$A$ and $B$ travel around a circular track at uniform speeds in opposite directions, starting from diametrically opposite points. If they start at the same time, meet first after $B$ has travelled $100$ yards, and meet a second time $60$ yards before $A$ completes one lap, then the circumference of the track in yards is $\textbf{(A) }400\qquad\textbf{(B) }440\qquad\textbf{(C) }480\qquad\textbf{(D) }560\qquad \textbf{(E) }880$

People from n different countries sit at a round table. Assume that for every two members of the same country their neighbours sitting next to them on the right hand side are from different countries. Find the largest possible number of people sitting around the table?

Given the positive numbers $a$ and $b$ and the natural number $n$, find the greatest among the $n + 1$ monomials in the binomial expansion of $\left(a+b\right)^n$.

Determine the greatest number of figures congruent to [img]https://cdn.artofproblemsolving.com/attachments/c/6/343f9197bcebf6794460ed1a74ba83ec18a377.png[/img] that can be placed in a $9 \times 9$ grid (without overlapping), such that each figure covers exactly $4$ unit squares. The figures can be rotated and flipped over. For example, the picture below shows that at least $3$ such figures can be placed in a $4 \times4$ grid. [img]https://cdn.artofproblemsolving.com/attachments/1/e/d38fc34b650a1333742bb206c29985c94146aa.png[/img]

Andrew the Antelope canters along the surface of a regular icosahedron, which has twenty equilateral triangle faces and edge length 4. If he wants to move from one vertex to the opposite vertex, the minimum distance he must travel can be expressed as $\sqrt{n}$ for some integer $n$. Compute $n$.

When three different numbers from the set $ \{-3,-2,-1, 4, 5\} $ are multiplied, the largest possible product is $ \text{(A)}\ 10\qquad\text{(B)}\ 20\qquad\text{(C)}\ 30\qquad\text{(D)}\ 40\qquad\text{(E)}\ 60 $

A bank charges $ \$6$ for a loan of $ \$120$. The borrower receives $ \$114$ and repays the loan in $ 12$ installments of $ \$10$ a month. The interest rate is approximately: $ \textbf{(A)}\ 5 \% \qquad \textbf{(B)}\ 6 \% \qquad \textbf{(C)}\ 7 \% \qquad \textbf{(D)}\ 9\% \qquad \textbf{(E)}\ 15 \%$

Let $V$ be a finite set of points in three-dimensional space. Let $S_1, S_2, S_3$ be the sets consisting of the orthogonal projections of the points of $V$ onto the $yz$-plane, $zx$-plane, $xy$-plane, respectively. Prove that $| V|^2 \leq | S1|\cdot|S2|\cdot |S3|$, where $| A|$ denotes the number of elements in the finite set $A.$

Let $n$ be a positive integer. Prove that for each factor $m$ of the number $1+2+\cdots+n$ such that $m\ge n$, the set $\{1,2,\ldots,n\}$ can be partitioned into disjoint subsets, the sum of the elements of each being equal to $m$. [b]Edit[/b]:Typographical error fixed.

Find all functions $f:\mathbb{R}\rightarrow \mathbb{R}$ satisfying \[f(x+f(y))=x+f(f(y))\] for all real numbers $x$ and $y$, with the additional constraint $f(2004)=2005$.

A moving particle starts at the point $\left(4,4\right)$ and moves until it hits one of the coordinate axes for the first time. When the particle is at the point $\left(a,b\right)$, it moves at random to one of the points $\left(a-1,b\right)$, $\left(a,b-1\right)$, or $\left(a-1,b-1\right)$, each with probability $\tfrac{1}{3}$, independently of its previous moves. The probability that it will hit the coordinate axes at $\left(0,0\right)$ is $\tfrac{m}{3^n}$, where $m$ and $n$ are positive integers, and $m$ is not divisible by $3$. Find $m+n$.

Cut a rectangle into $18$ rectangles so that no two adjacent ones form a rectangle.

[hide=D stands for Descartes, L stands for Leibniz]they had two problem sets under those two names[/hide] [b]L.10[/b] Given the following system of equations where $x, y, z$ are nonzero, find $x^2 + y^2 + z^2$. $$x + 2y = xy$$ $$3y + z = yz$$ $$3x + 2z = xz$$ [u]Set 4[/u] [b]L.16 / D.23[/b] Anson, Billiam, and Connor are looking at a $3D$ figure. The figure is made of unit cubes and is sitting on the ground. No cubes are floating; in other words, each unit cube must either have another unit cube or the ground directly under it. Anson looks from the left side and says, “I see a $5 \times 5$ square.” Billiam looks from the front and says the same thing. Connor looks from the top and says the same thing. Find the absolute difference between the minimum and maximum volume of the figure. [b]L.17[/b] The repeating decimal $0.\overline{MBMT}$ is equal to $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers, and $M, B, T$ are distinct digits. Find the minimum value of $q$. [b]L.18[/b] Annie, Bob, and Claire have a bag containing the numbers $1, 2, 3, . . . , 9$. Annie randomly chooses three numbers without replacement, then Bob chooses, then Claire gets the remaining three numbers. Find the probability that everyone is holding an arithmetic sequence. (Order does not matter, so $123$, $213$, and $321$ all count as arithmetic sequences.) [b]L.19[/b] Consider a set $S$ of positive integers. Define the operation $f(S)$ to be the smallest integer $n > 1$ such that the base $2^k$ representation of $n$ consists only of ones and zeros for all $k \in S$. Find the size of the largest set $S$ such that $f(S) < 2^{2019}$. [b]L.20 / D.25[/b] Find the largest solution to the equation $$2019(x^{2019x^{2019}-2019^2+2019})^{2019} = 2019^{x^{2019}+1}.$$ [u]Set 5[/u] [b]L.21[/b] Steven is concerned about his artistic abilities. To make himself feel better, he creates a $100 \times 100$ square grid and randomly paints each square either white or black, each with probability $\frac12$. Then, he divides the white squares into connected components, groups of white squares that are connected to each other, possibly using corners. (For example, there are three connected components in the following diagram.) What is the expected number of connected components with 1 square, to the nearest integer? [img]https://cdn.artofproblemsolving.com/attachments/e/d/c76e81cd44c3e1e818f6cf89877e56da2fc42f.png[/img] [b]L.22[/b] Let x be chosen uniformly at random from $[0, 1]$. Let n be the smallest positive integer such that $3^n x$ is at most $\frac14$ away from an integer. What is the expected value of $n$? [b]L.23[/b] Let $A$ and $B$ be two points in the plane with $AB = 1$. Let $\ell$ be a variable line through $A$. Let $\ell'$ be a line through $B$ perpendicular to $\ell$. Let X be on $\ell$ and $Y$ be on $\ell'$ with $AX = BY = 1$. Find the length of the locus of the midpoint of $XY$ . [b]L.24[/b] Each of the numbers $a_i$, where $1 \le i \le n$, is either $-1$ or $1$. Also, $$a_1a_2a_3a_4+a_2a_3a_4a_5+...+a_{n-3}a_{n-2}a_{n-1}a_n+a_{n-2}a_{n-1}a_na_1+a_{n-1}a_na_1a_2+a_na_1a_2a_3 = 0.$$ Find the number of possible values for $n$ between $4$ and $100$, inclusive. [b]L.25[/b] Let $S$ be the set of positive integers less than $3^{2019}$ that have only zeros and ones in their base $3$ representation. Find the sum of the squares of the elements of $S$. Express your answer in the form $a^b(c^d - 1)(e^f - 1)$, where $a, b, c, d, e, f$ are positive integers and $a, c, e$ are not perfect powers. PS. You should use hide for answers. D.1-15 / L1-9 problems have been collected [url=https://artofproblemsolving.com/community/c3h2790795p24541357]here [/url] and D.16-30/ L10-15 [url=https://artofproblemsolving.com/community/c3h2790818p24541688]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Find all solutions of $2^n + 7 = x^2$ in which n and x are both integers . Prove that there are no other solutions.

Find the greatest natural number $n$ such there exist natural numbers $x_{1}, x_{2}, \ldots, x_{n}$ and natural $a_{1}< a_{2}< \ldots < a_{n-1}$ satisfying the following equations for $i =1,2,\ldots,n-1$: \[x_{1}x_{2}\ldots x_{n}= 1980 \quad \text{and}\quad x_{i}+\frac{1980}{x_{i}}= a_{i}.\]

Let $1,2,3,\dots,2005,2006,2007,2009,2012,2016,\dots$ be a sequence defined by $x_{k}=k$ for $k=1,2\dots,2006$ and $x_{k+1}=x_{k}+x_{k-2005}$ for $k\ge 2006.$ Show that the sequence has 2005 consecutive terms each divisible by 2006.

You, your friend, and two strangers are sitting at a table. A standard $52$-card deck is randomly dealt into $4$ piles of $13$ cards each, and each person at the table takes a pile. You look through your hand and see that you have one ace. Compute the probability that your friend’s hand contains the three remaining aces.

We are given $n$ points in space. Some pairs of these points are connected by line segments so that the number of segments equals $[n^2/4],$ and a connected triangle exists. Prove that any point from which the maximal number of segments starts is a vertex of a connected triangle.

In triangle $ABC$, the altitude from $B$ is tangent to the circumcircle of $ABC$. Prove that the largest angle of the triangle is between $90^o$ and $135^o$. If the altitudes from both $B$ and from $C$ are tangent to the circumcircle, then what are the angles of the triangle?

Find all functions$ f, g : (0,\infty) \to (0,\infty)$ such that for all $x > 0$ we have the relations: $f(g(x)) = \frac{x}{xf(x) - 2}$ and $g(f(x)) = \frac{x}{xg(x) - 2}$ .

Represent the number $989 \cdot 1001 \cdot 1007 +320$ as a product of primes.

Given $n$ integers $a_1, a_2, ..., a_n$, which $a_1 = 1$ and $a_i \le a_{i + 1} \le 2a_i$ for each $i \in \{1,2,...,n-1\}$ . Prove that if $a_1 + a_2 +... + a_n$ is even, you do select some of the numbers so that their sum equals $\frac{a_1 + a_2 +... + a_n}{2}$ .

Let $ABC$ be a triangle and let $P$ be a point in its interior. Suppose $ \angle B A P = 10 ^ { \circ } , \angle A B P = 20 ^ { \circ } , \angle P C A = 30 ^ { \circ } $ and $ \angle P A C = 40 ^ { \circ } $. Find $ \angle P B C $.

Let $m$ and $n$ be fixed positive integers. Tsvety and Freyja play a game on an infinite grid of unit square cells. Tsvety has secretly written a real number inside of each cell so that the sum of the numbers within every rectangle of size either $m$ by $n$ or $n$ by $m$ is zero. Freyja wants to learn all of these numbers. One by one, Freyja asks Tsvety about some cell in the grid, and Tsvety truthfully reveals what number is written in it. Freyja wins if, at any point, Freyja can simultaneously deduce the number written in every cell of the entire infinite grid (If this never occurs, Freyja has lost the game and Tsvety wins). In terms of $m$ and $n$, find the smallest number of questions that Freyja must ask to win, or show that no finite number of questions suffice. [i]Nikolai Beluhov[/i]

Let $ABCD$ be a convex quadrilateral with $\angle CBD = 2 \angle ADB$, $\angle ABD = 2 \angle CDB$ and $AB = CB$. Prove that $AD = CD$.