For each positive integer $k$, let $g(k)$ be the maximum possible number of points in the plane such that pairwise distances between these points have only $k$ different values. Prove that there exists $k$ such that $g(k) > 2k + 2020$.

The following method of approximate measurement is known for distances. Suppose, for example, that the observer is on the river bank at point $C$ in order to measure its width. To do this, he fixes point $A$ on the opposite bank so that the angle between the shoreline and the line $CA$ is close to the line. Then the observer pulls forward the right hand with the raised thumb, closes left eye and aligns the raised finger with point $A$. Next, opens the left eye, closes right and estimates the distance between the point on the opposite bank to which the finger points, and point $A$. Multiply this distance by $10$ and get the approximate value of the distance to point $A$, ie the width of the river. Justify this method of measuring distance. [hide=original wording]Відомий наступний спосіб наближеного вимірювання відстані. Нехай, наприклад, спостерігач знаходиться на березі річки у точці C і має на меті виміряти її ширину. Для цього він фіксує точку A на протилежному березі так, щоб кут між лінією берега і прямою CA був близьким до прямого. Потім спостерігач витягує вперед праву руку з піднятим вгору великим пальцем, заплющує ліве око і суміщає піднятий палець з точкою A. Далі, відкриває ліве око, заплющує праве і оцінює відстань між точкою на протилежному березі, на яку вказує палець, і точкою A. Цю відстань множить на 10 і отримує наближене значення відстані до точки A, тобто ширини річки. Обґрунтуйте цей спосіб вимірювання відстані.[/hide]

The sum of the squares of five real numbers $a_1, a_2, a_3, a_4, a_5$ equals $1$. Prove that the least of the numbers $(a_i - a_j)^2$, where $i, j = 1, 2, 3, 4,5$ and $i \neq j$, does not exceed $\frac{1}{10}.$

Let $D$ be a point inside a triangle $ABC$ such that $|AC| -|AD| \geq 1$ and $|BC|- |BD| \geq 1.$ Prove that for any point $E$ on the segment $AB$, we have $|EC| -|ED| \geq 1.$

The $ABCDA_1B_1C_1D_1$ cube has unit length edges. Find the distance between two circumferences, one of those is inscribed into the $ABCD$ base, and another comes through points $A,C$ and $B_1$ .

Pete has marked several (three or more) points in the plane such that all distances between them are different. A pair of marked points $A,B$ will be called unusual if $A$ is the furthest marked point from $B$, and $B$ is the nearest marked point to $A$ (apart from $A$ itself). What is the largest possible number of unusual pairs that Pete can obtain?

In the triangle $ABC, I$ is the center of the inscribed circle, point $M$ lies on the side of $BC$, with $\angle BIM = 90^o$. Prove that the distance from point $M$ to line $AB$ is equal to the diameter of the circle inscribed in triangle $ABC$

Consider the equilateral triangle $ABC$ with center of gravity $G$. Let $M$ be a point, inside the triangle and $O$ be the midpoint of the segment $MG$. Three segments go through $M$, each parallel to one side of the triangle and with the ends on the other two sides of the given triangle. a) Show that $O$ is at equal distances from the midpoints of the three segments considered. b) Show that the midpoints of the three segments are the vertices of an equilateral triangle.

For a set $S$ of at least $3$ points in the plane, let $d_{\text{min}}$ denote the minimal distance between two different points in $S$ and $d_{\text{max}}$ the maximal distance between two different points in $S$. For a real $c>0$, a set $S$ will be called $c$-[i]balanced[/i] if \[\frac{d_{\text{max}}}{d_{\text{min}}}\leq c|S|\] Prove that there exists a real $c>0$ so that for every $c$-balanced set of points $S$, there exists a triangle with vertices in $S$ that contains at least $\sqrt{|S|}$ elements of $S$ in its interior or on its boundary.

The quadrilateral $ABCD$ is inscribed in a circle of radius $1$, so that $AB$ is a diameter of the circumference and $CD = 1$. A variable point $X$ moves along the semicircle determined by $AB$ that does not contain $C$ or $D$. Determine the position of $X$ for which the sum of the distances from $X$ to lines $BC, CD$ and $DA$ is maximum.

All faces of the tetrahedron $ABCD$ are acute-angled. Take a point $X$ in the interior of the segment $AB$, and similarly $Y$ in $BC, Z$ in $CD$ and $T$ in $AD$. [b]a.)[/b] If $\angle DAB+\angle BCD\ne\angle CDA+\angle ABC$, then prove none of the closed paths $XYZTX$ has minimal length; [b]b.)[/b] If $\angle DAB+\angle BCD=\angle CDA+\angle ABC$, then there are infinitely many shortest paths $XYZTX$, each with length $2AC\sin k$, where $2k=\angle BAC+\angle CAD+\angle DAB$.

Certain king of a certain state wants to build $n$ cities and $n-1$ roads, connecting them to provide a possibility to move from every city to every city. (Each road connects two cities, the roads do not intersect, and don't come through another city.) He wants also, to make the shortests distances between the cities, along the roads, to be $1,2,3,...,n(n-1)/2$ kilometres. Is it possible for a) $n=6$ b) $n=1986$ ?

Let $n$ be a positive integer. Define a chameleon to be any sequence of $3n$ letters, with exactly $n$ occurrences of each of the letters $a, b,$ and $c$. Define a swap to be the transposition of two adjacent letters in a chameleon. Prove that for any chameleon $X$ , there exists a chameleon $Y$ such that $X$ cannot be changed to $Y$ using fewer than $3n^2/2$ swaps.

$P$ is a point inside the triangle $ABC$. The perpendicular distances from $P$ to the three sides have product $p$. Show that $p \le \frac{ 8 S^3}{27abc}$, where $S =$ area $ABC$ and $a, b, c$ are the sides. Prove a similar result for a tetrahedron.

There are $N$ points in the plane such that the [b]total number[/b] of pairwise distances of these $N$ points is at most $n$. Prove that $N \le (n + 1)^2$.

Give a square of side $1$. Show that for each finite set of points of the sides of the square you can find a vertex of the square with the following property: the arithmetic mean of the squares of the distances from this vertex to the points of the set is greater than or equal to $3/4$.

Let $n$ distinct points in the plane be given. Prove that fewer than $2 n^{3 \slash 2}$ pairs of them are a unit distance apart.

Two circles $\omega_1$ and $\omega_2$ with centers $O_1$ and $O_2$ meet at points $A$ and $B$. Points $C$ and $D$ on $\omega_1$ and $\omega_2$, respectively, lie on the opposite sides of the line $AB$ and are equidistant from this line. Prove that $C$ and $D$ are equidistant from the midpoint of $O_1O_2$.

The distances from the centroid $G$ of a triangle $ABC$ to its sides $a,b,c$ are denoted $g_a,g_b,g_c$ respectively. Let $r$ be the inradius of the triangle. Prove that: a) $g_a,g_b,g_c \ge \frac{2}{3}r$ b) $g_a+g_b+g_c \ge 3r$

A clueless ant makes the following route: starting at point $ A $ goes $ 1$ cm north, then $ 2$ cm east, then $ 3$ cm south, then $ 4$ cm west, immediately $ 5$ cm north, continues $ 6$ cm east, and so on, finally $ 41$ cm north and ends in point $ B $. Calculate the distance between $ A $ and $ B $ (in a straight line).

Lyuba, Tanya, Lena and Ira ran across a flat field. At some point it turned out that among the pairwise distances between them there are distances of $1, 2, 3, 4$ and $5$ meters, and there are no other distances. Give an example of how this could be.

Can the distances from a certain point on the plane to the vertices of a certain square be equal to $1, 4, 7$, and $8$ ?

A village is constructed in the form of a square, consisting of $9$ blocks , each of side length $\ell$, in a $3 \times 3$ formation . Each block is bounded by a bitumen road . If we commence at a corner of the village, what is the smallest distance we must travel along bitumen roads , if we are to pass along each section of bitumen road at least once and finish at the same corner? (Muscovite folklore)

Let $n$ be a positive integer. Define a chameleon to be any sequence of $3n$ letters, with exactly $n$ occurrences of each of the letters $a, b,$ and $c$. Define a swap to be the transposition of two adjacent letters in a chameleon. Prove that for any chameleon $X$ , there exists a chameleon $Y$ such that $X$ cannot be changed to $Y$ using fewer than $3n^2/2$ swaps.

Can three persons, having one double motorcycle, overcome the distance of $70$ km in $3$ hours? Pedestrian speed is $5$ km / h and motorcycle speed is $50$ km / h.