Navvye/Olympiad-Problems-CSV huggingface dataset:
2006 Dutch Mathematical Olympiad
Given is triangle $ABC$ with an inscribed circle with center $M$ and radius $r$.

The tangent to this circle parallel to $BC$ intersects $AC$ in $D$ and $AB$ in $E$.

The tangent to this circle parallel to $AC$ intersects $AB$ in $F$ and $BC$ in $G$.

The tangent to this circle parallel to $AB$ intersects $BC$ in $H$ and $AC$ in $K$.

Name the centers of the inscribed circles of triangle $AED$, triangle $FBG$ and triangle $KHC$ successively $M_A, M_B, M_C$ and the rays successively $r_A, r_B$ and $r_C$.

Prove that $r_A + r_B + r_C = r$.
2006 Dutch Mathematical Olympiad
Player $A$ and player $B$ play the next game on an $8$ by $8$ square chessboard.

They in turn color a field that is not yet colored. One player uses red and the other blue. Player $A$ starts. The winner is the first person to color the four squares of a square of $2$ by $2$ squares with his color somewhere on the board.

Prove that player $B$ can always prevent player $A$ from winning.
2005 Dutch Mathematical Olympiad
In how many ways can one choose distinct numbers a and b from {1, 2, 3, ..., 2005} such that a + b is a multiple of 5?
2005 Dutch Mathematical Olympiad
Let $P_1P_2P_3\dots P_{12}$ be a regular dodecagon. Show that $$\left|P_1P_2\right|^2 + \left|P_1P_4\right|^2 + \left|P_1P_6\right|^2 + \left|P_1P_8\right|^2 + \left|P_1P_{10}\right|^2 + \left|P_1P_{12}\right|^2$$ is equal to $$\left|P_1P_3\right|^2 + \left|P_1P_5\right|^2 + \left|P_1P_7\right|^2 + \left|P_1P_9\right|^2 + \left|P_1P_{11}\right|^2.$$
2005 Dutch Mathematical Olympiad
Let $a_1,a_2,a_3,a_4,a_5$ be distinct real numbers. Consider all sums of the form $a_i + a_j$ where $i,j \in \{1,2,3,4,5\}$ and $i \neq j$. Let $m$ be the number of distinct numbers among these sums. What is the smallest possible value of $m$?
2005 Dutch Mathematical Olympiad
Let $ABCD$ be a quadrilateral with $AB \parallel CD$, $AB > CD$. Prove that the line passing through $AC \cap BD$ and $AD \cap BC$ passes through the midpoints of $AB$ and $CD$.
2005 Dutch Mathematical Olympiad
Consider an array of numbers of size $8 \times 8$. Each of the numbers in the array equals 1 or -1. "Doing a move" means that you pick any number in the array and you change the sign of all numbers which are in the same row or column as the number you picked. (This includes changing the sign of the "chosen" number itself.) Prove that one can transform any given array into an array containing numbers +1 only by performing this kind of moves repeatedly.
2004 Dutch Mathematical Olympiad
Determine the number of pairs of positive integers $(a, b)$, with $a \le b$, for which lcm  $(a, b) = 2004$.

lcm ($a, b$) means the least common multiple of $a$ and $b$. Example: lcm $(18, 24) = 72$.
2004 Dutch Mathematical Olympiad
Two circles $A$ and $B$, both with radius $1$, touch each other externally.

Four circles $P, Q, R$ and $S$, all four with the same radius $r$, lie such that

$P$ externally touches on $A, B, Q$ and $S$,

$Q$ externally touches on $P, B$ and $R$,

$R$ externally touches on $A, B, Q$ and $S$,

$S$ externally touches on $P, A$ and $R$.

Calculate the length of $r.$
2004 Dutch Mathematical Olympiad
Start with a stack of $100$ cards.

Now repeat the following: choose a stack of at least $2$ cards and split them into two smaller piles (at least $1$ card of each). Continue this until there are finally $100$ stacks of $1$ card each. Every time you split a pile into two stacks you get a number of points that is equal to the product of the number of cards in the two new stacks.

What is the maximum number of points that you can earn in total?
2004 Dutch Mathematical Olympiad
Two circles $C_1$ and $C_2$ touch each other externally in a point $P$. At point $C_1$ there is a point $Q$ such that the tangent line in $Q$ at $C_1$ intersects the circle $C_2$ at points $A$ and $B$. The line $QP$ still intersects $C_2$ at point $C$.

Prove that triangle $ABC$ is isosceles.
2004 Dutch Mathematical Olympiad
A right triangle with perpendicular sides $a$ and $b$ and hypotenuse $c$ has the following properties:

$a = p^m$ and $b = q^n$ with $p$ and $q$ prime numbers and $m$ and $n$ positive integers, $c = 2k +1$ with $k$ a positive integer.

Determine all possible values of $c$ and the associated values of $a$ and $b$.
2003 Dutch Mathematical Olympiad
A Pythagorean triangle is a right triangle whose three sides are integers.

The best known example is the triangle with rectangular sides $3$ and $4$ and hypotenuse $5$.

Determine all Pythagorean triangles whose area is twice the perimeter.
2003 Dutch Mathematical Olympiad
Two squares with side $12$ lie exactly on top of each other.

One square is rotated around a corner point through an angle of $30$ degrees relative to the other square.

Determine the area of the common piece of the two squares.
2003 Dutch Mathematical Olympiad
Determine all positive integers$ n$ that can be written as the product of two consecutive integers and as well as the product of four consecutive integers numbers.

In the formula: $n = a (a + 1) = b (b + 1) (b + 2) (b + 3)$.
2003 Dutch Mathematical Olympiad
In a circle with center $M$, two chords $AC$ and $BD$ intersect perpendicularly.

The circle of diameter $AM$ intersects the circle of diameter $BM$ besides  $M$ also in point $P$. The circle of diameter $BM$ intersects the circle with diameter $CM$ besides  $M$ also in point $Q$. The circle of diameter $CM$ intersects the circle of diameter $DM$ besides $M$ also in point $R$. The circle of diameter $DM$ intersects the circle of diameter $AM$ besides $M$ also in point $S$. Prove that quadrilateral $PQRS$ is a rectangle.
2003 Dutch Mathematical Olympiad
There are a number of cards on a table. A number is written on each card. The "pick and replace" operation involves the following: two random cards are taken from the table and replaced by one new card. If the numbers $a$ and $b$ appear on the two packed cards, the number $a + b + ab$ is set on the new card.

If we start with ten cards with the numbers $1, 2, 3, 4, 5, 6, 7, 8, 9$ and $10$ respectively, what value(s) can the number have that "grab and replace" nine times is on the only card still on the table? Prove your answer
2002 Dutch Mathematical Olympiad
The sides of a $10$ by $10$ square $ABCD$ are reflective on the inside. A beam of light enters the square via the vertex $A$ and heads to the point $P$ on $CD$ with $CP = 3$ and $PD = 7$. In $P$ it naturally reflects on the $CD$ side. The light beam can only leave the square via one of the angular points $A, B, C$ or $D$.

What is the distance that the light beam travels within the square before it leaves the square again?

By which vertex does that happen?
2002 Dutch Mathematical Olympiad
Determine all triplets $(x, y, z)$ of positive integers with $x \le y \le z$ that satisfy $\left(1+\frac1x \right)\left(1+\frac1y \right)\left(1+\frac1z \right) = 3$
2002 Dutch Mathematical Olympiad
$A, B$ and $C$ are points in the plane with integer coordinates.

The lengths of the sides of triangle $ABC$ are integer numbers.

Prove that the perimeter of the triangle is an even number.
2002 Dutch Mathematical Olympiad
Five pairs of cartoon characters, Donald and Katrien Duck, Asterix and Obelix, Suske and Wiske, Tom and Jerry, Heer Bommel and Tom Poes, sit around a round table with $10$ chairs. The two members of each pair ensure that they sit next to each other. In how many different ways can the ten seats be occupied?

Two ways are different if they cannot be transferred to each other by a rotation.
2002 Dutch Mathematical Olympiad
In triangle $ABC$, angle $A$ is twice as large as angle $B$. $AB = 3$ and $AC = 2$. Calculate $BC$.
2001 Dutch Mathematical Olympiad
In a tournament, every team plays exactly once against every other team. One won match earns $3$ points for the winner and $0$ for the loser. With a draw both teams receive $1$ point each. At the end of the tournament it appears that all teams together have achieved $15$ points. The last team on the final list scored exactly $1$ point. The second to last team has not lost a match.

a) How many teams participated in the tournament?

b) How many points did the team score in second place in the final ranking?
2001 Dutch Mathematical Olympiad
The function f has the following properties :

$f(x + y) = f(x) + f(y) + xy$ for all real $x$ and $y$

$f(4) = 10$

Calculate $f(2001)$.
2001 Dutch Mathematical Olympiad
A wooden beam $EFGH$ $ABCD$ is with three cuts in $8$ smaller ones sawn beams. Each cut is parallel to one of the three pair of opposit sides. Each pair of saw cuts is shown perpendicular to each other. The smaller bars at the corners $A, C, F$ and $H$ have a capacity of $9, 12, 8, 24$ respectively.(The proportions in the picture are not correct!!). Calculate content of the entire bar.
2001 Dutch Mathematical Olympiad
The function is given $f(x) = \frac{2x^3 -6x^2 + 13x + 10}{2x^2 - 9x}$.



Determine all positive integers $x$ for which $f(x)$ is an integer
2001 Dutch Mathematical Olympiad
If you take a subset of $4002$ numbers from the whole numbers $1$ to $6003$, then there is always a subset of $2001$ numbers within that subset with the following property:

If you order the $2001$ numbers from small to large, the numbers are alternately even and odd (or odd and even).

Prove this.
2000 Dutch Mathematical Olympiad
Let $a$ and $b$ be integers.

Define $a$ to be a power of $b$ if there exists a positive integer $n$ such that $a = b^n$.

Define $a$ to be a multiple of $b$ if there exists an integer $n$ such that $a = bn$.



Let $x$, $y$ and $z$ be positive integer such that $z$ is a power of both $x$ and $y$.



Decide for each of the following statements whether it is true or false. Prove your answers.



(a) The number $x + y$ is even.

(b) One of $x$ and $y$ is a multiple of the other one.

(c) One of $x$ and $y$ is a power of the other one.

(d) There exist an integer $v$ such that both $x$ and $y$ are powers of $v$

(e) For each power of $x$ and for each power of $y$, an integer $w$ can be found such that $w$ is a power of each of these powers.

(f) There exists a positive integer $k$ such that $x^k > y$.
2000 Dutch Mathematical Olympiad
Three boxes contain 600 balls each. The first box contains 600 identical red balls, the second box contains 600 identical white balls and the third box contains 600 identical blue balls. From these three boxes, 900 balls are chosen. In how many ways can the balls be chosen? For example, one can choose 250 red balls, 187 white balls and 463 balls, or one can choose 360 red balls and 540 blue balls.
2000 Dutch Mathematical Olympiad
Isosceles, similar triangles $QPA$ and $SPB$ are constructed (outwards) on the sides of parallelogram $PQRS$ (where $PQ = AQ$ and $PS = BS$). Prove that triangles $RAB$, $QPA$ and $SPB$ are similar.
2000 Dutch Mathematical Olympiad
Fifteen boys are standing on a field, and each of them has a ball. No two distances between two of the boys are equal. Each boy throws his ball to the boy standing closest to him.



(a) Show that one of the boys does not get any ball.

(b) Prove that none of the boys gets more than five balls.
2000 Dutch Mathematical Olympiad
Consider an infinite strip of unit squares. The squares are numbered "1", "2", "3", ... A pawn starts on one of the squares and it can move according to the following rules:

(1) from the square numbered "$n$" to the square numbered "$2n$", and vice versa;

(2) from the square numbered "$n$" to the square numbered "$3n + 1$", and vice versa.

Show that the pawn can reach the square numbered "$1$" in a finite number of moves.
1999 Dutch Mathematical Olympiad
Let $f: \mathbb{Z} \rightarrow \{-1,1\}$ be a function such that $$ f(mn) =f(m)f(n),\ \forall m,n \in \mathbb{Z}. $$ Show that there exists a positive integer $a$ such that $1 \leq a \leq 12$ and $f(a) = f(a + 1) = 1$.
1999 Dutch Mathematical Olympiad
A $9 \times 9$ square consists of $81$ unit squares. Some of these unit squares are painted black, and the others are painted white, such that each $2 \times 3$ rectangle and each $3 \times 2$ rectangle contain exactly 2 black unit squares and 4 white unit squares. Determine the number of black unit squares.
1999 Dutch Mathematical Olympiad
Let $ABCD$ be a square and let $\ell$ be a line. Let $M$ be the centre of the square. The diagonals of the square have length 2 and the distance from $M$ to $\ell$ exceeds 1. Let $A',B',C',D'$ be the orthogonal projections of $A,B,C,D$ onto $\ell$. Suppose that one rotates the square, such that $M$ is invariant. The positions of $A,B,C,D,A',B',C',D'$ change. Prove that the value of $AA'^2 + BB'^2 + CC'^2 + DD'^2$ does not change.
1999 Dutch Mathematical Olympiad
Consider a matrix of size $8 \times 8$, containing positive integers only. One may repeatedly transform the entries of the matrix according to the following rules:



-Multiply all entries in some row by 2.

-Subtract 1 from all entries in some column.



Prove that one can transform the given matrix into the zero matrix.
1999 Dutch Mathematical Olympiad
Let $c$ be a nonnegative integer, and define $a_n = n^2 + c$ (for $n \geq 1)$. Define $d_n$ as the greatest common divisor of $a_n$ and $a_{n + 1}$.

(a) Suppose that $c = 0$. Show that $d_n = 1,\ \forall n \geq 1$.

(b) Suppose that $c = 1$. Show that $d_n \in \{1,5\},\ \forall n \geq 1$.

(c) Show that $d_n \leq 4c + 1,\ \forall n \geq 1$.
1998 Dutch Mathematical Olympiad
Consider any permutation $\sigma$ of $\{0,1,2,\dots,9\}$ and for each of the 8 triples of consecutive numbers in this permutation, consider the sum of these three numbers. Let $M(\sigma)$ be the largest of these 8 sums. (For example, for the permutation $\sigma = (4, 6, 2, 9, 0, 1, 8, 5, 7, 3)$ we get the 8 sums 12, 17, 11, 10, 9, 14, 20, 15, and $M(\sigma) = 20$.)



(a) Find a permutation $\sigma_1$ such that $M(\sigma_1) = 13$.



(b) Does there exist a permutation $\sigma_2$ such that $M(\sigma_2) = 12$?
1998 Dutch Mathematical Olympiad
Let $TABCD$ be a pyramid with top vertex $T$, such that its base $ABCD$ is a square of side length 4. It is given that, among the triangles $TAB$, $TBC$, $TCD$ and $TDA$, one  can find an isosceles triangle and a right-angled triangle. Find all possible values for the volume of the pyramid.
1998 Dutch Mathematical Olympiad
Let $m$ and $n$ be positive integers such that $m - n = 189$ and such that the least common multiple of $m$ and $n$ is equal to $133866$. Find $m$ and $n$.
1998 Dutch Mathematical Olympiad
Let $ABCD$ be a convex quadrilateral such that $AC \perp BD$.



(a) Prove that $AB^2 + CD^2 = BC^2 + DA^2$.



(b) Let $PQRS$ be a convex quadrilateral such that $PQ = AB$, $QR = BC$, $RS = CD$ and $SP = DA$. Prove that $PR \perp QS$.
1998 Dutch Mathematical Olympiad
Find all real solutions of the following equation: $$ (x + 1995)(x + 1997)(x + 1999)(x + 2001) + 16 = 0. $$
1997 Dutch Mathematical Olympiad
For each positive integer $n$ we define $f (n)$ as the product of the sum of the digits of $n$ with $n$ itself.

Examples: $f (19) = (1 + 9) \times 19 = 190$, $f (97) = (9 + 7)  \times 97 = 1552$.

Show that there is no number $n$ with $f (n) = 19091997$.
1997 Dutch Mathematical Olympiad
The lines $AD , BE$ and $CF$ intersect in $S$ within a triangle $ABC$ .

It is given that $AS: DS = 3: 2$ and $BS: ES = 4: 3$ . Determine the ratio $CS: FS$ .
1997 Dutch Mathematical Olympiad
a. View the second-degree   quadratic equation $x^2+? x +? = 0$

Two players successively put an integer each at the location of a question mark. Show that the second player can always ensure that the quadratic gets two integer solutions.

Note: we say that the quadratic also has two integer solutions, even when they are equal (for example if they are both equal to $3$).



b.View the third-degree equation $x^3 +? x^2 +? x +? = 0$

Three players successively put an integer each at the location of a question mark. The equation appears to have three integer (possibly again the same) solutions. It is given that two players each put a $3$ in the place of a question mark. What number did the third player put? Determine that number and the place where it is placed and prove that only one number is possible.
1997 Dutch Mathematical Olympiad
We look at an octahedron, a regular octagon, one of the side surfaces painted red and the other seven surfaces blue. We throw the octahedron like a die. The surface that comes up is painted: if it is red it is painted blue and if it is blue it is painted red. Then we throw the octahedron again and paint it again according to the above rule. In total we throw the octahedron $10$ times.

How many different octahedra can we get after finishing the $10$th time?

(Two octahedrons are different if they cannot be rotated.)
1997 Dutch Mathematical Olympiad
Given is a triangle $ABC$ and a point $K$ within the triangle. The point $K$ is mirrored in the sides of the triangle: $P , Q$ and $R$ are the mirrorings of $K$ in $AB , BC$ and $CA$, respectively . $M$ is the center of the circle passing through the vertices  of triangle $PQR$. $M$ is mirrored again in the sides of triangle $ABC$: $P', Q'$ and $R'$ are the mirror of $M$ in $AB$ respectively, $BC$ and $CA$.

a. Prove that $K$ is the center of the circle passing through the vertices  of triangle $P'Q'R'$ .

b. Where should you choose $K$ within triangle $ABC$ so that $M$ and $K$ coincide? Prove your answer.
1996 Dutch Mathematical Olympiad
How many different (non similar) triangles are there whose angles have an integer number of degrees?
1996 Dutch Mathematical Olympiad
Investigate whether for two positive integers $m$ and $n$ the numbers $m^2 + n$ and $n^2 + m$ can be both squares of integers.
1996 Dutch Mathematical Olympiad
What is the largest number of horses that you can put on a chessboard without there being two horses that can beat each other?

a. Describe an arrangement with that maximum number.

b. Prove that a larger number is not possible.



(A chessboard consists of $8 \times 8$ spaces and a horse jumps from one field to another field according to the line "two squares vertically and one squared horizontally" or "one square vertically and two squares horizontally")
1996 Dutch Mathematical Olympiad
A line $l$ intersects the segment $AB$ perpendicular to $C$. Three circles are drawn successively with $AB, AC$ and $BC$ as the diameter. The largest circle intersects $l$ in $D$. The segments $DA$ and $DB$ still intersect the two smaller circles in $E$ and $F$.

a. Prove that quadrilateral $CFDE$ is a rectangle.

b. Prove that the line through $E$ and $F$ touches the circles with diameters $AC$ and $BC$ in $E$ and $F$.
1996 Dutch Mathematical Olympiad
For the positive integers $x , y$ and $z$ apply $\frac{1}{x}+\frac{1}{y}=\frac{1}{z}$ .

Prove that if the three numbers $x , y,$ and $z$ have no common divisor greater than $1$, $x + y$ is the square of an integer.
1995 Dutch Mathematical Olympiad
A kangaroo jumps from lattice poin to lattice point in the coordinate plane. It can make only two kinds of jumps: $ (A)$ $ 1$ to right and $ 3$ up, and $ (B)$ $ 2$ to the left and $ 4$ down.

$ (a)$ The start position of the kangaroo is $ (0,0)$. Show that it can jump to the point $ (19,95)$ and determine the number of jumps needed.

$ (b)$ Show that if the start position is $ (1,0)$, then it cannot reach $ (19,95)$.

$ (c)$ If the start position is $ (0,0)$, find all points $ (m,n)$ with $ m,n \ge 0$ which the kangaroo can reach.
1995 Dutch Mathematical Olympiad
For any point $ P$ on a segment $ AB$, isosceles and right-angled triangles $ AQP$ and $ PRB$ are constructed on the same side of $ AB$, with $ AP$ and $ PB$ as the bases. Determine the locus of the midpoint $ M$ of $ QR$ when $ P$ describes the segment $ AB$.
1995 Dutch Mathematical Olympiad
Let $ 101$ marbles be numbered from $ 1$ to $ 101$. The marbles are divided over two baskets $ A$ and $ B$. The marble numbered $ 40$ is in basket $ A$. When this marble is removed from basket $ A$ and put in $ B$, the averages of the numbers $ A$ and $ B$ both increase by $ \frac{1}{4}$. How many marbles were there originally in basket $ A?$
1995 Dutch Mathematical Olympiad
A number of spheres with radius $ 1$ are being placed in the form of a square pyramid. First, there is a layer in the form of a square with $ n^2$ spheres. On top of that layer comes the next layer with $ (n-1)^2$ spheres, and so on. The top layer consists of only one sphere. Compute the height of the pyramid.
1995 Dutch Mathematical Olympiad
An array $ (a_1,a_2,...,a_{13})$ of $ 13$ integers is called $ tame$ if for each $ 1 \le i \le 13$ the following condition holds: If $ a_i$ is left out, the remaining twelve integers can be divided into two groups with the same sum of elements. A tame array is called $ turbo$ $ tame$ if the remaining twelve numbers can always be divided in two groups of six numbers having the same sum.

$ (a)$ Give an example of a tame array of $ 13$ integers (not all equal).

$ (b)$ Prove that in a tame array all numbers are of the same parity.

$ (c)$ Prove that in a turbo tame array all numbers are equal.
1994 Dutch Mathematical Olympiad
A unit square is divided into two rectangles in such a way that the smaller rectangle can be put on the greater rectangle with every vertex of the smaller on exactly one of the edges of the greater. Calculate the dimensions of the smaller rectangle.
1994 Dutch Mathematical Olympiad
A sequence of integers $ a_1,a_2,a_3,...$ is such that $ a_1=2, a_2=3$, and



$ a_{n+1}=2a_{n-1}$ or $ 3a_n-2a_{n-1}$ for all $ n \ge 2$.



Prove that no number between $ 1600$ and $ 2000$ can be an element of the sequence.
1994 Dutch Mathematical Olympiad
$ (a)$ Prove that every multiple of $ 6$ can be written as a sum of four cubes.

$ (b)$ Prove that every integer can be written as a sum of five cubes.
1994 Dutch Mathematical Olympiad
Let $ P$ be a point on the diagonal $ BD$ of a rectangle $ ABCD$, $ F$ be the projection of $ P$ on $ BC$, and $ H \not= B$ be the point on $ BC$ such that $ BF=FH$. If lines $ PC$ and $ AH$ intersect at $ Q$, prove that the areas of triangles $ APQ$ and $ CHQ$ are equal.
1994 Dutch Mathematical Olympiad
Three real numbers $ a,b,c$ satisfy the inequality $ |ax^2+bx+c| \le 1$ for all $ x \in [-1,1]$. Prove that $ |cx^2+bx+a| \le 2$ for all $ x \in [-1,1]$.
1993 Dutch Mathematical Olympiad
Show that any subset of $ V= \{ 1,2,...,24,25 \}$ with $ 17$ or more elements contains at least two distinct numbers the product of which is a perfect square.
1993 Dutch Mathematical Olympiad
In a triangle $ ABC$ with $ \angle A=90^{\circ}$, $ D$ is the midpoint of $ BC$, $ F$ that of $ AB$, $ E$ that of $ AF$ and $ G$ that of $ FB$. Segment $ AD$ intersects $ CE,CF$ and $ CG$ in $ P,Q$ and $ R$, respectively. Determine the ratio: $ \frac{PQ}{QR}$.
1993 Dutch Mathematical Olympiad
A sequence of numbers is defined by $ u_1=a, u_2=b$ and $ u_{n+1}=\frac{u_n+u_{n-1}}{2}$ for $ n \ge 2$. Prove that $ \displaystyle\lim_{n\to\infty}u_n$ exists and express its value in terms of $ a$ and $ b$.
1993 Dutch Mathematical Olympiad
Let $ C$ be a circle with center $ M$ in a plane $ V$, and $ P$ be a point not on the circle $ C$.

$ (a)$ If $ P$ is fixed, prove that $ AP^2+BP^2$ is a constant for every diameter $ AB$ of the circle $ C$.

$ (b)$ Let $ AB$ be a fixed diameter of $ C$ and $ P$ a point on a fixed sphere $ S$ not intersecting $ V$. Determine the points $ P$ on $ S$ that minimize $ AP^2+BP^2$.
1993 Dutch Mathematical Olympiad
Eleven distinct points $ P_1,P_2,...,P_{11}$ are given on a line so that $ P_i P_j \le 1$ for every $ i,j$. Prove that the sum of all distances $ P_i P_j, 1 \le i <j \le 11$, is smaller than $ 30$.
1992 Dutch Mathematical Olympiad
Four dice are thrown. What is the probability that the product of the number equals $ 36?$
1992 Dutch Mathematical Olympiad
In the fraction below and its decimal notation (with period of length $ 4$) every letter represents a digit, and different letters denote different digits. The numerator and denominator are coprime. Determine the value of the fraction:



$ \frac{ADA}{KOK}=0.SNELSNELSNELSNEL...$



$ Note.$ Ada Kok is a famous dutch swimmer, and "snel" is Dutch for "fast".