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? $\textbf{(A) }140\qquad \textbf{(B) }175\qquad \textbf{(C) }210\qquad \textbf{(D) }245\qquad \textbf{(E) }280\qquad$

Find the real numbers $x$ and $y$ such that $$(x^2 -x +1)(3y^2-2y + 3) -2=0.$$

Six ants simultaneously stand on the six vertices of a regular octahedron, with each ant at a different vertex. Simultaneously and independently, each ant moves from its vertex to one of the four adjacent vertices, each with equal probability. What is the probability that no two ants arrive at the same vertex? $ \textbf{(A)}\ \frac {5}{256} \qquad \textbf{(B)}\ \frac {21}{1024} \qquad \textbf{(C)}\ \frac {11}{512} \qquad \textbf{(D)}\ \frac {23}{1024} \qquad \textbf{(E)}\ \frac {3}{128}$

Let $a$ and $b$ be two positive integers and $A$ be a subset of $\{1, 2,…, a + b \}$ that has more than $ \frac {a + b} {2}$ elements. Show that there are two numbers in $A$ whose difference is $a$ or $b$.

In assigning dormitory rooms, a college gives preference to pairs of students in this order: $$AA,\, AB ,\, AC, \, BB , \, BC ,\, AD , \, CC, \, BD, \, CD, \, DD$$ in which $AA$ means two seniors, $AB$ means a senior and a junior, etc. Determine numerical values to assign to $A,B,C$ and $D$ so that the set of numbers $A+A, A+B, A+C, B+B, \ldots $ corresponding to the order above will be in descending order. Find the general solution and the solution in least positive integers.

Let $ a$, $ b$, $ c$ are sides of a triangle. Find the least possible value $ k$ such that the following inequality always holds: $ \left|\frac{a\minus{}b}{a\plus{}b}\plus{}\frac{b\minus{}c}{b\plus{}c}\plus{}\frac{c\minus{}a}{c\plus{}a}\right|<k$ [i](Vitaly Lishunov)[/i]

Logan is constructing a scaled model of his town. The city's water tower stands $ 40$ meters high, and the top portion is a sphere that holds $ 100,000$ liters of water. Logan's miniature water tower holds $ 0.1$ liters. How tall, in meters, should Logan make his tower? $ \textbf{(A)}\ 0.04\qquad \textbf{(B)}\ \frac{0.4}{\pi}\qquad \textbf{(C)}\ 0.4\qquad \textbf{(D)}\ \frac{4}{\pi}\qquad \textbf{(E)}\ 4$

$8712$ is an integral multiple of its reversal, $2178$, as $8712=4 * 2178$. Find another $4$-digit number which is a non-trivial integral multiple of its reversal.

Max repeatedly throws a fair coin in a hurricane. For each throw, there is a $4\%$ chance that the coin gets blown away. He records the number of heads $H$ and the number of tails $T$ before the coin is lost. (If the coin is blown away on a toss, no result is recorded for that toss.) What is the expected value of $|H-T|$? [i]Proposed by Krit Boonsiriseth.[/i]

Let be a quadratic polynom that has the property that the modulus of the sum between the leading and the free coefficient is smaller than the modulus of the middle coefficient. Prove that this polynom admits two distinct real roots, one belonging to the interval $ (-1,1) , $ and the other belonging outside of the interval $ (-1,1). $

Given a set $C$ of points in the plane, it is called the distance of a point $P$ from the plane to the set $C$ at the smallest of the distances from $P$ to each of the points of $C$. Let the sets be $C = \{A,B\}$, with $A = (1, 0)$ and $B = (2, 0)$; and $C'= \{A',B'\}$ with $A' = (0, 1)$ and $B' = (0, 7)$, in an orthogonal reference system. Find and draw the set $M$ of points in the plane that are equidistant from $C$ and $C'$ . Study whether the function whose graph is the set $M$ previously obtained is derivable.

Prove that all real roots of the polynomial $$ P=X^{1985}-X^{1984}+1983\cdot X^{1983}+1994\cdot X^{992} -884064 $$ are positive.

[b]4.[/b] Call a subset $S$ of the set $\{1,\dots,n\}$ [i]exceptional[/i] if any pair of distinct elements of $S$ are coprime. Consider an exceptional set with a maximal sum of elements (among all exceptional sets for a fixed $n$). Prove that if $n$ is sufficiently large, then each element of $S$ has at most two distinct prime divisors. ([b]N.17[/b]) [P. Erdos]