Navvye/Olympiad-Problems-CSV huggingface dataset:
1965 Spain Mathematical Olympiad
We have an empty equilateral triangle with length of a side $l$. We put the triangle, horizontally, over a sphere of radius $r$. Clearly, if the triangle is small enough, the triangle is held by the sphere. Which is the distance between any vertex of the triangle and the centre of the sphere (as a function of $l$ and $r$)?
1965 Spain Mathematical Olympiad
A truncated cone has the bigger base of radius $r$ centimetres and the generatrix makes an angle, with that base, whose tangent equals $m$. The truncated cone is constructed of a material of density $d$ (g/cm$^3$) and the smaller base is covered by a special material of density $p$ (g/cm$^2$). Which is the height of the truncated cone that maximizes the total mass?
1965 Spain Mathematical Olympiad
Let be $\gamma_1$ a circumference of radius $r$ and $P$ an exterior point that is at distance $a$ from the centre of $\gamma_1$. We build two tangent lines $r,s$ to $\gamma_1$ from $P$ and $\gamma_2$ is constructed as a smaller circumference, tangent to both lines and, also, tangent to $\gamma_1$. We construct inductively $\gamma_{n+1}$ as a tangent circumference to $\gamma_{n}$ and, also, tangent to $r$ and $s$. Determine:



a) The radius of $\gamma_2$.

b) The radius of $\gamma_n$.

c) The sum of the lengths of $\gamma_1, \gamma_2, \gamma_3, ...$.
1964 Spain Mathematical Olympiad
Given the equation $x^2+ax+1=0$, determine:



a) The interval of possible values for $a$ where the solutions to the previous equation are not real.

b) The loci of the roots of the polynomial, when $a$ is in the previous interval.
1964 Spain Mathematical Olympiad
The RTP tax is a function $f(x)$, where $x$ is the total of the annual profits (in pesetas). Knowing that:



a) $f(x)$ is a continuous function

b) The derivative $\frac{df(x)}{dx}$ on the interval $0 \leq 6000$ is constant and equals zero; in the interval $6000< x < P$ is constant and equals $1$; and when $x>P$ is constant and equal 0.14.

c) $f(0)=0$ and $f(140000)=14000$.



Determine the value of the amount $P$ (in pesetas) and represent graphically the function $y=f(x)$.
1964 Spain Mathematical Olympiad
A convex polygon of $n$ sides is considered. All its diagonals are drawn and we suppose that any three of them can only intersect on a vertex and that there is no pair of parallel diagonals. Under these conditions, we wish to compute



a) The total number of intersection points of these diagonals, excluding the vertices.



b) How many points, of these intersections, lie inside the polygon and how many lie outside.
1964 Spain Mathematical Olympiad
We are given an equilateral triangle $ABC$, of side $a$, inside its circumscribed circle. We consider the smallest of the two portions of circle limited between $AB$ and the circumference. If we consider parallel lines to $BC$, some of them cut the portion of circle in a segment. Which is the maximum possible length for one of the segments?
1964 Spain Mathematical Olympiad
Given a regular pentagon, its five diagonals are drawn. How many triangles do appear in the figure? Classify the set of triangles in classes of equal triangles.
1964 Spain Mathematical Olympiad
Make a graphical representation of the function $y=\vert \vert \vert x-1 \vert -2 \vert -3 \vert$ on the interval $-8 \leq x \leq 8$.
1964 Spain Mathematical Olympiad
A table with 1000 cards on a line, numbered from 1 to 1000, is considered. The cards are ordered in the usual way. Now, we proceed in the following way.



The first card (which is 1) is put just before the last card (between 999 and 1000) and, after, the new first card (which is 2) is put after the last card (which was 1000). Show that after 1000 movements, the cards are ordered again in the usual way. Show that the analogous result ($n$ movements for $n$ cards) does not hold when $n$ is odd.
1964 Spain Mathematical Olympiad
The points $A$ and $B$ lie on a horizontal line over a vertical plane. We consider the semicircumference passing through $A$ and $B$ that lies under the horizontal line. A segment of length $a$, with the same diameter that the semicircumference, moves in a way that always contains the point $A$ and one of its extremes lies always on the semicircumference. Determine the value of the cosine of the angle between this segment and the horizontal line that makes the medium point of the segment to be as down as possible.
1962 Swedish Mathematical Competition
Find all polynomials $f(x)$ such that $f(2x) = f'(x) f''(x)$.
1962 Swedish Mathematical Competition
$ABCD$ is a square side $1$. $P$ and $Q$ lie on the side $AB$ and $R$ lies on the side $CD$. What are the possible values for the circumradius of $PQR$?
1962 Swedish Mathematical Competition
Find all pairs $(m, n)$ of integers such that $n^2 - 3mn + m - n = 0$.
1962 Swedish Mathematical Competition
Which of the following statements are true?

(A) $X$ implies $Y$, or $Y$ implies $X$, where $X$ is the statement, the lines $L_1, L_2, L_3$ lie in a plane, and $Y$ is the statement, each pair of the lines  $L_1, L_2, L_3$ intersect.

(B) Every sufficiently large integer $n$ satisfies $n = a^4 + b^4$ for some integers a, b.

(C) There are real numbers $a_1, a_2,... , a_n$ such that $a_1 \cos x + a_2 \cos 2x +... + a_n \cos nx > 0$ for all real $x$.
1962 Swedish Mathematical Competition
Find the largest cube which can be placed inside a regular tetrahedron with side $1$ so that one of its faces lies on the base of the tetrahedron.
1963 Swedish Mathematical Competition.
How many positive integers have square less than $10^7$?
1963 Swedish Mathematical Competition.
The squares of a chessboard have side $4$. What is the circumference of the largest circle that can be drawn entirely on the black squares of the board?
1963 Swedish Mathematical Competition.
What is the remainder on dividing $1234^{567} + 89^{1011}$ by $12$?
1963 Swedish Mathematical Competition.
Given the real number $k$, find all differentiable real-valued functions $f(x)$ defined on the reals such that $f(x+y) = f(x) + f(y) + f(kxy)$ for all $x, y$.
1963 Swedish Mathematical Competition.
A road has constant width. It is made up of finitely many straight segments joined by corners, where the inner corner is a point and the outer side is a circular arc. The direction of the straight sections is always between $NE$ ($45^o$) and $SSE$ ($157 1/2^o$). A person wishes to walk along the side of the road from point $A$ to point $B$ on the same side. He may only cross the street perpendicularly. What is the shortest route?



[figure missing]
1963 Swedish Mathematical Competition.
The real-valued function $f(x)$ is defined on the reals. It satisfies $|f(x)| \le A$, $|f''(x)| \le B$ for some positive $A, B$ (and all $x$). Show that $|f'(x)| \le  C$, for some fixed$ C$, which depends only on $A$ and $B$. What is the smallest possible value of $C$?
1964 Swedish Mathematical Competition
Find the side lengths of the triangle $ABC$ with area $S$ and $\angle BAC = x$ such that the side $BC$ is as short as possible.
1964 Swedish Mathematical Competition
Find all positive integers $m, n$ such that $n + (n+1) + (n+2) + ...+ (n+m) = 1000$.
1964 Swedish Mathematical Competition
Find a polynomial with integer coefficients which has $\sqrt2 + \sqrt3$ and $\sqrt2 + \sqrt[3]{3}$ as roots.
1964 Swedish Mathematical Competition
Points $H_1, H_2, ... , H_n$ are arranged in the plane so that each distance $H_iH_j \le 1$. The point $P$ is chosen to minimise $\max (PH_i)$. Find the largest possible value of $\max (PH_i)$ for $n = 3$. Find the best upper bound you can for $n = 4$.
1964 Swedish Mathematical Competition
$a_1, a_2, ... , a_n$ are constants such that $f(x) = 1 + a_1 cos x + a_2 cos 2x + ...+ a_n cos nx \ge 0$ for all $x$. We seek estimates of $a_1$. If $n = 2$, find the smallest and largest possible values of $a_1$. Find corresponding estimates for other values of $n$.
1965 Swedish Mathematical Competition
The feet of the altitudes in the triangle $ABC$ are $A', B', C'$. Find the angles of $A'B'C'$ in terms of the angles $A, B, C$. Show that the largest angle in $A'B'C'$ is at least as big as the largest angle in $ABC$. When is it equal?
1965 Swedish Mathematical Competition
Find all positive integers m, n such that $m^3 - n^3 = 999$.
1965 Swedish Mathematical Competition
Show that for every real $x \ge \frac12$ there is an integer $n$ such that $|x - n^2| \le \sqrt{x-\frac{1}{4}}$.
1965 Swedish Mathematical Competition
Find constants $A > B$ such that $\frac{f\left( \frac{1}{1+2x}\right) }{f(x)}$ is independent of $x$, where $f(x) = \frac{1 + Ax}{1 + Bx}$ for all real $x \ne - \frac{1}{B}$. Put $a_0 = 1$, $a_{n+1} = \frac{1}{1 + 2a_n}$. Find an expression for an by considering $f(a_0), f(a_1), ...$.
1965 Swedish Mathematical Competition
Let $S$ be the set of all real polynomials $f(x) = ax^3 + bx^2 + cx + d$ such that $|f(x)| \le 1$ for all $ -1 \le x \le 1$. Show that the set of possible $|a|$ for $f$ in $S$ is bounded above and find the smallest upper bound.
1966 Swedish Mathematical Competition
Let $\{x\}$ denote the fractional part of $x$, $x -  [x]$. The sequences $x_1, x_2, x_3, ...$ and $y_1, y_2, y_3, ...$ are such that $\lim \{x_n\} = \lim \{y_n\} = 0$. Is it true that $\lim \{x_n + y_n\} = 0$? $\lim \{x_n - y_n\} = 0$?
1966 Swedish Mathematical Competition
$a_1 + a_2 + ... + a_n = 0$, for some $k$ we have $a_j \le 0$ for $j \le  k$ and $a_j \ge 0$ for $j > k$. If ai are not all $0$, show that $a_1 + 2a_2 + 3a_3 + ... + na_n > 0$.
1966 Swedish Mathematical Competition
Show that an integer $= 7 \mod 8$ cannot be sum of three squares.
1966 Swedish Mathematical Competition
Let $f(x) = 1 + \frac{2}{x}$. Put $f_1(x) = f(x)$, $f_2(x) = f(f_1(x))$, $f_3(x) = f(f_2(x))$, $... $. Find the solutions to $x = f_n(x)$ for $n > 0$.
1966 Swedish Mathematical Competition
Let $f(r)$ be the number of lattice points inside the circle radius $r$, center the origin.

Show that $\lim_{r\to \infty} \frac{f(r)}{r^2}$ exists and find it. If the limit is $k$, put $g(r) = f(r) - kr^2$.

Is it true that $\lim_{r\to \infty} \frac{g(r)}{r^h} = 0$ for any $h < 2$?
1967 Swedish Mathematical Competition
$p$ parallel lines are drawn in the plane and $q$ lines perpendicular to them are also drawn. How many rectangles are bounded by the lines?
1967 Swedish Mathematical Competition
You are given a ruler with two parallel straight edges a distance $d$ apart. It may be used

(1) to draw the line through two points,

(2) given two points a distance $\ge d$ apart, to draw two parallel lines, one through each point,

(3) to draw a line parallel to a given line, a distance d away.

One can also (4) choose an arbitrary point in the plane, and (5) choose an arbitrary point on a line.

Show how to construct (A) the bisector of a given angle, and

(B) the perpendicular to the midpoint of a given line segment.
1967 Swedish Mathematical Competition
Show that there are only finitely many triples $(a, b, c)$ of positive integers such that $\frac{1}{a} + \frac{1}{b} + \frac{1}{c} = \frac{1}{1000}$.
1967 Swedish Mathematical Competition
The sequence $a_1, a_2, a_3, ...$ of positive reals is such that $\sum a_i$ diverges.

Show that there is a sequence $b_1, b_2, b_3, ...$ of positive reals such that $\lim b_n = 0$ and  $\sum a_ib_i$ diverges.
1967 Swedish Mathematical Competition
$a_1, a_2, a_3, ...$ are positive reals such that $a_n^2 \ge a_1 + a_2 +... + a_{n-1}$.

Show that for some $C > 0$ we have $a_n \ge C n$ for all $n$.
1967 Swedish Mathematical Competition
The vertices of a triangle are lattice points. There are no lattice points on the sides (apart from the vertices) and $n$ lattice points inside the triangle. Show that its area is $n + \frac12$. Find the formula for the general case where there are also $m$ lattice points on the sides (apart from the vertices).
1968 Swedish Mathematical Competition
Find the maximum and minimum values of $x^2 + 2y^2 + 3z^2$ for real $x, y, z$ satisfying $x^2 + y^2 + z^2 = 1$.
1968 Swedish Mathematical Competition
How many different ways (up to rotation) are there of labeling the faces of a cube with the numbers $1, 2,..., 6$?
1968 Swedish Mathematical Competition
Show that the sum of the squares of the sides of a quadrilateral is at least the sum of the squares of the diagonals. When does equality hold?
1968 Swedish Mathematical Competition
For $n\ne  0$, let f(n) be the largest $k$ such that $3^k$ divides $n$. If $M$ is a set of $n > 1$ integers, show that the number of possible values for $f(m-n)$, where $m, n$ belong to $M$ cannot exceed $n-1$.
1968 Swedish Mathematical Competition
Let $a, b$ be non-zero integers. Let $m(a, b)$ be the smallest value of $\cos ax + \cos bx$ (for real $x$).

Show that for some $r$, $m(a, b) \le r < 0$ for all $a, b$.
1969 Swedish Mathematical Competition
Find all integers m, n such that $m^3 = n^3 + n$.
1969 Swedish Mathematical Competition
Show that $\tan \frac{\pi}{3n}$ is irrational for all positive integers $n$.
1969 Swedish Mathematical Competition
$a_1 \le a_2 \le  ... \le  a_n$ is a sequence of reals  $b_1, _b2, b_3,..., b_n$ is any rearrangement of the sequence $B_1 \le B_2 \le  ...\le B_n$. Show that $\sum  a_ib_i \le  \sum a_i B_i$.
1969 Swedish Mathematical Competition
Define $g(x)$ as the largest value of$ |y^2 - xy|$ for $y$ in $[0, 1]$. Find the minimum value of $g$ (for real $x$).
1969 Swedish Mathematical Competition
Let $N = a_1a_2...a_n$ in binary. Show that if $a_1-a_2 + a_3 -... + (-1)^{n-1}a_n = 0$ mod $3$, then $N = 0$ mod $3$.
1969 Swedish Mathematical Competition
Given $3n$ points in the plane, no three collinear, is it always possible to form $n$ triangles (with vertices at the points), so that no point in the plane lies in more than one triangle?
1970 Swedish Mathematical Competition
Show that infinitely many positive integers cannot be written as a sum of three fourth powers of integers.
1970 Swedish Mathematical Competition
$6$ open disks in the plane are such that the center of no disk lies inside another. Show that no point lies inside all $6$ disks.
1970 Swedish Mathematical Competition
A polynomial with integer coefficients takes the value $5$ at five distinct integers. Show that it does not take the value $9$ at any integer.
1970 Swedish Mathematical Competition
Let $p(x) = (x- x_1)(x- x_2)(x- x_3)$, where $x_1, x_2$ and $x_3$ are real. Show that $p(x) p''(x) \le p'(x)^2$ for all $x$.
1970 Swedish Mathematical Competition
A $3\times 1$ paper rectangle is folded twice to give a square side $1$. The square is folded along a diagonal to give a right-angled triangle. A needle is driven through an interior point of the triangle, making $6$ holes in the paper. The paper is then unfolded. Where should the point be in order to maximise the smallest distance between any two holes?
1970 Swedish Mathematical Competition
Show that $\frac{(n - m)!}{m!} \le  \left(\frac{n}{2} + \frac{1}{2}\right)^{n-2m}$ for positive integers $m, n$ with $2m \le  n$.
1971 Swedish Mathematical Competition
Show that

$$

\left(1 + a + a^2\right)^2 < 3\left(1 + a^2 + a^4\right) 

$$for real $a \neq 1$.
1971 Swedish Mathematical Competition
An arbitrary number of lines divide the plane into regions. Show that the regions can be colored red and blue so that neighboring regions have different colors.
1971 Swedish Mathematical Competition
A table is covered by $15$ pieces of paper. Show that we can remove $7$ pieces so that the remaining $8$ cover at least $8/15$ of the table.
1971 Swedish Mathematical Competition
Find

$$

\frac{65533^3 + 65534^3 + 65535^3 + 65536^3 + 65537^3 + 65538^3+ 65539^3}{32765\cdot 32766 + 32767\cdot 32768 + 32768\cdot 32769 + 32770\cdot 32771}

$$
1971 Swedish Mathematical Competition
Show that  $$

\max\limits_{|x|\leq t} |1 - a \cos x| \geq \tan^2 \frac{t}{2} 

$$for $a$ positive and $t \in (0, \frac{\pi}{2})$.
1971 Swedish Mathematical Competition
$99$ cards each have a label chosen from $1,2,\dots,99$, such that no (non-empty) subset of the cards has labels with total divisible by $100$. Show that the labels must all be equal.
1972 Swedish Mathematical Competition
Find the largest real number $a$ such that $$\left\{ \begin{array}{l}

x - 4y = 1 \\
ax + 3y = 1\\
 \end{array} \right.

$$has an integer solution.
1972 Swedish Mathematical Competition
A rectangular grid of streets has $m$ north-south streets and $n$ east-west streets. For which $m, n > 1$ is it possible to start at an intersection and drive through each of the other intersections just once before returning to the start?
1972 Swedish Mathematical Competition
A steak temperature $5^\circ$ is put into an oven. After $15$ minutes, it has temperature $45^\circ$. After another $15$ minutes it has temperature $77^\circ$. The oven is at a constant temperature. The steak changes temperature at a rate proportional to the difference between its temperature and that of the oven. Find the oven temperature.
1972 Swedish Mathematical Competition
Put $x = \log_{10} 2$, $y = \log_{10} 3$. Then $15 < 16$ implies $1 - x + y < 4x$, so $1 + y < 5x$.

Derive similar inequalities from $80 < 81$ and $243 < 250$. Hence show that $$

0.47 < \log_{10} 3 < 0.482.

$$
1972 Swedish Mathematical Competition
Show that

$$

\int\limits_0^1 \frac{1}{(1+x)^n} dx > 1-\frac{1}{n}

$$for all positive integers $n$.
1972 Swedish Mathematical Competition
$a_1,a_2,a_3,\dots$ and $b_1,b_2,b_3,\dots$ are sequences of positive integers. Show that we can find $m < n$ such that $a_m \leq a_n$ and $b_m \leq b_n$.
1973 Swedish Mathematical Competition
$\log_8 2 = 0.2525$ in base $8$ (to $4$ places of decimals). Find $\log_8 4$ in base $8$ (to $4$ places of decimals).
1973 Swedish Mathematical Competition
The Fibonacci sequence $f_1,f_2,f_3,\dots$ is defined by $f_1=f_2=1$, $f_{n+2}=f_{n+1}+f_n$. Find all $n$ such that $f_n = n^2$.
1973 Swedish Mathematical Competition
$ABC$ is a triangle with $\angle A = 90^\circ$, $\angle B = 60^\circ$. The points $A_1$, $B_1$, $C_1$ on $BC$, $CA$, $AB$ respectively are such that $A_1B_1C_1$ is equilateral and the perpendiculars (to $BC$ at $A_1$, to $CA$ at $B_1$ and to $AB$ at $C_1$) meet at a point $P$ inside the triangle. Find the ratios $PA_1:PB_1:PC_1$.
1973 Swedish Mathematical Competition
$p$ is a prime. Find all relatively prime positive integers $m$, $n$ such that

$$

\frac{m}{n}+\frac{1}{p^2}=\frac{m+p}{n+p}

$$
1973 Swedish Mathematical Competition
$f(x)$ is a polynomial of degree $2n$. Show that all polynomials $p(x)$, $q(x)$ of degree at most $n$ such that $f(x)q(x)-p(x)$ has the form

$$

\sum\limits_{2n<k\leq 3n} (a^k + x^k)

$$have the same $p(x)/q(x)$.
1973 Swedish Mathematical Competition
$f(x)$ is a real valued function defined for $x \geq 0$ such that $f(0) = 0$, $f(x+1)=f(x)+\sqrt{x}$ for all $x$, and

$$

f(x) < \frac{1}{2}f\left(x - \frac{1}{2}\right)+\frac{1}{2}f\left(x + \frac{1}{2}\right) \quad \text{for all} \quad x \geq \frac{1}{2}

$$Show that $f\left(\frac{1}{2}\right)$ is uniquely determined.
1974 Swedish Mathematical Competition
Let $a_n = 2^{n-1}$ for $n > 0$. Let

$$

b_n = \sum\limits_{r+s \leq n} a_ra_s 

$$Find $b_n-b_{n-1}$, $b_n-2b_{n-1}$ and $b_n$.
1974 Swedish Mathematical Competition
Show that

$$

1 - \frac{1}{k} \leq n\left(\sqrt[n]{k}-1\right) \leq k - 1 

$$for all positive integers $n$ and positive reals $k$.
1974 Swedish Mathematical Competition
Let $a_1=1$, $a_2=2^{a_1}$, $a_3=3^{a_2}$, $a_4=4^{a_3}$, $\dots$, $a_9 = 9^{a_8}$. Find the last two digits of $a_9$.
1974 Swedish Mathematical Competition
Find all polynomials $p(x)$ such that $p(x^2) = p(x)^2$ for all $x$. Hence find all polynomials $q(x)$ such that

$$

q\left(x^2 - 2x\right) = q\left(x-2\right)^2

$$
1974 Swedish Mathematical Competition
Find the smallest positive real $t$ such that

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

 x_1 + x_3 = 2t x_2 \\ 

 x_2 + x_4 = 2t x_3 \\
 x_3 + x_5=2t x_4 \\  

 \end{array} \right.

$$has a solution $x_1$, $x_2$, $x_3$, $x_4$, $x_5$ in non-negative reals, not all zero.
1974 Swedish Mathematical Competition
For which $n$ can we find positive integers $a_1,a_2,\dots,a_n$ such that

$$

a_1^2+a_2^2+\cdots+a_n^2

$$is a square?
1975 Swedish Mathematical Competition
$A$ is the point $(1,0)$, $L$ is the line $y = kx$ (where $k > 0$). For which points $P(t,0)$ can we find a point $Q$ on $L$ such that $AQ$ and $QP$ are perpendicular?
1975 Swedish Mathematical Competition
Is there a positive integer $n$ such that the fractional part of

$$

\left(3+\sqrt{5}\right)^n >0.99 ?

$$
1975 Swedish Mathematical Competition
Show that

$$

a^n + b^n + c^n \geq ab^{n-1} + bc^{n-1} + ca^{n-1}

$$for real $a,b,c \geq 0$ and $n$ a positive integer.
1975 Swedish Mathematical Competition
$P_1$, $P_2$, $P_3$, $Q_1$, $Q_2$, $Q_3$ are distinct points in the plane. The distances $P_1Q_1$, $P_2Q_2$, $P_3Q_3$ are equal. $P_1P_2$ and $Q_2Q_1$ are parallel (not antiparallel), similarly $P_1P_3$ and $Q_3Q_1$, and $P_2P_3$ and $Q_3Q_2$. Show that $P_1Q_1$, $P_2Q_2$ and $P_3Q_3$ intersect in a point.
1975 Swedish Mathematical Competition
Show that $n$ divides $2^n + 1$ for infinitely many positive integers $n$.
1975 Swedish Mathematical Competition
$f(x)$ is defined for $0 \leq x \leq 1$ and has a continuous derivative satisfying $|f'(x)| \leq C|f(x)|$ for some positive constant $C$. Show that if $f(0) = 0$, then $f(x)=0$ for the entire interval.
1976 Swedish Mathematical Competition
In a tournament every team plays every other team just once. Each game is won by one of the teams (there are no draws). Each team loses at least once. Show that there must be three teams $A$, $B$, $C$ such that $A$ beat $B$, $B$ beat $C$ and $C$ beat $A$.
1976 Swedish Mathematical Competition
For which real $a$ are there distinct reals $x$, $y$ such that $x = a - y^2$ and $y = a - x^2$?
1976 Swedish Mathematical Competition
If $a$, $b$, $c$ are rational, show that

$$

\frac{1}{(b-c)^2}+\frac{1}{(c-a)^2}+\frac{1}{(a-b)^2}

$$is the square of a rational.
1976 Swedish Mathematical Competition
A number is placed in each cell of an $n \times n$ board so that the following holds:

(A) the cells on the boundary all contain 0;

(B) other cells on the main diagonal are each1 greater than the mean of the numbers to the left and right;

(C) other cells are the mean of the numbers to the left and right.

Show that (B) and (C) remain true if  ''left and right'' is replaced by  ''above and below''.
1976 Swedish Mathematical Competition
$f(x)$ is defined for $x \geq 0$ and has a continuous derivative. It satisfies $f(0)=1$, $f'(0)=0$ and $(1+f(x))f''(x)=1+x$. Show that $f$ is increasing and that $f(1) \leq 4/3$.
1976 Swedish Mathematical Competition
Show that there are only finitely many integral solutions to

$$

3^m - 1 = 2^n

$$and find them.
1977 Swedish Mathematical Competition
$p$ is a prime. Find the largest integer $d$ such that $p^d$ divides $p^4!$.
1977 Swedish Mathematical Competition
There is a point inside an equilateral triangle side $d$ whose distance from the vertices is $3, 4, 5$. Find $d$.
1977 Swedish Mathematical Competition
Show that the only integral solution to

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

xy + yz + zx = 3n^2 - 1\\
 x + y + z = 3n \\
 \end{array} \right.

$$with $x \geq y \geq z$ is $x=n+1$, $y=n$, $z=n-1$.
1977 Swedish Mathematical Competition
Show that if

$$

\frac{\cos x}{\cos y}+\frac{\sin x}{\sin y}=-1

$$then

$$

\frac{\cos^3 y}{\cos x}+\frac{\sin^3 y}{\sin x}=1

$$
1977 Swedish Mathematical Competition
The numbers $1, 2, 3, ... , 64$ are written in the cells of an $8 \times 8$ board (in some order, one per cell). Show that at least four $2 \times 2$ squares have sum greater than $100$.
1977 Swedish Mathematical Competition
Show that there are positive reals $a$, $b$, $c$ such that

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

 a^2 + b^2 + c^2 > 2 \\ 

 a^3 + b^3 + c^3 <2 \\
 a^4 + b^4 + c^4 > 2 \\ 

 \end{array} \right.

$$
1978 Swedish Mathematical Competition
Let $a,b,c,d$ be real numbers such that $a>b>c>d\geq 0$ and $a + d = b + c$. Show that

$$

x^a + x^d \geq x^b + x^c

$$for $x>0$.
1978 Swedish Mathematical Competition
Let $s_m$ be the number $66\cdots 6$ with $m$ digits $6$. Find

$$

s_1 + s_2 + \cdots + s_n

$$