In the cells of the $4\times N$ table, integers are written, modulo no more than $2024$ (i.e. numbers from the set $\{-2024, -2023,\dots , -2, -1, 0, 1, 2, 3,\dots , 2024\}$) so that in each of the four lines there are no two equal numbers. At what maximum $N$ could it turn out that in each column the sum of the numbers is equal to $2$? [i]G.M.Sharafetdinova[/i]

Let us name a move of the chess knight horizontal if it moves two cells horizontally and one vertically, and vertical otherwise. It is required to place the knight on a cell of a ${46} \times {46}$ board and alternate horizontal and vertical moves. Prove that if each cell is visited not more than once then the number of moves does not exceed 2024. Alexandr Gribalko

There are some cities in a country; one of them is the capital. For any two cities $A$ and $B$ there is a direct flight from $A$ to $B$ and a direct flight from $B$ to $A$, both having the same price. Suppose that all round trips with exactly one landing in every city have the same total cost. Prove that all round trips that miss the capital and with exactly one landing in every remaining city cost the same.

Find some nine-digit number $N$, consisting of different digits, such that among all the numbers obtained from $N$ by crossing out seven digits, there would be no more than one prime. Prove that the number found is correct. (If the number obtained by crossing out the digits starts at zero, then the zero is crossed out.)

Let $ ABCD$ be a convex quadrilateral, such that $ \angle CBD\equal{}2\cdot\angle ADB, \angle ABD\equal{}2\cdot\angle CDB$ and $ AB\equal{}CB$. Prove that quadrilateral $ ABCD$ is a kite.

Point $A$ lies inside an angle with vertex $M$. A ray issuing from point $A$ is reflected in one side of the angle at point $B$, then in the other side at point $C$ and then returns back to point $A$ (the ordinary rule of reflection holds). Prove that the center of the circle circumscribed about triangle $\triangle BCM$ lies on line $AM$.

In $\vartriangle ABC$, $AB = 3$, $BC = 5$, and $CA = 6$. Points $D$ and $E$ are chosen such that $ACDE$ is a square which does not overlap with $\vartriangle ABC$. The length of $BD$ can be expressed in the form $\sqrt{m + n\sqrt{p}}$, where $m$, $n$, and $p$ are positive integers and $p$ is not divisible by the square of a prime. Determine the value of $m + n + p$.

Alex, Bob, and Chris are driving cars down a road at distinct constant rates. All people are driving a positive integer number of miles per hour. All of their cars are $15$ feet long. It takes Alex $1$ second longer to completely pass Chris than it takes Bob to completely pass Chris. The passing time is defined as the time where their cars overlap. Find the smallest possible sum of their speeds, in miles per hour. [i]Proposed by Sammy Charney[/i]

Let $ABC$ be a triangle such that $\angle C=2\angle B$ and $\omega$ be its circumcircle. a tangent from $A$ to $\omega$ intersect $BC$ at $E$. $\Omega$ is a circle passing throw $B$ that is tangent to $AC$ at $C$. Let $\Omega\cap AB=F$. $K$ is a point on $\Omega$ such that $EK$ is tangent to $\Omega$ ($A,K$ aren't in one side of $BC$). Let $M$ be the midpoint of arc $BC$ of $\omega$ (not containing $A$). Prove that $AFMK$ is a cyclic quadrilateral. [asy] import graph; size(15.424606256655986cm); real labelscalefactor = 0.5; /* changes label-to-point distance */ pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); /* default pen style */ real xmin = -7.905629294221492, xmax = 11.618976962434495, ymin = -5.154837585051625, ymax = 4.0091473316396895; /* image dimensions */ pen uuuuuu = rgb(0.26666666666666666,0.26666666666666666,0.26666666666666666); /* draw figures */ draw(circle((1.4210145017438194,0.18096629151696939), 2.581514123077079)); draw(circle((1.4210145017438194,-1.3302878964546825), 2.8984706754484924)); draw(circle((-0.7076932767793396,-0.4161825262831505), 2.9101722408015513), linetype("4 4") + red); draw((3.996177869179178,0.)--(-3.839514259733819,0.)); draw((3.996177869179178,0.)--(0.07833180472267817,2.385828723227042)); draw((0.07833180472267817,2.385828723227042)--(-1.154148865691539,0.)); draw((-3.839514259733819,0.)--(-0.6807342461448075,-3.3262298939043657)); draw((0.07833180472267817,2.385828723227042)--(-3.839514259733819,0.)); /* dots and labels */ dot((3.996177869179178,0.),blue); label("$B$", (4.040279615036859,0.10218054796102663), NE * labelscalefactor,blue); dot((-1.154148865691539,0.),blue); label("$C$", (-1.3803811057738653,-0.14328333373606214), NE * labelscalefactor,blue); dot((1.4210145017438194,1.5681827789938092),linewidth(4.pt)); label("$F$", (1.4629088572174203,1.6465574703052102), NE * labelscalefactor); dot((0.07833180472267817,2.385828723227042),linewidth(3.pt) + blue); label("$A$", (-0.04055741817725232,2.5568193649319144), NE * labelscalefactor,blue); dot((-3.839514259733819,0.),linewidth(3.pt)); label("$E$", (-4.049800819229713,-0.06146203983703255), NE * labelscalefactor); dot((1.4210145017438194,-2.40054783156011),linewidth(4.pt) + uuuuuu); label("$M$", (1.4117705485305265,-2.6490604593938434), NE * labelscalefactor,uuuuuu); dot((-0.6807342461448075,-3.3262298939043657),linewidth(4.pt)); label("$K$", (-0.7871767250058992,-3.5490946922831688), NE * labelscalefactor); clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); /* end of picture */[/asy]

Does there exist a rectangle which can be cut into a hundred rectangles such that all of them are similar to the original one but no two are congruent? [i]Mikhail Murashkin[/i]

Ana baked 15 pasties. She placed them on a round plate in a circular way: 7 with cabbage, 7 with meat and 1 with cherries in that exact order and put the plate into a microwave. She doesn’t know how the plate has been rotated in the microwave. She wants to eat a pasty with cherries. Is it possible for Ana, by trying no more than three pasties, to find exactly where the pasty with cherries is?

Kate bakes a $20$-inch by $18$-inch pan of cornbread. The cornbread is cut into pieces that measure $2$ inches by $2$ inches. How many pieces of cornbread does the pan contain? $ \textbf{(A) }90 \qquad \textbf{(B) }100 \qquad \textbf{(C) }180 \qquad \textbf{(D) }200 \qquad \textbf{(E) }360 \qquad $

Determine all functions $f:(0,\infty)\to\mathbb{R}$ satisfying $$\left(x+\frac{1}{x}\right)f(y)=f(xy)+f\left(\frac{y}{x}\right)$$ for all $x,y>0$.

Find all natural numbers $n$ for which both $n^n + 1$ and $(2n)^{2n} + 1$ are prime numbers.

Let $f$ be a function $R \to R$ that satisfies the following equation: $$f (x)^2 + f (y)^2 = f (x^2 + y^2)+ f (0)$$ If there are $n$ possibilities for the function, find the sum of all values of $n \cdot f (12)$

Facundo and Luca have been given a cake that is shaped like the quadrilateral in the figure. [img]https://cdn.artofproblemsolving.com/attachments/3/2/630286edc1935e1a8dd9e704ed4c813c900381.png[/img] They are going to make two straight cuts on the cake, thus obtaining $4$ portions in the shape of a quadrilateral. Then Facundo will be left with two portions that do not share any side, the other two will be for Luca. Show how they can cut the cuts so that both children get the same amount of cake. Justify why cutting in this way achieves the objective.

How many pairs of real numbers $(x,y)$ are there such that $x^4+y^4 + 2x^2y + 2xy^2+ 2 = x^2 + y^2 + 2x + 2y$? $ \textbf{(A)}\ 6 \qquad\textbf{(B)}\ 5 \qquad\textbf{(C)}\ 4 \qquad\textbf{(D)}\ 3 \qquad\textbf{(E)}\ 2 $

Alison has an analog clock whose hands have the following lengths: $a$ inches (the hour hand), $b$ inches (the minute hand), and $c$ inches (the second hand), with $a < b < c$. The numbers $a$, $b$, and $c$ are consecutive terms of an arithmetic sequence. The tips of the hands travel the following distances during a day: $A$ inches (the hour hand), $B$ inches (the minute hand), and $C$ inches (the second hand). The numbers $A$, $B$, and $C$ (in this order) are consecutive terms of a geometric sequence. What is the value of $\frac{B}{A}$?

Define sequence $\{a_n\}$ as following: $a_0=0$, $a_1=1$, and $a_{i}=2a_{i-1}-a_{i-2}+2$ for all $i\geq 2$. Determine the value of $a_{1000}.$ [i] Proposed by Yannick Yao [/i]

Each of the numbers $1, 2, 3,... 25$ is arranged in a $5$ by $5$ table. In each row they appear in increasing order (left to right). Find the maximal and minimal possible sum of the numbers in the third column. (Folklore)

Triangle $ABC$ is assumed. The point $T$ is the second intersection of the symmedian of vertex $A$ with the circumcircle of the triangle $ABC$ and the point $D \neq A$ lies on the line $AC$ such that $BA=BD$. The line that at $D$ tangents to the circumcircle of the triangle $ADT$, intersects the circumcircle of the triangle $DCT$ for the second time at $K$. Prove that $\angle BKC = 90^{\circ}$(The symmedian of the vertex $A$, is the reflection of the median of the vertex $A$ through the angle bisector of this vertex).

Find all real numbers $x$ satisfying the equation \[\left\lfloor \frac{20}{x+18}\right\rfloor+\left\lfloor \frac{x+18}{20}\right\rfloor=1.\]

When a positive integer $x$ is divided by a positive integer $y$, the quotient is $u$ and the remainder is $v$, where $u$ and $v$ are integers. What is the remainder when $x+2uy$ is divided by $y$? ${{ \textbf{(A)}\ 0 \qquad\textbf{(B)}\ 2u \qquad\textbf{(C)}\ 3u \qquad\textbf{(D)}\ v }\qquad\textbf{(E)}\ 2v } $

Find the geometric locus of the center of a rectangle whose vertices lie on the perimeter of a given triangle.

Three real numbers $x$, $y$, and $z$ are chosen independently and uniformly at random from the interval $[0,1]$. Compute the probability that $x$, $y$, and $z$ can be the side lengths of a triangle.