What is the least positive integer $k$ such that, in every convex 1001-gon, the sum of any k diagonals is greater than or equal to the sum of the remaining diagonals?

Prove that for each $n \in \mathbb N$ there exist natural numbers $a_1<a_2<...<a_n$ such that $\phi(a_1)>\phi(a_2)>...>\phi(a_n)$. [i]Proposed by Amirhossein Gorzi[/i]

The side lengths of a trapezoid are $\sqrt[4]{3}, \sqrt[4]{3}, \sqrt[4]{3}$, and $2 \cdot \sqrt[4]{3}$. Its area is the ratio of two relatively prime positive integers, $m$ and $n$. Find $m + n$.

Let $ABC$ be a non-equilateral triangle and let $R$ be the radius of its circumcircle. The incircle of $ABC$ has $I$ as its centre and is tangent to side $CA$ in point $D$ and to side $CB$ in point $E$. Let $A_1$ be the point on line $EI$ such that $A_1I=R$, with $I$ being between $A_1$ and $E$. Let $B_1$ be the point on line $DI$ such that $B_1I=R$, with $I$ being between $B_1$ and $D$. Let $P$ be the intersection of lines $AA_1$ and $BB_1$. (a) Prove that $P$ belongs to the circumcircle of $ABC$. (b) Let us now also suppose that $AB=1$ and $P$ coincides with $C$. Determine the possible values of the perimeter of $ABC$.

Compute the sum of the irrational solutions of the equation $$\frac{x^2+16x+54}{x^2+11x+35}=\frac{x^2+13x+35}{x^2+14x+54}.$$

The plane is divided into domains by $ n$ straight lines in general position, where $ n \geq 3$. Determine the maximum and minimum possible number of angular domains among them. (We say that $ n$ lines are in general position if no two are parallel and no three are concurrent.)

Four mathletes and two coaches sit at a circular table. How many distinct arrangements are there of these six people if the two coaches sit opposite each other?

A regular hexagon, as shown in the attachment, is dissected into 54 congruent equilateral triangles by parallels to its sides. Within the figure we yield exactly 37 points which are vertices of at least one of those triangles. Those points are enumerated in an arbitrary way. A triangle is called [i]clocky[/i] if running in a clockwise direction from the vertex with the smallest assigned number, we pass a medium number and finally reach the vertex with the highest number. Prove that at least 19 out of 54 triangles are clocky.

Quadrilateral $ABCD$ is circumscribed around a circle with center $I$. Points $M$ and $N$ are the midpoints of diagonals $AC$ and $BD$. Prove that $ABCD$ is cyclic quadrilateral if and only if $IM : AC = IN : BD$. [i]Nikolai Beluhov and Aleksey Zaslavsky[/i]

Prove that $x^{12} - x^9 + x^4 - x + 1 > 0$ for all $x$.

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} \ \ $