Let $S=\{s_{0}, \ldots, s_{n}\}$ be a finite set of integers, and define $S+k=\{s_{0}+k, \ldots, s_{n}+k\}$. We say that $S$ and $T$ are equivalent, written $S \sim T$, if $T=S+k$ for some $k$. Given a (possibly infinite) set of integers $A$, we say that $S$ tiles $A$ if $A$ can be partitioned into subsets equivalent to $S$. Such a partition is called a tiling of $A$ by $S$.
Suppose that $S$ tiles the set of odd prime numbers. Prove that $S$ has only one element.
Let $S=\{s_{0}, \ldots, s_{n}\}$ be a finite set of integers, and define $S+k=\{s_{0}+k, \ldots, s_{n}+k\}$. We say that $S$ and $T$ are equivalent, written $S \sim T$, if $T=S+k$ for some $k$. Given a (possibly infinite) set of integers $A$, we say that $S$ tiles $A$ if $A$ can be partitioned into subsets equivalent to $S$. Such a partition is called a tiling of $A$ by $S$.
Find all sets $S$ with minimum element $1$ that tile $A=\{1, \ldots, 12\}$.
The volume of a cube (in cubic inches) plus three times the total length of its edges (in inches) is equal to twice its surface area (in square inches). How many inches long is its long diagonal?
Five people of different heights are standing in line from shortest to tallest. As it happens, the tops of their heads are all collinear; also, for any two successive people, the horizontal distance between them equals the height of the shorter person. If the shortest person is 3 feet tall and the tallest person is 7 feet tall, how tall is the middle person, in feet?
One hundred people are in line to see a movie. Each person wants to sit in the front row, which contains one hundred seats, and each has a favorite seat, chosen randomly and independently. They enter the row one at a time from the far right. As they walk, if they reach their favorite seat, they sit, but to avoid stepping over people, if they encounter a person already seated, they sit to that person's right. If the seat furthest to the right is already taken, they sit in a different row. What is the most likely number of people that will get to sit in the first row?
Triangle $ABC$ has $AB=1$, $BC=\sqrt{7}$, and $CA=\sqrt{3}$. Let $\ell_1$ be the line through $A$ perpendicular to $AB$, $\ell_2$ the line through $B$ perpendicular to $AC$, and $P$ the point of intersection of $\ell_1$ and $\ell_2$. Find $PC$.
Let $k$ be an integer such that $k \mid a$ and $k \mid b$. Prove that if an $m \times n$ rectangle is $(a, b)$-tileable, then $2k \mid m$ or $2k \mid n$.
Find
$$
\{\ln (1+e)\}+\left\{\ln \left(1+e^{2}\right)\right\}+\left\{\ln \left(1+e^{4}\right)\right\}+\left\{\ln \left(1+e^{8}\right)\right\}+\cdots,
$$
where $\{x\}=x-\lfloor x\rfloor$ denotes the fractional part of $x$.
In triangle $ABC$, $AC = 3 AB$. Let $AD$ bisect angle $A$ with $D$ lying on $BC$, and let $E$ be the foot of the perpendicular from $C$ to $AD$. Find $[ABD]/[CDE]$. (Here, $[XYZ]$ denotes the area of triangle $XYZ$.)
In plane, let be given an octagon $A_1A_2A_3A_4A_5A_6A_7A_8$ such that no three of its diagonals are concurrent. Each point of intersection of every two diagonals of the octagon is called a cross.
Consider the convex quadrilaterals whose vertices are vertices of the given octagon. Each such quadrilateral is called a subquadrilateral.
Find the least positive integer $n$ such that one can colour $n$ crosses so that for every $i, k \in \{1, 2, 3, 4, 5, 6, 7, 8\}$ and $i \ne k$, if we denote by $s(i, k)$ the number of
subquadrilaterals admitting $A_i, A_k$ as vertices and admitting a coloured cross as the point of intersection of the diagonals then the values $s(i, k)$ are all equal.
In plane, let be given a fixed circle $(O)$ with radius $R$, two fixed points $A$, $B$ on the circle $(O)$ such that $A$, $B$, $O$ are not collinear.
Let $C$ be a point on $(O)$ distinct from $A$ and $B$. Construct the circle $(O_1)$ passing through $A$, touching the line $BC$ at $C$, construct the circle $(O_2)$ passing through $B$, touching the line $AC$ at $C$. The circle $(O_1)$ and $(O_2)$ intersect again at $D$ distinct from $C$.
Prove that
$$
i) \, CD \le R;$$
$$ii) \text{The line } CD \text{passes through a fixed point when} C \text{moves on} (O) \text{ so that } C \text{ does not coincide with } A \text{ and } B.$$
($(O)$ denotes a circle with center $O$.)
Find the least value and the greatest value of the expression
$$
P = x + y
$$
where $x, y$ are real numbers satisfying the condition
$$
x - 3\sqrt{x + 1} = 3\sqrt{y + 2} - y.
$$
Find all functions $f$ defined on $\mathbb{R}$, taking values in $\mathbb{R}$ so that
$$
f(f(x - y)) = f(x)f(y) - f(x) + f(y) - xy
$$
for all real numbers $x, y$.