Navvye/Olympiad-Problems-CSV huggingface dataset:
1994 Baltic Way
Show that for any integer $a\ge 5$ there exist integers $b$ and $c$, $c\ge b\ge a$, such that $a,b,c$ are the lengths of the sides of a right-angled triangle.
1994 Baltic Way
Find all pairs of positive integers $(a,b)$ such that $2^a+3^b$ is the square of an integer.
1994 Baltic Way
How many positive integers satisfy the following three conditions:



a) All digits of the number are from the set $\{1,2,3,4,5\}$;

b) The absolute value of the difference between any two consecutive digits is $1$;

c) The integer has $1994$ digits?
1994 Baltic Way
Let $NS$ and $EW$ be two perpendicular diameters of a circle $\mathcal{C}$. A line $\ell$ touches $\mathcal{C}$ at point $S$. Let $A$ and $B$ be two points on $\mathcal{C}$, symmetric with respect to the diameter $EW$. Denote the intersection points of $\ell$ with the lines $NA$ and $NB$ by $A'$ and $B'$, respectively. Show that $|SA'|\cdot |SB'|=|SN|^2$.
1994 Baltic Way
The inscribed circle of the triangle $A_1A_2A_3$ touches the sides $A_2A_3,A_3A_1,A_1A_2$ at points $S_1,S_2,S_3$, respectively. Let $O_1,O_2,O_3$ be the centres of the inscribed circles of triangles $A_1S_2S_3, A_2S_3S_1,A_3S_1S_2$, respectively. Prove that the straight lines $O_1S_1,O_2S_2,O_3S_3$ intersect at one point.
1994 Baltic Way
Find the smallest number $a$ such that a square of side $a$ can contain five disks of radius $1$, so that no two of the disks have a common interior point.
1994 Baltic Way
Let $\alpha,\beta,\gamma$ be the angles of a triangle opposite to its sides with lengths $a,b,c$ respectively. Prove the inequality

$$a\left(\frac{1}{\beta}+\frac{1}{\gamma}\right)+b\left(\frac{1}{\gamma}+\frac{1}{\alpha}\right)+c\left(\frac{1}{\alpha}+\frac{1}{\beta}\right)\ge2\left(\frac{a}{\alpha}+\frac{b}{\beta}+\frac{c}{\gamma}\right)$$
1994 Baltic Way
Does there exist a triangle such that the lengths of all its sides and altitudes are integers and its perimeter is equal to $1995$?
1994 Baltic Way
The Wonder Island is inhabited by Hedgehogs. Each Hedgehog consists of three segments of unit length having a common endpoint, with all three angles between them $120^{\circ}$. Given that all Hedgehogs are lying flat on the island and no two of them touch each other, prove that there is a finite number of Hedgehogs on Wonder Island.
1994 Baltic Way
In a certain kingdom, the king has decided to build $25$ new towns on $13$ uninhabited islands so that on each island there will be at least one town. Direct ferry connections will be established between any pair of new towns which are on different islands. Determine the least possible number of these connections.
1994 Baltic Way
There are $n>2$ lines given in the plane. No two of the lines are parallel and no three of them intersect at one point. Every point of intersection of these lines is labelled with a natural number between $1$ and $n-1$. Prove that, if and only if $n$ is even, it is possible to assign the labels in such a way that every line has all the numbers from $1$ to $n-1$ at its points of intersection with the other $n-1$ lines.
1994 Baltic Way
The Wonder Island Intelligence Service has $16$ spies in Tartu. Each of them watches on some of his colleagues. It is known that if spy $A$ watches on spy $B$, then $B$ does not watch on $A$. Moreover, any $10$ spies can numbered in such a way that the first spy watches on the second, the second watches on the third and so on until the tenth watches on the first. Prove that any $11$ spies can also be numbered is a similar manner.
1994 Baltic Way
An equilateral triangle is divided into $9000000$ congruent equilateral triangles by lines parallel to its sides. Each vertex of the small triangles is coloured in one of three colours. Prove that there exist three points of the same colour being the vertices of a triangle with its sides parallel to the lines of the original triangle.
1993 Baltic Way
$a_1a_2a_3$ and $a_3a_2a_1$ are two three-digit decimal numbers, with $a_1$ and $a_3$ different non-zero digits. Squares of these numbers are five-digit numbers $b_1b_2b_3b_4b_5$ and $b_5b_4b_3b_2b_1$ respectively. Find all such three-digit numbers.
1993 Baltic Way
Do there exist positive integers $a>b>1$ such that for each positive integer $k$ there exists a positive integer $n$ for which $an+b$ is a $k$-th power of a positive integer?
1993 Baltic Way
Let’s call a positive integer interesting if it is a product of two (distinct or equal) prime numbers. What is the greatest number of consecutive positive integers all of which are interesting?
1993 Baltic Way
Determine all integers $n$ for which

$$\sqrt{\frac{25}{2}+\sqrt{\frac{625}{4}-n}}+\sqrt{\frac{25}{2}-\sqrt{\frac{625}{4}-n}}$$

is an integer.
1993 Baltic Way
Prove that for any odd positive integer $n$, $n^{12}-n^8-n^4+1$ is divisible by $2^9$.
1993 Baltic Way
Suppose two functions $f(x)$ and $g(x)$ are defined for all $x$ with $2<x<4$ and satisfy: $2<f(x)<4,2<g(x)<4,f(g(x))=g(f(x))=x,f(x)\cdot g(x)=x^2$ for all $2<x<4$.

Prove that $f(3)=g(3)$.
1993 Baltic Way
Solve the system of equations in integers:

$$\begin{cases}z^x=y^{2x}\\ 2^z=4^x\\ x+y+z=20.\end{cases}$$
1993 Baltic Way
Compute the sum of all positive integers whose digits form either a strictly increasing or strictly decreasing sequence.
1993 Baltic Way
Solve the system of equations

$$\begin{cases}x^5=y+y^5\\ y^5=z+z^5\\ z^5=t+t^5\\ t^5=x+x^5.\end{cases}$$
1993 Baltic Way
Let $a_1,a_2,\ldots,a_n$ and $b_1,b_2,\ldots,b_n$ be two finite sequences consisting of $2n$ real different numbers. Rearranging each of the sequences in increasing order we obtain $a_1',a_2',\ldots,a_n'$ and $b_1',b_2',\ldots,b_n'$. Prove that

$$\max_{1\le i\le n}|a_i-b_i|\ge\max_{1\le i\le n}|a_i'-b_i'|.$$
1993 Baltic Way
An equilateral triangle is divided into $n^2$ congruent equilateral triangles. A spider stands at one of the vertices, a fly at another. Alternately each of them moves to a neighbouring vertex. Prove that the spider can always catch the fly.
1993 Baltic Way
There are $13$ cities in a certain kingdom. Between some pairs of the cities a two-way direct bus, train or plane connections are established. What is the least possible number of connections to be established so that choosing any two means of transportation one can go from any city to any other without using the third kind of vehicle?
1993 Baltic Way
An equilateral triangle $ABC$ is divided into $100$ congruent equilateral triangles. What is the greatest number of vertices of small triangles that can be chosen so that no two of them lie on a line that is parallel to any of the sides of the triangle $ABC$?
1993 Baltic Way
A square is divided into $16$ equal squares, obtaining the set of $25$ different vertices. What is the least number of vertices one must remove from this set, so that no $4$ points of the remaining set are the vertices of any square with sides parallel to the sides of the initial square?
1993 Baltic Way
On each face of two dice some positive integer is written. The two dice are thrown and the numbers on the top face are added. Determine whether one can select the integers on the faces so that the possible sums are $2,3,4,5,6,7,8,9,10,11,12,13$, all equally likely?
1993 Baltic Way
Two circles, both with the same radius $r$, are placed in the plane without intersecting each other. A line in the plane intersects the first circle at the points $A,B$ and the other at points $C,D$, so that $|AB|=|BC|=|CD|=14\text{cm}$. Another line intersects the circles at $E,F$, respectively $G,H$ so that $|EF|=|FG|=|GH|=6\text{cm}$. Find the radius $r$.
1993 Baltic Way
Let’s consider three pairwise non-parallel straight constant lines in the plane. Three points are moving along these lines with different nonzero velocities, one on each line (we consider the movement to have taken place for infinite time and continue infinitely in the future). Is it possible to determine these straight lines, the velocities of each moving point and their positions at some “zero” moment in such a way that the points never were, are or will be collinear?
1993 Baltic Way
In the triangle $ABC$, $|AB|=15,|BC|=12,|AC|=13$. Let the median $AM$ and bisector $BK$ intersect at point $O$, where $M\in BC,K\in AC$. Let $OL\perp AB,L\in AB$. Prove that $\angle OLK=\angle OLM$.
1993 Baltic Way
A convex quadrangle $ ABCD$ is inscribed in a circle with center $ O$. The angles $ AOB, BOC, COD$ and $ DOA$, taken in some order, are of the same size as the angles of the quadrangle $ ABCD$.

Prove that $ ABCD$ is a square
1993 Baltic Way
Let $ \mathcal Q$ be a unit cube. We say that a tetrahedron is good if all its edges are equal and all of its vertices lie on the boundary of $ \mathcal Q$. Find all possible volumes of good tetrahedra.
1992 Baltic Way
Let $p,q$ be two consecutive odd prime numbers. Prove that $p+q$ is a product of at least $3$ natural numbers greater than $1$ (not necessarily different).
1992 Baltic Way
Denote by $ d(n)$ the number of all positive divisors of a natural number $ n$ (including $ 1$ and $ n$). Prove that there are infinitely many $ n$, such that $ n/d(n)$ is an integer.
1992 Baltic Way
Find an infinite non-constant arithmetic progression of natural numbers such that each term is neither a sum of two squares, nor a sum of two cubes (of natural numbers).
1992 Baltic Way
Is it possible to draw a hexagon with vertices in the knots of an integer lattice so that the squares of the lengths of the sides are six consecutive positive integers?
1992 Baltic Way
It is given that $ a^2+b^2+(a+b)^2=c^2+d^2+(c+d)^2$. Prove that $ a^4+b^4+(a+b)^4=c^4+d^4+(c+d)^4$.
1992 Baltic Way
Prove that the product of the 99 numbers $ \frac{k^3-1}{k^3+1},k=2,3,\ldots,100$ is greater than $ 2/3$.
1992 Baltic Way
Let $ a=\sqrt[1992]{1992}$. Which number is greater

$$ \underbrace{a^{a^{a^{\ldots^{a}}}}}_{1992}\quad\text{or}\quad 1992?

$$
1992 Baltic Way
Find all integers satisfying the equation $ 2^x\cdot(4-x)=2x+4$.
1992 Baltic Way
A polynomial $ f(x)=x^3+ax^2+bx+c$ is such that $ b<0$ and $ ab=9c$. Prove that the polynomial $ f$ has three different real roots.
1992 Baltic Way
Find all fourth degree polynomial $ p(x)$ such that the following four conditions are satisfied:

(i) $ p(x)=p(-x)$ for all $ x$,

(ii) $ p(x)\ge0$ for all $ x$,

(iii) $ p(0)=1$

(iv) $ p(x)$ has exactly two local minimum points $ x_1$ and $ x_2$ such that $ |x_1-x_2|=2$.
1992 Baltic Way
Let $ Q^+$ denote the set of positive rational numbers. Show that there exists one and only one function $f: Q^+\to Q^+$ satisfying the following conditions:

(i) If $ 0<q<1/2$ then $ f(q)=1+f(q/(1-2q))$,

(ii) If $ 1<q\le2$ then $ f(q)=1+f(q-1)$,

(iii) $ f(q)\cdot f(1/q)=1$ for all $ q\in Q^+$.
1992 Baltic Way
Let $ N$ denote the set of natural numbers. Let $ \phi: N\rightarrow N$ be a bijective function and assume that there exists a finite limit

$$ \lim_{n\rightarrow\infty}\frac{\phi(n)}{n}=L.

$$ What are the possible values of $ L$?
1992 Baltic Way
Prove that for any positive $ x_1,x_2,\ldots,x_n,y_1,y_2,\ldots,y_n$ the inequality

$$ \sum_{i=1}^n\frac1{x_iy_i}\ge\frac{4n^2}{\sum_{i=1}^n(x_i+y_i)^2}

$$ holds.
1992 Baltic Way
There is a finite number of towns in a country. They are connected by one direction roads. It is known that, for any two towns, one of them can be reached from another one. Prove that there is a town such that all remaining towns can be reached from it.
1992 Baltic Way
Noah has 8 species of animals to fit into 4 cages of the ark. He plans to put species in each cage. It turns out that, for each species, there are at most 3 other species with which it cannot share the accomodation. Prove that there is a way to assign the animals to their cages so that each species shares with compatible species.
1992 Baltic Way
All faces of a convex polyhedron are parallelograms. Can the polyhedron have exactly 1992 faces?
1992 Baltic Way
Quadrangle $ ABCD$ is inscribed in a circle with radius 1 in such a way that the diagonal $ AC$ is a diameter of the circle, while the other diagonal $ BD$ is as long as $ AB$. The diagonals intersect at $ P$. It is known that the length of $ PC$ is $ 2/5$. How long is the side $ CD$?
1992 Baltic Way
Show that in a non-obtuse triangle the perimeter of the triangle is always greater than two times the diameter of the circumcircle.
1992 Baltic Way
Let $ C$ be a circle in plane. Let $ C_1$ and $ C_2$ be nonintersecting circles touching $ C$ internally at points $ A$ and $ B$ respectively. Let $ t$ be a common tangent of $ C_1$ and $ C_2$ touching them at points $ D$ and $ E$ respectively, such that both $ C_1$ and $ C_2$ are on the same side of $ t$. Let $ F$ be the point of intersection of $ AD$ and $ BE$. Show that $ F$ lies on $ C$.
1992 Baltic Way
Let $ a\le b\le c$ be the sides of a right triangle, and let $ 2p$ be its perimeter. Show that

$$ p(p - c) = (p - a)(p - b) = S,

$$

where $ S$ is the area of the triangle.
1991 Baltic Way
Find the smallest positive integer $n$ having the property that for any $n$ distinct integers $a_1, a_2, \dots , a_n$ the product of all differences $a_i-a_j$ $(i < j)$ is divisible by $1991$.
1991 Baltic Way
Prove that $102^{1991} + 103^{1991}$ is not a proper power of an integer.
1991 Baltic Way
There are $20$ cats priced from $\$12$ to $\$15$ and $20$ sacks priced from $10$ cents to $\$1$ for sale, all of different prices. Prove that John and Peter can each buy a cat in a sack paying the same amount of money.
1991 Baltic Way
A polynomial $p$ with integer coefficients is such that $p(-n) < p(n) < n$ for some integer $n$. Prove that $p(-n) < -n$.
1991 Baltic Way
For any positive numbers $a, b, c$ prove the inequalities

$$\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\ge \frac{2}{a+b}+\frac{2}{b+c}+\frac{2}{c+a}\ge \frac{9}{a+b+c}.$$
1991 Baltic Way
Solve the equation $[x] \cdot \{x\} = 1991x$. (Here $[x]$ denotes the greatest integer less than or equal to $x$, and $\{x\}=x-[x]$.)
1991 Baltic Way
If $\alpha,\beta,\gamma$ are the angles of an acute-angled triangle, prove that

$$\sin \alpha + \sin \beta > \cos \alpha + \cos\beta  + \cos\gamma.$$
1991 Baltic Way
Let $a, b, c, d, e$ be distinct real numbers. Prove that the equation

$$(x - a)(x - b)(x - c)(x - d) + (x - a)(x - b)(x - c)(x - e)$$

$$+(x - a)(x - b)(x - d)(x - e) + (x - a)(x - c)(x - d)(x - e)$$

$$+(x - b)(x - c)(x - d)(x - e) = 0$$

has four distinct real solutions.
1991 Baltic Way
Find the number of real solutions of the equation $a e^x = x^3$, where $a$ is a real parameter.
1991 Baltic Way
Express the value of $\sin 3^\circ$ in radicals.
1991 Baltic Way
The integers from $1$ to $1000000$ are divided into two groups consisting of numbers with odd or even sums of digits respectively. Which group contains more numbers?
1991 Baltic Way
The vertices of a convex $1991$-gon are enumerated with integers from $1$ to $1991$. Each side and diagonal of the $1991$-gon is colored either red or blue. Prove that, for an arbitrary renumeration of vertices, one can find integers $k$ and $l$ such that the segment connecting the vertices numbered $k$ and $l$ before the renumeration has the same color as the segment connecting the vertices numbered $k$ and $l$ after the renumeration.
1991 Baltic Way
An equilateral triangle is divided into $25$ equal equilateral triangles labelled by $1$ through $25$. Prove that one can find two triangles having a common side whose labels differ by more than $3$.
1991 Baltic Way
A castle has a number of halls and $n$ doors. Every door leads into another hall or outside. Every hall has at least two doors. A knight enters the castle. In any hall, he can choose any door for exit except the one he

just used to enter that hall. Find a strategy allowing the knight to get outside after visiting no more than $2n$ halls (a hall is counted each time it is entered).
1991 Baltic Way
In each of the squares of a chessboard an arbitrary integer is written. A king starts to move on the board. Whenever the king moves to some square, the number in that square is increased by $1$. Is it always possible to make the numbers on the chessboard:

(a) all even;

(b) all divisible by $3$;

(c) all equal?
1991 Baltic Way
Two circles $C_1$ and $C_2$ with radii $r_1$ and $r_2$ touch each other externally and both touch a line $l$. A circle $C_3$ with radius $r_3 < r_1, r_2$ is tangent to $l$ and externally to $C_1$ and $C_2$. Prove that

$$\frac{1}{\sqrt{r_3}}=\frac{1}{\sqrt{r_2}}+\frac{1}{\sqrt{r_2}}.$$
1991 Baltic Way
Let the coordinate planes have the reflection property. A ray falls onto one of them. How does the final direction of the ray after reflecting from all three coordinate planes depend on its initial direction?
1991 Baltic Way
Is it possible to place two non-intersecting tetrahedra of volume $\frac{1}{2}$ into a sphere with radius $1$?
1991 Baltic Way
Three circles in the plane, whose interiors have no common point, meet each other at three pairs of points: $A_1$ and $A_2$, $B_1$ and $B_2$, and $C_1$ and $C_2$, where points $A_2,B_2,C_2$ lie inside the triangle $A_1B_1C_1$. Prove that

$$A_1B_2 \cdot B_1C_2 \cdot C_1A_2 = A_1C_2 \cdot C_1B_2 \cdot B_1A_2 .$$
1991 Baltic Way
Consider two points $A(x_1, y_1)$ and $B(x_2, y_2)$ on the graph of the function $y = \frac{1}{x}$ such that $0 < x_1 < x_2$ and $AB = 2 \cdot OA$, where $O = (0, 0)$. Let $C$ be the midpoint of the segment $AB$. Prove that the angle between the $x$-axis and the ray $OA$ is equal to three times the angle between the $x$-axis and the ray $OC$.
1990 Baltic Way
Numbers $1, 2, \dots , n$ are written around a circle in some order. What is the smallest possible sum of the absolute differences of adjacent numbers?
1990 Baltic Way
The squares of a squared paper are enumerated as shown on the picture.

$$\begin{array}{|c|c|c|c|c|c}

\ddots &&&&&\\ \hline

10&\ddots&&&&\\ \hline

6&9&\ddots&&&\\ \hline

3&5&8&12&\ddots&\\ \hline

1&2&4&7&11&\ddots\\ \hline

\end{array}$$

Devise a polynomial $p(m, n)$ in two variables such that for any $m, n \in \mathbb{N}$ the number written in the square with coordinates $(m, n)$ is equal to $p(m, n)$.
1990 Baltic Way
Given $a_0 > 0$ and $c > 0$, the sequence $(a_n)$ is defined by

$$a_{n+1}=\frac{a_n+c}{1-ca_n}\quad\text{for }n=1,2,\dots$$

Is it possible that $a_0, a_1, \dots , a_{1989}$ are all positive but $a_{1990}$ is negative?
1990 Baltic Way
Prove that, for any real numbers $a_1, a_2, \dots , a_n$,

$$ \sum_{i,j=1}^n \frac{a_ia_j}{i+j-1}\ge 0.$$
1990 Baltic Way
Let $*$ be an operation, assigning a real number $a * b$ to each pair of real numbers $(a, b)$. Find an equation which is true (for all possible values of variables) provided the operation $*$ is commutative or associative and which can be false otherwise.
1990 Baltic Way
Let $ABCD$ be a quadrilateral with $AD = BC$ and $\angle DAB + \angle ABC = 120^\circ$. An equilateral triangle $DPC$ is erected in the exterior of the quadrilateral. Prove that the triangle $APB$ is also equilateral.
1990 Baltic Way
The midpoint of each side of a convex pentagon is connected by a segment with the centroid of the triangle formed by the remaining three vertices of the pentagon. Prove that these five segments have a common point.
1990 Baltic Way
It is known that for any point $P$ on the circumcircle of a triangle $ABC$, the orthogonal projections of $P$ onto $AB,BC,CA$ lie on a line, called a Simson line of $P$. Show that the Simson lines of two diametrically opposite points $P_1$ and $P_2$ are perpendicular.
1990 Baltic Way
Two congruent triangles are inscribed in an ellipse. Are they necessarily symmetric with respect to an axis or the center of the ellipse?
1990 Baltic Way
A segment $AB$ is marked on a line $t$. The segment is moved on the plane so that it remains parallel to $t$ and that the traces of points $A$ and $B$ do not intersect. The segment finally returns onto $t$. How far can point $A$ now be from its initial position?
1990 Baltic Way
Prove that the modulus of an integer root of a polynomial with integer coefficients cannot exceed the maximum of the moduli of the coefficients.
1990 Baltic Way
Let $m$ and $n$ be positive integers. Show that $25m+ 3n$ is divisible by $83$ if and only if so is $3m+ 7n$.
1990 Baltic Way
Show that the equation $x^2-7y^2 = 1$ has infinitely many solutions in natural numbers.
1990 Baltic Way
Do there exist $1990$ pairwise coprime positive integers such that all sums of two or more of these numbers are composite numbers?
1990 Baltic Way
Prove that none of the numbers $2^{2^n}+ 1$, $n = 0, 1, 2, \dots$ is a perfect cube.
1990 Baltic Way
A closed polygonal line is drawn on a unit squared paper so that its vertices lie at lattice points and its sides have odd lengths. Prove that its number of sides is divisible by $4$.
1990 Baltic Way
There are two piles with $72$ and $30$ candies. Two students alternate taking candies from one of the piles. Each time the number of candies taken from a pile must be a multiple of the number of candies in the other pile. Which student can always assure taking the last candy from one of the piles?
1990 Baltic Way
Numbers $1, 2,\dots , 101$ are written in the cells of a $101\times 101$ square board so that each number is repeated $101$ times. Prove that there exists either a column or a row containing at least $11$ different numbers.
1990 Baltic Way
What is the largest possible number of subsets of the set $\{1, 2, \dots , 2n+1\}$ such that the intersection of any two subsets consists of one or several consecutive integers?
1990 Baltic Way
A creative task: propose an original competition problem together with its solution.
2021 Benelux
(a) Prove that for all $a, b, c, d \in \mathbb{R}$ with $a + b + c + d = 0$,

$$

\max(a, b) + \max(a, c) + \max(a, d) + \max(b, c) + \max(b, d) + \max(c, d) \geqslant 0.

$$(b) Find the largest non-negative integer $k$ such that it is possible to replace $k$ of the six maxima in this inequality by minima in such a way that the inequality still holds for all $a, b, c, d \in \mathbb{R}$ with $a + b + c + d = 0$.
2021 Benelux
Pebbles are placed on the squares of a $2021\times 2021$ board in such a way that each square contains at most one pebble. The pebble set of a square of the board is the collection of all pebbles which are in the same row or column as this square. (A pebble belongs to the pebble set of the square in which it is placed.) What is the least possible number of pebbles on the board if no two squares have the same pebble set?
2021 Benelux
A cyclic quadrilateral $ABXC$ has circumcentre $O$. Let $D$ be a point on line $BX$ such that $AD = BD$. Let $E$ be a point on line $CX$ such that $AE = CE$. Prove that the circumcentre of triangle $\triangle DEX$ lies on the perpendicular bisector of $OA$.
2021 Benelux
A sequence $a_1, a_2, a_3, \ldots$ of positive integers satisfies $a_1 > 5$ and $a_{n+1} = 5 + 6 + \cdots + a_n$ for all positive integers $n$. Determine all prime numbers $p$ such that, regardless of the value of $a_1$, this sequence must contain a multiple of $p$.
2020 Benelux
Find all positive integers $d$ with the following property:  there exists a polynomial $P$ of degree $d$ with integer coefficients such that $\left|P(m)\right|=1$ for at least $d+1$ different integers $m$.
2020 Benelux
Let $N$ be a positive integer. A collection of $4N^2$ unit tiles with two segments drawn on them as shown is assembled into a $2N\times2N$ board. Tiles can be rotated.

[asy]size(1.5cm);draw((0,0)--(1,0)--(1,1)--(0,1)--cycle);draw((0,0.5)--(0.5,0),red);draw((0.5,1)--(1,0.5),red);[/asy]

The  segments  on  the  tiles  define  paths  on  the  board. Determine  the  least  possible  number  and  the largest possible number of such paths.



For example, there are $9$ paths on the $4\times4$ board shown below.

[asy]size(4cm);draw((0,0)--(4,0)--(4,4)--(0,4)--cycle);draw((0,1)--(4,1));draw((0,2)--(4,2));draw((0,3)--(4,3));draw((1,0)--(1,4));draw((2,0)--(2,4));draw((3,0)--(3,4));draw((0,3.5)--(0.5,4),red);draw((0,2.5)--(1.5,4),red);draw((3.5,0)--(4,0.5),red);draw((2.5,0)--(4,1.5),red);draw((0.5,0)--(0,0.5),red);draw((2.5,4)--(3,3.5)--(3.5,4),red);draw((4,3.5)--(3.5,3)--(4,2.5),red);draw((0,1.5)--(1,2.5)--(1.5,2)--(0.5,1)--(1.5,0),red);draw((1.5,3)--(2,3.5)--(3.5,2)--(2,0.5)--(1.5,1)--(2.5,2)--cycle,red);[/asy]