Find all positive integers $m$ and $n$ such that the inequality: \[ [ (m+n) \alpha ] + [ (m+n) \beta ] \geq [ m \alpha ] + [n \beta] + [ n(\alpha+\beta)] \] is true for any real numbers $\alpha$ and $\beta$. Here $[x]$ denote the largest integer no larger than real number $x$.

Let $X$ be a set with $n$ elements. Find the largest number of subsets of $X$, each with $3$ elements, so that no two of them are disjoint.

Point $D$ lies in $\triangle ABC$ and $BD=CD$,$\angle BDC=120$. Point $E$ lies outside $ABC$ and $AE=CE,\angle AEC=60$. Points $B$ and $E$ lies on different sides of $AC$. $F$ is midpoint $BE$. Prove, that $\angle AFD=90$

Two circles intersect at $X$ and $Y$ . The line through the centers of the circles intersect the first circle at $A$ and $C$, and intersect the second circle at $B$ and $D$ so that $A, B, C, D$ lie in this order. The common chord $XY$ cuts $BC$ at $P$, and a point $O$ is arbitrarily chosen on segment $XP$. Lines $CO$ and $BO$ are extended to intersect the first and second circles at $M$ and $N$, respectively. If lines $AM$ and $DN$ intersect at $Z$, prove that $X, Y$ and $Z$ lie on the same line.

Let be distinct points on a plane, four of which form a quadrangle, and three of which are in the interior or boundary of this quadrangle. Show that the diagonals of this quadrangle are longer than the double of the minimum of the distances between any two of these seven points. [i]Paul Erdős[/i] [hide=Side note]If the quadrangle is convex, the constant from the inequality can be improved from $ 2 $ to $ \sqrt{\frac{3\pi}{2}}. $[/hide]

For every positive integer $n\ge 3$, let $\phi_n$ be the set of all positive integers less than and coprime to $n$. Consider the polynomial: $$P_n(x)=\sum_{k\in\phi_n} {x^{k-1}}.$$ a. Prove that $P_n(x)=(x^{r_n}+1)Q_n(x)$ for some positive integer $r_n$ and polynomial $Q_n(x)\in\mathbb{Z}[x]$ (not necessary non-constant polynomial). b. Find all $n$ such that $P_n(x)$ is irreducible over $\mathbb{Z}[x]$.

On the team round, an LMT team of six students wishes to divide itself into two distinct groups of three, one group to work on part $1,$ and one group to work on part $2.$ In addition, a captain of each group is designated. In how many ways can this be done?

Let $ABC$ be a triangle with $BC=a$ $AC=b$ $AB=c$. For each line $\Delta$ we denote $d_{A}, d_{B}, d_{C}$ the distances from $A,B, C$ to $\Delta$ and we consider the expresion $E(\Delta)=ad_{A}^{2}+bd_{B}^{2}+cd_{C}^{2}$. Prove that if $E(\Delta)$ is minimum, then $\Delta$ passes through the incenter of $\Delta ABC$.

Let $d_n$ be the last nonzero digit of the decimal representation of $n!$. Prove that $d_n$ is aperiodic; that is, there do not exist $T$ and $n_0$ such that for all $n \geq n_0, d_{n+T} = d_n.$

Find the number of sequences $(a_n)_{n=1}^\infty$ of integers satisfying $a_n \ne -1$ and \[a_{n+2} =\frac{a_n + 2006}{a_{n+1} + 1}\] for each $n \in \mathbb{N}$.

Let $K_1, K_2, K_3, K_4, K_5$ be 5 distinguishable keys, and let $D_1, D_2, D_3, D_4, D_5$ be $5$ distinguishable doors. For $1 \leq i \leq 5$, key $K_i$ opens doors $D_{i}$ and $D_{i+1}$ (where $D_6 = D_1$) and can only be used once. The keys and doors are placed in some order along a hallway. Key\$ha walks into the hallway, picks a key and opens a door with it, such that she never obtains a key before all the doors in front of it are unlocked. In how many orders can the keys and doors be placed such that Key\$ha can open all of the doors? [i]Author: Mitchell Lee[/i] [hide="Clarifications"] [list=1][*]The doors and keys are in series. In other words, the doors aren't lined up along the side of the hallway. They are blocking Key\$ha's path to the end, and the only way she can get past them is by getting the appropriate keys along the hallway. [*]The doors and keys appear consecutively along the hallway. For example, she might find $K_1 D_1 K_2 D_2 K_3 D_3 K_4 D_4 K_5 D_5$ down the hallway in that order. Also, by "she never obtains a key before all the doors in front of it are unlocked," we mean that she cannot obtain a key before all the doors appearing before the key are unlocked. In essence, it merely states that locked doors cannot be passed. [*]The doors and keys do not need to alternate down the hallway.[/list][/hide]

Let $ABC$ be an acute-angled triangle. Suppose that the altitude of $\vartriangle ABC$ at $B$ intersects the circle with diameter $AC$ at $P$ and $Q$, and the altitude at $C$ intersects the circle with diameter $AB$ at $M$ and $N$. Prove that $P, Q, M$ and $N$ lie on a circle.

Let $n$ be a positive integer and let $p, q>n$ be odd primes. Prove that the positive integers $1, 2, \ldots, n$ can be colored in $2$ colors, such that for any $x \neq y$ of the same color, $xy-1$ is not divisible by $p$ and $q$.

Given an ordered list of $3N$ real numbers, we can trim it to form a list of $N$ numbers as follows: We divide the list into $N$ groups of $3$ consecutive numbers, and within each group, discard the highest and lowest numbers, keeping only the median. \\ Consider generating a random number $X$ by the following procedure: Start with a list of $3^{2021}$ numbers, drawn independently and unfiformly at random between $0$ and $1$. Then trim this list as defined above, leaving a list of $3^{2020}$ numbers. Then trim again repeatedly until just one number remains; let $X$ be this number. Let $\mu$ be the expected value of $\left|X-\frac{1}{2} \right|$. Show that \[ \mu \ge \frac{1}{4}\left(\frac{2}{3} \right)^{2021}. \]

Determine, with proof, the smallest positive integer $c$ such that for any positive integer $n$, the decimal representation of the number $c^n+2014$ has digits all less than $5$. [i]Proposed by Evan Chen[/i]

Inside the square $ABCD$ there is the square $A'B' C'D'$ so that the segments $AA', BB', CC'$ and $DD'$ do not intersect each other neither the sides of the smaller square (the sides of the larger and the smaller square do not need to be parallel). Prove that the sum of areas of the quadrangles $AA'B' B$ and $CC'D'D$ is equal to the sum of areas of the quadrangles $BB'C'C$ and $DD'A'A$.

Compute \[\sum_{i=0}^{\frac{q-1}{2}}\left\lfloor\frac{ip}{q}\right\rfloor+\sum_{j=0}^{\frac{p-1}{2}}\left\lfloor\frac{jq}{p}\right\rfloor\] if $p=51$ and $q=81$. [i]2018 CCA Math Bonanza Team Round #7[/i]

Let $f(n)=\dfrac{x_1+x_2+...+x_n}{n}$, where $n$ is a positive integer. If $x_k=(-1)^k,k=1,2,...,n$, the set of possible values of $f(n)$ is: $\textbf{(A)}\ \{0\} \qquad \textbf{(B)}\ \{\dfrac{1}{n}\} \qquad \textbf{(C)}\ \{0,-\dfrac{1}{n}\} \qquad \textbf{(D)}\ \{0,\dfrac{1}{n}\} \qquad \textbf{(E)}\ \{1,\dfrac{1}{n}\}$

Six regular hexagons surround a regular hexagon of side length $1$ as shown. What is the area of $\triangle ABC$? [asy] for(int i = 0; i < 6; ++i){ for(int j = 0; j < 6; ++j){ draw(sqrt(3)*dir(60*i+30)+dir(60*j)--sqrt(3)*dir(60*i+30)+dir(60*j+60)); } } draw(2*dir(60)--2*dir(180)--2*dir(300)--cycle); label("A",2*dir(180),dir(180)); label("B",2*dir(60),dir(60)); label("C",2*dir(300),dir(300)); [/asy] $ \textbf {(A) } 2\sqrt{3} \qquad \textbf {(B) } 3\sqrt{3} \qquad \textbf {(C) } 1+3\sqrt{2} \qquad \textbf {(D) } 2+2\sqrt{3} \qquad \textbf {(E) } 3+2\sqrt{3} $

A box whose shape is a parallelepiped can be completely filled with cubes of side $1.$ If we put in it the maximum possible number of cubes, each of volume $2$, with the sides parallel to those of the box, then exactly $40$ percent of the volume of the box is occupied. Determine the possible dimensions of the box.

Find the largest real number $ C_1 $ and the smallest real number $ C_2 $, such that, for all reals $ a,b,c,d,e $, we have \[ C_1 < \frac{a}{a+b}+\frac{b}{b+c}+\frac{c}{c+d}+\frac{d}{d+e}+\frac{e}{e+a} < C_2 \]

Suppose $l(M, XYZ)$ is a Simson line of the triangle $XYZ$ that passes through $M$. Suppose $ABCDEF$ is a cyclic hexagon such that $l(A, BDF), l(B, ACE), l(D, ABF), l(E, ABC)$ intersect at a single point. Prove that $CDEF$ is a rectangle. [color=blue]Should the first sentence read: Suppose $l(M, XYZ)$ is a Simson line of the triangle $XYZ$ [u]with respect to[/u] $M$. ? Since it appears weird that a Simson line that passes a point is to be constructed. However, this is Unsolved after all, so I'm not sure.[/color]

Find all the positive integers $n$ so that $n$ has the same number of digits as its number of different prime factors and the sum of these different prime factors is equal to the sum of exponents of all these primes in factorization of $n$.

Investigate the behaviour as $t\to+\infty$ of solutions of the systems \[\begin{cases} \dot{x}=y\\ \dot{y}=2\sin y-y-x\end{cases}\] \[\begin{cases} \dot{x}=y\\ \dot{y}=2x-x^{3}-x^{2}-\epsilon y\end{cases}\] where $\epsilon\ll 1$.

David drives from his home to the airport to catch a flight. He drives $35$ miles in the first hour, but realizes that he will be $1$ hour late if he continues at this speed. He increases his speed by $15$ miles per hour for the rest of the way to the airport and arrives $30$ minutes early. How many miles is the airport from his home? $\textbf{(A) }140\qquad \textbf{(B) }175\qquad \textbf{(C) }210\qquad \textbf{(D) }245\qquad \textbf{(E) }280\qquad$