Let $a,b$ and $c$ be positive integers that $$\frac{a\sqrt{3}+b}{b\sqrt3+c}$$ is a rational number, show that $$\frac{a^2+b^2+c^2}{a+b+ c}$$ is an integer. [i](Anonymous314)[/i]

The increasing sequence $3, 15, 24, 48, \ldots$ consists of those positive multiples of 3 that are one less than a perfect square. What is the remainder when the 1994th term of the sequence is divided by 1000?

Prove that there is a constant $c>0$ and infinitely many positive integers $n$ with the following property: there are infinitely many positive integers that cannot be expressed as the sum of fewer than $cn\log(n)$ pairwise coprime $n$th powers. [i]Canada[/i]

Simplify \[\sum_{k=0}^n \frac{(2n)!}{(k!)^2((n-k)!)^2}.\]

Call a positive integer $n$ quixotic if the value of \[\operatorname{lcm}(1,2,...,n)\cdot\left(\frac11+\frac12+\frac13+\dots+\frac1n\right)\]is divisible by 45. Compute the tenth smallest quixotic integer.

[b]p1.[/b] If $a \diamond b = ab - a + b$, find $(3 \diamond 4) \diamond 5$ [b]p2.[/b] If $5$ chickens lay $5$ eggs in $5$ days, how many chickens are needed to lay $10$ eggs in $10$ days? [b]p3.[/b] As Alissa left her house to go to work one hour away, she noticed that her odometer read $16261$ miles. This number is a "special" number for Alissa because it is a palindrome and it contains exactly $1$ prime digit. When she got home that evening, it had changed to the next greatest "special" number. What was Alissa's average speed, in miles per hour, during her two hour trip? [b]p4.[/b] How many $1$ in by $3$ in by $8$ in blocks can be placed in a $4$ in by $4$ in by $9$ in box? [b]p5.[/b] Apple loves eating bananas, but she prefers unripe ones. There are $12$ bananas in each bunch sold. Given any bunch, if there is a $\frac13$ probability that there are $4$ ripe bananas, a $\frac16$ probability that there are $6$ ripe bananas, and a $\frac12$ probability that there are $10$ ripe bananas, what is the expected number of unripe bananas in $12$ bunches of bananas? [b]p6.[/b] The sum of the digits of a $3$-digit number $n$ is equal to the same number without the hundreds digit. What is the tens digit of $n$? [b]p7.[/b] How many ordered pairs of positive integers $(a, b)$ satisfy $a \le 20$, $b \le 20$, $ab > 15$? [b]p8.[/b] Let $z(n)$ represent the number of trailing zeroes of $n!$. What is $z(z(6!))?$ (Note: $n! = n\cdot (n-1) \cdot\cdot\cdot 2 \cdot 1$) [b]p9.[/b] On the Cartesian plane, points $A = (-1, 3)$, $B = (1, 8)$, and $C = (0, 10)$ are marked. $\vartriangle ABC$ is reflected over the line $y = 2x + 3$ to obtain $\vartriangle A'B'C'$. The sum of the $x$-coordinates of the vertices of $\vartriangle A'B'C'$ can be expressed as $\frac{a}{b}$ for relatively prime positive integers $a$, $b$. Compute $a + b$. [b]p10.[/b] How many ways can Bill pick three distinct points from the figure so that the points form a non-degenerate triangle? [img]https://cdn.artofproblemsolving.com/attachments/6/a/8b06f70d474a071b75556823f70a2535317944.png[/img] [b]p11.[/b] Say piece $A$ is attacking piece $B$ if the piece $B$ is on a square that piece $A$ can move to. How many ways are there to place a king and a rook on an $8\times 8$ chessboard such that the rook isn't attacking the king, and the king isn't attacking the rook? Consider rotations of the board to be indistinguishable. (Note: rooks move horizontally or vertically by any number of squares, while kings move $1$ square adjacent horizontally, vertically, or diagonally). [b]p12.[/b] Let the remainder when $P(x) = x^{2020} - x^{2017} - 1$ is divided by $S(x) = x^3 - 7$ be the polynomial $R(x) = ax^2 + bx + c$ for integers $a$, $b$, $c$. Find the remainder when $R(1)$ is divided by $1000$. [b]p13.[/b] Let $S(x) = \left \lfloor \frac{2020}{x} \right\rfloor + \left \lfloor \frac{2020}{x + 1} \right\rfloor$. Find the number of distinct values $S(x)$ achieves for integers $x$ in the interval $[1, 2020]$. [b]p14.[/b] Triangle $\vartriangle ABC$ is inscribed in a circle with center $O$ and has sides $AB = 24$, $BC = 25$, $CA = 26$. Let $M$ be the midpoint of $\overline{AB}$. Points $K$ and $L$ are chosen on sides $\overline{BC}$ and $\overline{CA}$, respectively such that $BK < KC$ and $CL < LA$. Given that $OM = OL = OK$, the area of triangle $\vartriangle MLK$ can be expressed as $\frac{a\sqrt{b}}{c}$ where $a, b, c$ are positive integers, $gcd(a, c) = 1$ and $b$ is not divisible by the square of any prime. Find $a + b + c$. [b]p15.[/b] Euler's totient function, $\phi (n)$, is defined as the number of positive integers less than $n$ that are relatively prime to $n$. Let $S(n)$ be the set of composite divisors of $n$. Evaluate $$\sum^{50}_{k=1}\left( k - \sum_{d\in S(k)} \phi (d) \right)$$ PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

A positive integer is called a repunit, if it is written only by ones. The repunit with $n$ digits will be denoted as $\underbrace{{11\cdots1}}_{n}$ . Prove that: α) the repunit $\underbrace{{11\cdots1}}_{n}$is divisible by $37$ if and only if $n$ is divisible by $3$ b) there exists a positive integer $k$ such that the repunit $\underbrace{{11\cdots1}}_{n}$ is divisible by $41$ if $n$ is divisible by $k$

Let $m$ be a positive integer. Define the sequence $a_0, a_1, a_2, \cdots$ by $a_0 = 0,\; a_1 = m,$ and $a_{n+1} = m^2a_n - a_{n-1}$ for $n = 1,2,3,\cdots$. Prove that an ordered pair $(a,b)$ of non-negative integers, with $a \leq b$, gives a solution to the equation \[ {\displaystyle \frac{a^2 + b^2}{ab + 1} = m^2} \] if and only if $(a,b)$ is of the form $(a_n,a_{n+1})$ for some $n \geq 0$.

Find all functions $f:\mathbb{Z} \rightarrow \mathbb{Z}$ such that $f(1) \neq f(-1)$ and $$f(m+n)^2 \mid f(m)-f(n)$$ for all integers $m, n$. [i]Proposed by Liam Baker, South Africa[/i]

Call a number "prime-looking" if it is composite but not divisible by 2, 3, or 5. The three smallest prime-looking numbers are 49, 77, and 91. There are 168 prime numbers less than 1000. How many prime-looking numbers are there less than 1000? $ \textbf{(A)}\ 100 \qquad \textbf{(B)}\ 102 \qquad \textbf{(C)}\ 104 \qquad \textbf{(D)}\ 106 \qquad \textbf{(E)}\ 108$

Find all ordered triples $(a,b, c)$ of positive integers which satisfy $5^a + 3^b - 2^c = 32$

Find all positive integers $a$ so that for any $\left \lfloor \frac{a+1}{2} \right \rfloor$-digit number that is composed of only digits $0$ and $2$ (where $0$ cannot be the first digit) is not a multiple of $a$.

Prove that there exist infinitely many integers \(n\) which satisfy \(2017^2 | 1^n + 2^n + ... + 2017^n\).

We are given a non-infinite sequence $a_1,a_2…a_n$ of natural numbers. While it is possible, on each turn are chosen two arbitrary indexes $i<j$ such that $a_i \nmid a_j$, and then $a_i$ and $a_j$ are changed with their $gcd$ and $lcm$. Prove that this process is non-infinite and the created sequence doesn’t depend on the made choices.

$a,b$ are naturals and $$a \times GCD(a,b)+b \times LCM(a,b)<2.5 ab$$. Prove that $b|a$

Determine all triples of prime numbers $(p, q, r)$ that satisfy \[p2^q + r^2 = 2025.\] Proposed by [i]Ilija Jovcevski[/i]

Let $a,b$ be positive integers such that $a+b=10$. Let $\tfrac{p}{q}$ be the difference between the maximum and minimum possible values of $\tfrac{1}{a}+\tfrac{1}{b}$, where $p$ and $q$ are relatively prime positive integers. Compute $p+q$.

Find all non-negative integers $x, y$ and primes $p$ such that $$3^x+p^2=7 \cdot 2^y.$$

In the set of integers solve the equation $\frac{1}{x}+\frac{1}{y}=\frac{1}{p}$, where $p$ is a prime number.

Show that for every integer $k \ge 1$ there is a positive integer $n$ such that the decimal representation of $2^n$ contains a block of exactly $k$ zeros, i.e. $2^n = \dots a00 \dots 0b \cdots$ with $k$ zeros and $a, b \ne 0$.

Let $a, b, c$ be positive integers whose greatest common divisor is $1$. Determine whether there always exists a positive integer $n$ such that, for every positive integer $k$, the number $2^n$ is not a divisor of $a^k+b^k+c^k$.

For a positive integer n, let $w(n)$ denote the number of distinct prime divisors of n. Determine the least positive integer k such that $2^{w(n)} \leq k \sqrt[4]{n}$ for all positive integers n.

Find all natural numbers which are divisible by $30$ and which have exactly $30$ different divisors. (M Levin)

Let $Q(x)$ be a cubic polynomial with integer coefficients. Suppose that a prime $p$ divides $Q(x_j)$ for $j = 1$ ,$2$ ,$3$ ,$4$ , where $x_1 , x_2 , x_3 , x_4$ are distinct integers from the set $\{0,1,\cdots, p-1\}$. Prove that $p$ divides all the coefficients of $Q(x)$.

Let $\phi(n)$ be the number of positive integers less than $n$ that are relatively prime to $n$, where $n$ is a positive integer. Find all pairs of positive integers $(m,n)$ such that \[2^n + (n-\phi(n)-1)! = n^m+1.\]