Let $S$ be a randomly selected four-element subset of $\{1, 2, 3, 4, 5, 6, 7, 8\}$. Let $m$ and $n$ be relatively prime positive integers so that the expected value of the maximum element in $S$ is $\dfrac{m}{n}$. Find $m + n$.

Prove that (a) if the natural number $n$ can be represented as $n =4k+1$ (where $k$ is an integer), then there exist $n$ odd positive integers whose sum is equal to their product, (b) if $n$ cannot be represented in this form then such a set does not exist. (M. Kontsevich)

Suppose $ n$ is a product of four distinct primes $ a,b,c,d$ such that: $ (i)$ $ a\plus{}c\equal{}d;$ $ (ii)$ $ a(a\plus{}b\plus{}c\plus{}d)\equal{}c(d\minus{}b);$ $ (iii)$ $ 1\plus{}bc\plus{}d\equal{}bd$. Determine $ n$.

If $m, n$, and $p$ are three different natural numbers, each between $2$ and $9$, what then are all the possible integer value(s) of the expression $\frac{m+n+p}{m+n}$?

A right parallelepiped (i.e. a parallelepiped one of whose edges is perpendicular to a face) is given. Its vertices have integral coordinates, and no other points with integral coordinates lie on its faces or edges. Prove that the volume of this parallelepiped is a sum of three perfect squares. [i]Proposed by A. Golovanov[/i]

[b]p1.[/b] Nine coins are placed in a row, alternating between heads and tails as follows: $H T H T H T H T H$. A legal move consists of turning over any two adjacent coins. (a) Give a sequence of legal moves that changes the configuration into $H H H H H H H H H$. (b) Prove that there is no sequence of legal moves that changes the original configuration into $T T T T T T T T T$. [b]p2.[/b] Find (with proof) all integers $k $that satisfy the equation $$\frac{k - 15}{2000}+\frac{k - 12}{2003}+\frac{k - 9}{2006}+\frac{k - 6}{2009}+\frac{k - 3}{2012} = \frac{k - 2000}{15}+\frac{k - 2003}{12}+\frac{k - 2006}{9}+\frac{k - 2009}{6}+\frac{k - 2012}{3}.$$ [b]p3.[/b] Some (not necessarily distinct) natural numbers from $1$ to $2015$ are written on $2015$ lottery tickets, with exactly one number written on each ticket. It is known that the sum of the numbers on any nonempty subset of tickets (including the set of all tickets) is not divisible by $2016$. Prove that the same number is written on all of the tickets. [b]p4.[/b] A set of points $A$ is called distance-distinct if every pair of points in $A$ has a different distance. (a) Show that for all infinite sets of points $B$ on the real line, there exists an infinite distance-distinct set A contained in $B$. (b) Show that for all infinite sets of points $B$ on the real plane, there exists an infinite distance-distinct set A contained in $B$. [b]p5.[/b] Let $ABCD$ be a (not necessarily regular) tetrahedron and consider six points $E, F, G, H, I, J$ on its edges $AB$, $BC$, $AC$, $AD$, $BD$, $CD$, respectively, such that $$|AE| \cdot |EB| = |BF| \cdot |FC| = |AG| \cdot |GC| = |AH| \cdot |HD| = |BI| \cdot |ID| = |CJ| \cdot |JD|.$$ Prove that the points $E, F, G, H, I$, and $J$ lie on the surface of a sphere. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Is it possible to represent number $2011... 2011$, where number $2011$ is written $20112011$ times, as a product of some number and sum of its digits?

A [i]semiprime [/i] is a positive integer that is a product of two prime numbers. For example, $9$ and $10$ are semiprimes. How many semiprimes less than $100$ are there?

Two people play this game. At the beginning there are numbers 1, 2, 3, 4 in a circle. With each move, the first one adds 1 to two adjacent numbers, and the second swaps any two adjacent numbers. The first one wins if all numbers become equal. Can the second one interfere with him?

Let $S$ be a finite set of integers. Prove that there exists a number $c$ depending on $S$ such that for each non-constant polynomial $f$ with integer coefficients the number of integers $k$ satisfying $f(k)\in S$ does not exceed $\max(\deg f,c)$.

Let $a_1,a_2,\dots, a_n$ be positive integers with product $P,$ where $n$ is an odd positive integer. Prove that $$\gcd(a_1^n+P,a_2^n+P,\dots, a_n^n+P)\le 2\gcd(a_1,\dots, a_n)^n.$$ [i]Proposed by Daniel Liu[/i]

[b]a.)[/b] For which $n>2$ is there a set of $n$ consecutive positive integers such that the largest number in the set is a divisor of the least common multiple of the remaining $n-1$ numbers? [b]b.)[/b] For which $n>2$ is there exactly one set having this property?

The function $ f : \mathbb{N} \to \mathbb{Z}$ is defined by $ f(0) \equal{} 2$, $ f(1) \equal{} 503$ and $ f(n \plus{} 2) \equal{} 503f(n \plus{} 1) \minus{} 1996f(n)$ for all $ n \in\mathbb{N}$. Let $ s_1$, $ s_2$, $ \ldots$, $ s_k$ be arbitrary integers not smaller than $ k$, and let $ p(s_i)$ be an arbitrary prime divisor of $ f\left(2^{s_i}\right)$, ($ i \equal{} 1, 2, \ldots, k$). Prove that, for any positive integer $ t$ ($ t\le k$), we have $ 2^t \Big | \sum_{i \equal{} 1}^kp(s_i)$ if and only if $ 2^t | k$.

Determine all integers $ a$ for which the equation $ x^2\plus{}axy\plus{}y^2\equal{}1$ has infinitely many distinct integer solutions $ x,y$.

Show that there exists a positive integer $n$ such that the decimal representations of $3^n$ and $7^n$ both start with the digits $10$.

[u]Set 1[/u] [b]G1.[/b] The Daily Challenge office has a machine that outputs the number $2.75$ when operated. If it is operated $12$ times, then what is the sum of all $12$ of the machine outputs? [b]G2.[/b] A car traveling at a constant velocity $v$ takes $30$ minutes to travel a distance of $d$. How long does it take, in minutes, for it travel $10d$ with a constant velocity of $2.5v$? [b]G3.[/b] Andy originally has $3$ times as many jelly beans as Andrew. After Andrew steals 15 of Andy’s jelly beans, Andy now only has $2$ times as many jelly beans as Andrew. Find the number of jelly beans Andy originally had. [u]Set 2[/u] [b]G4.[/b] A coin is weighted so that it is $3$ times more likely to come up as heads than tails. How many times more likely is it for the coin to come up heads twice consecutively than tails twice consecutively? [b]G5.[/b] There are $n$ students in an Areteem class. When 1 student is absent, the students can be evenly divided into groups of $5$. When $8$ students are absent, the students can evenly be divided into groups of $7$. Find the minimum possible value of $n$. [b]G6.[/b] Trapezoid $ABCD$ has $AB \parallel CD$ such that $AB = 5$, $BC = 4$ and $DA = 2$. If there exists a point $M$ on $CD$ such that $AM = AD$ and $BM = BC$, find $CD$. [u]Set 3[/u] [b]G7.[/b] Angeline has $10$ coins (either pennies, nickels, or dimes) in her pocket. She has twice as many nickels as pennies. If she has $62$ cents in total, then how many dimes does she have? [b]G8.[/b] Equilateral triangle $ABC$ has side length $6$. There exists point $D$ on side $BC$ such that the area of $ABD$ is twice the area of $ACD$. There also exists point $E$ on segment $AD$ such that the area of $ABE$ is twice the area of $BDE$. If $k$ is the area of triangle $ACE$, then find $k^2$. [b]G9.[/b] A number $n$ can be represented in base $ 6$ as $\underline{aba}_6$ and base $15$ as $\underline{ba}_{15}$, where $a$ and $b$ are not necessarily distinct digits. Find $n$. PS. You should use hide for answers. Sets 4-6 have been posted [url=https://artofproblemsolving.com/community/c3h3131305p28367080]here[/url] and 7-9 [url=https://artofproblemsolving.com/community/c3h3131308p28367095]here[/url].Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Show that exists a sequence of $ 100$ terms such that: 1)Every term is a perfect square 2) every term is greater than the one before it ( it is strictly increasing) 3)Every two terms of the sequence are relative prime 4) The average between two consecutive terms is also a perfect square Daniel

Let $n = 6901$. There are $6732$ positive integers less than or equal to $n$ that are also relatively prime to $n$. Find the sum of the distinct prime factors of $n$.

[b]p1.[/b] Compute $$\sqrt{(\sqrt{63} +\sqrt{112} +\sqrt{175})(-\sqrt{63} +\sqrt{112} +\sqrt{175})(\sqrt{63}-\sqrt{112} +\sqrt{175})(\sqrt{63} +\sqrt{112} -\sqrt{175})}$$ [b]p2.[/b] Consider the set $S = \{0, 1, 2, 3, 4, 5, 6, 7, 8, 9\}$. How many distinct $3$-element subsets are there such that the sum of the elements in each subset is divisible by $3$? [b]p3.[/b] Let $a^2$ and $b^2$ be two integers. Consider the triangle with one vertex at the origin, and the other two at the intersections of the circle $x^2 + y^2 = a^2 + b^2$ with the graph $ay = b|x|$. If the area of the triangle is numerically equal to the radius of the circle, what is this area? [b]p4.[/b] Suppose $f(x) = x^3 + x - 1$ has roots $a$, $b$ and $c$. What is $$\frac{a^3}{1-a}+\frac{b^3}{1-b}+\frac{c^3}{1-c} ?$$ [b]p5.[/b] Lisa has a $2D$ rectangular box that is $48$ units long and $126$ units wide. She shines a laser beam into the box through one of the corners such that the beam is at a $45^o$ angle with respect to the sides of the box. Whenever the laser beam hits a side of the box, it is reflected perfectly, again at a $45^o$ angle. Compute the distance the laser beam travels until it hits one of the four corners of the box. [b]p6.[/b] How many ways can we form a group with an odd number of members (plural) from $99$ people total? Express your answer in the form $a^b + c$, where $a$, $b$, and $c$ are integers, and $a$ is prime. [b]p7.[/b] Let $$S = \log_2 9 \log_3 16 \log_4 25 ...\log_{999} 1000000.$$ Compute the greatest integer less than or equal to $\log_2 S$. [b]p8.[/b] A prison, housing exactly four hundred prisoners in four hundred cells numbered $1$-$400$, has a really messed-up warden. One night, when all the prisoners are asleep and all of their doors are locked, the warden toggles the locks on all of their doors (that is, if the door is locked, he unlocks the door, and if the door is unlocked, he locks it again), starting at door $1$ and ending at door $400$. The warden then toggles the lock on every other door starting at door $2$ ($2$, $4$, $6$, etc). After he has toggled the lock on every other door, the warden then toggles every third door (doors $3$, $6$, $9$, etc.), then every fourth door, etc., finishing by toggling every $400$th door (consisting of only the $400$th door). He then collapses in exhaustion. Compute the number of prisoners who go free (that is, the number of unlocked doors) when they wake up the next morning. [b]p9.[/b] Let $A$ and $B$ be fixed points on a $2$-dimensional plane with distance $AB = 1$. An ant walks on a straight line from point $A$ to some point $C$ on the same plane and finds that the distance from itself to $B$ always decreases at any time during this walk. Compute the area of the locus of points where point $C$ could possibly be located. [b]p10.[/b] A robot starts in the bottom left corner of a $4 \times 4$ grid of squares. How many ways can it travel to each square exactly once and then return to its start if it is only allowed to move to an adjacent (not diagonal) square at each step? [b]p11.[/b] Assuming real values for $p$, $q$, $r$, and $s$, the equation $$x^4 + px^3 + qx^2 + rx + s$$ has four non-real roots. The sum of two of these roots is $4 + 7i$, and the product of the other two roots is $3 - 4i$. Find $q$. [b]p12.[/b] A cube is inscribed in a right circular cone such that one face of the cube lies on the base of the cone. If the ratio of the height of the cone to the radius of the cone is $2 : 1$, what fraction of the cone's volume does the cube take up? Express your answer in simplest radical form. [b]p13.[/b] Let $$y =\dfrac{1}{1 +\dfrac{1}{9 +\dfrac{1}{5 +\dfrac{1}{9 +\dfrac{1}{5 +...}}}}}$$ If $y$ can be represented as $\frac{a\sqrt{b} + c}{d}$, where $b$ is not divisible by the square of any prime, and the greatest common divisor of $a$ and $d$ is $1$, find the sum $a + b + c + d$. [b]p14.[/b] Alice wants to paint each face of an octahedron either red or blue. She can paint any number of faces a particular color, including zero. Compute the number of ways in which she can do this. Two ways of painting the octahedron are considered the same if you can rotate the octahedron to get from one to the other. [b]p15.[/b] Find $n$ in the equation $$133^5 + 110^5 + 84^5 + 27^5 = n^5,$$ where $n$ is an integer less than $170$. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

A positive integer $x$ is $k$-[i]equivocal [/i] if there exists two positive integers $b$, $b'$ such that when $x$ is represented in base $b$ and base $b'$, the two representations have digit sequences of length $k$ that are permutations of each other. The smallest $2$-equivocal number is $7$, since $7$ is $21$ in base $3$ and $12$ in base $5$. Find the smallest $3$-equivocal number.

The solution to the equation $\log_{3x} 4 = \log_{2x} 8$, where $x$ is a positive real number other than $\tfrac{1}{3}$ or $\tfrac{1}{2}$, can be written as $\tfrac {p}{q}$ where $p$ and $q$ are relatively prime positive integers. What is $p + q$? $\textbf{(A) } 5 \qquad \textbf{(B) } 13 \qquad \textbf{(C) } 17 \qquad \textbf{(D) } 31 \qquad \textbf{(E) } 35 $

Given positive odd number $m$ and integer ${a}.$ Proof: For any real number $c,$ $$\#\left\{x\in\mathbb Z\cap [c,c+\sqrt m]\mid x^2\equiv a\pmod m\right\}\le 2+\log_2m.$$ [i]Proposed by Yinghua Ai[/i]

For a positive integer $n$, let $\tau(n)$ and $\sigma(n)$ be the number of positive divisors of $n$ and the sum of positive divisors of $n$, respectively. let $a$ and $b$ be positive integers such that $\sigma(a^n)$ divides $\sigma(b^n)$ for all $n\in \mathbb{N}$. Prove that each prime factor of $\tau(a)$ divides $\tau(b)$. Proposed by MohammadAmin Sharifi

Does there exist a sequence of positive integers $a_1,a_2,...$ such that every positive integer occurs exactly once and that the number $\tau (na_{n+1}^n+(n+1)a_n^{n+1})$ is divisible by $n$ for all positive integer. Here $\tau (n)$ denotes the number of positive divisor of $n$.

Find all positive integers $x$ such that the product of all digits of $x$ is given by $x^2 - 10 \cdot x - 22.$