Let $n$ and $k$ be positive integers. Consider $n$ infinite arithmetic progressions of nonnegative integers with the property that among any $k$ consecutive nonnegative integers, at least one of $k$ integers belongs to one of the $n$ arithmetic progressions. Let $d_1,d_2,\ldots,d_n$ denote the differences of the arithmetic progressions, and let $d=\min\{d_1,d_2,\ldots,d_n\}$. In terms of $n$ and $k$, what is the maximum possible value of $d$?

A fair coin is repeatedly flipped until $2019$ consecutive coin flips are the same. Compute the probability that the first and last flips of the coin come up differently.

Seven cookies of radius $1$ inch are cut from a circle of cookie dough, as shown. Neighboring cookies are tangent, and all except the center cookie are tangent to the edge of the dough. The leftover scrap is reshaped to form another cookie of the same thickness. What is the radius in inches of the scrap cookie? [asy] draw(circle((0,0),3)); draw(circle((0,0),1)); draw(circle((1,sqrt(3)),1)); draw(circle((-1,sqrt(3)),1)); draw(circle((-1,-sqrt(3)),1)); draw(circle((1,-sqrt(3)),1)); draw(circle((2,0),1)); draw(circle((-2,0),1)); [/asy] $\textbf{(A) } \sqrt{2} \qquad \textbf{(B) } 1.5 \qquad \textbf{(C) } \sqrt{\pi} \qquad \textbf{(D) } \sqrt{2\pi} \qquad \textbf{(E) } \pi$

Prove that the line segment, joining the orthocenter and the intersection point of the medians of the acute-angled triangle $ABC$ is parallel to the side $AB$ iff $\tan \angle A \cdot \tan \angle B = 3$.

Mr Green measures his rectangular garden by walking two of the sides and finds that it is 15 steps by 20 steps. Each or Mr Green's steps is two feet long. Mr Green expect half a pound of potatoes per square foot from his garden. How many pounds of potatoes does Mr Green expect from his garden? $ \textbf{(A) }600\qquad\textbf{(B) }800\qquad\textbf{(C) }1000\qquad\textbf{(D) }1200\qquad\textbf{(E) }1400 $

Let $S$ be a set of $n$ points in the plane such that any two points of $S$ are at least $1$ unit apart. Prove there is a subset $T$ of $S$ with at least $\frac{n}{7}$ points such that any two points of $T$ are at least $\sqrt{3}$ units apart.

In a triangle $ABC$ it is given that $2AB=AC+BC$. Prove that the incentre of $\triangle ABC$, the circumcentre of $\triangle ABC$, and the midpoints of $AC$ and $BC$ are concyclic.

Let $n \ge 1$. Prove that it is possible to select some of the integers $1,2,...,2^n$ so that for each $p = 0,1,...,n - 1$ the sum of the $p$-th powers of the selected numbers is equal to the sum of the $p$-th powers of the remaining numbers.

Let $ABCD$ be a rhombus and suppose $E$ and $F$ are the midpoints of $\overline{AD}$ and $\overline{EF}$ are the midpoints of $\overline{AD}$ and $\overline{BC},$ respectively. If $G$ is the intersection of $\overline{AC}$ and $\overline{EF},$ find the ratio of the area of $AEG$ to the area of $AGFB.$

How many distinct integral solutions of the form $(x, y)$ exist to the equation$ 21x + 22y = 43$ such that $1 < x < 11$ and $y < 22$?

Equilateral triangle $ABC$ has side length $840$. Point $D$ lies on the same side of line $BC$ as $A$ such that $\overline{BD} \perp \overline{BC}$. The line $\ell$ through $D$ parallel to line $BC$ intersects sides $\overline{AB}$ and $\overline{AC}$ at points $E$ and $F$, respectively. Point $G$ lies on $\ell$ such that $F$ is between $E$ and $G$, $\triangle AFG$ is isosceles, and the ratio of the area of $\triangle AFG$ to the area of $\triangle BED$ is $8:9$. Find $AF$. [asy] pair A,B,C,D,E,F,G; B=origin; A=5*dir(60); C=(5,0); E=0.6*A+0.4*B; F=0.6*A+0.4*C; G=rotate(240,F)*A; D=extension(E,F,B,dir(90)); draw(D--G--A,grey); draw(B--0.5*A+rotate(60,B)*A*0.5,grey); draw(A--B--C--cycle,linewidth(1.5)); dot(A^^B^^C^^D^^E^^F^^G); label("$A$",A,dir(90)); label("$B$",B,dir(225)); label("$C$",C,dir(-45)); label("$D$",D,dir(180)); label("$E$",E,dir(-45)); label("$F$",F,dir(225)); label("$G$",G,dir(0)); label("$\ell$",midpoint(E--F),dir(90)); [/asy]

Let $n$ be a positive integer. In how many ways can we mark cells on an $n\times n$ board such that no two rows and no two columns have the same number of marked cells? [i]Selim Bahadir, Turkey[/i]

Two people throw coins. One throws his coin $10$ times, the other throws his $11$ times . What is the probability that the second coin fell showing "heads" a greater number of times than the first? (For those not familiar with Probability Theory this question could have been reformulated thus : Consider various arrangements of a $21$ digit number in which each digit must be a " $1$ " or a "$2$" . Among all such numbers what fraction of them will have more occurrences of the digit "$2$" among the last $11$ digits than among the first $10$?) (S. Fomin , Leningrad)

In a game of Among Us, there are $10$ players and $12$ colors. Each player has a "default" color that they will automatically get if nobody else has that color. Otherwise, they get a random color that is not selected. If $10$ random players with random default colors join a game one by one, the expected number of players to get their default color can be expressed as $\frac{m}{n}$. Compute $m+n$. Note that the default colors are not necessarily distinct. [i]Proposed by Jeff Lin[/i]

In a $2\times n$ array we have positive reals s.t. the sum of the numbers in each of the $n$ columns is $1$. Show that we can select a number in each column s.t. the sum of the selected numbers in each row is at most $\frac{n+1}4$.

An $18.5$ g sample of tin $\text{(M = 118.7)}$ combines with $10.0$ g of sulfur $\text{(M = 32.07)}$ to form a compound. What is the empirical formula of this compound? $ \textbf{(A) }\ce{SnS}\qquad\textbf{(B) }\ce{SnS2}\qquad\textbf{(C) }\ce{Sn2S}\qquad\textbf{(D) }\ce{Sn2S3}\qquad $

Let $n,d$ be natural numbers. Prove that there is an arithmetic sequence of positive integers. $$a_1,a_2,...,a_n$$ with common difference of $d$ and $a_i$ with prime factor greater than or equal to $i$ for all values $i=1,2,...,n$. [i](nooonuii)[/i]

Three circles, $\omega_1$, $\omega_2$, $\omega_3$ are drawn, with $\omega_3$ externally tangent to $\omega_1$ at $C$ and internally tangent to $\omega_2$ at $D$. Say also that $\omega_1$, $\omega_2$ intersect at points $A, B$. Suppose the radius of $\omega_1$ is $20$, the radius of $\omega_2$ is $15$, and the radius of $\omega_3$ is $6$. Draw line $CD$, and suppose it meets $AB$ at point $X$. If $AB = 24$, then $CX$ can be written in the form $\frac{a \sqrt{b}}{c}$, where$ a, b, c$ are positive integers where $b$ is square-free, and $a, c$ are relatively prime. Find $a + b + c$.

Triangle $\vartriangle ABC$ has $AB = 8$, $AC = 10$, and $AD =\sqrt{33}$, where $D$ is the midpoint of $BC$. Perpendiculars are drawn from $D$ to meet $AB$ and $AC$ at $E$ and $F$, respectively. The length of $EF$ can be expressed as $\frac{a\sqrt{b}}{c}$ , where $a, c$ are relatively prime and $b$ is square-free. Compute $a + b + c$.

Let $ a$, $ b$ and $ c$ be the lengths of the sides of a triangle. Prove that \[ a^{2}b(a \minus{} b) \plus{} b^{2}c(b \minus{} c) \plus{} c^{2}a(c \minus{} a)\ge 0. \] Determine when equality occurs.

The rectangle is cut into $10$ squares as shown in the figure on the right. Find its sides if the side of the smallest square is $3$.[img]https://cdn.artofproblemsolving.com/attachments/e/5/1fe3a0e41b2d3182338a557d3d44ff5ef9385d.png[/img]

Two magicians show the following trick. The first magician goes out of the room. The second magician takes a deck of $100$ cards labelled by numbers $1,2,\ldots ,100$ and asks three spectators to choose in turn one card each. The second magician sees what card each spectator has taken. Then he adds one more card from the rest of the deck. Spectators shuffle these $4$ cards, call the first magician and give him these $4$ cards. The first magician looks at the $4$ cards and “guesses” what card was chosen by the first spectator, what card by the second and what card by the third. Prove that the magicians can perform this trick.

Let $n \ge 3$ be a positive integer. For every set $S$ with $n$ distinct positive integers, prove that there exists a bijection $f: \{1,2, \cdots n\} \rightarrow S$ which satisfies the following condition. For all $1 \le i < j < k \le n$, $f(j)^2 \neq f(i) \cdot f(k)$.

A finite number of circles are placed into a $1 \times 1$ square. Let $C$ be the sum of the perimeters of the circles. For how many $C$s from $C=\dfrac {43}5$, $9$, $\dfrac{91}{10}$, $\dfrac{19}{2}$, $10$, we can definitely say there exists a line cutting four of the circles? $ \textbf{(A)}\ 0 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 2 \qquad\textbf{(D)}\ 3 \qquad\textbf{(E)}\ 4 $

Cyclic pentagon $ ABCDE$ has side lengths $ AB\equal{}BC\equal{}5$, $ CD\equal{}DE\equal{}12$, and $ AE \equal{} 14$. Determine the radius of its circumcircle.