What is the dimension of the space of solutions continuous on $x^2+y^2\ge1$ of the problem \[\Delta u=0\text{ for }x^2+y^2>1\] \[\partial u/\partial n=0\text{ for }x^2+y^2=1\]

Prove that, if $a$ and $b$ are given integers, the system of equatìons $$x + y + 2z + 2t = a$$ $$2x - 2y + z- t = b$$ has a solution in integers $x, y,z,t$.

Let $ABCD$ be a parallelogram. A variable line $g$ through the vertex $A$ intersects the rays $BC$ and $DC$ at the points $X$ and $Y$, respectively. Let $K$ and $L$ be the $A$-excenters of the triangles $ABX$ and $ADY$. Show that the angle $\measuredangle KCL$ is independent of the line $g$. [i]Proposed by Vyacheslev Yasinskiy, Ukraine[/i]

Bessie and her $2015$ bovine buddies work at the Organic Milk Organization, for a total of $2016$ workers. They have a hierarchy of bosses, where obviously no cow is its own boss. In other words, for some pairs of employees $(A, B)$, $B$ is the boss of $A$. This relationship satisfies an obvious condition: if $B$ is the boss of $A$ and $C$ is the boss of $B$, then $C$ is also a boss of $A$. Business has been slow, so Bessie hires an outside organizational company to partition the company into some number of groups. To promote growth, every group is one of two forms. Either no one in the group is the boss of another in the group, or for every pair of cows in the group, one is the boss of the other. Let $G$ be the minimum number of groups needed in such a partition. Find the maximum value of $G$ over all possible company structures. [i]Proposed by Yang Liu[/i]

Find the smallest number of squares on an $8\times8$ board that should be colored so that every $L$-tromino on the board contains at least one colored square.

Pentagon $ABCDE$ with $CD\parallel BE$ is inscribed in circle $\omega$. Tangent to $\omega$ through $B$ intersects line $AC$ at $F$ in a way that $A$ lies between $C$ and $F$. Lines $BD$ and $AE$ intersect at $G$. Prove that $FG$ is tangent to the circumcircle of $\triangle ADG$.

Let $a_1, \ldots, a_n$ be $n$ positive numbers and $0 < q < 1.$ Determine $n$ positive numbers $b_1, \ldots, b_n$ so that: [i]a.)[/i] $ a_{k} < b_{k}$ for all $k = 1, \ldots, n,$ [i]b.)[/i] $q < \frac{b_{k+1}}{b_{k}} < \frac{1}{q}$ for all $k = 1, \ldots, n-1,$ [i]c.)[/i] $\sum \limits^n_{k=1} b_k < \frac{1+q}{1-q} \cdot \sum \limits^n_{k=1} a_k.$

Given $10$ points in the space such that each $4$ points are not lie on a plane. Connect some points with some segments such that there are no triangles or quadrangles. Find the maximum number of the segments.

Points $A$ and $B$ on the rays $CX$ and $CY$ respectively of an acute angle $XCY$ are given so that $CX < CA = CB < CY$. Construct a line meeting the ray $CX$ and the segments $AB,BC$ at $K,L,M$, respectively, such that $KA \cdot YB = XA \cdot MB = LA\cdot LB \ne 0$.

A pair $(f,g)$ of degree $2$ real polynomials is called [i]foolish[/i] if $f(g(x)) = f(x) \cdot g(x)$ for all real $x.$ How many positive integers less than $2023$ can be a root of $g(x)$ for some foolish pair $(f,g)$?