Cities $A$ and $B$ are $45$ miles apart. Alicia lives in $A$ and Beth lives in $B$. Alicia bikes towards $B$ at 18 miles per hour. Leaving at the same time, Beth bikes toward $A$ at 12 miles per hour. How many miles from City $A$ will they be when they meet? $\textbf{(A) }20\qquad\textbf{(B) }24\qquad\textbf{(C) }25\qquad\textbf{(D) }26\qquad\textbf{(E) }27$

Find all primes $p$ and $q$ such that $p^2 + 7pq + q^2$ is a perfect square.

Find all functions $f:\mathbb{R^{+}}\to\mathbb{R^+}$ such that for all $x,y\in\mathbb{R^+}$ it holds that $$f\left(xy\left(\frac{1}{x}+\frac{1}{y}+\frac{1}{x+y}\right)\right)=f\left(xy\left(\frac{1}{x}+\frac{1}{y}\right)\right)+f(x)f\left(\frac{y}{x+y}\right).$$

Is it possible to partition three-dimensional space into tetrahedra (not necessarily regular) such that there exists a plane that intersects the edges of each tetrahedron at exactly 4 or 0 points? Proposed by Arvin Taheri

Let $f: [0, 1] \to \mathbb{R}$ be a continuous strictly increasing function such that \[ \lim_{x \to 0^+} \frac{f(x)}{x}=1. \] (a) Prove that the sequence $(x_n)_{n \ge 1}$ defined by \[ x_n=f \left(\frac{1}{1} \right)+f \left(\frac{1}{2} \right)+\cdots+f \left(\frac{1}{n} \right)-\int_1^n f \left(\frac{1}{x} \right) \mathrm dx \] is convergent. (b) Find the limit of the sequence $(y_n)_{n \ge 1}$ defined by \[ y_n=f \left(\frac{1}{n+1} \right)+f \left(\frac{1}{n+2} \right)+\cdots+f \left(\frac{1}{2021n} \right). \]

Let $n$ be a natural number. An integer $a>2$ is called $n$-decomposable, if $a^n-2^n$ is divisible by all the numbers of the form $a^d+2^d$, where $d\neq n$ is a natural divisor of $n$. Find all composite $n\in \mathbb{N}$, for which there's an $n$-decomposable number.

[b]p1.[/b] Given that $7 \times 22 \times 13 = 2002$, compute $14 \times 11 \times 39$. [b]p2.[/b] Ariel the frog is on the top left square of a $8 \times 10$ grid of squares. Ariel can jump from any square on the grid to any adjacent square, including diagonally adjacent squares. What is the minimum number of jumps required so that Ariel reaches the bottom right corner? [b]p3.[/b] The distance between two floors in a building is the vertical distance from the bottom of one floor to the bottom of the other. In Evans hall, the distance from floor $7$ to floor $5$ is $30$ meters. There are $12$ floors on Evans hall and the distance between any two consecutive floors is the same. What is the distance, in meters, from the first floor of Evans hall to the $12$th floor of Evans hall? [b]p4.[/b] A circle of nonzero radius $ r$ has a circumference numerically equal to $\frac13$ of its area. What is its area? [b]p5.[/b] As an afternoon activity, Emilia will either play exactly two of four games (TwoWeeks, DigBuild, BelowSaga, and FlameSymbol) or work on homework for exactly one of three classes (CS61A, Math 1B, Anthro 3AC). How many choices of afternoon activities does Emilia have? [b]p6.[/b] Matthew wants to buy merchandise of his favorite show, Fortune Concave Decagon. He wants to buy figurines of the characters in the show, but he only has $30$ dollars to spend. If he can buy $2$ figurines for $4$ dollars and $5$ figurines for $8$ dollars, what is the maximum number of figurines that Matthew can buy? [b]p7.[/b] When Dylan is one mile from his house, a robber steals his wallet and starts to ride his motorcycle in the direction opposite from Dylan’s house at $40$ miles per hour. Dylan dashes home at $10$ miles per hour and, upon reaching his house, begins driving his car at $60$ miles per hour in the direction of the robber’s motorcycle. How long, starting from when the robber steals the wallet, does it take for Dylan to catch the robber? Express your answer in minutes. [b]p8.[/b] Deepak the Dog is tied with a leash of $7$ meters to a corner of his $4$ meter by $6$ meter rectangular shed such that Deepak is outside the shed. Deepak cannot go inside the shed, and the leash cannot go through the shed. Compute the area of the region that Deepak can travel to. [img]https://cdn.artofproblemsolving.com/attachments/f/8/1b9563776325e4e200c3a6d31886f4020b63fa.png[/img] [b]p9.[/b] The quadratic equation $a^2x^2 + 2ax -3 = 0$ has two solutions for x that differ by $a$, where $a > 0$. What is the value of $a$? [b]p10.[/b] Find the number of ways to color a $2 \times 2$ grid of squares with $4$ colors such that no two (nondiagonally) adjacent squares have the same color. Each square should be colored entirely with one color. Colorings that are rotations or reflections of each other should be considered different. [b]p11[/b]. Given that $\frac{1}{y^2+5} - \frac{3}{y^4-39} = 0$, and $y \ge 0$, compute $y$. [b]p12.[/b] Right triangle $ABC$ has $AB = 5$, $BC = 12$, and $CA = 13$. Point $D$ lies on the angle bisector of $\angle BAC$ such that $CD$ is parallel to $AB$. Compute the length of $BD$. [img]https://cdn.artofproblemsolving.com/attachments/c/3/d5cddb0e8ac43c35ddfc94b2a74b8d022292f2.png[/img] [b]p13.[/b] Let $x$ and $y$ be real numbers such that $xy = 4$ and $x^2y + xy^2 = 25$. Find the value of $x^3y +x^2y^2 + xy^3$. [b]p14.[/b] Shivani is planning a road trip in a car with special new tires made of solid rubber. Her tires are cylinders that are $6$ inches in width and have diameter $26$ inches, but need to be replaced when the diameter is less than $22$ inches. The tire manufacturer says that $0.12\pi$ cubic inches will wear away with every single rotation. Assuming that the tire manufacturer is correct about the wear rate of their tires, and that the tire maintains its cylindrical shape and width (losing volume by reducing radius), how many revolutions can each tire make before she needs to replace it? [b]p15.[/b] What’s the maximum number of circles of radius $4$ that fit into a $24 \times 15$ rectangle without overlap? [b]p16.[/b] Let $a_i$ for $1 \le i \le 10$ be a finite sequence of $10$ integers such that for all odd $i$, $a_i = 1$ or $-1$, and for all even $i$, $a_i = 1$, $-1$, or $0$. How many sequences a_i exist such that $a_1+a_2+a_3+...+a_{10} = 0$? [b]p17.[/b] Let $\vartriangle ABC$ be a right triangle with $\angle B = 90^o$ such that $AB$ and $BC$ have integer side lengths. Squares $ABDE$ and $BCFG$ lie outside $\vartriangle ABC$. If the area of $\vartriangle ABC$ is $12$, and the area of quadrilateral $DEFG$ is $38$, compute the perimeter of $\vartriangle ABC$. [img]https://cdn.artofproblemsolving.com/attachments/b/6/980d3ba7d0b43507856e581476e8ad91886656.png[/img] [b]p18.[/b] What is the smallest positive integer $x$ such that there exists an integer $y$ with $\sqrt{x} +\sqrt{y} = \sqrt{1025}$ ? [b]p19. [/b]Let $a =\underbrace{19191919...1919}_{19\,\, is\,\,repeated\,\, 3838\,\, times}$. What is the remainder when $a$ is divided by $13$? [b]p20.[/b] James is watching a movie at the cinema. The screen is on a wall and is $5$ meters tall with the bottom edge of the screen $1.5$ meters above the floor. The floor is sloped downwards at $15$ degrees towards the screen. James wants to find a seat which maximizes his vertical viewing angle (depicted below as $\theta$ in a two dimensional cross section), which is the angle subtended by the top and bottom edges of the screen. How far back from the screen in meters (measured along the floor) should he sit in order to maximize his vertical viewing angle? [img]https://cdn.artofproblemsolving.com/attachments/1/5/1555fb2432ee4fe4903accc3b74ea7215bc007.png[/img] PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

$a, b, c, d$ are integers. Show that $x^2 + ax + b = y^2 + cy + d$ has infinitely many integer solutions iff $a^2 - 4b = c^2 - 4d$.

Given that: $\tfrac{6}{11}-\tfrac{10}{19}=\tfrac{9}{19}-\tfrac{n}{11}$, find $n$.

Determine all integers $n\geqslant 3$ such that there exist $n{}$ pairwise distinct real numbers $a_1,\ldots,a_n$ for which the sums $a_i+a_j$ over all $1\leqslant i<j\leqslant n$ form an arithmetic progression.

Give an example of ten different noncoplanar points $ P_1,\ldots ,P_5,Q_1,\ldots ,Q_5$ in $ 3$-space such that connecting each $ P_i$ to each $ Q_j$ by a rigid rod results in a rigid system. [i]L. Lovasz[/i]

Find all injective functions$f:\mathbb{Z}\to \mathbb{Z}$ that satisfy: $\left| f\left( x \right)-f\left( y \right) \right|\le \left| x-y \right|$ ,for any $x,y\in \mathbb{Z}$.

In a math competition, $14$ schools participate, each sending $14$ students. The students are separated into $14$ groups of $14$ so that no two students from the same school are in the same group. The tournament organizers noted that, from the competitors, exactly $15$ have participated in the competition before. The organizers want to select two representatives, with the conditions that they must be former participants, must come from different schools, and must also be in different groups. It turns out that there are $ n$ ways to do this. What is the minimum possible value of $n$?

Let $X$ be normally distributed with mean $\mu$ and variance $\sigma^2>0$. What is the variance of $e^X$?

One of the following numbers is not divisible by any prime number less than 10. Which is it? (A) $2^{606} - 1 \ \ $ (B) $2^{606} + 1 \ \ $ (C) $2^{607} - 1 \ \ $ (D) $2^{607} + 1 \ \ $ (E) $2^{607} + 3^{607} \ \ $

Define a regular $n$-pointed star to be a union of $n$ lines segments $P_1P_2, P_2P_3, ..., P_nP_1$ such that $\bullet$ the points $P_1,P_2,...,P_n$ are coplanar and no three of them are collinear, $\bullet$ each of the $n$ line segments intersects at least one of the other line segments at a point other than an endpoint, $\bullet$ all of the angles at $P_1, P_2,..., P_n$ are congruent , $\bullet$ all of the $n$ line segments $P_1P_2, P_2P_3, ..., P_nP_1$ are congruent, and $\bullet$ the path $P_1P_2...P_nP_1$ turns counterclockwise at an angle less than $180^o$ at each vertex. There are no regular $3$-pointed, $4$-pointed, or $6$-pointed stars. All regular $5$-pointed star are similar, but there are two non-similar regular $7$-pointed stars. Find all possible values of $n$ such that there are exactly $29$ non-similar regular $n$-pointed stars.

Suppose that $1000$ students are standing in a circle. Prove that there exists an integer $k$ with $100 \leq k \leq 300$ such that in this circle there exists a contiguous group of $2k$ students, for which the first half contains the same number of girls as the second half. [i]Proposed by Gerhard Wöginger, Austria[/i]

Lines containing the medians of the triangle $ABC$ intersect its circumscribed circle for a second time at the points $A_1, B_1, C_1$. The straight lines passing through $A,B,C$ parallel to opposite sides intersect it at points $A_2, B_2, C_2$. Prove that lines $A_1A_2,B_1B_2,C_1C_2$ intersect at one point.

Let $k\geq 2$ be an integer. We want to determine the weight of $n$ balls. One try consists of choosing two balls, and we are given the sum of the weights of the two chosen balls. We know that at most $k$ of the answers can be wrong. Let $f_k(n)$ denote the smallest number for which it is true that we can always find the weights of the balls with $f_k(n)$ tries (the tries don't have to be decided in advance). Prove that there exist numbers $a_k$ and $b_k$ for which $|f_k(n)-a_kn|\leq b_k$ holds. [i]Proposed by Surányi László, Budapest and Bálint Virág, Toronto[/i]

Find the number of ordered triples of positive integers $\left(a,b,c\right)$ such that $a\times b\times c=2008^2$.

In a certain city there are $n$ straight streets, such that every two streets intersect, and no three streets pass through the same intersection. The City Council wants to organize the city by designating the main and the side street on every intersection. Prove that this can be done in such way that if one goes along one of the streets, from its beginning to its end, the intersections where this street is the main street, and the ones where it is not, will apear in alternating order. [i]Proposed by Serbia[/i]

Snow White has, in her huge basket, $2021$ apples, and she knows that exactly one of them has a deadly poison, capable of killing a human being hours after ingesting just a measly piece. Contrary to what the fairy tales say, Snow White is more malevolent than the Evil Queen, and doesn't care about the lives of the seven dwarfs. Therefore, she decided to use them to find out which apple is poisonous. To this end, at the beginning of each day, Snow White prepares some apple salads (each salad is a mixture of pieces of some apples chosen by her), and forces some of the dwarfs (possibly all) to eat a salad each. At the end of the day, she notes who died and who survived, and the next day she again prepares some apple salads, forcing some of the surviving dwarves (possibly all) to eat a salad each. And she continues to do this, day after day, until she discovers the poisoned apple or until all the dwarves die. (a) Prove that there is a strategy in which Snow White manages to discover the poisoned apple after a few days. (b) What is the minimum number of days Snow White needs to find the poisoned apple, no matter how lucky she is?

Xenia and Sergey play the following game. Xenia thinks of a positive integer $N$ not exceeding $5000$. Then she fixes $20$ distinct positive integers $a_1, a_2, \cdots, a_{20}$ such that, for each $k = 1,2,\cdots,20$, the numbers $N$ and $a_k$ are congruent modulo $k$. By a move, Sergey tells Xenia a set $S$ of positive integers not exceeding $20$, and she tells him back the set $\{a_k : k \in S\}$ without spelling out which number corresponds to which index. How many moves does Sergey need to determine for sure the number Xenia thought of? [i]Sergey Kudrya, Russia[/i]

It is known that a $n$-vertex contains within itself a polyhedron $M$ with a center of symmetry at some point $Q$ and is itself contained in a polyhedron homothetic to $M$ with a homothety center at a point $Q$ and coefficient $k$. Find the smallest value of $k$ if a) $n = 4$, b) $n = 5$.

In Thailand there are $n$ cities. Each pair of cities is connected by a one-way street which can be borrowed, depending on its type, only by bike or by car. Show that there is a city from which you can reach any other city, either by bike or by car. [i]Remark : It is not necessary to use the same means of transport for each city[/i]