Let $N$ be the orthocentre of triangle $ABC$ (i .e. the point where the altitudes meet). Prove that the circumscribed circles of triangles $ABN, ACN$ and $BCN$ each have equal radius.

A real symmetric $2018\times 2018$ matrix $A=(a_{ij})$ satisfies $|a_{ij}-2018|\leq 1$ for every $1\leq i,j\leq 2018$. Denote the largest eigenvalue of $A$ by $\lambda(A)$. Find maximum and minumum value of $\lambda(A)$.

Two integers $n, k$ satisfies $n \ge 2$ and $k \ge \frac{5}{2}n-1$. Prove that whichever $k$ lattice points with $x$ and $y$ coordinate no less than $1$ and no more than $n$ we pick, there must be a circle passing through at least four of these points.

Show that one can color all the points of a plane using only two colors such that no line segment has all points of the same color.

Let $p = 1+\frac{1}{2}+\frac{1}{2^2}+\frac{1}{2^3}+\frac{1}{2^4}+\frac{1}{2^5}. $ For nonnegative reals $x, y,z$ satisfying $(x-1)^2 + (y-1)^2 + (z-1)^2 = 27,$ find the maximum value of $x^p + y^p + z^p.$

Let $w_1, w_2$ be non-intersecting, congruent circles with centers $O_1, O_2$ and let $P$ be in the exterior of both of them. The tangents from $P$ to $w_1$ meet $w_1$ at $A_1, B_1$ and define $A_2, B_2$ similarly. If lines $A_1B_1, A_2B_2$ meet at $Q$ show that the midpoint of $PQ$ is equidistant from $O_1, O_2$.

Let $ABCD$ be a rectangle and denote by $M$ and $N$ the midpoints of $AD$ and $BC$ respectively. The point $P$ is on $(CD$ such that $D\in (CP)$, and $PM$ intersects $AC$ in $Q$. Prove that $m(\angle{MNQ})=m(\angle{MNP})$.

Consider a hexagon with vertices labeled $M$, $M$, $A$, $T$, $H$, $S$ in that order. Clayton starts at the $M$ adjacent to $M$ and $A$, and writes the letter down. Each second, Clayton moves to an adjacent vertex, each with probability $\frac{1}{2}$, and writes down the corresponding letter. Clayton stops moving when the string he's written down contains the letters $M, A, T$, and $H$ in that order, not necessarily consecutively (for example, one valid string might be $MAMMSHTH$.) What is the expected length of the string Clayton wrote? [i]Proposed by Andrew Milas and Andrew Wu[/i]

Find the smallest real number $A$, such that there are two different triangles, with integer sidelengths and so that the area of each be $A$.

We have $ n \geq 2$ lamps $ L_{1}, . . . ,L_{n}$ in a row, each of them being either on or off. Every second we simultaneously modify the state of each lamp as follows: if the lamp $ L_{i}$ and its neighbours (only one neighbour for $ i \equal{} 1$ or $ i \equal{} n$, two neighbours for other $ i$) are in the same state, then $ L_{i}$ is switched off; – otherwise, $ L_{i}$ is switched on. Initially all the lamps are off except the leftmost one which is on. $ (a)$ Prove that there are infinitely many integers $ n$ for which all the lamps will eventually be off. $ (b)$ Prove that there are infinitely many integers $ n$ for which the lamps will never be all off.

[b]4.[/b] Find all positive integers $\alpha , \beta (\alpha >1)$ and all prime numbers $p, q, r$ which satisfy the equation $p^{\alpha}= q^{\beta}+r^{\alpha}$ ($\alpha , \beta , p, q, r$ need not necessarily be different). [b](N. 12)[/b]

Find the largest constant $c>0$ such that for every positive integer $n\ge 2$, there always exist a positive divisor $d$ of $n$ such that $$d\le \sqrt{n}\hspace{0.5cm} \text{and} \hspace{0.5cm} \tau(d)\ge c\sqrt{\tau(n)}$$ where $\tau(n)$ is the number of divisors of $n$. [i]Proposed by Mohd. Suhaimi Ramly[/i]

Iris has an infinite chessboard, in which an $8\times 8$ subboard is marked as Sacred. In order to preserve the Sanctity of this chessboard, her friend Rosabel wishes to place some indistinguishable Holy Knights on the chessboard (not necessarily within the Sacred subboard) such that: [list] [*] No two Holy Knights occupy the same square; [*] Each Holy Knight attacks at least one Sacred square; [*] Each Sacred square is attacked by exactly one Holy Knight. [/list] In how many ways can Rosabel protect the Sanctity of Iris' chessboard? (A Holy Knight works in the same way as a knight piece in chess, that is, it attacks any square that is two squares away in one direction and one square away in a perpendicular direction. Note that a Holy Knight does \emph{not} attack the square it is on.) [i]Proposed by Yannick Yao[/i]

An infinitely long slab of glass is rotated. A light ray is pointed at the slab such that the ray is kept horizontal. If $\theta$ is the angle the slab makes with the vertical axis, then $\theta$ is changing as per the function \[ \theta(t) = t^2, \]where $\theta$ is in radians. Let the $\emph{glassious ray}$ be the ray that represents the path of the refracted light in the glass, as shown in the figure. Let $\alpha$ be the angle the glassious ray makes with the horizontal. When $\theta=30^\circ$, what is the rate of change of $\alpha$, with respect to time? Express your answer in radians per second (rad/s) to 3 significant figures. Assume the index of refraction of glass to be $1.50$. Note: the second figure shows the incoming ray and the glassious ray in cyan. [asy] fill((6,-2-sqrt(3))--(3, -2)--(3+4/sqrt(3),2)--(6+4/sqrt(3),2-sqrt(3))--cycle, gray(0.7)); draw((3+2/sqrt(3),0)--(10,0),linetype("4 4")+grey); draw((0,0)--(3+2/sqrt(3),0)); draw((3+2/sqrt(3), 0)--(7+2/sqrt(3), -1)); arrow((6.5+2/sqrt(3), -7/8), dir(180-14.04), 9); draw((3,-2)--(3,2), linetype("4 4")); draw((6,-2-sqrt(3))--(3,-2)--(3+2/sqrt(3),0)--(3+4/sqrt(3), 2)--(6+4/sqrt(3), 2-sqrt(3))); draw(anglemark((3+2/sqrt(3),0),(3,-2),(3,0),15)); label("$\theta$", (3.2, -1.6), N, fontsize(8)); label("$\alpha$", (6, -0.2), fontsize(8)); [/asy] [asy] fill((6,-2-sqrt(3))--(3, -2)--(3+4/sqrt(3),2)--(6+4/sqrt(3),2-sqrt(3))--cycle, gray(0.7)); draw((3+2/sqrt(3),0)--(10,0),linetype("4 4")+grey); draw((0,0)--(3+2/sqrt(3),0), cyan); draw((3+2/sqrt(3), 0)--(7+2/sqrt(3), -1), cyan); arrow((6.5+2/sqrt(3), -7/8), dir(180-14.04), 9, cyan); draw((3,-2)--(3,2), linetype("4 4")); draw((6,-2-sqrt(3))--(3,-2)--(3+2/sqrt(3),0)--(3+4/sqrt(3), 2)--(6+4/sqrt(3), 2-sqrt(3))); draw(anglemark((3+2/sqrt(3),0),(3,-2),(3,0),15)); label("$\theta$", (3.2, -1.6), N, fontsize(8)); label("$\alpha$", (6, -0.2), fontsize(8)); [/asy] [i]Problem proposed by Ahaan Rungta[/i]

Consider the addition $\begin{tabular}{cccc} & O & N & E \\ + & T & W & O \\ \hline F & O & U & R \\ \end{tabular}$ where different letters represent different nonzero digits. What is the smallest possible value of the four-digit number $FOUR$?

Pedro and Cecilia play the following game: Pedro chooses a positive integer number $a$ and Cecilia wins if she finds a positive integrer number $b$, prime with $a$, such that, in the factorization of $a^3+b^3$ will appear three different prime numbers. Prove that Cecilia can always win.

[b]p1.[/b] Consider a right triangle with legs of lengths $a$ and $b$ and hypotenuse of length $c$ such that the perimeter of the right triangle is numerically (ignoring units) equal to its area. Prove that there is only one possible value of $a + b - c$, and determine that value. [b]p2.[/b] Last August, Jennifer McLoud-Mann, along with her husband Casey Mann and an undergraduate David Von Derau at the University of Washington, Bothell, discovered a new tiling pattern of the plane with a pentagon. This is the fifteenth pattern of using a pentagon to cover the plane with no gaps or overlaps. It is unknown whether other pentagons tile the plane, or even if the number of patterns is finite. Below is a portion of this new tiling pattern. [img]https://services.artofproblemsolving.com/download.php?id=YXR0YWNobWVudHMvOS8xLzM4M2RjZDEzZTliYTlhYTJkZDU4YTA4ZGMwMTA0MzA5ODk1NjI0LnBuZw==&rn=bW1wYyAyMDE1LnBuZw==[/img] Determine the five angles (in degrees) of the pentagon $ABCDE$ used in this tiling. Explain your reasoning, and give the values you determine for the angles at the bottom. [b]p3.[/b] Let $f(x) =\sqrt{2019 + 4\sqrt{2015}} +\sqrt{2015} x$. Find all rational numbers $x$ such that $f(x)$ is a rational number. [b]p4.[/b] Alice has a whiteboard and a blackboard. The whiteboard has two positive integers on it, and the blackboard is initially blank. Alice repeats the following process. $\bullet$ Let the numbers on the whiteboard be $a$ and $b$, with $a \le b$. $\bullet$ Write $a^2$ on the blackboard. $\bullet$ Erase $b$ from the whiteboard and replace it with $b - a$. For example, if the whiteboard began with 5 and 8, Alice first writes $25$ on the blackboard and changes the whiteboard to $5$ and $3$. Her next move is to write $9$ on the blackboard and change the whiteboard to $2$ and $3$. Alice stops when one of the numbers on the whiteboard is 0. At this point the sum of the numbers on the blackboard is $2015$. a. If one of the starting numbers is $1$, what is the other? b. What are all possible starting pairs of numbers? [b]p5.[/b] Professor Beatrix Quirky has many multi-volume sets of books on her shelves. When she places a numbered set of $n$ books on her shelves, she doesn’t necessarily place them in order with book $1$ on the left and book $n$ on the right. Any volume can be placed at the far left. The only rule is that, except the leftmost volume, each volume must have a volume somewhere to its left numbered either one more or one less. For example, with a series of six volumes, Professor Quirky could place them in the order $123456$, or $324561$, or $564321$, but not $321564$ (because neither $4$ nor $6$ is to the left of $5$). Let’s call a sequence of numbers a [i]quirky [/i] sequence of length $n$ if: 1. the sequence contains each of the numbers from $1$ to $n$, once each, and 2. if $k$ is not the first term of the sequence, then either $k + 1$ or $k - 1$ occurs somewhere before $k$ in the sequence. Let $q_n$ be the number of quirky sequences of length $n$. For example, $q_3 = 4$ since the quirky sequences of length $3$ are $123$, $213$, $231$, and $321$. a. List all quirky sequences of length $4$. b. Find an explicit formula for $q_n$. Prove that your formula is correct. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

In the trapezoid $ABCD$ we know that $CD \perp BC, $ and $CD \perp AD .$ Circle $w$ with diameter $AB$ intersects $AD$ in points $A$ and $P,$ tangent from $P$ to $w$ intersects $CD$ at $M.$ The second tangent from $M$ to $w$ touches $w$ at $Q.$ Prove that midpoint of $CD$ lies on $BQ.$

Let $n{}$ be a natural number and $f{}$ be polynomial with integer coefficients. It is known that for any integer $m{}$ there is an integer $k{}$ such that $f(k)-m$ is divisible by $n{}$. Prove that there exists a polynomial $g{}$ with integer coefficients such that $f(g(m))-m$ is divisible by $n{}$ for any integer $m{}$. [i]From the folklore[/i]

Let $f:[0,1]\rightarrow\mathbb{R}$ be an integrable function such that: \[0<\left\vert \int_{0}^{1}f(x)\, \text{d}x\right\vert\le 1.\] Show that there exists $x_1\not= x_2, x_1,x_2\in [0,1]$, such that: \[\int_{x_1}^{x_2}f(x)\, \text{d}x=(x_1-x_2)^{2002}\]

Does the set $\{1,2,3,...,3000\}$ contain a subset $ A$ consisting of 2000 numbers that $x\in A$ implies $2x \notin A$ ?!! :?:

Prove that if the angles $ A $, $ B $, $ C $ of a triangle satisfy the equation $$\cos 3A + \cos 3B + \cos 3C = 1,$$ then one of these angles equals $120^\circ $.

Let $T$ be the set of all positive integer divisors of $2004_{100}$. What is the largest possible number of elements that a subset $S$ of $T$ can have if no element of $S$ is an integer multiple of any other element of $S$?

Define a sequence $ < a_n > _{n\geq0}$ by $ a_0 \equal{} 0$, $ a_1 \equal{} 1$ and \[ a_n \equal{} 2a_{n \minus{} 1} \plus{} a_{n \minus{} 2},\] for $ n\geq2.$ $ (a)$ For every $ m > 0$ and $ 0\leq j\leq m,$ prove that $ 2a_m$ divides $ a_{m \plus{} j} \plus{} ( \minus{} 1)^ja_{m \minus{} j}$. $ (b)$ Suppose $ 2^k$ divides $ n$ for some natural numbers $ n$ and $ k$. Prove that $ 2^k$ divides $ a_n.$

Positive integers $a, b, c$ satisfying the equality $a^2 + b^2 = c^2$. Show that the number $\frac12(c - a) (c - b)$ is square of an integer.