For each of $k=1,2,3,4,5$ find necessary and sufficient conditions on $a>0$ such that there exists a tetrahedron with $k$ edges length $a$ and the remainder length $1$.

What is the greatest positive integer $m$ such that $ n^2(1+n^2-n^4)\equiv 1\pmod{2^m} $ for all odd integers $n$?

Let $\mathbb{F}_{13} = {\overline{0}, \overline{1}, \cdots, \overline{12}}$ be the finite field with $13$ elements (with sum and product modulus $13$). Find how many matrix $A$ of size $5$ x $5$ with entries in $\mathbb{F}_{13}$ exist such that $$A^5 = I$$ where $I$ is the identity matrix of order $5$

Three numbered tiles are arranged in a tray as shown: [img]https://cdn.artofproblemsolving.com/attachments/d/0/d449364f92b7fae971fd348a82bafd25aa8ea1.jpg[/img] Show that we cannot interchange the $1$ and the $3$ by a sequence of moves where we slide a tile to the adjacent vacant space.

A stage course is attended by $n \ge 4$ students. The day before the final exam, each group of three students conspire against another student to throw him/her out of the exam. Prove that there is a student against whom there are at least $\sqrt[3]{(n-1)(n- 2)} $conspirators.

All positive integers are distributed among two disjoint sets $N_{1}$ and $N_{2}$ such that no difference of two numbers belonging to the same set is a prime greater than 100. Find all such distributions. [i]Proposed by N. Sedrakyan[/i]

Find all positive integer pairs $(a,b)$ that satisfy the equation$$a^2b+ab^2+73=8ab+9a+9b.$$

Consider an equilateral triangle $\vartriangle ABC$ with side length $1$. Let $D$ and $E$ lie on segments $AB$ and $AC$ respectively such that $\angle ADE = 30^o$ and $DE$ is tangent to the incircle of $\vartriangle ABC$. Compute the perimeter of $\vartriangle ADE$.

One of Michael's responsibilities in organizing the family vacation is to call around and find room rates for hotels along the root the Kubik family plans to drive. While calling hotels near the Grand Canyon, a phone number catches Michael's eye. Michael notices that the first four digits of $987-1234$ descend $(9-8-7-1)$ and that the last four ascend in order $(1-2-3-4)$. This fact along with the fact that the digits are split into consecutive groups makes that number easier to remember. Looking back at the list of numbers that Michael called already, he notices that several of the phone numbers have the same property: their first four digits are in descending order while the last four are in ascending order. Suddenly, Michael realizes that he can remember all those numbers without looking back at his list of hotel phone numbers. "Wow," he thinks, "that's good marketing strategy." Michael then wonders to himself how many businesses in a single area code could have such phone numbers. How many $7$-digit telephone numbers are there such that all seven digits are distinct, the first four digits are in descending order, and the last four digits are in ascending order?

Let $ABC$ be a triangle, and let $M$ be the midpoint of side $AB$. If $AB$ is $17$ units long and $CM$ is $8$ units long, find the maximum possible value of the area of $ABC$.

Find all funtions $f:\mathbb R\to\mathbb R$ such that: $$f(xy-1)+f(x)f(y)=2xy-1$$ for all $x,y\in \mathbb{R}$.

Let $ABC$ be a triangle with $AB=5$, $BC=7$, $CA=8$, and circumcircle $\omega$. Let $P$ be a point inside $ABC$ such that $PA:PB:PC=2:3:6$. Let rays $\overrightarrow{AP}$, $\overrightarrow{BP}$, and $\overrightarrow{CP}$ intersect $\omega$ again at $X$, $Y$, and $Z$, respectively. The area of $XYZ$ can be expressed in the form $\dfrac{p\sqrt q}{r}$ where $p$ and $r$ are relatively prime positive integers and $q$ is a positive integer not divisible by the square of any prime. What is $p+q+r$? [i]Proposed by James Lin[/i]

There is a unique differentiable function $f$ from $\mathbb{R}$ to $\mathbb{R}$ satisfying $f(x) + (f(x))^3 = x + x^7$ for all real $x.$ The derivative of $f(x)$ at $x = 2$ can be expressed as a common fraction $a/b.$ Compute $a + b.$

Two squirrels, Bushy and Jumpy, have collected $2021$ walnuts for the winter. Jumpy numbers the walnuts from 1 through $2021$, and digs $2021$ little holes in a circular pattern in the ground around their favourite tree. The next morning Jumpy notices that Bushy had placed one walnut into each hole, but had paid no attention to the numbering. Unhappy, Jumpy decides to reorder the walnuts, and Bushy decides to interfere with Jumpy. The two take turns to reorder the walnuts. Each time, Bushy chooses $1232$ walnuts and reorders them and then Jumpy chooses $n$ walnuts to reorder. Find the least positive integer $n$ such that whatever Bushy does, Jumpy can ensure that the $i$th hole from the left has the $i$th walnut [i]Ift0501[/i]

Twenty cubes have been coloured in the following way: There are two red faces opposite each other, two blue faces opposite each other and two green faces opposite each other. The cubes have been glued together as shown in the figure. Two faces that are glued together always have the same colour. The figure shows the colours of some of the faces. Which colours are possible for the face marked with the symbol $\times$? [img]https://cdn.artofproblemsolving.com/attachments/8/2/6127db5bfdce7a749d730fe3626499582f62ba.png[/img]

There are $n$ matches lying on a table, forming $n$ one-match piles. Adam wants to combine them into a single pile of $n$ matches. He will do this using $n-1$ operations, each of which consists of combining two piles into one. Adam has made a deal with Bartek that every time he combines a pile of $a$ matches with a pile of $b$ matches, he will receive $a\cdot b$ candies from Bartek. What is the greatest number of candies that Adam can receive after performing $n-1$ operations? Justify your answer.

Three ants are at three corners of a rectangle. It is assumed that each ant moves only when the other two are stopped and always parallel to the line defined by them. Will be is it possible that the three ants are simultaneously at midpoints on the sides of the rectangle?

Let $\ell$ and $m$ be two non-coplanar lines in space, and let $P_1$ be a point on $\ell.$ Let $P_2$ be the point on $,m$ closest to $P_1,$ $P_3$ be the point on $\ell$ closest to $P_3,$ $P_4$ be the point on $m$ closest to $P_3,$ and $P_5$ be the point on $\ell$ closest to $P_4.$ Given that $P_1P_2=5, P_2P_3=3,$ and $P_3P_4=2,$ compute $P_4P_5.$

A hexagon $A_1A_2A_3A_4A_5A_6$ has no four concyclic vertices, and its diagonals $A_1A_4$, $A_2A_5$ and $A_3A_6$ concur. Let $l_i $ be the radical axis of circles $A_iA_{i+1}A_{i-2} $ and $A_iA_{i-1}A_{i+2} $ (the points $A_i $ and $A_{i+6} $ coincide). Prove that $l_i, i=1,\cdots,6$, concur.

Given polynomial $P(x) = a_{0}x^{n}+a_{1}x^{n-1}+\dots+a_{n-1}x+a_{n}$. Put $m=\min \{ a_{0}, a_{0}+a_{1}, \dots, a_{0}+a_{1}+\dots+a_{n}\}$. Prove that $P(x) \ge mx^{n}$ for $x \ge 1$. [i]A. Khrabrov [/i]

Prove that if the family of integral curves of the differential equation $$ \frac{dy}{dx} +p(x) y= q(x),$$ where $p(x) q(x) \ne 0$, is cut by the line $x=k$ the tangents at the points of intersection are concurrent.

Start with a positive integer. Double it, subtract $4,$ halve it, then subtract the original integer to get $x.$ What is $x?$

Let $\alpha \leq -2$ be an integer. Prove that for every pair $(\beta_0, \beta_1)$ of integers there exists a uniquely determined sequence $0\leq q_0, \ldots, q_k<\alpha ^ 2 - \alpha$ of integers, such that $q_k\neq 0$ if $(\beta_0, \beta 1)\neq (0,0)$ and $$\beta_i=\sum_{j=0}^k q_j(\alpha - i)^j,\text{ for }i=0,1$$

Let $n \ge 2$ be an integer. There are $n$ houses in a town. All distances between pairs of houses are different. Every house sends a visitor to the house closest to it. Find all possible values of $n$ (with full justification) for which we can design a town with $n$ houses where every house is visited.

Shown is a segment of length $19$, marked with $20$ points dividing the segment into $19$ segments of length $1$. Draw $20$ semicircular arcs, each of whose endpoints are two of the $20$ marked points, satisfying all of the following conditions: [list=1] [*] When the drawing is complete, there will be: [list] [*] $8$ arcs with diameter $1$, [/*] [*] $6$ arcs with diameter $3$, [/*] [*] $4$ arcs with diameter $5$, [/*] [*] $2$ arcs with diameter $7$. [/*] [/list] [/*] [*] Each marked point is the endpoint of exactly two arcs: one above the segment and one below the segment. [/*] [*] No two distinct arcs can intersect except at their endpoints. [/*] [*] No two distinct arcs can connect the same pair of points. (That is, there can be no full circles.) [/*] [/list] Three arcs have already been drawn for you. [asy] size(10cm); draw((0,0)--(19,0)); for(int i=0;i<20;++i){ dot((i,0)); } draw((7,0){down}..{up}(8,0)); draw((12,0){down}..{up}(13,0)); draw((5,0){up}..{down}(10,0)); [/asy]