The rows and columns of a $7\times 7$ grid are each numbered $1,2,\ldots, 7$. In how many ways can one choose 8 cells of this grid such that for every two chosen cells $X$ and $Y$, either the positive difference of their row numbers is at least $3$, or the positive difference of their column numbers is at least $3$? [hide="Clarifications"] [list] [*] The ``or'' here is inclusive (as by convention, despite the ``either''), i.e. $X$ and $Y$ are permitted if and only if they satisfy the row condition, the column condition, or both.[/list][/hide] [i]Ray Li[/i]

For a positive integer $k$, denote by $f(k)$ the number of positive integer $m$ such that the remainder of $km$ modulo $2019^3$ is greater than $m$. Find the amount of different numbers among $f(1), f(2), ..., f(2019^3)$.

Circle $\omega$ lies inside the circle $\Omega$ and touches it internally at point $P$. Point $S$ is taken on $\omega$ and the tangent to $\omega$ is drawn through it. This tangent meets $\Omega$ at points $A$ and $B$. Let $I$ be the centre of $\omega$. Find the locus of circumcentres of triangles $AIB$.

Let $ABC$ be a triangle with $AB > AC$. Let $D$ be a point on the side $AB$ such that $DB = DC$ and let $M$ be the midpoint of $AC$. The line parallel to $BC$ passing through $D$ intersects the line $BM$ in $K$. Show that $\angle KCD = \angle DAC.$

Patio blocks that are hexagons $1$ unit on a side are used to outline a garden by placing the blocks edge to edge with $n$ on each side. The diagram indicates the path of blocks around the garden when $n=5.$ [asy] size(250);int i,j; real r=sqrt(3); for(i=0; i<6; i=i+1) { for(j=0; j<4; j=j+1) { draw(shift(((j*r)*dir(60*i+150)).x, ((j*r)*dir(60*i+150)).y)*shift((4r*dir(60i+30)).x,(4r*dir(60i+30)).y)*polygon(6)); }}[/asy] If $n=202,$ then the area of the garden enclosed by the path, not including the path itself, is $m(\sqrt{3}/2)$ square units, where $m$ is a positive integer. Find the remainder when $m$ is divided by $1000.$

Inscribed in a circle is a quadrilateral having sides of lengths $25,~39,~52,$ and $60$ taken consecutively. The diameter of this circle has length $\textbf{(A) }62\qquad\textbf{(B) }63\qquad\textbf{(C) }65\qquad\textbf{(D) }66\qquad \textbf{(E) }69$

Steph Curry is playing the following game and he wins if he has exactly $5$ points at some time. Flip a fair coin. If heads, shoot a $3$-point shot which is worth $3$ points. If tails, shoot a free throw which is worth $1$ point. He makes $\frac12$ of his $3$-point shots and all of his free throws. Find the probability he will win the game. (Note he keeps flipping the coin until he has exactly $5$ or goes over $5$ points)

Find all ordered triples $(a,b, c)$ of positive integers which satisfy $5^a + 3^b - 2^c = 32$

Show that there exists an integer polynomial $P$ such that $P(1) = 2024$ and the set of prime divisors of {$P(2^k)$},$k=0,1,2,.....$ is an infinite set.

Consider the sequence $(a_n)$ satisfying $a_1=\dfrac{1}{2},a_{n+1}=\sqrt[3]{3a_{n+1}-a_n}$ and $0\le a_n\le 1,\forall n\ge 1.$ a. Prove that the sequence $(a_n)$ is determined uniquely and has finite limit. b. Let $b_n=(1+2.a_1)(1+2^2a_2)...(1+2^na_n), \forall n\ge 1.$ Prove that the sequence $(b_n)$ has finite limit.

A natural number $n$ is said to be $good$ if $n$ is the sum or $r$ consecutive positive integers, for some $r \geq 2 $. Find the number of good numbers in the set $\{1,2 \dots , 100\}$.

25. Chris and Paul each rent a different room of a hotel from rooms $1 - 60$. However, the hotel manager mistakes them for one person and gives ”Chris Paul” a room with Chris’s and Paul’s room concatenated. For example, if Chris had $15$ and Paul had $9$, ”Chris Paul” has $159$. If there are $360$ rooms in the hotel, what is the probability that ”Chris Paul” has a valid room? 26. Find the number of ways to choose two nonempty subsets $X$ and $Y$ of $\{1, 2, \ldots , 2001\}$, such that $|Y| = 1001$ and the smallest element of $Y$ is equal to the largest element of $X$. 27. Let $r_1, r_2, r_3, r_4$ be the four roots of the polynomial $x^4 - 4x^3 + 8x^2 - 7x + 3$. Find the value of $$\frac{r_1^2}{r_2^2+r_3^2+r_4^2}+\frac{r_2^2}{r_1^2+r_3^2+r_4^2}+\frac{r_3^2}{r_1^2+r_2^2+r_4^2}+\frac{r_4^2}{r_1^2+r_2^2+r_3^2}$$

Let $ABCD$ be a rectangle with $AB=10$ and $AD=5.$ Suppose points $P$ and $Q$ are on segments $CD$ and $BC,$ respectively, such that the following conditions hold: [list] [*] $BD \parallel PQ$ [*] $\angle APQ=90^{\circ}.$ [/list] What is the area of $\triangle CPQ?$ [i]Proposed by Kyle Lee[/i]

Lennart and Eddy are playing a betting game. Lennart starts with $7$ dollars and Eddy starts with $3$ dollars. Each round, both Lennart and Eddy bet an amount equal to the amount of the player with the least money. For example, on the first round, both players bet $3$ dollars. A fair coin is then tossed. If it lands heads, Lennart wins all the money bet; if it lands tails, Eddy wins all the money bet. They continue playing this game until one person has no money. What is the probability that Eddy ends with $10$ dollars?

The positive integers are partitioned into 2 sequences $a_1<a_2<\dots$ and $b_1<b_2<\dots$ such that $b_n=a_n+n$ for every positive integer $n$. Show that $a_n+b_n=a_{b_n}$.

Given a convex polygon M invariant under a $90^\circ$ rotation, show that there exist two circles, the ratio of whose radii is $\sqrt2$, one containing M and the other contained in M. [i]A. Khrabrov[/i]

[b]p1.[/b] Solve for $x$ and $y$ if $\frac{1}{x^2}+\frac{1}{xy}=\frac{1}{9}$ and $\frac{1}{y^2}+\frac{1}{xy}=\frac{1}{16}$ [b]p2.[/b] Find positive integers $p$ and $q$, with $q$ as small as possible, such that $\frac{7}{10} <\frac{p}{q} <\frac{11}{15}$. [b]p3.[/b] Define $a_1 = 2$ and $a_{n+1} = a^2_n -a_n + 1$ for all positive integers $n$. If $i > j$, prove that $a_i$ and $a_j$ have no common prime factor. [b]p4.[/b] A number of points are given in the interior of a triangle. Connect these points, as well as the vertices of the triangle, by segments that do not cross each other until the interior is subdivided into smaller disjoint regions that are all triangles. It is required that each of the givien points is always a vertex of any triangle containing it. Prove that the number of these smaller triangular regions is always odd. [b]p5.[/b] In triangle $ABC$, let $\angle ABC=\angle ACB=40^o$ is extended to $D$ such that $AD=BC$. Prove that $\angle BCD=10^o$. [img]https://cdn.artofproblemsolving.com/attachments/6/c/8abfbf0dc38b76f017b12fa3ec040849e7b2cd.png[/img] PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $a$ and $d$ be two positive integers. Prove that there exists a constant $K$ such that every set of $K$ consecutive elements of the arithmetic progression $\{a+nd\}_{n=1}^\infty$ contains at least one number which is not prime.

Equilateral triangle $ABC$ has circumcircle $\Omega$. Points $D$ and $E$ are chosen on minor arcs $AB$ and $AC$ of $\Omega$ respectively such that $BC=DE$. Given that triangle $ABE$ has area $3$ and triangle $ACD$ has area $4$, find the area of triangle $ABC$.

Let $ABCD$ be a cyclic quadrilateral. Let $P$, $Q$, $R$ be the feet of the perpendiculars from $D$ to the lines $BC$, $CA$, $AB$, respectively. Show that $PQ=QR$ if and only if the bisectors of $\angle ABC$ and $\angle ADC$ are concurrent with $AC$.

Two points $J$ and $H$ lie $26$ units apart on a given plane. Let $M$ be the locus of points $T$ on this plane such that $JT^2 + HT^2 = 2020$. Then, M encloses a region on the plane with area $a$ and perimeter $p$. If $q$ and $r$ are coprime positive integers and $\frac{a}{p} = \frac{q}{r}$ , then compute $q + r$.

Evaluate the sum \[1^2+2^2-3^2-4^2+5^2+6^2-7^2-8^2+\cdots-1000^2+1001^2\]

Prove that for any $a>0,b>0,c>0$ we have $8a^2 b^2 c^2 \geq (a^2 + ab + ac - bc)(b^2 + ba + bc - ac)(c^2 + ca + cb - ab)$.

Sarah pours four ounces of coffee into an eight-ounce cup and four ounces of cream into a second cup of the same size. She then transfers half the coffee from the first cup to the second and, after stirring thoroughly, transfers half the liquid in the second cup back to the first. What fraction of the liquid in the first cup is now cream? $ \textbf{(A)}\ 1/4 \qquad \textbf{(B)}\ 1/3 \qquad \textbf{(C)}\ 3/8 \qquad \textbf{(D)}\ 2/5 \qquad \textbf{(E)}\ 1/2$

Let $n$ and $k$ be fixed positive integers , and $a$ be arbitrary nonnegative integer . Choose a random $k$-element subset $X$ of $\{1,2,...,k+a\}$ uniformly (i.e., all k-element subsets are chosen with the same probability) and, independently of $X$, choose random n-elements subset $Y$ of $\{1,2,..,k+a+n\}$ uniformly. Prove that the probability $P\left( \text{min}(Y)>\text{max}(X)\right)$ does not depend on $a$.