Find the digits left and right of the decimal point in the decimal form of the number \[ (\sqrt{2} + \sqrt{3})^{1980}. \]

Anna and Bob play the following game. In the beginning, Bob writes down the numbers $1, 2, ... , 2022$ on a piece of paper, such that half of the numbers are on the left and half on the right. Furthermore, we assume that the $1011$ numbers on both sides are written in some order. After Bob does this, Anna has the opportunity to swap the positions of the two numbers lying on different sides of the paper if they have different parity. Anna wins if, after finitely many moves, all odd numbers end up on the left, in increasing order, and all even ones end up on the right, in increasing order. Can Bob write down a arrangement of numbers for which Anna cannot win? For example, Bob could write down numbers in the following way: $$4, 2, 5, 7, 9, ... , 2021\,\,\,\,\,\,\,\,\,\,,\, \,\,\,\,\,\,\,\,\,\,,\, 3, 1, 6, 8, 10, ... , 2022$$ Then Anna could swap the numbers $1, 4$ and then swap $2, 3$ to win. However, if Anna swapped the pairs $3, 4$ and $1, 2$, the resulting numbers on the left and on the right would not be in increasing order, and hence Anna would not win.

Seven students in a class compare their marks in 12 subjects studied and observe that no two of the students have identical marks in all 12 subjects. Prove that we can choose 6 subjects such that any two of the students have different marks in at least one of these subjects.

How many ways are there to paint the squares of a $6 \times 6$ board black or white such that within each $2\times 2$ square on the board, the number of black squares is odd?

Five squirrels together have a supply of 2022 nuts. On the first day 2 nuts are added, on the second day 4 nuts, on the third day 6 nuts and so on, i.e. on each further day 2 nuts more are added than on the day before. At the end of any day the squirrels divide the stock among themselves. Is it possible that they all receive the same number of nuts and that no nut is left over?

We define the sequence $x_n$ so that \[x_1=a, x_2=b, x_n=\frac{{x_{n-1}}^2+{x_{n-2}}^2}{x_{n-1}+x_{n-2}} \quad \forall n \geq 3.\] Where $a,b >1$ are relatively prime numbers. Show that $x_n$ is not an integer for $n \geq 3$.

In triangle $ABC$, ($AB \ne AC$), let the orthocenter be $H$, circumcenter be $O$, and the midpoint of $BC$ be $M$. Let $HM \cap AO = D$. Let $P,Q,R,S$ be the midpoints of $AB,CD,AC,BD$. Let $X = PQ\cap RS$. Find $AH/OX$.

Suppose that $P(x)$ is a polynomial with degree $10$ and integer coefficients. Prove that, there is an infinite arithmetic progression (open to bothside) not contain value of $P(k)$ with $k\in\mathbb{Z}$

Let $G$ be the centroid of a triangle $ABC$. Prove that if $AB+GC = AC+GB$, then the triangle is isosceles

Let $n \ge 3$ be an integer. What is the largest possible number of interior angles greater than $180^\circ$ in an $n$-gon in the plane, given that the $n$-gon does not intersect itself and all its sides have the same length?

Six chairs sit in a row. Six people randomly seat themselves in the chairs. Each person randomly chooses either to set their feet on the floor, to cross their legs to the right, or to cross their legs to the left. There is only a problem if two people sitting next to each other have the person on the right crossing their legs to the left and the person on the left crossing their legs to the right. The probability that this will [b]not[/b] happen is given by $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

Set $S_n = \sum_{p=1}^n (p^5+p^7)$. Determine the greatest common divisor of $S_n$ and $S_{3n}.$

The area of the triangle whose altitudes have lengths $36.4$, $39$, and $42$ can be written as $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

We have “bricks” made in the following way: we take a unit cube and glue to three of its faces which have a common vertex three more cubes in such a way that the faces glued together coincide. Is it possible to construct from these bricks an $11 \times 12 \times 13$ box? (A Andjans, Riga )

The set $X$ has $1983$ members. There exists a family of subsets $\{S_1, S_2, \ldots , S_k \}$ such that: [b](i)[/b] the union of any three of these subsets is the entire set $X$, while [b](ii)[/b] the union of any two of them contains at most $1979$ members. What is the largest possible value of $k ?$

The 40 unit squares of the 9 9-table (see below) are labeled. The horizontal or vertical row of 9 unit squares is good if it has more labeled unit squares than unlabeled ones. How many good (horizontal and vertical) rows totally could have the table?

Find all integer solutions to \begin{align*} x^2+2y^2+3z^2&=36,\\ 3x^2+2y^2+z^2&=84,\\ xy+xz+yz&=-7. \end{align*}

A right rectangular prism of silly powder has dimensions $20 \times 24 \times 25.$ Jerry the wizard applies $10$ bouts of highdroxylation to the box, each of which increases one dimension of the silly powder by $1$ and decreases a different dimension of the silly powder by $1,$ with every possible choice of dimensions equally likely to be chosen and independent of all previous choices. Compute the expected volume of the silly powder after Jerry’s routine.

Let $ P$ be a probability distribution defined on the Borel sets of the real line. Suppose that $ P$ is symmetric with respect to the origin, absolutely continuous with respect to the Lebesgue measure, and its density function $ p$ is zero outside the interval $ [\minus{}1,1]$ and inside this interval it is between the positive numbers $ c$ and $ d$ ($ c < d$). Prove that there is no distribution whose convolution square equals $ P$. [i]T. F. Mori, G. J. Szekely[/i]

Let $A,B,C,D$ be four points in space, and $M$ and $N$ be the midpoints of $AC$ and $BD$, respectively. Show that $$AB^2+BC^2+CD^2+DA^2 = AC^2+BD^2+4MN^2$$

Let $n$ and $k$ be positive integers. Evaluate the following sum $$\sum_{j=0}^k\binom kj^2\binom{n+2k-j}{2k}$$where $\binom nk=\frac{n!}{k!(n-k)!}$.

Two circles with radii $71$ and $100$ are externally tangent. Compute the largest possible area of a right triangle whose vertices are each on at least one of the circles.

An $ABC$ triangle divide by a $d$ line such that, new two pieces' areas and perimeters are equal. Prove that $ABC$'s incenter lies $d$

In $\triangle ABC$, we have $\angle ABC=20^\circ$. In addition, $D$ is drawn on $\overline{AB}$ such that $AC=CD=BD$. Compute $\angle ACD$ in degrees.

Let $f(x)$ be a polynomial with positive integer coefficients. For every $n\in\mathbb{N}$, let $a_{1}^{(n)}, a_{2}^{(n)}, \dots , a_{n}^{(n)}$ be fixed positive integers that give pairwise different residues modulo $n$ and let \[g(n) = \sum\limits_{i=1}^{n} f(a_{i}^{(n)}) = f(a_{1}^{(n)}) + f(a_{2}^{(n)}) + \dots + f(a_{n}^{(n)})\] Prove that there exists a constant $M$ such that for all integers $m>M$ we have $\gcd(m, g(m))>2023^{2023}$.