We wish to construct a matrix with $19$ rows and $86$ columns, with entries $x_{ij} \in \{0, 1, 2\} \ (1 \leq i \leq 19, 1 \leq j \leq 86)$, such that: [i](i)[/i] in each column there are exactly $k$ terms equal to $0$; [i](ii)[/i] for any distinct $j, k \in \{1, . . . , 86\}$ there is $i \in \{1, . . . , 19\}$ with $x_{ij} + x_{ik} = 3.$ For what values of $k$ is this possible?

Each member of a set of circles in the $xy$-plane is tangent to the $x$-axis and no two of the circles intersect. Show that (a) the points of tangency can include all rational points on the axis. (b) the points of tangency cannot include all the irrational points.

An $n$ by $n$ grid, where every square contains a number, is called an $n$-code if the numbers in every row and column form an arithmetic progression. If it is sufficient to know the numbers in certain squares of an $n$-code to obtain the numbers in the entire grid, call these squares a key. [b]a.) [/b]Find the smallest $s \in \mathbb{N}$ such that any $s$ squares in an $n-$code $(n \geq 4)$ form a key. [b]b.)[/b] Find the smallest $t \in \mathbb{N}$ such that any $t$ squares along the diagonals of an $n$-code $(n \geq 4)$ form a key.

Find all real $x$ such that $$\lfloor x \lceil x \rceil \rfloor = 2022.$$ Express your answer in interval notation.

Maddy and Lara see a list of numbers written on a blackboard. Maddy adds $3$ to each number in the list and finds that the sum of her new numbers is $45$. Lara multiplies each number in the list by $3$ and finds that the sum of her new numbers is also $45$. How many numbers are written on the blackboard? $\textbf{(A) }10\qquad\textbf{(B) }5\qquad\textbf{(C) }6\qquad\textbf{(D) }8\qquad\textbf{(E) }9$

Suppose that $n$ people each know exactly one piece of information, and all $n$ pieces are different. Every time person $A$ phones person $B$, $A$ tells $B$ everything that $A$ knows, while $B$ tells $A$ nothing. What is the minimum number of phone calls between pairs of people needed for everyone to know everything? Prove your answer is a minimum.

Determine the unique pair of real numbers $(x,y)$ that satisfy the equation \[(4 x^2+ 6 x + 4)(4 y^2 - 12 y + 25 ) = 28 .\]

Let $n\ge 2$ be a positive integer, and consider an $n\times n$ grid of $n^2$ equilateral triangles. Two triangles are adjacent if they share at least one vertex. Each triangle is colored red or blue, splitting the grid into regions. Find, with proof, the minimum number of triangles in the largest region. [i]Rohan Bodke[/i]

Hong and Song each have a shuffled deck of eight cards, four red and four black. Every turn, each player places down the two topmost cards of their decks. A player can thus play one of three pairs: two black cards, two red cards, or one of each color. The probability that Hong and Song play exactly the same pairs as each other for all four turns is $\frac{m}{n}$, where $m$ and $n$ are positive integers with $\gcd(m,n)=1$. Compute $100m+n$. [i]Proposed by Sean Li[/i]

We call a rectangle of the size $1 \times 2$ a domino. Rectangle of the $2 \times 3$ removing two opposite (under center of rectangle) corners we call tetramino. These figures can be rotated. It requires to tile rectangle of size $2008 \times 2010$ by using dominoes and tetraminoes. What is the minimal number of dominoes should be used?

Find all triples $(a,b,c)$ of real numbers for which there exists a function $f:\mathbb{Z}^{+}\rightarrow\mathbb{Z}^{+}$ satisfying $af(n)+bf(n+1)+cf(n+2)<0$ for every $n\in\mathbb{Z}^{+}$ ($\mathbb{Z}^{+}$ denotes the set of positive integers). Proposed by [i]András Imolay[/i], Budapest

Let $ABC$ be an acute-angled triangle, with $\angle B \neq \angle C$ . Let $M$ be the midpoint of side $BC$, and $E,F$ be the feet of the altitude from $B,C$ respectively. Denote by $K,L$ the midpoints of segments $ME,MF$, respectively. Suppose $T$ is a point on the line $KL$ such that $AT//BC$. Prove that $TA=TM$ .

Let $ABC$ be a triangle with $AB+AC=3BC$. The $B$-excircle touches side $AC$ and line $BC$ at $E$ and $D$, respectively. The $C$-excircle touches side $AB$ at $F$. Let lines $CF$ and $DE$ meet at $P$. Prove that $\angle PBC = 90^{\circ}$. [i]Ray Li[/i]

Let $ABC$ be an acute, scalene triangle with orthocentre $H$. Let $\ell_a$ be the line through the reflection of $B$ with respect to $CH$ and the reflection of $C$ with respect to $BH$. Lines $\ell_b$ and $\ell_c$ are defined similarly. Suppose lines $\ell_a$, $\ell_b$, and $\ell_c$ determine a triangle $\mathcal T$. Prove that the orthocentre of $\mathcal T$, the circumcentre of $\mathcal T$, and $H$ are collinear. [i]Fedir Yudin, Ukraine[/i]

Let $ \left( a_n\right)_{n\ge 1} $ be a sequence defined by $ a_1>0 $ and $ \frac{a_{n+1}}{a}=\frac{a_n}{a}+\frac{a}{a_n} , $ with $ a>0. $ Calculate $ \lim_{n\to\infty} \frac{a_n}{\sqrt{n+a}} . $ [i]Florin Rotaru[/i]

Let $A,B,C,D \in \mathcal{M}_n(\mathbb{C}),$ $A$ and $C$ invertible. Prove that if $A^k B = C^k D$ for any positive integer $k,$ then $B=D.$

In connection with the formation of a stable government, the President invited all $240$ Members of Parliament to three separate consultations, where each member participated in exactly one consultation, and at each consultation there has been at least one member present. Discussions between pairs of members are to take place to discuss the consultations. Is it possible for these discussions to occur in such a way that there exists a non-negative integer $k$, such that for every two members who participated in different consultations, there are exactly $k$ members who participated in the remaining consultation, with whom each of the two members has a conversation, and exactly $k$ members who participated in the remaining consultation, with whom neither of the two has a conversation? If yes, then find all possible values of $k$.

A triangle of sides $\frac{3}{2}, \frac{\sqrt{5}}{2}, \sqrt{2}$ is folded along a variable line perpendicular to the side of $\frac{3}{2}.$ Find the maximum value of the coincident area.

Find all the functions $\varphi:\mathbb{Z}[x]\to\mathbb{Z}[x]$ such that $\varphi(x)=x,$ any integer polynomials $f, g$ satisfy $\varphi(f+g)=\varphi(f)+\varphi(g)$ and $\varphi(f)$ is a perfect power if and only if $f{}$ is a perfect power. [i]Note:[/i] A polynomial $f\in \mathbb{Z}[x]$ is a perfect power if $f = g^n$ for some $g\in \mathbb{Z}[x]$ and $n\geqslant 2.$ [i]Proposed by Pavel Ciurea[/i]

There are positive integers $x$ and $y$ that satisfy the system of equations \begin{align*} \log_{10} x + 2 \log_{10} (\gcd(x,y)) &= 60 \\ \log_{10} y + 2 \log_{10} (\text{lcm}(x,y)) &= 570. \end{align*} Let $m$ be the number of (not necessarily distinct) prime factors in the prime factorization of $x$, and let $n$ be the number of (not necessarily distinct) prime factors in the prime factorization of $y$. Find $3m+2n$.

Prove that \[\displaystyle \sum_{\{i,j,k\}=\{1,2,3\}} \csc ^{13} \frac{2^i \pi}{7}\csc ^{14} \frac{2^j \pi}{7}\csc ^{2005} \frac{2^k\pi}{7}\] is rational. Here, $(i,j,k)$ is summed over all possible permutations of $(1,2,3)$.

$$problem 2$$:A point $P$ is taken in the interior of a right triangle$ ABC$ with $\angle C = 90$ such hat $AP = 4, BP = 2$, and$ CP = 1$. Point $Q$ symmetric to $P$ with respect to $AC$ lies on the circumcircle of triangle $ABC$. Find the angles of triangle $ABC$.

Let $n$ be a fixed positive integer and let $b(n)$ be the minimum value of $$k+\frac{n}{k},$$ where $k$ is allowed to range through all positive integers. Prove that $\lfloor b(n) \rfloor= \lfloor \sqrt{4n+1} \rfloor.$

Points $W$, $X$, $Y,$ and $Z$ are chosen inside a regular octagon so that four congruent rhombuses are formed, as shown in the diagram below. If the side length of the octagon is $1$, compute the area of quadrilateral $WXY Z$. [img]https://cdn.artofproblemsolving.com/attachments/9/6/bb12385cbd9fd802b3f3960b5e449268be45d4.png[/img]

Let $F_m$ be the $m$'th Fibonacci number, defined by $F_1=F_2=1$ and $F_m = F_{m-1}+F_{m-2}$ for all $m \geq 3$. Let $p(x)$ be the polynomial of degree 1008 such that $p(2n+1)=F_{2n+1}$ for $n=0,1,2,\ldots,1008$. Find integers $j$ and $k$ such that $p(2019) = F_j - F_k$.