Navvye/Olympiad-Problems-CSV huggingface dataset:
1997 Spain Mathematical Olympiad
The exact quantity of gas needed for a car to complete a single loop around a track is distributed among $n$ containers placed along the track. Prove that there exists a position starting at which the car, beginning with an empty tank of gas,

can complete a loop around the track without running out of gas. The tank of gas is assumed to be large enough.
1996 Spain Mathematical Olympiad
The natural numbers $a$ and $b$ are such that $ \frac{a+1}{b}+ \frac{b+1}{a}$ is an integer. Show that the greatest common divisor of a and b is not greater than $\sqrt{a+b}$.
1996 Spain Mathematical Olympiad
Let $G$ be the centroid of a triangle $ABC$. Prove that if $AB+GC = AC+GB$, then the triangle is isosceles
1996 Spain Mathematical Olympiad
Consider the functions $ f(x) = ax^{2} + bx + c $ , $ g(x) = cx^{2} + bx + a $, where a, b, c are real numbers. Given that $ |f(-1)| \leq 1 $, $ |f(0)| \leq 1 $, $ |f(1)| \leq 1 $, prove that $ |f(x)| \leq \frac{5}{4} $ and $ |g(x)|  \leq 2 $ for $ -1 \leq  x \leq 1 $.
1996 Spain Mathematical Olympiad
For each real value of $p$, find all real solutions of the equation $\sqrt{x^2 - p}+2\sqrt{x^2-1} = x$.
1996 Spain Mathematical Olympiad
At Port Aventura there are $16$ secret agents, each of whom is watching one or more other agents. It is known that if agent $A$ is watching agent $B$, then $B$ is not watching $A$. Moreover, any $10$ agents can be ordered so that the first is watching the second, the second is watching the third, etc, the last is watching the first. Show that any $11$ agents can also be so ordered.
1996 Spain Mathematical Olympiad
A regular pentagon is constructed externally on each side of a regular pentagon of side $1$. The figure is then folded and the two edges of the external pentagons meeting at each vertex of the original pentagon are glued together. Find the volume of water that can be poured into the obtained container.
1995 Spain Mathematical Olympiad
Consider all sets $A$ of one hundred different natural numbers with the property that any three elements $a,b,c \in A$ (not necessarily different) are the sides of a non-obtuse triangle. Denote by $S(A)$ the sum of the perimeters of all such

triangles. Compute the smallest possible value of $S(A)$.
1995 Spain Mathematical Olympiad
Several paper-made disks (not necessarily equal) are put on the table so that there is some overlapping, but no disk is entirely inside another. The parts that overlap are cut off and removed. Show that the remaining parts cannot be assembled so as to form different disks.
1995 Spain Mathematical Olympiad
A line through the centroid G of the triangle ABC intersects the side AB at P and the side AC at Q Show that $\frac{PB}{PA} \cdot \frac{QC}{QA} \leq \frac{1}{4}$.



Sorry for Triple-Posting. If possible, please merge the solutions to one document.



I think there was an error because it may have automatically triple-posted.
1995 Spain Mathematical Olympiad
Given a prime number $p$, find all integer solutions of $p(x+y) = xy$.
1995 Spain Mathematical Olympiad
Prove that if the equations

$x^3+mx-n = 0$

$nx^3-2m^2x^2 -5mnx-2m^3-n^2 = 0$

have one root in common ($n \ne 0$), then the first equation has two equal roots, and find the roots of the equations in terms of $n$.
1995 Spain Mathematical Olympiad
Let $C$ be a variable interior point of a fixed segment $AB$. Equilateral triangles $ACB' $ and $CBA'$  are constructed on the same side and $ABC' $ on the other side of the line $AB$.

(a) Prove that the lines $AA' ,BB'$ , and $CC'$ meet at some point $P$.

(b) Find the locus of $P$ as $C$ varies.

(c) Prove that the centers $A'' ,B'' ,C''$  of the three triangles form an equilateral triangle.

(d) Prove that $A'' ,B'',C''$ , and $P$ lie on a circle.
1994 Spain Mathematical Olympiad
Prove that if an arithmetic progression contains a perfect square, then it contains infinitely many perfect squares.
1994 Spain Mathematical Olympiad
Let $Oxyz$ be a trihedron whose edges $x,y, z$ are mutually perpendicular. Let $C$ be the point on the ray $z$ with $OC = c$. Points $P$ and $Q$ vary on the rays $x$ and $y$ respectively in such a way that $OP+OQ = k$ is constant. For every $P$ and $Q$, the circumcenter of the sphere through $O,C,P,Q$ is denoted by $W$.

Find the locus of the projection of $W$ on the plane O$xy$.

Also find the locus of points $W$.
1994 Spain Mathematical Olympiad
A tourist office was investigating the numbers of sunny and rainy days in a year in each of six regions. The results are partly shown in the following table:



Region  ,    sunny or rainy  ,   unclassified

$A  \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, 336 \,\,\,\,\,\,\,\,\,\,\,\,\,\, \,\,\,\,\,\,\,29$

$B   \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,  321 \,\,\,\,\,\,\,\,\,\,\,\,\,\, \,\,\,\,\,\,\,44$

$C  \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,   335 \,\,\,\,\,\,\,\,\,\,\,\,\,\, \,\,\,\,\,\,\,30$

$D   \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,  343 \,\,\,\,\,\,\,\,\,\,\,\,\,\, \,\,\,\,\,\,\,22$

$E   \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,  329 \,\,\,\,\,\,\,\,\,\,\,\,\,\, \,\,\,\,\,\,\,36$

$F  \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,   330 \,\,\,\,\,\,\,\,\,\,\,\,\,\, \,\,\,\,\,\,\,35$



Looking at the detailed data, an officer observed that if one region is excluded, then the total number of rainy days in the other regions equals one third of the total number of sunny days in these regions. Determine which region is excluded.
1994 Spain Mathematical Olympiad
In a triangle $ABC$ with $ \angle A = 36^o$ and $AB = AC$, the bisector of the angle at $C$ meets the oposite side at $D$. Compute the angles of $\triangle BCD$. Express the length of side $BC$ in terms of the length $b$ of side $AC$ without using trigonometric functions.
1994 Spain Mathematical Olympiad
Let $21$ pieces, some white and some black, be placed on the squares of a $3\times 7$ rectangle. Prove that there always exist four pieces of the same color standing at the vertices of a rectangle.
1994 Spain Mathematical Olympiad
A convex $n$-gon is dissected into $m$ triangles such that each side of each triangle is either a side of another triangle or a side of the polygon. Prove that $m+n$ is even. Find the number of sides of the triangles inside the square and the number of vertices inside the square in terms of $m$ and $n$.
1993 Spain Mathematical Olympiad
There is a reunion of $201$ people from $5$ different countries. It is known that in each group of $6$ people, at least two have the same age. Show that there must be at least $5$ people with the same country, age and sex.
1993 Spain Mathematical Olympiad
In the arithmetic triangle below each number (apart from those in the first row) is the sum of the two numbers immediately above.

$0 \, 1\, 2\, 3 \,4\,  ... \,1991 \,1992\, 1993$

$\,\,1\, 3\, 5 \,7\, ......\,\,\,\,3983 \,3985$

$\,\,\,4 \,8 \,12\, .......... \,\,\,7968$

·······································

Prove that the bottom number is a multiple of $1993$.
1993 Spain Mathematical Olympiad
Prove that in every triangle the diameter of the incircle is not greater than the radius of the circumcircle.
1993 Spain Mathematical Olympiad
Prove that for each prime number distinct from $2$ and $5$ there exist infinitely many multiples of $p$ of the form $1111...1$.
1993 Spain Mathematical Olympiad
Given a 4×4 grid of points, the points at two opposite corners are denoted $A$ and $D$. We need to choose two other points $ B$ and $C$ such that the six pairwise distances of these four points are all distinct.

(a) How many such quadruples of points are there?

(b) How many such quadruples of points are non-congruent?

(c) If each point is assigned a pair of coordinates $(x_i,y_i)$, prove that the sum of the expressions $|x_i-x_j |+|y_i-y_j|$ over all six pairs of points in a quadruple is constant.
1993 Spain Mathematical Olympiad
A game in a casino uses the diagram shown. At the start a ball appears at $S$. Each time the player presses a button,

the ball moves to one of the adjacent letters with equal probability. The game ends when one of the following two things happens:

(i) The ball returns to $S$, the player loses.

(ii) The ball reaches $G$, the player wins.

Find the probability that the player wins and the expected duration of a game.
1992 Spain Mathematical Olympiad
Determine the smallest number N, multiple of 83, such that N^2 has 63 positive divisors.
1992 Spain Mathematical Olympiad
Given two circles of radii $r$ and $r'$ exterior to each other, construct a line parallel to a given line and intersecting the two circles in chords with the sum of lengths $\ell$.
1992 Spain Mathematical Olympiad
Prove that if $a,b,c,d$ are nonnegative integers satisfying $(a+b)^2+2a+b= (c+d)^2+2c+d$, then $a = c $ and $b = d$.

Show that the same is true if $a,b,c,d$ satisfy $(a+b)^2+3a+b=(c+d)^2+3c+d$, but show that there exist $a,b,c,d $ with $a \ne c$ and $b \ne d$ satisfying $(a+b)^2+4a+b = (c+d)^2+4c+d$.
1992 Spain Mathematical Olympiad
Prove that the arithmetic progression $3,7,11,15,...$. contains infinitely many prime numbers.
1992 Spain Mathematical Olympiad
Given a triangle $ABC$, show how to construct the point $P$ such that $\angle PAB= \angle PBC= \angle PCA$.

Express this angle in terms of $\angle A,\angle B,\angle C$ using trigonometric functions.
1992 Spain Mathematical Olympiad
For a positive integer $n$, let $S(n) $be the set of complex numbers $z = x+iy$ ($x,y \in R$) with $ |z| = 1$ satisfying

$(x+iy)^n+(x-iy)^n = 2x^n$ .

(a) Determine $S(n)$ for $n = 2,3,4$.

(b) Find an upper bound (depending on $n$) of the number of elements of $S(n)$ for $n > 5$.
1991 Spain Mathematical Olympiad
In the coordinate plane, consider the set of all segments of integer lengths whose endpoints have integer coordinates. Prove that no two of these segments form an angle of $45^o$. Are there such segments in coordinate space?
1991 Spain Mathematical Olympiad
Given two distinct elements $a,b \in \{-1,0,1\}$, consider the matrix $A$ .

Find a subset $S$ of the set of the rows of $A$, of minimum size, such that every other row of $A$ is a linear combination of the rows in $S$ with integer coefficients.
1991 Spain Mathematical Olympiad
What condition must be satisfied by the coefficients $u,v,w$ if the roots of the polynomial $x^3 -ux^2+vx-w$ are the sides of a triangle
1991 Spain Mathematical Olympiad
The incircle of $ABC$ touches the sides $BC,CA,AB$ at $A' ,B' ,C'$ respectively. The line $A' C'$  meets the angle bisector of $\angle A$ at $D$. Find $\angle ADC$.
1991 Spain Mathematical Olympiad
For a positive integer $n$, let $s(n)$ denote the sum of the binary digits of $n$. Find the sum $s(1)+s(2)+s(3)+...+s(2^k)$ for each positive integer $k$.
1991 Spain Mathematical Olympiad
Find the integer part of $ \frac{1}{\sqrt1}+\frac{1}{\sqrt2}+\frac{1}{\sqrt3}+...+\frac{1}{\sqrt{1000}} $
1990 Spain Mathematical Olympiad
Prove that $\sqrt{x}+\sqrt{y}+\sqrt{xy}$ is equal to$ \sqrt{x}+\sqrt{y+xy+2y\sqrt{x}}$



and compare the numbers $\sqrt{3}+\sqrt{10+2\sqrt{3}}$ and $\sqrt{5+\sqrt{22}}+\sqrt{8-

\sqrt{22}+2\sqrt{15-3\sqrt{22}}}$.
1990 Spain Mathematical Olympiad
Every point of the plane is painted with one of three colors. Can we always find two points a distance $1$ cm apart which are of the same color?
1990 Spain Mathematical Olympiad
Prove that $ \lfloor{(4+\sqrt11)^{n}}\rfloor $ is odd for every natural number n.
1990 Spain Mathematical Olympiad
Prove that the sum $\sqrt[3]{\frac{a+1}{2}+\frac{a+3}{6}\sqrt{ \frac{4a+3}{3}}} +\sqrt[3]{\frac{a+1}{2}-\frac{a+3}{6}\sqrt{ \frac{4a+3}{3}}}$

is independent of $a$ for  $ a \ge - \frac{3}{4}$  and evaluate it.
1990 Spain Mathematical Olympiad
On the sides $BC,CA$ and $AB$ of a triangle $ABC$ of area $S$ are taken points $A' ,B' ,C'$ respectively such that $AC' /AB = BA' /BC = CB' /CA = p$, where $0 < p < 1$ is variable.

(a) Find the area of triangle $A' B' C'$  in terms of $ p$.

(b) Find the value of $p$ which minimizes this area.

(c) Find the locus of the intersection point $P$ of the lines through $A' $ and $C'$ parallel to $AB$ and $AC$ respectively.
1990 Spain Mathematical Olympiad
There are $n$ points in the plane so that no two pairs are equidistant. Each point is connected to the nearest point by a segment. Show that no point is connected to more than five points.
1989 Spain Mathematical Olympiad
An exam at a university consists of one question randomly selected from the$ n$ possible questions. A student knows only one question, but he can take the exam $n$ times. Express as a function of $n$ the probability $p_n$ that the student will pass the exam. Does $p_n$ increase or decrease as $n$ increases? Compute $lim_{n\to \infty}p_n$. What is the largest lower bound of the probabilities $p_n$?
1989 Spain Mathematical Olympiad
Points $A' ,B' ,C'$ on the respective sides $BC,CA,AB$ of triangle $ABC$ satisfy $\frac{AC' }{AB} = \frac{BA' }{BC} = \frac{CB' }{CA} = k$. The lines $AA' ,BB' ,CC' $ form a triangle $A_1B_1C_1$ (possibly degenerate). Given $k$ and the area $S$ of $\triangle ABC$, compute the area of $\triangle A_1B_1C_1$.
1989 Spain Mathematical Olympiad
Prove $  \frac{1}{10\sqrt2}<\frac{1}{2}\frac{3}{4}\frac{5}{6}...\frac{99}{100}<\frac{1}{10} $
1989 Spain Mathematical Olympiad
Show that the number $1989$ as well as each of its powers $1989^n$ ($n \in N$), can be expressed as a sum of two positive squares in at least two ways.
1989 Spain Mathematical Olympiad
Consider the set $D$ of all complex numbers of the form $a+b\sqrt{-13}$ with $a,b \in Z$. The number $14 = 14+0\sqrt{-13}$ can be written as a product of two elements of $D$: $14 = 2 \cdot 7$. Find all possible ways to express $14$ as a product of two elements of $D$.
1989 Spain Mathematical Olympiad
Prove that among any seven real numbers there exist two,$ a$ and $b$, such that $\sqrt3|a-b|\le |1+ab|$.

Give an example of six real numbers not having this property.
1988 Spain Mathematical Olympiad
A sequence of integers $(x_n)_{n=1}^{\infty}$ satisfies $x_1 = 1$ and $xn < x_{n+1} \le  2n$ for all $n$.

Show that for every positive integer $k$ there exist indices $r, s$ such that $x_r-x_s = k$.
1988 Spain Mathematical Olympiad
We choose $n > 3$ points on a circle and number them $1$ to $ n$ in some order. We say that two non-adjacent points $A$ and $B$ are related if, in one of the arcs $AB$, all the points are marked with numbers less than those at $A,B$. Show that the number of pairs of related points is exactly $n-3$.
1988 Spain Mathematical Olympiad
Prove that if one of the numbers $25x+31y, 3x+7y$ (where $x,y \in Z$) is a multiple of $41$, then so is the other.
1988 Spain Mathematical Olympiad
The Fibonacci sequence is given by $a_1 = 1, a_2 = 2$ and $a_{n+1} = a_n +a_{n-1}$ for $n > 1$. Express $a_{2n}$ in terms of only $a_{n-1},a_n,a_{n+1}$.
1988 Spain Mathematical Olympiad
A well-known puzzle asks for a partition of a cross into four parts which are to be reassembled into a square. One

solution is exhibited on the picture.



Show that there are infinitely many solutions. (Some solutions split the cross into four equal parts!)
1988 Spain Mathematical Olympiad
For all integral values of parameter $t$, find all integral solutions $(x,y)$ of the equation

$$ y^2 = x^4-22x^3+43x^2-858x+t^2+10452(t+39)$$.
1987 Spain Mathematical Olympiad
Let $a, b, c$ be the side lengths of a scalene triangle and let $O_a, O_b$ and $O_c$ be three concentric circles with radii $a, b$ and $c$ respectively.

(a) How many equilateral triangles with different areas can be constructed such that the lines containing the sides are tangent to the circles?

(b) Find the possible areas of such triangles.
1987 Spain Mathematical Olympiad
Show that for each natural number $n > 1$

$1 \cdot \sqrt{{n \choose 1}}+ 2 \cdot \sqrt{{n \choose 2}}+...+n \cdot \sqrt{{n \choose n}} <\sqrt{2^{n-1}n^3}$
1987 Spain Mathematical Olympiad
A given triangle is divided into $n$ triangles in such a way that any line segment which is a side of a tiling triangle is either a side of another tiling triangle or a side of the given triangle. Let $s$ be the total number of sides and $v$ be the total number of vertices of the tiling triangles (counted without multiplicity).

(a) Show that if $n$ is odd then such divisions are possible, but each of them has the same number $v$ of vertices and the same number $s$ of sides. Express $v$ and $s$ as functions of $n$.

(b) Show that, for $n$ even, no such tiling is possible
1987 Spain Mathematical Olympiad
If $a$ and $b$ are distinct real numbers, solve the systems

(a) $\begin{cases} x+y = 1  \\ (ax+by)^2 \le  a^2x+b^2y \end{cases}$ and (b) $\begin{cases} x+y = 1  \\ (ax+by)^4 \le  a^4x+b^4y \end{cases}$
1987 Spain Mathematical Olympiad
In a triangle $ABC, D$ lies on $AB, E$ lies on $AC$ and $ \angle ABE = 30^o, \angle EBC = 50^o, \angle ACD = 20^o$, $\angle DCB = 60^o$. Find $\angle EDC$.
1987 Spain Mathematical Olympiad
For all natural numbers $n$, consider the polynomial $P_n(x) = x^{n+2}-2x+1$.

(a) Show that the equation $P_n(x)=0$ has exactly one root $c_n$ in the open interval $(0,1)$.

(b) Find $lim_{n \to \infty}c_n$.
1986 Spain Mathematical Olympiad
Define the distance between real numbers $x$ and $y$ by $d(x,y) =\sqrt{([x]-[y])^2+(\{x\}-\{y\})^2}$ .

Determine (as a union of intervals) the set of real numbers whose distance from $3/2$ is less than $202/100$ .
1986 Spain Mathematical Olympiad
A segment $d$ is said to divide a segment $s$ if there is a natural number $n$ such that $s = nd = d+d+ ...+d$ ($n$ times).

(a) Prove that if a segment $d$ divides segments $s$ and $s'$ with $s < s'$, then it also  divides their difference $s'-s$.

(b) Prove that no segment divides the side $s$ and the diagonal $s'$ of a regular pentagon (consider the pentagon formed by the diagonals of the given pentagon without explicitly computing the ratios).
1986 Spain Mathematical Olympiad
Find all natural numbers $n$ such that $5^n+3$ is a power of $2$
1986 Spain Mathematical Olympiad
Denote by $m(a,b)$ the arithmetic mean of positive real numbers $a,b$. Given a positive real function $g$ having positive derivatives of the first and second order, define $\mu (a,b)$ the mean value of $a$ and $b$ with respect to $g$ by $2g(\mu (a,b)) = g(a)+g(b)$. Decide which of the two mean values $m$ and $\mu$ is larger.
1986 Spain Mathematical Olympiad
Consider the curve $\Gamma$ defined by the equation $y^2 = x^3 +bx+b^2$, where $b$ is a nonzero rational constant. Inscribe in the curve $\Gamma$ a triangle whose vertices have rational coordinates.
1986 Spain Mathematical Olympiad
Evaluate $$\prod_{k=1}^{14} cos \big(\frac{k\pi}{15}\big)$$
1985 Spain Mathematical Olympiad
Let $f : P\to P$ be a bijective map from a plane $P$ to itself such that:

(i) $f (r)$ is a line for every line $r$,

(ii) $f (r) $ is parallel to $r$ for every line $r$.

What possible transformations can $f$ be?
1985 Spain Mathematical Olympiad
Determine if there exists a subset $E$ of $Z \times Z$ with the properties:

(i) $E$ is closed under addition,

(ii) $E$ contains $(0,0),$

(iii) For every $(a,b) \ne (0,0), E$ contains exactly one of $(a,b)$ and $-(a,b)$.



Remark: We define $(a,b)+(a',b') = (a+a',b+b')$ and $-(a,b) = (-a,-b)$.
1985 Spain Mathematical Olympiad
Solve the equation $tan^2 2x+2 tan2x tan3x = 1$
1985 Spain Mathematical Olympiad
Prove that for each positive integer $k $ there exists a triple $(a,b,c)$ of positive integers such that $abc = k(a+b+c)$. In all such cases prove that $a^3+b^3+c^3$ is not a prime.
1985 Spain Mathematical Olympiad
Find the equation of the circle in the complex plane determined by the roots of the equation $z^3 +(-1+i)z^2+(1-i)z+i= 0$.
1985 Spain Mathematical Olympiad
Let $OX$ and $OY$ be non-collinear rays. Through a point $A$ on $OX$, draw two lines $r_1$ and $r_2$ that are antiparallel with respect to $\angle XOY$. Let $r_1$ cut $OY$ at $M$ and $r_2$ cut $OY$ at $N$. (Thus, $\angle OAM = \angle ONA$). The bisectors of $ \angle AMY$ and $\angle ANY$ meet at $P$. Determine the location of $P$.
1985 Spain Mathematical Olympiad
Find the values of $p$ for which the equation $x^5 - px-1 = 0$ has two roots $r$ and $s$ which are the roots of equation $x^2-ax+b= 0$ for some integers $a,b$.
1985 Spain Mathematical Olympiad
A square matrix is sum-magic if the sum of all elements in each row, column and major diagonal is constant. Similarly, a square matrix is product-magic if the product of all elements in each row, column and major diagonal is constant.

Determine if there exist $3\times 3$ matrices of real numbers which are both sum-magic and product-magic.
1984 Spain Mathematical Olympiad
At a position $O$ of an airport in a plateau there is a gun which can rotate arbitrarily. Two tanks moving along two given segments $AB$ and $CD$ attack the airport. Determine, by a ruler and a compass, the reach of the gun, knowing that the total length of the parts of the trajectories of the two tanks reachable by the gun is equal to a given length $\ell$.
1984 Spain Mathematical Olympiad
Find the number of five-digit numbers whose square ends in the same five digits in the same order.
1984 Spain Mathematical Olympiad
If $p$ and $q$ are positive numbers with $p+q = 1$,

knowing that any real numbers $x,y$ satisfy $(x-y)^2 \ge  0$, show that



$\frac{x+y}{2} \ge  \sqrt{xy}$,



$\frac{x^2+y^2}{2} \ge \big(\frac{x+y}{2}\big)^2$,



$\big(p+\frac{1}{p}\big)^2+\big(q+\frac{1}{q}\big)^2 \ge \frac{25}{2}$
1984 Spain Mathematical Olympiad
Evaluate $lim_{n\to \infty} cos\frac{x}{2}cos\frac{x}{2^2} cos\frac{x}{2^3}...cos\frac{x}{2^n}$
1984 Spain Mathematical Olympiad
Let $A$ and $A' $ be fixed points on two equal circles in the plane and let $AB$ and $A' B'$  be arcs of these circles of the same length $x$. Find the locus of the midpoint of segment $BB'$ when $x$ varies:

(a) if the arcs have the same direction,

(b) if the arcs have opposite directions.
1984 Spain Mathematical Olympiad
Consider the circle $\gamma$ with center at point $(0,3)$ and radius $3$, and a line $r$ parallel to the axis $Ox$ at a distance $3$ from the origin. A variable line through the origin meets  $\gamma$ at point $M$ and $r$ at point $P$. Find the locus of the intersection point of the lines through $M$ and $P$ parallel to $Ox$ and $Oy$ respectively.
1984 Spain Mathematical Olympiad
Consider the natural numbers written in the decimal system.

(a) Find the smallest number which decreases five times when its first digit is erased. Which form do all numbers with this property have?

(b) Prove that there is no number that decreases $12$ times when its first digit is erased.

(c) Find the necessary and sufficient condition on $k$ for the existence of a natural number which is divided by $k$ when its first digit is erased.
1984 Spain Mathematical Olympiad
Find the remainder upon division by $x^2-1$ of the determinant



$$\begin{vmatrix} 

x^3+3x & 2 & 1 & 0

\\ x^2+5x & 3 & 0 & 2

\\x^4 +x^2+1 & 2  & 1 & 3

\\x^5 +1 & 1 & 2  & 3

\\ \end{vmatrix}$$

.
1965 Spain Mathematical Olympiad
We consider an equilateral triangle with its circumscribed circle, of center $O$, and radius $4$cm. We rotate the triangle $90º$ around $O$. Compute the common area that was covered by the previous position of the triangle and is also covered by the new one.
1965 Spain Mathematical Olympiad
How many numbers of $3$ digits have their central digit greater than any of the other two? How many of them have also three different digits?
1965 Spain Mathematical Olympiad
A disk in a record turntable makes $100$ revolutions per minute and it plays during $24$ minutes and $30$ seconds. The recorded line over the disk is a spiral with a diameter that decreases uniformly from $29$cm to $11.5$cm. Compute the length of the recorded line.
1965 Spain Mathematical Olympiad
Find all the intervals $I$ where any element of the interval $x \in I$ satisfies $$\cos x +\sin x >1.$$Do the same computation when $x$ satisfies $$\cos x +\vert \sin x \vert>1.$$
1965 Spain Mathematical Olympiad
It is well-known that if $\frac{p}{q}=\frac{r}{s}$, both of the expressions are also equal to $\frac{p-r}{q-s}$. Now we write the equality $$\frac{3x-b}{3x-5b}=\frac{3a-4b}{3a-8b}.$$The previous property shows that both fractions should be equal to $$\frac{3x-b-3a+4b}{3x-5b-3a+8b}=\frac{3x-3a+3b}{3x-3a+3b}=1.$$However, the initial fractions given may not be equal to $1$. Explain what is going on.