$\boxed{\text{A5}}$ Let $x,y,z$ be positive reals. Prove that $(x^2+y+1)(x^2+z+1)(y^2+x+1)(y^2+z+1)(z^2+x+1)(z^2+y+1)\geq (x+y+z)^6$

Find all integers $n$ for which there exists a table with $n$ rows, $2022$ columns, and integer entries, such that subtracting any two rows entry-wise leaves every remainder modulo $2022$. [i]Proposed by Tony Wang[/i]

Define a [i]beautiful number[/i] to be an integer of the form $a^n$, where $a\in\{3,4,5,6\}$ and $n$ is a positive integer. Prove that each integer greater than $2$ can be expressed as the sum of pairwise distinct beautiful numbers. [i]Proposed by Matthew Babbitt[/i]

In the future, each country in the world produces its Olympic athletes via cloning and strict training programs. Therefore, in the finals of the 200 m free, there are two indistinguishable athletes from each of the four countries. How many ways are there to arrange them into eight lanes?

A circle $\omega$ is inscribed in a quadrilateral $ABCD$. Let $I$ be the center of $\omega$. Suppose that \[ (AI + DI)^2 + (BI + CI)^2 = (AB + CD)^2. \] Prove that $ABCD$ is an isosceles trapezoid.

Compute the number of ordered quadruples of complex numbers $(a,b,c,d)$ such that \[ (ax+by)^3 + (cx+dy)^3 = x^3 + y^3 \] holds for all complex numbers $x, y$. [i]Proposed by Evan Chen[/i]

In $\triangle ABC$, we have three edges with lengths $BC=a, \, CA=b \, AB=c$, and $c<b<a<2c$. $P$ and $Q$ are two points of the edges of $\triangle ABC$, and the straight line $PQ$ divides $\triangle ABC$ into two parts with the same area. Find the minimum value of the length of the line segment $PQ$.

The graphs of the lines $$y=x+2, \quad y=3x+4, \quad y=5x+6,\quad y=7x+8,\quad y=9x+10,\quad y=11x+12$$ are drawn. These six lines divide the plane into several regions. Compute the number of regions the plane is divided into.

The points $K$ and $N$ lie on the hypotenuse $AB$ of a right triangle $ABC$. Prove that orthocenters the triangles $BCK$ and $ACN$ coincide if and only if $\frac{BN}{AK}=\tan^2 A.$

Let $a, b$ and $c$ be nonzero digits. Let $p$ be a prime number which divides the three digit numbers $\overline{abc}$ and $\overline{cba}.$ Show that $p$ divides at least one of the numbers $a+b+c, a-b+c$ and $a-c.$

Determine the real numbers $a$ such that the equation $a 3^x + 3^{-x} = 3$ has exactly one solution $x.$

Find all polynomials of the form $P(x) = (-1)^nx^n + a_1x^{n-1} + a_2x^{n-2} + ...+ a_{n-1}x + a_n$ with the following two properties: (i) $\{a_1, a_2, . . . , a_n-1, a_n\} =\{0, 1\}$, and (ii) all roots of $P(x)$ are distinct real numbers

Frist Campus Center is located $1$ mile north and $1$ mile west of Fine Hall. The area within $5$ miles of Fine Hall that is located north and east of Frist can be expressed in the form $\frac{a}{b} \pi - c$, where $a, b, c$ are positive integers and $a$ and $b$ are relatively prime. Find $a + b + c$.

Let $M$ be a set of points in the Cartesian plane, and let $\left(S\right)$ be a set of segments (whose endpoints not necessarily have to belong to $M$) such that one can walk from any point of $M$ to any other point of $M$ by travelling along segments which are in $\left(S\right)$. Find the smallest total length of the segments of $\left(S\right)$ in the cases [b]a.)[/b] $M = \left\{\left(-1,0\right),\left(0,0\right),\left(1,0\right),\left(0,-1\right),\left(0,1\right)\right\}$. [b]b.)[/b] $M = \left\{\left(-1,-1\right),\left(-1,0\right),\left(-1,1\right),\left(0,-1\right),\left(0,0\right),\left(0,1\right),\left(1,-1\right),\left(1,0\right),\left(1,1\right)\right\}$. In other words, find the Steiner trees of the set $M$ in the above two cases.

A set $X$ of positive integers is said to be [i]iberic[/i] if $X$ is a subset of $\{2, 3, \dots, 2018\}$, and whenever $m, n$ are both in $X$, $\gcd(m, n)$ is also in $X$. An iberic set is said to be [i]olympic[/i] if it is not properly contained in any other iberic set. Find all olympic iberic sets that contain the number $33$.

Let the sequence $\left \{ a_n \right \}_{n = -2}^\infty$ satisfy $a_{-1} = a_{-2} = 0, a_0 = 1$, and for all non-negative integers $n$, $$n^2 = \sum_{k = 0}^n a_{n - k}a_{k - 1} + \sum_{k = 0}^n a_{n - k}a_{k - 2}$$ Given $a_{2018}$ is rational, find the maximum integer $m$ such that $2^m$ divides the denominator of the reduced form of $a_{2018}$.

Positive numbers $\alpha ,\beta , x_1, x_2,\ldots, x_n$ ($n \geq 1$) satisfy $x_1+x_2+\cdots+x_n = 1$. Prove that \[\sum_{i=1}^{n} \frac{x_i^3}{\alpha x_i+\beta x_{i+1}} \geq \frac{1}{n(\alpha+\beta)}.\] [b]Note.[/b] $x_{n+1}=x_1$.

Alice and Bob play a game with a string of $2015$ pearls. In each move, one player cuts the string between two pearls and the other player chooses one of the resulting parts of the string while the other part is discarded. In the first move, Alice cuts the string, thereafter, the players take turns. A player loses if he or she obtains a string with a single pearl such that no more cut is possible. Who of the two players does have a winning strategy? (Theresia Eisenkölbl)

Prove that if $n$ is a natural number such that $1 + 2^n + 4^n$ is prime then $n = 3^k$ for some $k \in N_0$.

Find all ordered pairs of positive integers$ (x, y)$ such that:$$x^3+y^3=x^2+42xy+y^2.$$

Prove that the smallest dihedral angle between faces of an arbitrary tetrahedron is not greater than the dihedral angle between faces of a regular tetrahedron. (S. Shosman, O. Ogievetsky)

Let $S$ be the statement "If the sum of the digits of the whole number $n$ is divisible by 6, then $n$ is divisible by 6." A value of $n$ which shows $S$ to be false is $ \textbf{(A)}\ 30 \qquad \textbf{(B)}\ 33 \qquad \textbf{(C)}\ 40 \qquad \textbf{(D)}\ 42 \qquad \textbf{(E)}\ \text{None of these} $

Let $n \geq 4$ be an integer. Consider a regular $2n-$gon for which to every vertex, an integer is assigned, which we call the value of said vertex. If four distinct vertices of this $2n-$gon form a rectangle, we say that the sum of the values of these vertices is a rectangular sum. Determine for which (not necessarily positive) integers $m$ the integers $m + 1, m + 2, . . . , m + 2n$ can be assigned to the vertices (in some order) in such a way that every rectangular sum is a prime number. (Prime numbers are positive by definition.)

What is the least possible ratio of two isosceles triangles areas, if three vertices of the first one belong to three different sides of the second one?

On a large piece of paper, Dana creates a “rectangular spiral” by drawing line segments of lengths, in cm, of 1, 1, 2, 2, 3, 3, 4, 4, ... as shown. Dana’s pen runs out of ink after the total of all the lengths he has drawn is 3000 cm. What is the length of the longest line segment that Dana draws? $\textbf{(A)}\ 38 \qquad \textbf{(B)}\ 39 \qquad \textbf{(C)}\ 54 \qquad \textbf{(D)}\ 55 \qquad \textbf{(E)}\ 30$