Consider an arrangement of tokens in the plane, not necessarily at distinct points. We are allowed to apply a sequence of moves of the following kind: Select a pair of tokens at points $A$ and $B$ and move both of them to the midpoint of $A$ and $B$.
We say that an arrangement of $n$ tokens is collapsible if it is possible to end up with all $n$ tokens at the same point after a finite number of moves. Prove that every arrangement of $n$ tokens is collapsible if and only if $n$ is a power of 2.
Let five points on a circle be labelled $A$, $B$, $C$, $D$, and $E$ in clockwise order. Assume $A E = D E$ and let $P$ be the intersection of $A C$ and $B D$. Let $Q$ be the point on the line through $A$ and $B$ such that $A$ is between $B$ and $Q$ and $A Q = D P$. Similarly, let $R$ be the point on the line through $C$ and $D$ such that $D$ is between $C$ and $R$ and $D R = A P$. Prove that $P E$ is perpendicular to $Q R$.
Two positive integers $a$ and $b$ are prime-related if $a = p b$ or $b = p a$ for some prime $p$. Find all positive integers $n$, such that $n$ has at least three divisors, and all the divisors can be arranged without repetition in a circle so that any two adjacent divisors are prime-related.
Find all polynomials $p(x)$ with real coefficients that have the following property: There exists a polynomial $q(x)$ with real coefficients such that
$$
p(1)+p(2)+p(3)+\cdots+p(n)=p(n) q(n)
$$
for all positive integers $n$.
Let $k$ be a given even positive integer. Sarah first picks a positive integer $N$ greater than $1$ and proceeds to alter it as follows: every minute, she chooses a prime divisor $p$ of the current value of $N$, and multiplies the current $N$ by $p^{k}-p^{-1}$ to produce the next value of $N$. Prove that there are infinitely many even positive integers $k$ such that, no matter what choices Sarah makes, her number $N$ will at some point be divisible by $2018$.
Prove that
$$
3abc + a + b + c \ge 2(ab + bc + ca)
$$
holds for all real numbers $a, b, c \ge 1$.
Determine all cases for which the equality is obtained.
One hundred people attended the party, some of whom were previously acquainted. All acquaintances were mutual and no new were made during the party.
A gong rang $100$ times during the party. After the first sounding of the gong, all the people not acquainted with anyone left the party. After the second sounding of the gong, all the people with exactly one acquaintance (among the remaining people) left. It continued all the way, meaning that after the $k$th sounding of the gong, all the people acquainted with exactly $k-1$ remaining people left the party ($k = 1, \dots, 100$).
At the end of the party, there were $n$ people still present. Find all possible values of $n$.
Let $ABCD$ be a square, and let $k$ be the circle centred at $B$ passing through $A$, $C$ and the point $T$ inside the square. Tangent on $k$ at $T$ intersects the segments $\overline{CD}$ and $\overline{DA}$ at $E$ and $F$, respectively. Let $G$ and $H$ be the intersections of the lines $BE$ and $BF$ with the segment $\overline{AC}$, respectively.
Prove that the lines $BT$, $EH$ and $FG$ are passing through the same point.
Let $a$, $b$ and $c$ be positive real numbers such that $a + b + c = 2$. Prove that
$$
\frac{(a-1)^2}{b} + \frac{(b-1)^2}{c} + \frac{(c-1)^2}{a} \ge \frac{1}{4} \left( \frac{a^2+b^2}{a+b} + \frac{b^2+c^2}{b+c} + \frac{c^2+a^2}{c+a} \right).
$$
Let $n$ be a positive integer. A good word is a sequence of $3n$ letters, in which each of the letters $A$, $B$ and $C$ appears exactly $n$ times. Prove that for every good word $X$ there exists a good word $Y$ such that $Y$ cannot be obtained from $X$ by swapping neighbouring letters fewer than $\frac{3}{2}n^2$ times. (IMO Shortlist 2017)
Let $k$ be a circle centred at $O$. Let $\overline{AB}$ be a chord of that circle and $M$ its midpoint. Tangents on $k$ at points $A$ and $B$ intersect at $T$. The line $\ell$ goes through $T$, intersects the shorter arc $\overarc{AB}$ at the point $C$ and the longer arc $\overarc{AB}$ at the point $D$, so that $|BC| = |BM|$.
Prove that the circumcentre of the triangle $ADM$ is the reflection of $O$ across the line $AD$.
Let $n$ be a positive integer. Points $A_1, A_2, \dots, A_n$ are located on the inside of a circle, and points $B_1, B_2, \dots, B_n$ are on the circle, so that the segments $\overline{A_1B_1}$, $\overline{A_2B_2}$, \dots, $\overline{A_nB_n}$ are mutually disjoint. A grasshopper can jump from point $A_i$ to point $A_j$ (for $i, j \in \{1, \dots, n\}, i \neq j$) if and only if the segment $\overline{A_iA_j}$ does not go through any of the inner points of the segments $\overline{A_1B_1}, \overline{A_2B_2}, \dots, \overline{A_nB_n}$.
Show that the grasshopper can take a series of jumps to get from any point $A_i$ to any point $A_j$. (Russia 1994)
Let $\triangle ABC$ be an acute-angled triangle such that $|AB| \lt |AC|$. Point $D$ is the midpoint of the shorter arc $\widehat{BC}$ of the circumcircle of the triangle $ABC$. Point $I$ is the incentre of the triangle $ABC$, and point $J$ is the reflection of $I$ across the line $BC$. Line $DJ$ intersects the circumcircle of the triangle $ABC$ at the point $E$ which lies on the shorter arc $\widehat{AB}$.
Prove that $|AI| = |IE|$ holds. (Romania 2017)
Determine all triples $(x, y, z)$ of real numbers that satisfy
$$
\begin{align*}
x + y - z &= -1 \\
x^2 - y^2 + z^2 &= 1 \\
-x^3 + y^3 + z^3 &= -1.
\end{align*}
$$
Let $D_0, D_1, \dots, D_{2018}$ be points on the segment $\overline{AB}$ such that $D_0 = A$, $D_{2018} = B$ and
$$
|D_0 D_1| = |D_1 D_2| = \dots = |D_{2017} D_{2018}|.
$$
If $C$ is a point such that $\angle BCA = 90^\circ$, prove that
$$
|CD_0|^2 + |CD_1|^2 + \dots + |CD_{2018}|^2 = |AD_1|^2 + |AD_2|^2 + \dots + |AD_{2018}|^2.
$$
(Ivan Krijan)
Let $p$ be a prime number, and let $n \ge p-1$ be an integer. If $np+1$ is a perfect square, prove that $n+1$ can be represented as a sum of squares of exactly $p$ positive integers.
Let $\triangle ABC$ be a triangle such that $\angle CAB = 2\angle ABC$. A point $D$ is given in the interior of the triangle $\triangle ABC$, such that $|AD| = |BD|$ and $|CD| = |AC|$. Prove that $\angle ACB = 3\angle DCB$.