Let $m, n > 1$ be coprime odd integers. Show that $$\big \lfloor \frac{m^{\phi (n)+1} + n^{\phi (m)+1}}{mn} \rfloor$$ is an even integer, where $\phi$ is Euler’s totient function.

Find all pairs $\left(m,n\right)$ of positive integers, with $m,n\geq2$, such that $a^n-1$ is divisible by $m$ for each $a\in \left\{1,2,3,\ldots,n\right\}$.

Determine all pairs $(p, q)$, $p, q \in \mathbb {N}$, such that $(p + 1)^{p - 1} + (p - 1)^{p + 1} = q^q$.

Prove that there exist infinitely many positive integers $k$ such that the sequence $\{x_n\}$ satisfying $$ x_1=1, x_2=k+2, x_{n+2}-(k+1)x_{n+1}+x_n=0(n \ge 0)$$ does not contain any prime number.

The sequence ($a_n$) of natural numbers is defined by $a_1 = 2$ and $a_{n+1}$ equals the sum of tenth powers of the decimal digits of $a_n$ for all $n \ge 1$. Are there numbers which appear twice in the sequence ($a_n$)?

We say that a natural number $ n\ge 4 $ is [i]unusual[/i] if, for any $ n\times n $ array of real numbers, the sum of the numbers from any $ 3\times 3 $ compact subarray is negative, and the sum of the numbers from any $ 4\times 4 $ compact subarray is positive. Find all unusual numbers.

Prove that there are infinitely many positive integers $ n$ such that $ n^{2} \plus{} 1$ has a prime divisor greater than $ 2n \plus{} \sqrt {2n}$. [i]Author: Kestutis Cesnavicius, Lithuania[/i]

Every integer $> 2024$ is given a color, white or black. The product of any two white integers is a black integer. Prove that there are two black integers that have a difference of one.

The cells of a $100 \times 100$ table contain non-zero numbers. It turned out that all $100$ hundred-digit numbers written horizontally are divisible by 11. Could it be that exactly $99$ hundred-digit numbers written vertically are also divisible by $11$?

[b]p1.[/b] Len's Spanish class has four tests in the first term. Len scores $72$, $81$, and $78$ on the first three tests. If Len wants to have an 80 average for the term, what is the minimum score he needs on the last test? [b]p2.[/b] In $1824$, the Electoral College had $261$ members. Andrew Jackson won $99$ Electoral College votes and John Quincy Adams won $84$ votes. A plurality occurs when no candidate has more than $50\%$ of the votes. Should a plurality occur, the vote goes to the House of Representatives to break the tie. How many more votes would Jackson have needed so that a plurality would not have occurred? [b]p3.[/b] $\frac12 + \frac16 + \frac{1}{12} + \frac{1}{20} + \frac{1}{30}= 1 - \frac{1}{n}$. Find $n$. [b]p4.[/b] How many ways are there to sit Samuel, Esun, Johnny, and Prat in a row of $4$ chairs if Prat and Johnny refuse to sit on an end? [b]p5.[/b] Find an ordered quadruple $(w, x, y, z)$ that satisfies the following: $$3^w + 3^x + 3^y = 3^z$$ where $w + x + y + z = 2017$. [b]p6.[/b] In rectangle $ABCD$, $E$ is the midpoint of $CD$. If $AB = 6$ inches and $AE = 6$ inches, what is the length of $AC$? [b]p7.[/b] Call an integer interesting if the integer is divisible by the sum of its digits. For example, $27$ is divisible by $2 + 7 = 9$, so $27$ is interesting. How many $2$-digit interesting integers are there? [b]p8.[/b] Let $a\#b = \frac{a^3-b^3}{a-b}$ . If $a, b, c$ are the roots of the polynomial $x^3 + 2x^2 + 3x + 4$, what is the value of $a\#b + b\#c + c\#a$? [b]p9.[/b] Akshay and Gowri are examining a strange chessboard. Suppose $3$ distinct rooks are placed into the following chessboard. Find the number of ways that one can place these rooks so that they don't attack each other. Note that two rooks are considered attacking each other if they are in the same row or the same column. [img]https://cdn.artofproblemsolving.com/attachments/f/1/70f7d68c44a7a69eb13ce12291c0600d11027c.png[/img] [b]p10.[/b] The Earth is a very large sphere. Richard and Allen have a large spherical model of Earth, and they would like to (for some strange reason) cut the sphere up with planar cuts. If each cut intersects the sphere, and Allen holds the sphere together so it does not fall apart after each cut, what is the maximum number of pieces the sphere can be cut into after $6$ cuts? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Prove that a $46$-element set of integers contains two distinct doubletons $\{u, v\}$ and $\{x,y\}$ such that $u + v \equiv x + y$ (mod $2016$).

For any odd prime $p$ and any integer $n,$ let $d_p (n) \in \{ 0,1, \dots, p-1 \}$ denote the remainder when $n$ is divided by $p.$ We say that $(a_0, a_1, a_2, \dots)$ is a [i]p-sequence[/i], if $a_0$ is a positive integer coprime to $p,$ and $a_{n+1} =a_n + d_p (a_n)$ for $n \geqslant 0.$ (a) Do there exist infinitely many primes $p$ for which there exist $p$-sequences $(a_0, a_1, a_2, \dots)$ and $(b_0, b_1, b_2, \dots)$ such that $a_n >b_n$ for infinitely many $n,$ and $b_n > a_n$ for infinitely many $n?$ (b) Do there exist infinitely many primes $p$ for which there exist $p$-sequences $(a_0, a_1, a_2, \dots)$ and $(b_0, b_1, b_2, \dots)$ such that $a_0 <b_0,$ but $a_n >b_n$ for all $n \geqslant 1?$ [I]United Kingdom[/i]

Let $ p_1,\dots,p_k$ be prime numbers, and let $ S$ be the set of those integers whose all prime divisors are among $ p_1,\dots,p_k$. For a finite subset $ A$ of the integers let us denote by $ \mathcal G(A)$ the graph whose vertices are the elements of $ A$, and the edges are those pairs $ a,b\in A$ for which $ a \minus{} b\in S$. Does there exist for all $ m\geq 3$ an $ m$-element subset $ A$ of the integers such that (i) $ \mathcal G(A)$ is complete? (ii) $ \mathcal G(A)$ is connected, but all vertices have degree at most 2?

A math contest consists of $9$ objective type questions and $6$ fill in the blanks questions. From a school some number of students took the test and it was noticed that all students had attempted exactly $14$ out of $15$ questions. Let $O_1, O_2, \dots , O_9$ be the nine objective questions and $F_1, F_2, \dots , F_6$ be the six fill inthe blanks questions. Let $a_{ij}$ be the number of students who attemoted both questions $O_i$ and $F_j$. If the sum of all the $a_{ij}$ for $i=1, 2,\dots , 9$ and $j=1, 2,\dots , 6$ is $972$, then find the number of students who took the test in the school.

[b]p1.[/b] Ana, Bob, Cho, Dan, and Eve want to use a microwave. In order to be fair, they choose a random order to heat their food in (all orders have equal probability). Ana's food needs $5$ minutes to cook, Bob's food needs $7$ minutes, Cho's needs $1$ minute, Dan's needs $12$ minutes, and Eve's needs $5$ minutes. What is the expected number of minutes Bob has to wait for his food to be done? [b]p2.[/b] $ABC$ is an equilateral triangle. $H$ lies in the interior of $ABC$, and points $X$, $Y$, $Z$ lie on sides $AB, BC, CA$, respectively, such that $HX\perp AB$, $HY \perp BC$, $HZ\perp CA$. Furthermore, $HX =2$, $HY = 3$, $HZ = 4$. Find the area of triangle $ABC$. [b]p3.[/b] Amy, Ben, and Chime play a dice game. They each take turns rolling a die such that the $first$ person to roll one of his favorite numbers wins. Amy's favorite number is $1$, Ben's favorite numbers are $2$ and $3$, and Chime's are $4$, $5$, and $6$. Amy rolls first, Ben rolls second, and Chime rolls third. If no one has won after Chime's turn, they repeat the sequence until someone has won. What's the probability that Chime wins the game? [b]p4.[/b] A point $P$ is chosen randomly in the interior of a square $ABCD$. What is the probability that the angle $\angle APB$ is obtuse? [b]p5.[/b] Let $ABCD$ be the quadrilateral with vertices $A = (3, 9)$, $B = (1, 1)$, $C = (5, 3)$, and $D = (a, b)$, all of which lie in the first quadrant. Let $M$ be the midpoint of $AB$, $N$ the midpoint of $BC$, $O$ the midpoint of $CD$, and $P$ the midpoint of $AD$. If $MNOP$ is a square, find $(a, b)$. [b]p6.[/b] Let $M$ be the number of positive perfect cubes that divide $60^{60}$. What is the prime factorization of $M$? [b]p7.[/b] Given that $x$, $y$, and $z$ are complex numbers with $|x|=|y| =|z|= 1$, $x + y + z = 1$ and $xyz = 1$, find $|(x + 2)(y + 2)(z + 2)|$. [b]p8.[/b] If $f(x)$ is a polynomial of degree $2008$ such that $f(m) = \frac{1}{m}$ for $m = 1, 2, ..., 2009$, find $f(2010)$. [b]p9.[/b] A drunkard is randomly walking through a city when he stumbles upon a $2 \times 2$ sliding tile puzzle. The puzzle consists of a $2 \times 2$ grid filled with a blank square, as well as $3$ square tiles, labeled $1$, $2$, and $3$. During each turn you may fill the empty square by sliding one of the adjacent tiles into it. The following image shows the puzzle's correct state, as well as two possible moves you can make: [img]https://cdn.artofproblemsolving.com/attachments/c/6/7ddd9305885523deeee2a530dc90505875d1cc.png[/img] Assuming that the puzzle is initially in an incorrect (but solvable) state, and that the drunkard will make completely random moves to try and solve it, how many moves is he expected to make before he restores the puzzle to its correct state? [b]p10.[/b] How many polynomials $p(x)$ exist such that the coeffients of $p(x)$ are a rearrangement of $\{0, 1, 2, .., deg \, p(x)\}$ and all of the roots of $p(x)$ are rational? (Note that the leading coefficient of $p(x)$ must be nonzero.) PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Prove that if one of the numbers $25x+31y, 3x+7y$ (where $x,y \in Z$) is a multiple of $41$, then so is the other.

Determine all pairs $(p,m)$ consisting of a prime number $p$ and a positive integer $m$, for which $p^3 + m(p + 2) = m^2 + p + 1$ holds.

In the system of base $n^2 + 1$ find a number $N$ with $n$ different digits such that: [b](i)[/b] $N$ is a multiple of $n$. Let $N = nN'.$ [b](ii)[/b] The number $N$ and $N'$ have the same number $n$ of different digits in base $n^2 + 1$, none of them being zero. [b] (iii)[/b] If $s(C)$ denotes the number in base $n^2 + 1$ obtained by applying the permutation $s$ to the $n$ digits of the number $C$, then for each permutation $s, s(N) = ns(N').$

Jamar bought some pencils costing more than a penny each at the school bookstore and paid $\$1.43$. Sharona bought some of the same pencils and paid $\$1.87$. How many more pencils did Sharona buy than Jamar? $\textbf{(A)}\hspace{.05in}2 \qquad \textbf{(B)}\hspace{.05in}3 \qquad \textbf{(C)}\hspace{.05in}4 \qquad \textbf{(D)}\hspace{.05in}5 \qquad \textbf{(E)}\hspace{.05in}6 $

Let $a$, $b$ be positive integers such that $b^n+n$ is a multiple of $a^n+n$ for all positive integers $n$. Prove that $a=b$. [i]Proposed by Mohsen Jamali, Iran[/i]

For any positive integer $n$, show that there exists a positive integer $m$ such that $n$ divides $2016^m + m$.

Solve equation $(3a-bc)(3b-ac)(3c-ab)=1000$ in integers. (V.Brayman)

Determine all integers $k$ for which there exists a function $f: \mathbb{Z}_{>0} \to \mathbb{Z}$ such that $f(2023) = 2024$ and $f(ab) = f(a) + f(b) + kf(\gcd(a,b))$ for all positive integers $a$ and $b$.

Find all polynomials $f(x)$ with integer coefficients such that $f(n)$ and $f(2^{n})$ are co-prime for all natural numbers $n$.

Let $d = a^{1999} + b^{1999} + c^{1999}$ , where $a, b$ and $c$ are integers such that $a + b + c = 0$. (a) May it happen that $d = 2$? (b) May it happen that $d$ is prime? (V Senderov)