The variables $a,b,c,d,$ traverse, independently from each other, the set of positive real values. What are the values which the expression \[ S= \frac{a}{a+b+d} + \frac{b}{a+b+c} + \frac{c}{b+c+d} + \frac{d}{a+c+d} \] takes?

Let $ABC$ be a triangle, let $O$ be its circumcenter, and let $H$ be its orthocenter. Let $P$ be a point on the segment $OH$. Prove that $6r\leq PA+PB+PC\leq 3R$, where $r$ is the inradius and $R$ the circumradius of triangle $ABC$. [b]Moderator edit:[/b] This is true only if the point $P$ lies inside the triangle $ABC$. (Of course, this is always fulfilled if triangle $ABC$ is acute-angled, since in this case the segment $OH$ completely lies inside the triangle $ABC$; but if triangle $ABC$ is obtuse-angled, then the condition about $P$ lying inside the triangle $ABC$ is really necessary.)

Five circles in a row are each labeled with a positive integer. As shown in the diagram, each circle is connected to its adjacent neighbor(s). The integers must be chosen such that the sum of the digits of the neighbor(s) of a given circle is equal to the number labeling that point. In the example, the second number $23 = (1+8)+(5+9)$, but the other four numbers do not have the needed value. [img]https://services.artofproblemsolving.com/download.php?id=YXR0YWNobWVudHMvMi9lL2M2MzVkMmMyYTRlZjliNWEzYWNkOTM2OGVmY2NkOGZmOWVkN2VmLnBuZw==&rn=MjAxMSBCQU1PIDIucG5n[/img] What is the smallest possible sum of the five numbers? How many possible arrangements of the five numbers have this sum? Justify your answers.

Let $A$ be a $3 \times 9$ matrix. All elements of $A$ are positive integers. We call an $m\times n$ submatrix of $A$ "ox" if the sum of its elements is divisible by $10$, and we call an element of $A$ "carboxylic" if it is not an element of any "ox" submatrix. Find the largest possible number of "carboxylic" elements in $A$.

1. Given $n \geq 1$, let $A_{n}$ denote the set of the first $n$ positive integers. We say that a bijection $f: A_{n} \rightarrow A_{n}$ has a hump at $m \in A_{n} \backslash\{1, n\}$ if $f(m)>f(m+1)$ and $f(m)>f(m-1)$. We say that $f$ has a hump at 1 if $f(1)>f(2)$, and $f$ has a hump at $n$ if $f(n)>f(n-1)$. Let $P_{n}$ be the probability that a bijection $f: A_{n} \rightarrow A_{n}$, when selected uniformly at random, has exactly one hump. For how many positive integers $n \leq 2020$ is $P_{n}$ expressible as a unit fraction?

Let the number of ways for a rook to return to its original square on a $4\times 4$ chessboard in 8 moves if it starts on a corner be $k$. Find the number of positive integers that are divisors of $k$. A "move" counts as shifting the rook by a positive number of squares on the board along a row or column. Note that the rook may return back to its original square during an intermediate step within its 8-move path. [i]Proposed by Bradley Guo[/i]

Prove that the volume of a tetrahedron inscribed in a right circular cylinder of volume $1$ does not exceed $\frac{2}{3 \pi}.$

Let $f : P\to P$ be a bijective map from a plane $P$ to itself such that: (i) $f (r)$ is a line for every line $r$, (ii) $f (r) $ is parallel to $r$ for every line $r$. What possible transformations can $f$ be?

Let $ABCD$ be a non-isosceles trapezoid. Define a point $A1$ as intersection of circumcircle of triangle $BCD$ and line $AC$. (Choose $A_1$ distinct from $C$). Points $B_1, C_1, D_1$ are defined in similar way. Prove that $A_1B_1C_1D_1$ is a trapezoid as well.

On circle $k$ there are clockwise points $A$, $B$, $C$, $D$ and $E$ such that $\angle ABE = \angle BEC = \angle ECD = 45^{\circ}$. Prove that $AB^2 + CE^2 = BE^2 + CD^2$

A traffic light runs repeatedly through the following cycle: green for $ 30$ seconds, then yellow for $ 3$ seconds, and then red for $ 30$ seconds. Leah picks a random three-second time interval to watch the light. What is the probability that the color changes while she is watching? $ \textbf{(A)}\ \frac {1}{63}\qquad \textbf{(B)}\ \frac {1}{21}\qquad \textbf{(C)}\ \frac {1}{10}\qquad \textbf{(D)}\ \frac {1}{7}\qquad \textbf{(E)}\ \frac {1}{3}$

Compute the sum of all positive integers $50 \leq n \leq 100$ such that $2n+3 \nmid 2^{n!}-1$.

Let $P$ be a point outside a circle $\omega$. The tangents from $P$ to $\omega$ are drawn touching $\omega$ at points $A$ and $B$. Let $M$ and $N$ be the midpoints of $AP$ and $AB$, respectively. Line $MN$ is extended to cut $\omega$ at $C$ so that $N$ lies between $M$ and $C$. Line $PC$ intersects $\omega$ again at $D$, and lines $ND$ and $PB$ intersect at $O$. Prove that $MNOP$ is a rhombus.

Find the natural numbers $ n $ which have the property that $$ 2011=\left| \mathbb{Q}\cap\bigcup_{k=1}^n\left\{ x\in\mathbb{R} | 1+k^2x^2=2k\left( x-\lfloor x\rfloor \right) \right\} \right| . $$ [i]Marian Teler[/i]

Define a function $f$ on the positive integers as follows: $f(n) = m$, where $m$ is the least positive integer such that $n$ is a factor of $m^2$. Find the smallest integer $M$ such that $\sqrt{M}$ is both a product of prime numbers, of which there are at least $3$, and a factor of $$\sum_{ d|M} f(d),$$ the sum of $f(d)$ for all positive integers $d$ that divide $M$.

Let $n>1$ be a positive integer and $\mathcal S$ be the set of $n^{\text{th}}$ roots of unity. Suppose $P$ is an $n$-variable polynomial with complex coefficients such that for all $a_1,\ldots,a_n\in\mathcal S$, $P(a_1,\ldots,a_n)=0$ if and only if $a_1,\ldots,a_n$ are all different. What is the smallest possible degree of $P$? [i]Adam Ardeishar and Michael Ren[/i]