented as the sum of $15.1^4$)

$\textbf{N5.}$ Find all $f: \mathbb{N} \rightarrow \mathbb{N}$ such that for all $a,b,c \in \mathbb{N}$ $f(a)+f(b)+f(c)-ab-bc-ca \mid af(a)+bf(b)+cf(c)-3abc$

Find all functions $ f: \mathbb Q\longrightarrow\mathbb Q$ such that: $ f(x)+f(\frac1x)=1$ $ 2f(f(x))=f(2x)$

Let $n$ be a positive integer.Consider the polynomial $p(x)=x^2+x+1$. What is the remainder of $ x^3$ when divided by $x^2+x+1$.For what positive integers values of $n$ is $ x^{2n}+x^n+1$ divisible by $p(x)$? Post no:[size=300]$100$[/size]

Let $b,n > 1$ be integers. Suppose that for each $k > 1$ there exists an integer $a_k$ such that $b - a^n_k$ is divisible by $k$. Prove that $b = A^n$ for some integer $A$. [i]Author: Dan Brown, Canada[/i]

[b]p1.[/b] What is the largest number of five dollar footlongs Jimmy can buy with 88 dollars? [b]p2.[/b] Austin, Derwin, and Sylvia are deciding on roles for BMT $2021$. There must be a single Tournament Director and a single Head Problem Writer, but one person cannot take on both roles. In how many ways can the roles be assigned to Austin, Derwin, and Sylvia? [b]p3.[/b] Sofia has$ 7$ unique shirts. How many ways can she place $2$ shirts into a suitcase, where the order in which Sofia places the shirts into the suitcase does not matter? [b]p4.[/b] Compute the sum of the prime factors of $2021$. [b]p5.[/b] A sphere has volume $36\pi$ cubic feet. If its radius increases by $100\%$, then its volume increases by $a\pi$ cubic feet. Compute $a$. [b]p6.[/b] The full price of a movie ticket is $\$10$, but a matinee ticket to the same movie costs only $70\%$ of the full price. If $30\%$ of the tickets sold for the movie are matinee tickets, and the total revenue from movie tickets is $\$1001$, compute the total number of tickets sold. [b]p7.[/b] Anisa rolls a fair six-sided die twice. The probability that the value Anisa rolls the second time is greater than or equal to the value Anisa rolls the first time can be expressed as $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Compute $m + n$. [b]p8.[/b] Square $ABCD$ has side length $AB = 6$. Let point $E$ be the midpoint of $\overline{BC}$. Line segments $\overline{AC}$ and $\overline{DE}$ intersect at point $F$. Compute the area of quadrilateral ABEF. [b]p9.[/b] Justine has a large bag of candy. She splits the candy equally between herself and her $4$ friends, but she needs to discard three candies before dividing so that everyone gets an equal number of candies. Justine then splits her share of the candy between herself and her two siblings, but she needs to discard one candy before dividing so that she and her siblings get an equal number of candies. If Justine had instead split all of the candy that was originally in the large bag between herself and $14$ of her classmates, what is the fewest number of candies that she would need to discard before dividing so that Justine and her $14$ classmates get an equal number of candies? [b]p10.[/b] For some positive integers $a$ and $b$, $a^2 - b^2 = 400$. If $a$ is even, compute $a$. [b]p11.[/b] Let $ABCDEFGHIJKL$ be the equilateral dodecagon shown below, and each angle is either $90^o$ or $270^o$. Let $M$ be the midpoint of $\overline{CD}$, and suppose $\overline{HM}$ splits the dodecagon into two regions. The ratio of the area of the larger region to the area of the smaller region can be expressed as $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Compute $m + n$. [img]https://cdn.artofproblemsolving.com/attachments/3/e/387bcdf2a6c39fcada4f21f24ceebd18a7f887.png[/img] [b]p12.[/b] Nelson, who never studies for tests, takes several tests in his math class. Each test has a passing score of $60/100$. Since Nelson's test average is at least $60/100$, he manages to pass the class. If only nonnegative integer scores are attainable on each test, and Nelson gets a dierent score on every test, compute the largest possible ratio of tests failed to tests passed. Assume that for each test, Nelson either passes it or fails it, and the maximum possible score for each test is 100. [b]p13.[/b] For each positive integer $n$, let $f(n) = \frac{n}{n+1} + \frac{n+1}{n}$ . Then $f(1)+f(2)+f(3)+...+f(10)$ can be expressed as $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Compute $m + n$. [b]p14.[/b] Triangle $\vartriangle ABC$ has point $D$ lying on line segment $\overline{BC}$ between $B$ and $C$ such that triangle $\vartriangle ABD$ is equilateral. If the area of triangle $\vartriangle ADC$ is $\frac14$ the area of triangle $\vartriangle ABC$, then $\left( \frac{AC}{AB}\right)^2$ can be expressed as $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Compute $m + n$. [b]p15.[/b] In hexagon $ABCDEF$, $AB = 60$, $AF = 40$, $EF = 20$, $DE = 20$, and each pair of adjacent edges are perpendicular to each other, as shown in the below diagram. The probability that a random point inside hexagon $ABCDEF$ is at least $20\sqrt2$ units away from point $D$ can be expressed in the form $\frac{a-b\pi}{c}$ , where $a$, $b$, $c$ are positive integers such that gcd$(a, b, c) = 1$. Compute $a + b + c$. [img]https://cdn.artofproblemsolving.com/attachments/3/c/1b45470265d10a73de7b83eff1d3e3087d6456.png[/img] [b]p16.[/b] The equation $\sqrt{x} +\sqrt{20-x} =\sqrt{20 + 20x - x^2}$ has $4$ distinct real solutions, $x_1$, $x_2$, $x_3$, and $x_4$. Compute $x_1 + x_2 + x_3 + x_4$. [b]p17.[/b] How many distinct words with letters chosen from $B, M, T$ have exactly $12$ distinct permutations, given that the words can be of any length, and not all the letters need to be used? For example, the word $BMMT$ has $12$ permutations. Two words are still distinct even if one is a permutation of the other. For example, $BMMT$ is distinct from $TMMB$. [b]p18.[/b] We call a positive integer binary-okay if at least half of the digits in its binary (base $2$) representation are $1$'s, but no two $1$s are consecutive. For example, $10_{10} = 1010_2$ and $5_{10} = 101_2$ are both binary-okay, but $16_{10} = 10000_2$ and $11_{10} = 1011_2$ are not. Compute the number of binary-okay positive integers less than or equal to $2020$ (in base $10$). [b]p19.[/b] A regular octahedron (a polyhedron with $8$ equilateral triangles) has side length $2$. An ant starts on the center of one face, and walks on the surface of the octahedron to the center of the opposite face in as short a path as possible. The square of the distance the ant travels can be expressed as $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Compute $m + n$. [img]https://cdn.artofproblemsolving.com/attachments/f/8/3aa6abe02e813095e6991f63fbcf22f2e0431a.png[/img] [b]p20.[/b] The sum of $\frac{1}{a}$ over all positive factors $a$ of the number $360$ can be expressed in the form $\frac{m}{n}$ ,where $m$ and $n$ are relatively prime positive integers. Compute $m + n$. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Find the largest positive real number \( c \) such that for any positive integer \( n \), satisfies \(\{ \sqrt{7n} \} \geq \frac{c}{\sqrt{7n}}\).

Find the smallest positive integer $n$ such that both $n^3-n$ and $(n+1)^3-(n+1)$ are divisible by $2025$. [i]V. Kamianetski[/i]

For $p$, $q$, $l$ strictly positive real numbers, consider the following problem: for $y \geq 0$ fixed, determine the values $x \geq 0$ such that $x^p - lx^q \leq y$. Denote by $S(y)$ the set of solutions of this problem. Prove that if one has $p < q$, $\epsilon \in (0, l^\frac{1}{p-q})$, $0 \leq x \leq \epsilon$ and $x \in S(y)$, then $$x\leq ky^{\delta},\ \text{where}\ k=\epsilon\left(\epsilon^p-l\epsilon^q\right)^{-\frac{1}{p}}\ \text{and}\ \delta=\frac{1}{p}$$

Find all prime numbers $p$ for which $p^2 +11$ has exactly six positive divisors.

How many integral solutions are there of the equation $x^5 -31x+2015=0$ ? [list=1] [*] 2 [*] 4 [*] 1 [*] None of these [/list]

For a given a positive integer $n$, find all integers $x_1, x_2,... , x_n$ subject to $0 < x_1 < x_2 < ...< x_n < x_{n+1}$ and $$x_nx_{n+1} \le 2(x_1 + x_2 + ... + x_n).$$

Let $n>1$ be a fixed positive integer, and call an $n$-tuple $(a_1,a_2,\ldots,a_n)$ of integers greater than $1$ [i]good[/i] if and only if $a_i\Big|\left(\frac{a_1a_2\cdots a_n}{a_i}-1\right)$ for $i=1,2,\ldots,n$. Prove that there are finitely many good $n$-tuples. [i]Mitchell Lee.[/i]

The vertices of a triangle have integer coordinates and one of its sides is of length $\sqrt{n}$, where $n$ is a square-free natural number. Prove that the ratio of the circumradius and the inradius is an irrational number.