Navvye/Olympiad-Problems-CSV huggingface dataset:
1978 Swedish Mathematical Competition
Two satellites are orbiting the earth in the equatorial plane at an altitude $h$ above the surface. The distance between the satellites is always $d$, the diameter of the earth. For which $h$ is there always a point on the equator at which the two satellites subtend an angle of $90^\circ$?
1978 Swedish Mathematical Competition
$b_0, b_1, b_2, \dots$ is a sequence of positive reals such that the sequence $b_0,c b_1, c^2b_2,c^3b_3,\dots$ is convex for all $c > 0$. (A sequence is convex if each term is at most the arithmetic mean of its two neighbors.) Show that $\ln b_0, \ln b_1, \ln b_2, \dots$ is convex.
1978 Swedish Mathematical Competition
$k > 1$ is fixed. Show that for $n$ sufficiently large for every partition of $\{1,2,\dots,n\}$ into $k$ disjoint subsets we can find $a \neq b$ such that $a$ and $b$ are in the same subset and $a+1$ and $b+1$ are in the same subset. What is the smallest $n$ for which this is true?
1978 Swedish Mathematical Competition
$p(x)$ is a polynomial of degree $n$ with leading coefficient $c$, and $q(x)$ is a polynomial of degree $m$ with leading coefficient $c$, such that

$$

p(x)^2 = \left(x^2 - 1\right)q(x)^2 + 1

$$Show that $p'(x) = nq(x)$.
1979 Swedish Mathematical Competition
Solve the equations:

$$\left\{ \begin{array}{l}

 x_1 + 2 x_2 + 3 x_3 + \cdots + (n-1) x_{n-1} + n x_n = n \\ 

 2 x_1 + 3 x_2 + 4 x_3 + \cdots + n x_{n-1} + x_n = n-1 \\
 3 x_1 + 4 x_2 + 5 x_3 + \cdots + x_{n-1} + 2 x_n = n-2 \\
 \cdots \cdots \cdots \cdots\cdot\cdots \cdots \cdots \cdots\cdot\cdots \cdots \cdots \cdots\cdot  \\
 (n-1) x_1 + n x_2 + x_3 + \cdots + (n-3) x_{n-1} + (n-2) x_n = 2  \\
 n x_1 + x_2 + 2 x_3 + \cdots + (n-2) x_{n-1} + (n-1) x_n = 1  

 \end{array} \right.

$$
1979 Swedish Mathematical Competition
Find rational $x$ in $(3,4)$ such that $\sqrt{x-3}$ and $\sqrt{x+1}$ are rational.
1979 Swedish Mathematical Competition
Express

$$

x^{13} + \frac{1}{x^{13}}

$$as a polynomial in $y = x +  \frac{1}{x}$.
1979 Swedish Mathematical Competition
$f(x)$ is continuous on the interval $[0, \pi]$ and satisfies

$$

\int\limits_0^\pi f(x)dx=0, \qquad \int\limits_0^\pi f(x)\cos x dx=0

$$Show that $f(x)$ has at least two zeros in the interval $(0, \pi)$.
1979 Swedish Mathematical Competition
Find the smallest positive integer $a$ such that for some integers $b$, $c$ the polynomial $ax^2 - bx + c$ has two distinct zeros in the interval $(0,1)$.
1979 Swedish Mathematical Competition
Find the sharpest inequalities of the form $a\cdot AB < AG < b\cdot AB$ and $c\cdot AB < BG < d\cdot AB$ for all triangles $ABC$ with centroid $G$ such that $GA > GB > GC$.
1980 Swedish Mathematical Competition
Show that $\log_{10} 2$ is irrational.
1980 Swedish Mathematical Competition
$a_1$, $a_2$, $a_3$, $a_4$, $a_5$, $a_6$, $a_7$ and $b_1$, $b_2$, $b_3$, $b_4$, $b_5$, $b_6$, $b_7$ are two permutations of $1, 2, 3, 4, 5, 6, 7$. Show that $|a_1 - b_1|$, $|a_2 - b_2|$, $|a_3 - b_3|$, $|a_4 - b_4|$, $|a_5 - b_5|$, $|a_6 - b_6|$, $|a_7 - b_7|$ are not all different.
1980 Swedish Mathematical Competition
Let $T(n)$ be the number of dissimilar (non-degenerate) triangles with all side lengths integral and $\leq n$. Find $T(n+1)-T(n)$.
1980 Swedish Mathematical Competition
The functions $f$ and $g$ are positive and continuous. $f$ is increasing and $g$ is decreasing. Show that

$$

\int\limits_0^1 f(x)g(x) dx \leq \int\limits_0^1 f(x)g(1-x) dx

$$
1980 Swedish Mathematical Competition
A  word is a string of the symbols $a, b$ which can be formed by repeated application of the following:

(1) $ab$ is a word;

(2) if $X$ and $Y$ are words, then so is $XY$;

(3) if $X$ is a word, then so is $aXb$.

How many words have $12$ letters?
1980 Swedish Mathematical Competition
Find the smallest constant $c$ such that for every $4$ points in a unit square there are two a distance $\leq c$ apart.
1981 Swedish Mathematical Competition
Let $N = 11\cdots 122 \cdots 25$, where there are $n$ $1$s and $n+1$ $2$s. Show that $N$ is a perfect square.
1981 Swedish Mathematical Competition
Does

$$\left\{ \begin{array}{l}

 x^y = z \\ 

 y^z = x \\
 z^x = y \\  

 \end{array} \right.

$$have any solutions in positive reals apart from $x = y = z= 1$?
1981 Swedish Mathematical Competition
Find all polynomials $p(x)$ of degree $5$ such that $p(x) + 1$ is divisible by $(x-1)^3$ and $p(x) - 1$ is divisible by $(x+1)^3$.
1981 Swedish Mathematical Competition
A cube side $5$ is divided into $125$ unit cubes. $N$ of the small cubes are black and the rest white. Find the smallest $N$ such that there must be a row of $5$ black cubes parallel to one of the edges of the large cube.
1981 Swedish Mathematical Competition
$ABC$ is a triangle. $X$, $Y$, $Z$ lie on $BC$, $CA$, $AB$ respectively. Show that area $XYZ$ cannot be smaller than each of area $AYZ$, area $BZX$, area $CXY$.
1981 Swedish Mathematical Competition
Show that there are infinitely many triangles with side lengths $a$, $b$, $c$, where $a$ is a prime, $b$ is a power of $2$ and $c$ is the square of an odd integer.
1982 Swedish Mathematical Competition
How many solutions does

$$

x^2 - [x^2] = \left(x - [x]\right)^2

$$have satisfying $1 \leq x \leq n$?
1982 Swedish Mathematical Competition
Show that

$$

abc \geq (a+b-c)(b+c-a)(c+a-b)

$$for positive reals $a$, $b$, $c$.
1982 Swedish Mathematical Competition
Show that there is a point $P$ inside the quadrilateral $ABCD$ such that the triangles $PAB$, $PBC$, $PCD$, $PDA$ have equal area. Show that $P$ must lie on one of the diagonals.
1982 Swedish Mathematical Competition
$ABC$ is a triangle with $AB = 33$, $AC = 21$ and $BC = m$, an integer. There are points $D$, $E$ on the sides $AB$, $AC$ respectively such that $AD = DE = EC = n$, an integer. Find $m$.
1982 Swedish Mathematical Competition
Each point in a $12 \times 12$ array is colored red, white or blue. Show that it is always possible to find $4$ points of the same color forming a rectangle with sides parallel to the sides of the array.
1982 Swedish Mathematical Competition
Show that

$$

(2a-1) \sin x + (1-a) \sin(1-a)x \geq 0 

$$for $0 \leq a \leq 1$ and $0 \leq x \leq  \pi$.
1983 Swedish Mathematical Competition
The positive integers are grouped as follows: $1, 2+3, 4+5+6, 7+8+9+10,\dots$. Find the value of the $n$-th sum.
1983 Swedish Mathematical Competition
Show that

$$

\cos x^2 + \cos y^2 - \cos xy < 3

$$for reals $x$, $y$.
1983 Swedish Mathematical Competition
The systems of equations

$$\left\{ \begin{array}{l}

 2x_1 - x_2 = 1 \\ 

  -x_1 + 2x_2 - x_3 = 1 \\
 -x_2 + 2x_3 - x_4 = 1 \\
-x_3 + 3x_4 - x_5 =1 \\
\cdots\cdots\cdots\cdots\\
-x_{n-2} + 2x_{n-1} - x_n = 1 \\
-x_{n-1} + 2x_n = 1 \\     

 \end{array} \right.

$$has a solution in positive integers $x_i$. Show that $n$ must be even.
1983 Swedish Mathematical Competition
$C$, $C'$ are concentric circles with radii $R$, $R'$. A rectangle has two adjacent vertices on $C$ and the other two vertices on $C'$. Find its sides if its area is as large as possible.
1983 Swedish Mathematical Competition
Show that a unit square can be covered with three equal disks with radius less than $\frac{1}{\sqrt{2}}$.

What is the smallest possible radius?
1983 Swedish Mathematical Competition
Show that the only real solution to

$$\left\{ \begin{array}{l}

 x(x+y)^2 = 9 \\ 

 x(y^3 - x^3) = 7  \\
 \end{array} \right.

$$is $x = 1$, $y = 2$.
1984 Swedish Mathematical Competition
Let $A$ and $B$ be two points inside a circle $C$. Show that there exists a circle that contains $A$ and $B$ and lies completely inside $C$.
1984 Swedish Mathematical Competition
The squares in a $3\times 7$ grid are colored either blue or yellow. Consider all $m\times n$ rectangles in this grid, where $m \in \{2,3\}$, $n \in \{2,...,7\}$. Prove that at least one of these rectangles has all four corner squares the same color.
1984 Swedish Mathematical Competition
Prove that if $a,b$ are positive numbers, then

$$\left( \frac{a+1}{b+1}\right)^{b+1} \ge  \left( \frac{a}{b}\right)^{b}$$
1984 Swedish Mathematical Competition
Find all positive integers $p$ and $q$ such that all the roots of the polynomial $(x^2 - px+q)(x^2 -qx+ p)$ are positive integers.
1984 Swedish Mathematical Competition
Solve in natural numbers $a,b,c$ the system $$\left\{ \begin{array}{l}a^3 -b^3  -c^3 = 3abc \\
a^2 = 2(a+b+c)\\
 \end{array} \right.

$$
1984 Swedish Mathematical Competition
Assume $a_1,a_2,...,a_{14}$ are positive integers such that $\sum_{i=1}^{14}3^{a_i} = 6558$.

Prove that the numbers $a_1,a_2,...,a_{14}$ consist of the numbers $1,...,7$, each taken twice.
1985 Swedish Mathematical Competition
If $a > b > 0$, prove the inequality  $$\frac{(a-b)^2}{8a}< \frac{a+b}{2}- \sqrt{ab} < \frac{(a-b)^2}{8b}.$$
1985 Swedish Mathematical Competition
Find the least natural number such that if the first digit (in the decimal system) is placed last, the new number is $7/2 $ times as large as the original number.
1985 Swedish Mathematical Competition
Points $A,B,C$ with $AB = BC$ are given on a circle with radius $r$, and $D$ is a point inside the circle such that the triangle $BCD$ is equilateral. The line $AD$ meets the circle again at $E$. Show that $DE = r$.
1985 Swedish Mathematical Competition
Let $p(x)$ be a polynomial of degree $n$ with real coefficients such that $p(x) \ge  0$ for all $x$. Prove that $p(x)+ p'(x)+ p''(x)+...+ p^{(n)}(x) \ge 0$.
1985 Swedish Mathematical Competition
In a rectangular coordinate system, $O$ is the origin and $A(a,0)$, $B(0,b)$ and $C(c,d)$ the vertices of a triangle. Prove that $AB+BC+CA \ge  2CO$.
1985 Swedish Mathematical Competition
X-wich has a vibrant club-life. For every pair of inhabitants there is exactly one club to which they both belong. For every pair of clubs there is exactly one person who is a member of both. No club has fewer than $3$ members, and at least one club has $17$ members. How many people live in X-wich?
1986 Swedish Mathematical Competition
Show that the polynomial $x^6 -x^5 +x^4 -x^3 +x^2 -x+\frac34$ has no real zeroes.
1986 Swedish Mathematical Competition
The diagonals $AC$ and $BD$ of a quadrilateral $ABCD$ intersect at $O$. If $S_1$ and $S_2$ are the areas of triangles $AOB$ and $COD$ and S that of $ABCD$, show that $\sqrt{S_1} + \sqrt{S_2} \le \sqrt{S}$. Prove that equality holds if and only if $AB$ and $CD$ are parallel.
1986 Swedish Mathematical Competition
Let $N \ge 3$ be a positive integer. For every pair $(a,b)$ of integers with $1 \le a <b \le N$ consider the quotient $q = b/a$. Show that the pairs with $q < 2$ are equally numbered as those with $q > 2$.
1986 Swedish Mathematical Competition
Prove that $x = y = z = 1$ is the only positive solution of the system $$\left\{ \begin{array}{l}

x+y^2 +z^3 = 3\\
y+z^2 +x^3 = 3\\
z+x^2 +y^3 = 3\\
 \end{array} \right.

$$
1986 Swedish Mathematical Competition
In the arrangement of $pn$ real numbers below, the difference between the greatest and smallest numbers in each row is at most $d$, $d > 0$.

$$ \begin{array}{l} a_{11} \,\,  a_{12}  \,\,  ...  \,\, a_{1n}\\
a_{21} \,\, a_{22} \,\,   ...  \,\,  a_{2n}\\
 \,\, .   \,\, \,\, \,\,  \,\, . \,\,   \,\,  \,\,  \,\,  \,\,  \,\,  \,\,  \,\,   .\\
 \,\, .   \,\, \,\, \,\,  \,\, . \,\,   \,\,  \,\,  \,\,  \,\,  \,\,  \,\,  \,\,   .\\
 \,\, .   \,\, \,\, \,\,  \,\, . \,\,   \,\,  \,\,  \,\,  \,\,  \,\,  \,\,  \,\,   .\\
a_{n1} \,\,  a_{n2}  \,\,  ...  \,\, a_{nn}\\
 \end{array} 

$$Prove that, when the numbers in each column are rearranged in decreasing order, the difference between the greatest and smallest numbers in each row will still be at most d.
1986 Swedish Mathematical Competition
The interval $[0,1]$ is covered by a finite number of intervals. Show that one can choose a number of these intervals which are pairwise disjoint and have the total length at least $1/2$.
1987 Swedish Mathematical Competition
Sixteen real numbers are arranged in a magic square of side $4$ so that the sum of numbers in each row, column or main diagonal equals $k$. Prove that the sum of the numbers in the four corners of the square is also $k$.
1987 Swedish Mathematical Competition
A circle of radius $R$ is divided into two parts of equal area by an arc of another circle. Prove that the length of this arc is greater than $2R$.
1987 Swedish Mathematical Competition
Ten closed intervals, each of length $1$, are placed in the interval $[0,4]$. Show that there is a point in the larger interval that belongs to at least four of the smaller intervals.
1987 Swedish Mathematical Competition
A differentiable function $f$ with $f(0) = f(1) = 0$ is defined on the interval $[0,1]$.

Prove that there exists a point $y \in  [0,1]$ such that $| f' (y)| = 4 \int _0^1 | f(x)|dx$.
1987 Swedish Mathematical Competition
Show that there exists a positive number t such that for all positive numbers $a,b,c,d$ with $abcd = 1$,

$$\frac{1}{1+a}+\frac{1}{1+b}+\frac{1}{1+c}+\frac{1}{1+d}> t.$$and find the largest $t$ with this property.
1987 Swedish Mathematical Competition
A baker with access to a number of different spices bakes ten cakes. He uses more than half of the different kinds of spices in each cake, but no two of the combinations of spices are exactly the same. Show that there exist three spices $a,b,c$ such that every cake contains at least one of these.
1988 Swedish Mathematical Competition
Let $a > b > c$ be sides of a triangle and $h_a,h_b,h_c$ be the corresponding altitudes.

Prove that $a+h_a > b+h_b > c+h_c$.
1988 Swedish Mathematical Competition
Six ducklings swim on the surface of a pond, which is in the shape of a circle with radius $5$ m. Show that at every moment, two of the ducklings swim on the distance of at most $5$ m from each other.
1988 Swedish Mathematical Competition
Show that if $x_1+x_2+x_3 = 0$ for real numbers $x_1,x_2,x_3$, then $x_1x_2+x_2x_3+x_3x_1\le 0$.



Find all $n \ge 4$ for which $x_1+x_2+...+x_n = 0$ implies $x_1x_2+x_2x_3+...+x_{n-1}x_n+x_nx_1 \le  0$.
1988 Swedish Mathematical Competition
A polynomial $P(x)$ of degree $3$ has three distinct real roots.

Find the number of real roots of the equation $P'(x)^2 -2P(x)P''(x) = 0$.
1988 Swedish Mathematical Competition
Show that there exists a constant $a > 1$ such that, for any positive integers $m$ and $n$, $\frac{m}{n} < \sqrt7$ implies that $$7-\frac{m^2}{n^2} \ge \frac{a}{n^2} .$$
1988 Swedish Mathematical Competition
The sequence $(a_n)$ is defined by $a_1 = 1$ and $a_{n+1} = \sqrt{a_n^2  +\frac{1}{a_n}}$ for $n \ge 1$.

Prove that there exists $a$ such that $\frac{1}{2} \le \frac{a_n}{n^a} \le 2$ for $n \ge 1$.
1989 Swedish Mathematical Competition
Let $n$ be a positive integer. Prove that the numbers $n^2(n^2 + 2)^2$ and $n^4(n^2 + 2)^2$ are written in base $n^2 +1$ with the same digits but in opposite order.
1989 Swedish Mathematical Competition
Find all continuous functions $f$ such that $f(x)+ f(x^2) = 0$ for all real numbers $x$.
1989 Swedish Mathematical Competition
Find all positive integers $n$ such that $n^3 - 18n^2 + 115n - 391$ is the cube of a positive intege
1989 Swedish Mathematical Competition
Let $ABCD$ be a regular tetrahedron. Find the positions of point $P$ on the edge $BD$ such that the edge $CD$ is tangent to the sphere with diameter $AP$.
1989 Swedish Mathematical Competition
Assume $x_1,x_2,..,x_5$ are positive numbers such that $x_1 < x_2$ and $x_3,x_4, x_5$ are all greater than $x_2$. Prove that if $a > 0$, then

$$\frac{1}{(x_1 +x_3)^a}+ \frac{1}{(x_2 +x_4)^a}+ \frac{1}{(x_2 +x_5)^a} <\frac{1}{(x_1 +x_2)^a}+ \frac{1}{(x_2 +x_3)^a}+ \frac{1}{(x_4 +x_5)^a}$$
1989 Swedish Mathematical Competition
On a circle $4n$ points are chosen ($n \ge 1$). The points are alternately colored yellow and blue. The yellow points are divided into $n$ pairs and the points in each pair are connected with a yellow line segment. In the same manner the blue points are divided into $n$ pairs and the points in each pair are connected with a blue segment. Assume that no three of the segments pass through a single point. Show that there are at least $n$ intersection points of blue and yellow segments.
1990 Swedish Mathematical Competition
Let $d_1, d_2, ... , d_k$ be the positive divisors of $n = 1990!$. Show that $\sum \frac{d_i}{\sqrt{n}} = \sum \frac{\sqrt{n}}{d_i}$.
1990 Swedish Mathematical Competition
The points $A_1, A_2,.. , A_{2n}$ are equally spaced in that order along a straight line with $A_1A_2 = k$. $P$ is chosen to minimise $\sum PA_i$. Find the minimum.
1990 Swedish Mathematical Competition
Find all $a, b$ such that $\sin x + \sin a\ge  b \cos x$ for all $x$.
1990 Swedish Mathematical Competition
$ABCD$ is a quadrilateral. The bisectors of $\angle A$ and $\angle B$ meet at $E$. The line through $E$ parallel to $CD$ meets $AD$ at $L$ and $BC$ at $M$. Show that $LM = AL + BM$.
1990 Swedish Mathematical Competition
Find all monotonic positive functions $f(x)$ defined on the positive reals such that $f(xy) f\left( \frac{f(y)}{x}\right) = 1$ for all $x, y$.
1990 Swedish Mathematical Competition
Find all positive integers $m, n$ such that $\frac{117}{158} > \frac{m}{n} > \frac{97}{131}$ and $n \le 500$.
1991 Swedish Mathematical Competition
Find all positive integers $m, n$ such that $\frac{1}{m} + \frac{1}{n} - \frac{1}{mn} =\frac{2}{5}$.
1991 Swedish Mathematical Competition
$x, y$ are positive reals such that $x - \sqrt{x} \le  y - 1/4 \le x + \sqrt{x}$. Show that $y - \sqrt{y} \le x - 1/4 \le y + \sqrt{y}$.
1991 Swedish Mathematical Competition
The sequence $x_0, x_1, x_2, ...$ is defined by $x_0 = 0$, $x_{k+1} = [(n - \sum_0^k x_i)/2]$. Show that $x_k = 0$ for all sufficiently large $k$ and that the sum of the non-zero terms $x_k$ is $n-1$.
1991 Swedish Mathematical Competition
$x_1, x_2, ... , x_8$ is a permutation of $1, 2, ..., 8$. A move is to take $x_3$ or $x_8$ and place it at the start to from a new sequence. Show that by a sequence of moves we can always arrive at $1, 2, ..., 8$.
1991 Swedish Mathematical Competition
Show that there are infinitely many odd positive integers $n$ such that in binary $n$ has more $1$s than $n^2$.
1991 Swedish Mathematical Competition
Given any triangle, show that we can always pick a point on each side so that the three points form an equilateral triangle with area at most one quarter of the original triangle.
1992 Swedish Mathematical Competition
Is $\frac{19^{92} - 91^{29}}{90}$ an integer?
1992 Swedish Mathematical Competition
The squares in a $9\times 9$ grid are numbered from $11$ to $99$, where the first digit is the row and the second the column. Each square is colored black or white. Squares $44$ and $49$ are black. Every black square shares an edge with at most one other black square, and each white square shares an edge with at most one other white square. What color is square $99$?
1992 Swedish Mathematical Competition
Solve:

$2x_1 - 5x_2 + 3x_3 \ge  0$

$2x_2 - 5x_3 + 3x4  \ge  0$

$...$

$2x_{23} - 5x_{24} + 3x_{25}  \ge  0$

$2x_{24} - 5x_{25} + 3x_1  \ge  0$

$2x_{25} - 5x_1 + 3x_2  \ge  0$
1992 Swedish Mathematical Competition
Find all positive integers $a, b, c$ such that $a < b$, $a < 4c$, and $b c^3 \le a c^3 + b$.
1992 Swedish Mathematical Competition
A triangle has sides $a, b, c$ with longest side $c$, and circumradius $R$. Show that if $a^2 + b^2 = 2cR$, then the triangle is right-angled.
1992 Swedish Mathematical Competition
$(x_1, y_1), (x_2, y_2), (x_3, y_3)$ lie on a straight line and on the curve $y^2 = x^3$. Show that $\frac{x_1}{y_1} + \frac{x_2}{y_2}+\frac{x_3}{y_3} = 0$.
1993 Swedish Mathematical Competition
An integer $x$ has the property that the sums of the digits of $x$ and of $3x$ are the same. Prove that $x$ is divisible by $9$.
1993 Swedish Mathematical Competition
A railway line is divided into ten sections by the stations $A,B,C,D,E,F, G,H,I,J,K$. The length of each section is an integer number of kilometers and the distacne between $A$ and $K$ is $56$ km. A trip along two successive sections never exceeds $12$ km, but a trip along three successive sections is at least $17$ km. What is the distance between $B$ and $G$?
1993 Swedish Mathematical Competition
Assume that $a$ and $b$ are integers. Prove that the equation $a^2 +b^2 +x^2 = y^2$ has an integer solution $x,y$ if and only if the product $ab$ is even.
1993 Swedish Mathematical Competition
To each pair of nonzero real numbers $a$ and $b$ a real number $a*b$ is assigned so that $a*(b*c) = (a*b)c$ and $a*a = 1$ for all $a,b,c$. Solve the equation $x*36 = 216$.
1993 Swedish Mathematical Competition
A triangle with sides $a,b,c$ and perimeter $2p$ is given. Is possible, a new triangle with sides $p-a$, $p-b$, $p-c$ is formed. The process is then repeated with the new triangle. For which original triangles can this process be repeated indefinitely?
1993 Swedish Mathematical Competition
For real numbers $a$ and $b$ define $f(x) = \frac{1}{ax+b}$. For which $a$ and $b$ are there three distinct real numbers $x_1,x_2,x_3$ such that $f(x_1) = x_2$, $f(x_2) = x_3$ and $f(x_3) = x_1$?
1994 Swedish Mathematical Competition
$x\sqrt8 + \frac{1}{x\sqrt8} = \sqrt8$ has two real solutions $x_1, x_2$. The decimal expansion of $x_1$ has the digit $6$ in place $1994$. What digit does $x_2$ have in place $1994$?
1994 Swedish Mathematical Competition
In the triangle $ABC$, the medians from $B$ and $C$ are perpendicular. Show that $\cot B + \cot C \ge  \frac23$.
1994 Swedish Mathematical Competition
The vertex $B$ of the triangle $ABC$ lies in the plane $P$. The plane of the triangle meets the plane in a line $L$. The angle between $L$ and $AB$ is a, and the angle between $L$ and $BC$ is $b$. The angle between the two planes is $c$. Angle $ABC$ is $90^o$. Show that $\sin^2c = \sin^2a + \sin^2b$.
1994 Swedish Mathematical Competition
Find all integers $m, n$ such that $2n^3 - m^3 = mn^2 + 11$.
1994 Swedish Mathematical Competition
The polynomial $x^k + a_1x^{k-1} + a_2x^{k-2} +... + a_k$ has $k$ distinct real roots. Show that $a_1^2 > \frac{2ka_2}{k-1}$.
1994 Swedish Mathematical Competition
Let $N$ be the set of non-negative integers. The function $f:N\to N$ satisfies $f(a+b) = f(f(a)+b)$ for all $a, b$ and $f(a+b) = f(a)+f(b)$ for $a+b < 10$. Also $f(10) = 1$. How many three digit numbers $n$ satisfy $f(n) = f(N)$, where $N$ is the "tower" $2, 3, 4, 5$, in other words, it is $2^a$, where $a = 3^b$, where $b = 4^5$?
1995 Swedish Mathematical Competition
All pages of a magazine are numbered and printed on both sides. One sheet with two sides is missing. The numbers of the remaining pages sum to $963$. How many pages did the magazine have originally and which pages are missing?
1995 Swedish Mathematical Competition
Botvid left home between $4$ and $5$ for a short visit to Amanda. When he came back between $5$ and $6$, he found that the hands of the clock had changed places. What time was it?
1995 Swedish Mathematical Competition
Let $a,b,x,y$ be positive numbers with $a+b+x+y < 2$. Given that $a+b^2 = x+y^2$ and $a^2 +b = x^2 +y$, show that $a = x$ and $b = y$
1995 Swedish Mathematical Competition
The product of three positive numbers is $1$ and their sum is greater than the sum of their inverses. Prove that one of these numbers is greater than $1$, while the other two are smaller than $1$.
1995 Swedish Mathematical Competition
On a circle with center $O$ and radius $r$ are given points $A,B,C,D$ in this order such that $AB, BC$ and $CD$ have the same length $s$ and the length of $AD$ is $s+ r$.Assume that $s < r$. Determine the angles of quadrilateral $ABCD$.
1995 Swedish Mathematical Competition
Signals used for communication are binary sequences of length $10$. Unfortunately, the receiving device got broken so that it cannot distinguish between two signals unless those differ in more than five places. What is the largest possible number of signals that can still be used to prevent ambiguities?
1996 Swedish Mathematical Competition
Through an arbitrary point inside a triangle, lines parallel to the sides of the triangle are drawn, dividing the triangle into three triangles with areas $T_1,T_2,T_3$ and three parallelograms. If $T$ is the area of the original triangle, prove that

$$T=(\sqrt{T_1}+\sqrt{T_2}+\sqrt{T_3})^2$$.