The permutohedron of order $3$ is the hexagon determined by points $(1, 2, 3)$, $(1, 3, 2)$, $(2, 1, 3)$, $(2, 3, 1)$, $(3, 1, 2)$, and $(3, 2, 1)$. The pyramid determined by these six points and the origin has a unique inscribed sphere of maximal volume. Determine its radius.

Let $ABC$ be an acute triangle and $\omega$ be its circumcircle. The bisector of $\angle BAC$ intersects $\omega$ at [another point] $M$. Let $P$ be a point on $AM$ and inside $\triangle ABC$. Lines passing $P$ that are parallel to $AB$ and $AC$ intersects $BC$ on $E, F$ respectively. Lines $ME, MF$ intersects $\omega$ at points $K, L$ respectively. Prove that $AM, BL, CK$ are concurrent.

Suppose $ABCD$ is a convex quadrilateral satisfying $AB=BC$, $AC=BD$, $\angle ABD = 80^\circ$, and $\angle CBD = 20^\circ$. What is $\angle BCD$ in degrees?

What is the value of $(625^{\log_{5}{2015}})^{\frac{1}{4}}$? $\textbf{(A) }5\qquad\textbf{(B) }\sqrt[4]{2015}\qquad\textbf{(C) }625\qquad\textbf{(D) }2015\qquad\textbf{(E) }\sqrt[4]{5^{2015}}$

Rein solved a test on mathematics that consisted of questions on algebra, geometry and logic. After checking the results, it occurred that Rein had answered correctly $50\%$ of questions on algebra, $70\%$ of questions on geometry and $80\%$ of questions on logic. Thereby, Rein had answered correctly altogether $62\%$ of questions on algebra and logic, and altogether $74\%$ of questions on geometry and logic. What was the percentage of correctly answered questions throughout all the test by Rein?

The fraction $\frac1{10}$ can be expressed as the sum of two unit fraction in many ways, for example, $\frac1{30}+\frac1{15}$ and $\frac1{60}+\frac1{12}$. Find the number of ways that $\frac1{2007}$ can be expressed as the sum of two distinct positive unit fractions.

Show that $\frac{(x - y)^7 + (y - z)^7 + (z - x)^7 - (x - y)(y - z)(z - x) ((x - y)^4 + (y - z)^4 + (z - x)^4)} {(x - y)^5 + (y - z)^5 + (z - x)^5} \ge 3$ holds for all triples of distinct integers $x, y, z$. When does equality hold?

Larry initially has a one character string that is either `a', `b', `c', or `d'. Every minute, he chooses a character in the string and: [list] [*] if it's an `a' he can replace it with `ac' or `da', [*] if it's a `b' he can replace it with `cb' or `bd', [*] if it's a `c' he can replace it with `cc' or `ba', [*] if it's a `d' he can replace it with `dd' or `ab'. [/list] Larry does the above process for $10$ minutes. Find the number of possible strings he can end up with that are a permutation of `aabbccccddd'. [i]Individual #14[/i]

For every positive real numbers $a$ and $b$ prove the inequality \[\displaystyle \sqrt{ab} \leq \dfrac{1}{3} \sqrt{\dfrac{a^2+b^2}{2}}+\dfrac{2}{3} \dfrac{2}{\dfrac{1}{a}+\dfrac{1}{b}}.\] [i]A. Khabrov[/i]

(a) Let $a$ and $b$ be positive integers such that $M(a, b) = a - \frac1b +b(b + \frac3a)$ is an integer. Prove that $M(a,b)$ is a square. (b) Find nonzero integers $a$ and $b$ such that $M(a,b)$ is a positive integer, but not a square.

Professor Oak is feeding his $100$ Pokémon. Each Pokémon has a bowl whose capacity is a positive real number of kilograms. These capacities are known to Professor Oak. The total capacity of all the bowls is $100$ kilograms. Professor Oak distributes $100$ kilograms of food in such a way that each Pokémon receives a non-negative integer number of kilograms of food (which may be larger than the capacity of the bowl). The [i]dissatisfaction level[/i] of a Pokémon who received $N$ kilograms of food and whose bowl has a capacity of $C$ kilograms is equal to $\lvert N-C\rvert$. Find the smallest real number $D$ such that, regardless of the capacities of the bowls, Professor Oak can distribute food in a way that the sum of the dissatisfaction levels over all the $100$ Pokémon is at most $D$. [i]Oleksii Masalitin, Ukraine[/i]

Let $ A $ be a ring, $ a\in A, $ and let $ n,k\ge 2 $ be two natural numbers such that $ n\vdots\text{char} (A) $ and $ 1+a=a^k. $ Show that the following propositions are true: [b]a)[/b] $ \forall s\in\mathbb{N}\quad \exists p_0,p_1,\ldots ,p_{k-1}\in\mathbb{Z}_{\ge 0}\quad a^s=\sum_{i=0}^{k-1} p_ia^{i} . $ [b]b)[/b] $ \text{ord} (a)\neq\infty . $

$(BUL 3)$ One hundred convex polygons are placed on a square with edge of length $38 cm.$ The area of each of the polygons is smaller than $\pi cm^2,$ and the perimeter of each of the polygons is smaller than $2\pi cm.$ Prove that there exists a disk with radius $1$ in the square that does not intersect any of the polygons.

Prove that the points $ A_1, A_2, \ldots, A_n $ ($ n \geq 7 $) located on the surface of the sphere lie on a circle if and only if the planes tangent to the surface of the sphere at these points have a common point or are parallel to one straight line.

For positive integers $a$ and $b$, $gcd (a, b)$ denote their greatest common divisor and $lcm (a, b)$ their least common multiple. Determine the number of ordered pairs (a,b) of positive integers satisfying the equation $ab + 63 = 20\, lcm (a, b) + 12\, gcd (a,b)$

$N$ lines lie on a plane, no two of which are parallel and no three of which are concurrent. Prove that there exists a non-self-intersecting broken line $A_0A_1A_2A_3...A_N$ with $N$ parts, such that on each of the $N$ lines lies exactly one of the $N$ segments of the line.

In $xyz$ space, find the volume of the solid expressed by $x^2+y^2\leq z\le \sqrt{3}y+1.$

Given the graph $G$ and cycle $C$ in it, we can perform the following operation: add another vertex $v$ to the graph, connect it to all vertices in $C$ and erase all the edges from $C$. Prove that we cannot perform the operation indefinitely on a given graph.

A medial line parallel to the side $AC$ of triangle $ABC$ meets its circumcircle at points at $X$ and $Y$. Let $I$ be the incenter of triangle $ABC$ and $D$ be the midpoint of arc $AC$ not containing $B$.A point $L$ lie on segment $DI$ in such a way that $DL= BI/2$. Prove that $\angle IXL = \angle IYL$.

The geometric mean of any set of $m$ non-negative numbers is the $m$-th root of their product. $\quad (\text{i})\quad$ For which positive integers $n$ is there a finite set $S_n$ of $n$ distinct positive integers such that the geometric mean of any subset of $S_n$ is an integer? $\quad (\text{ii})\quad$ Is there an infinite set $S$ of distinct positive integers such that the geometric mean of any finite subset of $S$ is an integer?

The diagram represents a $ 7$-foot-by-$ 7$-foot floor that is tiled with $ 1$-square-foot black tiles and white tiles. Notice that the corners have white tiles. If a $ 15$-foot-by-$ 15$-foot floor is to be tiled in the same manner, how many white tiles will be needed? [asy]unitsize(10); draw((0,0)--(7,0)--(7,7)--(0,7)--cycle); draw((1,7)--(1,0)); draw((6,7)--(6,0)); draw((5,7)--(5,0)); draw((4,7)--(4,0)); draw((3,7)--(3,0)); draw((2,7)--(2,0)); draw((0,1)--(7,1)); draw((0,2)--(7,2)); draw((0,3)--(7,3)); draw((0,4)--(7,4)); draw((0,5)--(7,5)); draw((0,6)--(7,6)); fill((1,0)--(2,0)--(2,7)--(1,7)--cycle,black); fill((3,0)--(4,0)--(4,7)--(3,7)--cycle,black); fill((5,0)--(6,0)--(6,7)--(5,7)--cycle,black); fill((0,5)--(0,6)--(7,6)--(7,5)--cycle,black); fill((0,3)--(0,4)--(7,4)--(7,3)--cycle,black); fill((0,1)--(0,2)--(7,2)--(7,1)--cycle,black);[/asy] $ \textbf{(A)}\ 49 \qquad \textbf{(B)}\ 57 \qquad \textbf{(C)}\ 64 \qquad \textbf{(D)}\ 96 \qquad \textbf{(E)}\ 126$

Let $\mathbb R_{>0}$ be the set of positive real numbers. Determine all functions $f \colon \mathbb R_{>0} \to \mathbb R_{>0}$ such that \[x \big(f(x) + f(y)\big) \geqslant \big(f(f(x)) + y\big) f(y)\] for every $x, y \in \mathbb R_{>0}$.

In an $2023\times 2023$ grid we fill in numbers $1,2,\cdots,2023^2$ without duplicating. Find the largest integer $M$ such that there exists a way to fill the numbers, satisfying that any two adjacent numbers has a difference at least $M$ (two squares $(x_1,y_1),(x_2,y_2)$ are adjacent if $x_1=x_2$ and $y_1-y_2\equiv \pm1\pmod{2023}$ or $y_1=y_2$ and $x_1-x_2\equiv \pm1\pmod{2023}$). [i]Proposed by chengbilly.[/i]

The large square in the diagram below with sides of length $8$ is divided into $16$ congruent squares. Find the area of the shaded region. [img]https://cdn.artofproblemsolving.com/attachments/6/e/cf828197aa2585f5eab2320a43b80616072135.png[/img]

Let $p$ and $q$ be relatively prime positive odd integers such that $1 < p < q$. Let $A$ be a set of pairs of integers $(a, b)$, where $0 \le a \le p - 1, 0 \le b \le q - 1$, containing exactly one pair from each of the sets $$\{(a, b),(a + 1, b + 1)\}, \{(a, q - 1), (a + 1, 0)\}, \{(p - 1,b),(0, b + 1)\}$$ whenever $0 \le a \le p - 2$ and $0 \le b \le q - 2$. Show that $A$ contains at least $(p - 1)(q + 1)/8$ pairs whose entries are both even. Agnijo Banerjee and Joe Benton, United Kingdom