What is the largest integer $n$ such that $n$ is divisible by every integer less than $\sqrt[3]{n}$?

Fix an integer $n \ge 2$, let $Q_n$ be the graph consisting of all vertices and all edges of an $n$-cube, and let $T$ be a spanning tree in $Q_n$. Show that $Q_n$ has an edge whose adjunction to $T$ produces a simple cycle of length at least $2n$.

Let $m,n$ be positive integers greater than $1$. We define the sets $P_m=\left\{\frac{1}{m},\frac{2}{m},\cdots,\frac{m-1}{m}\right\}$ and $P_n=\left\{\frac{1}{n},\frac{2}{n},\cdots,\frac{n-1}{n}\right\}$. Find the distance between $P_m$ and $P_n$, that is defined as \[\min\{|a-b|:a\in P_m,b\in P_n\}\]

Let $n$ be a positive integer. Daniel and Merlijn are playing a game. Daniel has $k$ sheets of paper lying next to each other on a table, where $k$ is a positive integer. On each of the sheets, he writes some of the numbers from $1$ up to $n$ (he is allowed to write no number at all, or all numbers). On the back of each of the sheets, he writes down the remaining numbers. Once Daniel is finished, Merlijn can flip some of the sheets of paper (he is allowed to flip no sheet at all, or all sheets). If Merlijn succeeds in making all of the numbers from $1$ up to n visible at least once, then he wins. Determine the smallest $k$ for which Merlijn can always win, regardless of Daniel’s actions.

Given is a nonzero integer $k$. Prove that equation $k =\frac{x^2 - xy + 2y^2}{x + y}$ has an odd number of ordered integer pairs $(x, y)$ just when $k$ is divisible by seven.

For $t \ge 2$, define $S(t)$ as the number of times $t$ divides into $t!$. We say that a positive integer $t$ is a [i]peak[/i] if $S(t) > S(u)$ for all values of $u < t$. Prove or disprove the following statement: For every prime $p$, there is an integer $k$ for which $p$ divides $k$ and $k$ is a peak.

Symedians of an acute-angled non-isosceles triangle $ABC$ intersect at a point at point $L$, and $AA_1$, $BB_1$ and $CC_1$ are its altitudes. Prove that you can construct equilateral triangles $A_1B_1C'$, $B_1C_1A'$ and $C_1A_1B'$ not lying in the plane $ABC$, so that lines $AA' , BB'$ and $CC'$ and also perpendicular to the plane $ABC$ at point $L$ intersected at one point.

The map below shows an east/west road connecting the towns of Acorn, Centerville, and Midland, and a north/south road from Centerville to Drake. The distances from Acorn to Centerville, from Centerville to Midland, and from Centerville to Drake are each 60 kilometers. At noon Aaron starts at Acorn and bicycles east at 17 kilometers per hour, Michael starts at Midland and bicycles west at 7 kilometers per hour, and David starts at Drake and bicycles at a constant rate in a straight line across an open field. All three bicyclists arrive at exactly the same time at a point along the road from Centerville to Midland. Find the number of kilometers that David bicycles. For the map go to http://www.purplecomet.org/welcome/practice

Find all triples $(a,b,c)$ of positive integers such that if $n$ is not divisible by any prime less than $2014$, then $n+c$ divides $a^n+b^n+n$. [i]Proposed by Evan Chen[/i]

Given a positive integer $n$, determine all functions $f$ from the first $n$ positive integers to the positive integers, satisfying the following two conditions: [b](1)[/b] $\sum_{k=1}^{n}{f(k)}=2n$; and [b](2)[/b] $\sum_{k\in K}{f(k)}=n$ for no subset $K$ of the first $n$ positive integers.

Let $ ABC $ be an acute triangle. Denote by $ D $ the foot of the perpendicular line drawn from the point $ A $ to the side $ BC $, by $M$ the midpoint of $ BC $, and by $ H $ the orthocenter of $ ABC $. Let $ E $ be the point of intersection of the circumcircle $ \Gamma $ of the triangle $ ABC $ and the half line $ MH $, and $ F $ be the point of intersection (other than $E$) of the line $ ED $ and the circle $ \Gamma $. Prove that $ \tfrac{BF}{CF} = \tfrac{AB}{AC} $ must hold. (Here we denote $XY$ the length of the line segment $XY$.)

Let $m$ and $n$ be fixed positive integers. Tsvety and Freyja play a game on an infinite grid of unit square cells. Tsvety has secretly written a real number inside of each cell so that the sum of the numbers within every rectangle of size either $m$ by $n$ or $n$ by $m$ is zero. Freyja wants to learn all of these numbers. One by one, Freyja asks Tsvety about some cell in the grid, and Tsvety truthfully reveals what number is written in it. Freyja wins if, at any point, Freyja can simultaneously deduce the number written in every cell of the entire infinite grid (If this never occurs, Freyja has lost the game and Tsvety wins). In terms of $m$ and $n$, find the smallest number of questions that Freyja must ask to win, or show that no finite number of questions suffice. [i]Nikolai Beluhov[/i]

Let $a_0,a_1,\dots,a_n \in \mathbb N$. Prove that there exist positive integers $b_0,b_1,\dots,b_n$ such that for $0 \leq i \leq n : a_i \leq b_i \leq 2a_i$ and polynomial \[P(x) = b_0 + b_1 x + \dots + b_n x^n\] is irreducible over $\mathbb Q[x]$. (10 points)

Let $n>100$ be a positive integer and originally the number $1$ is written on the blackboard. Petya and Vasya play the following game: every minute Petya represents the number of the board as a sum of two distinct positive fractions with coprime nominator and denominator and Vasya chooses which one to delete. Show that Petya can play in such a manner, that after $n$ moves, the denominator of the fraction left on the board is at most $2^n+50$, no matter how Vasya acts.

Republic has $n \geq 2$ cities, between some pairs of cities there are non-directed flight routes. From each city it is possible to get to any other city, and we will call the minimal number of flights required to do that the [i]distance[/i] between the cities. For every city consider the biggest distance to another city. It turned out that for every city this number is equal to $m$. Find all values $m$ can attain for given $n$

Find all continuous real functions $ f,g$ and $ h$ defined on the set of positive real numbers and satisfying the relation \[ f(x\plus{}y)\plus{}g(xy)\equal{}h(x)\plus{}h(y)\] for all $ x>0$ and $ y>0$. [i]Z. Daroczy[/i]

Outside triangle $ ABC $ equilateral triangles $ BMC $, $ CNA $, and $ APB $ are constructed. Prove that the centers $ S $, $ T $, $ U $ of these triangles form an equilateral triangle.

Let $n$ be a positive integer. Let $S=\{a_1,\ldots, a_{k}\}$ be a finite collection of at least $n$ not necessarily distinct positive real numbers. Let $$f(S)=\left(\sum_{i=1}^{k} a_{i}\right)^{n}$$ and $$g(S)=\sum_{1\leq i_{1}<\ldots<i_{n} \leq k} a_{i_{1}}\cdot\ldots\cdot a_{i_{n}}.$$ Determine $\sup_{S} \frac{g(S)}{f(S)}$.

Over all real numbers $x$, let $k$ be the minimum possible value of the expression $$\sqrt{x^2 + 9} +\sqrt{x^2 - 6x + 45}.$$ Determine $k^2$.

Let $n\geq1$ be an integer and let $t_1<t_2<\dots<t_n$ be positive integers. In a group of $t_n+1$ people, some games of chess are played. Two people can play each other at most once. Prove that it is possible for the following two conditions to hold at the same time: (i) The number of games played by each person is one of $t_1,t_2,\dots,t_n$. (ii) For every $i$ with $1\leq i\leq n$, there is someone who has played exactly $t_i$ games of chess.

There are $14$ students with different heights. At each step, two adjacent students will be swapped. Whatever the first arrangement is, in at least how many steps the students can be lined up? $\textbf{(A)}\ 42 \qquad\textbf{(B)}\ 43 \qquad\textbf{(C)}\ 45 \qquad\textbf{(D)}\ 52 \qquad\textbf{(E)}\ \text{None}$

A picture with an area of $160$ square inches is surrounded by a $2$ inch border. The picture with its border is a rectangle twice as long as it is wide. How many inches long is that rectangle?

What is the smallest non-square positive integer that is the product of four prime numbers (not necessarily distinct)?

In the addition problem, each digit has been replaced by a letter. If different letters represent different digits then $C=$ [asy] unitsize(18); draw((-1,0)--(3,0)); draw((-3/4,1/2)--(-1/4,1/2)); draw((-1/2,1/4)--(-1/2,3/4)); label("$A$",(0.5,2.1),N); label("$B$",(1.5,2.1),N); label("$C$",(2.5,2.1),N); label("$A$",(1.5,1.1),N); label("$B$",(2.5,1.1),N); label("$A$",(2.5,0.1),N); label("$3$",(0.5,-.1),S); label("$0$",(1.5,-.1),S); label("$0$",(2.5,-.1),S); [/asy] $\text{(A)}\ 1 \qquad \text{(B)}\ 3 \qquad \text{(C)}\ 5 \qquad \text{(D)}\ 7 \qquad \text{(E)}\ 9$

Real numbers $x_1, x_2, \dots, x_{2018}$ satisfy $x_i + x_j \geq (-1)^{i+j}$ for all $1 \leq i < j \leq 2018$. Find the minimum possible value of $\sum_{i=1}^{2018} ix_i$.