Navvye/Olympiad-Problems-CSV huggingface dataset:
2011 Spain Mathematical Olympiad
Let $A$, $B$, $C$, $D$ be four points in space not all lying on the same plane. The segments $AB$, $BC$, $CD$, and $DA$ are tangent to the same sphere. Prove that their four points of tangency are coplanar.
2011 Spain Mathematical Olympiad
In triangle $ABC$, $\angle B=2\angle C$ and $\angle A>90^\circ$. Let $D$ be the point on the line $AB$ such that $CD$ is perpendicular to $AC$, and let $M$ be the midpoint of $BC$. Prove that $\angle AMB=\angle DMC$.
2011 Spain Mathematical Olympiad
Each rational number is painted either white or red. Call such a coloring of the rationals sanferminera if for any distinct rationals numbers $x$ and $y$ satisfying one of the following three conditions:



$xy=1$,

$x+y=0$,

$x+y=1$,



we have $x$ and $y$ painted different colors. How many sanferminera colorings are there?
2011 Spain Mathematical Olympiad
The sequence $S_0,S_1,S_2,\ldots$ is defined by



$S_n=1$ for $0\le n\le 2011$, and

$S_{n+2012}=S_{n+2011}+S_n$ for $n\ge 0$.



Prove that $S_{2011a}-S_a$ is a multiple of $2011$ for all nonnegative integers $a$.
2010 Spain Mathematical Olympiad
A pucelana sequence is an increasing sequence of $16$ consecutive odd numbers whose sum is a perfect cube. How many pucelana sequences are there with $3$-digit numbers only?
2010 Spain Mathematical Olympiad
Let $\mathbb{N}_0$ and $\mathbb{Z}$ be the set of all non-negative integers and the set of all integers, respectively. Let $f:\mathbb{N}_0\rightarrow\mathbb{Z}$ be a function defined as

$$f(n)=-f\left(\left\lfloor\frac{n}{3}\right\rfloor \right)-3\left\{\frac{n}{3}\right\} $$

where $\lfloor x \rfloor$ is the greatest integer smaller than or equal to $x$ and $\{ x\}=x-\lfloor x \rfloor$. Find the smallest integer $n$ such that $f(n)=2010$.
2010 Spain Mathematical Olympiad
Let $ABCD$ be a convex quadrilateral. $AC$ and $BD$ meet at $P$, with $\angle APD=60^{\circ}$. Let $E,F,G$, and $H$ be the midpoints of $AB,BC,CD$ and $DA$ respectively. Find the greatest positive real number $k$ for which

$$EG+3HF\ge kd+(1-k)s $$

where $s$ is the semi-perimeter of the quadrilateral $ABCD$ and $d$ is the sum of the lengths of its diagonals. When does the equality hold?
2010 Spain Mathematical Olympiad
Let $a,b,c$ be three positive real numbers. Show that

$$ \frac {a+b+3c}{3a+3b+2c}+\frac {a+3b+c}{3a+2b+3c}+\frac {3a+b+c}{2a+3b+3c} \ge \frac {15}{8}$$
2010 Spain Mathematical Olympiad
In a triangle $ABC$, let $P$ be a point on the bisector of $\angle BAC$ and let $A',B'$ and $C'$ be points on lines $BC,CA$ and $AB$ respectively such that $PA'$ is perpendicular to $BC,PB'\perp AC$, and $PC'\perp AB$. Prove that $PA'$ and $B'C'$ intersect on the median $AM$, where $M$ is the midpoint of $BC$.
2010 Spain Mathematical Olympiad
Let $p$ be a prime number and $A$ an infinite subset of the natural numbers. Let $f_A(n)$ be the number of different solutions of $x_1+x_2+\ldots +x_p=n$, with $x_1,x_2,\ldots  x_p\in A$. Does there exist a number $N$ for which $f_A(n)$ is constant for all $n<N$?
2009 Spain Mathematical Olympiad
Find all the finite sequences with $ n$ consecutive natural numbers $ a_1, a_2,..., a_n$, with $ n\geq3$ such that $ a_1+ a_2+...+ a_n=2009$.
2009 Spain Mathematical Olympiad
Let $ ABC$ be an acute triangle with the incircle $ C(I,r)$ and the circumcircle $ C(O,R)$ . Denote



$ D\in BC$ for which $ AD\perp BC$ and $ AD = h_a$ . Prove that $ DI^2 = (2R - h_a)(h_a - 2r)$ .
2009 Spain Mathematical Olympiad
Some edges are painted in red. We say that a coloring of this kind is good, if for each vertex of the polyhedron, there exists an edge which concurs in that vertex and is not painted red. Moreover, we say that a coloring where some of the edges of a regular polyhedron is completely good, if in addition to being good, no face of the polyhedron has all its edges painted red. What regular polyhedrons is  equal the maximum number of edges that can be painted in a good color and a completely good? Explain your answer.
2009 Spain Mathematical Olympiad
Find all the integer pairs $ (x,y)$ such that:

$$ x^2-y^4=2009$$
2009 Spain Mathematical Olympiad
Let, $ a,b,c$ real positive numbers with $ abc = 1$

Prove:



$ (\frac {a}{1 + ab})^2 + (\frac {b}{1 + bc})^2 + (\frac {c}{1 + ca})^2\geq \frac {3}{4}$



Thanks!
2009 Spain Mathematical Olympiad
Inside a circle of center $ O$ and radius $ r$, take two points $ A$ and $ B$ symmetrical about $ O$. We consider a variable point $ P$ on the circle and draw the chord $ \overline{PP'}\perp \overline{AP}$. Let $ C$ is the symmetric of $ B$ about $ \overline{PP'}$ ($ \overline{PP}'$ is the axis of symmetry) . Find the locus of point $ Q = \overline{PP'}\cap\overline{AC}$ when we change $ P$ in the circle.
2008 Spain Mathematical Olympiad
Find two positive integers $a$ and $b$, when their sum and their least common multiple is given. Find the numbers when the sum is $3972$ and the least common multiple is $985928$.
2008 Spain Mathematical Olympiad
Let $a$ and $b$ be two real numbers, with $0<a,b<1$. Prove that

$$\sqrt{ab^2+a^2b}+\sqrt{(1-a)(1-b)^2+(1-a)^2(1-b)}<\sqrt{2}$$
2008 Spain Mathematical Olympiad
Let $p\ge 3$ be a prime number. Each side of a triangle is divided into $p$ equal parts, and we draw a line from each division point to the opposite vertex. Find the maximum number of regions, every two of them disjoint, that are formed inside the triangle.
2008 Spain Mathematical Olympiad
Let $p$ and $q$ be two different prime numbers. Prove that there are two positive integers, $a$ and $b$, such that the arithmetic mean of the divisors of $n=p^aq^b$ is an integer.
2008 Spain Mathematical Olympiad
Given a circle, two fixed points $A$ and $B$ and a variable point $P$, all of them on the circle, and a line $r$, $PA$ and $PB$ intersect $r$ at $C$ and $D$, respectively. Find two fixed points on $r$, $M$ and $N$, such that $CM\cdot DN$ is constant for all $P$.
2008 Spain Mathematical Olympiad
Every point in the plane is coloured one of seven distinct colours. Is there an inscribed trapezoid whose vertices are all of the same colour?
2007 Spain Mathematical Olympiad
Let $a_0, a_1, a_2, a_3, a_4$ be five positive numbers in the arithmetic progression with a difference $d$. Prove that $a^3_2 \leq \frac{1}{10}(a^3_0 + 4a^3_1 + 4a^3_3 + a^3_4).$
2007 Spain Mathematical Olympiad
Determine all the possible non-negative integer values that are able to satisfy the expression:

$\frac{(m^2+mn+n^2)}{(mn-1)}$

if $m$ and $n$ are non-negative integers such that $mn \neq 1$.
2007 Spain Mathematical Olympiad
$O$ is the circumcenter of triangle $ABC$. The bisector from $A$ intersects the opposite side in point $P$. Prove that the following is satisfied:

$$AP^2 + OA^2 - OP^2 = bc.$$
2007 Spain Mathematical Olympiad
What are the positive integer numbers that we are able to obtain in $2007$ distinct ways, when the sum is at least out of two positive consecutive integers? What is the smallest of all of them?

Example: the number 9 is written in exactly two such distinct ways:

$9 = 4 + 5$

$9 = 2 + 3 + 4.$
2007 Spain Mathematical Olympiad
Let $a \neq 1$ and be a real positive number and $n$ be an integer greater than $1.$ Demonstrate that $n^2 < \frac{(a^n + a^{-n} -2)}{(a + a^{-1} -2)}.$
2007 Spain Mathematical Olympiad
Given a halfcircle of diameter $AB = 2R$, consider a chord $CD$ of length $c$. Let $E$ be the intersection of $AC$ with $BD$ and $F$ the inersection of $AD$ with $BC$.

Prove that the segment $EF$ has a constant length and direction when varying the chord $CD$ about the halfcircle.
2006 Spain Mathematical Olympiad
Let $P(x)$ be a polynomial with integer coefficients. Prove that if there is an integer $k$ such that none of the integers $P(1),P(2), ..., P(k)$ is divisible by $k$, then $P(x)$  does not have integer roots.
2006 Spain Mathematical Olympiad
The dimensions of a wooden octahedron  are natural numbers. We painted all its surface (the six faces), cut it by planes parallel to the cubed faces of an edge unit and observed that exactly half of the cubes did not have any painted faces. Prove that the number of octahedra with such property is finite.



(It may be useful to keep in mind that $\sqrt[3]{\frac{1}{2}}=1,79 ... <1,8$).



PS. Original text in Spanish



Quote:

Las dimensiones de un ortoedro de madera son enteras. Pintamos toda su superficie (las seis caras), lo cortamos mediante planos paralelos a las caras en cubos de una unidad de arista y observamos que exactamente la mitad de los cubos no tienen ninguna cara pintada. Probar que el número de ortoedros con tal propiedad es finito
2006 Spain Mathematical Olympiad
$ABC$ is an isosceles triangle with $AB = AC$. Let $P$ be any point of a circle tangent to the sides $AB$ in $B$ and to AC in C. Denote $a$, $b$ and $c$ to the distances from $P$ to the sides $BC, AC$ and $AB$ respectively. Prove that: $a^2=bc$
2006 Spain Mathematical Olympiad
Find all the functions $f:(0,+\infty) \to R $ that satisfy the equation



$$f(x)f(y)+f\big(\frac{\lambda}{x})f(\frac{\lambda}{y})=2f(xy)$$

for all pairs of $x,y$ real and positive numbers, where $\lambda$ is a positive real number such that $f(\lambda )=1$
2006 Spain Mathematical Olympiad
Prove that the product of four consecutive natural numbers can not be neither square nor perfect cube.
2006 Spain Mathematical Olympiad
The diagonals $AC$ and $BD$ of a convex quadrilateral $ABCD$ intersect at $E$. Denotes by $S_1,S_2$ and $S$ the areas of the triangles $ABE$, $CDE$ and the quadrilateral $ABCD$ respectively. Prove that $\sqrt{S_1}+\sqrt{S_2}\le \sqrt{S}$ . When equality is reached?
2005 Spain Mathematical Olympiad
Let $a$ and $b$ be integers. Demonstrate that the equation $$(x-a)(x-b)(x-3) +1 = 0$$has an integer solution.
2005 Spain Mathematical Olympiad
Is it possible to color points in the Cartesian Plane $(x,y)$ with integer coordinates with three colors, such that each color appears infinitely many times on infinitely many lines parallel to the $x$-axis and that any three points, each of a different color, are not in a line? Justify your answer.
2005 Spain Mathematical Olympiad
We will say that a triangle is multiplicative if the product of the heights of two of its sides is equal to the length of the third side. Given $ABC \dots XYZ$ is a regular polygon with $n$ sides of length $1$. The $n-3$ diagonals that go out from vertex $A$ divide the triangle $ZAB$ in $n-2$ smaller triangles. Prove that each one of these triangles is multiplicative.
2005 Spain Mathematical Olympiad
Prove that for every positive integer $n$, the decimal expression of $\frac{1}{n}+\frac{1}{n+1}+\frac{1}{n+2}$ is periodic .
2005 Spain Mathematical Olympiad
Let $r,s,u,v$ be real numbers. Prove that:



$$min\{r-s^2,s-u^2, u-v^2,v-r^2\}\le \frac{1}{4}$$
2005 Spain Mathematical Olympiad
In a triangle with sides $a, b, c$ the side $a$ is the arithmetic mean of $b$ and $c$. Prove that:

a) $0^o \le  A \le 60^o$.

b) The height relative to side $a$ is three times the inradius $r$.

c) The distance from the circumcenter to side $a$ is $R - r$, where $R$ is the circumradius.
2004 Spain Mathematical Olympiad
We have a set of ${221}$ real numbers whose sum is ${110721}$. It is deemed that the numbers form a rectangular table such that every row as well as the first and last columns are arithmetic progressions of more than one element. Prove that the sum of the elements in the four corners is equal to ${2004}$.
2004 Spain Mathematical Olympiad
${ABCD}$ is a quadrilateral, ${P}$ and ${Q}$ are midpoints of the diagonals ${BD}$ and ${AC}$, respectively. The lines parallel to the diagonals originating from ${P}$ and ${Q}$ intersect in the point ${O}$. If we join the four midpoints of the sides, ${X}$, ${Y}$, ${Z}$, and ${T}$, to ${O}$, we form four quadrilaterals: ${OXBY}$, ${OYCZ}$, ${OZDT}$, and ${OTAX}$. Prove that the four newly formed quadrilaterals have the same areas.
2004 Spain Mathematical Olympiad
Represent for $\mathbb {Z}$ the set of all integers. Find all of the functions ${f:}$ $ \mathbb{Z} \rightarrow \mathbb{Z}$ such that for any ${x,y}$ integers, they satisfy:

${f(x + f(y)) = f(x) - y.}$
2004 Spain Mathematical Olympiad
Does there exist such a power of ${2}$, that when written in the decimal system its digits are all different than zero and it is possible to reorder the other digits to form another power of ${2}$? Justify your answer.
2004 Spain Mathematical Olympiad
Demonstrate that the condition necessary so that, in triangle ${ABC}$, the median from ${B}$ is divided into three equal parts by the inscribed circumference of a circle is:

${A/5 = B/10 = C/13}$.
2004 Spain Mathematical Olympiad
We put, forming a circumference of a circle, ${2004}$ bicolored files: white on one side of the file and black on the other. A movement consists in choosing a file with the black side upwards and flipping three files: the one chosen, the one to its right, and the one to its left. Suppose that initially there was only one file with its black side upwards. Is it possible, repeating the movement previously described, to get all of the files to have their white sides upwards? And if we were to have ${2003}$ files, between which exactly one file began with the black side upwards?
2003 Spain Mathematical Olympiad
Prove that for any prime ${p}$, different than ${2}$ and ${5}$, there exists such a multiple of ${p}$ whose digits are all nines. For example, if ${p = 13}$, such a multiple is ${999999 = 13 * 76923}$.
2003 Spain Mathematical Olympiad
Does there exist such a finite set of real numbers ${M}$ that has at least two distinct elements and has the property that for two numbers, ${a}$, ${b}$, belonging to ${M}$, the number ${2a - b^2}$ is also an element in ${M}$?
2003 Spain Mathematical Olympiad
The altitudes of the triangle ${ABC}$ meet in the point ${H}$. You know that ${AB = CH}$. Determine the value of the angle $\widehat{BCA}$.
2003 Spain Mathematical Olympiad
Let ${x}$ be a real number such that ${x^3 + 2x^2 + 10x = 20.}$ Demonstrate that both ${x}$ and ${x^2}$ are irrational.
2003 Spain Mathematical Olympiad
How many possible areas are there in a convex hexagon with all of its angles being equal and its sides having lengths $1, 2, 3, 4, 5$ and $6,$ in any order?
2003 Spain Mathematical Olympiad
We string $2n$ white balls and $2n$ black balls, forming a continuous chain. Demonstrate that, in whatever order the balls are placed, it is always possible to cut a segment of the chain to contain exactly $n$ white balls and $n$ black balls.
2002 Spain Mathematical Olympiad
Find all the polynomials $P(t)$ of one variable that fullfill the following for all real numbers $x$ and $y$:

$P(x^2-y^2) = P(x+y)P(x-y)$.
2002 Spain Mathematical Olympiad
In the triangle $ABC$, $A'$ is the foot of the altitude to $A$, and $H$ is the orthocenter.

$a)$ Given a positive real number $k = \frac{AA'}{HA'}$ , find the relationship between the angles $B$ and $C$, as a function of $k$.

$b)$ If $B$ and $C$ are fixed, find the locus of the vertice $A$ for any value of $k$.
2002 Spain Mathematical Olympiad
The function $g$ is defined about the natural numbers and satisfies the following conditions:

$g(2) = 1$

$g(2n) = g(n)$

$g(2n+1) = g(2n) +1.$

Where $n$ is a natural number such that $1 \leq n \leq 2002$.



Find the maximum value $M$ of $g(n).$ Also, calculate how many values of $n$ satisfy the condition of $g(n) = M.$
2002 Spain Mathematical Olympiad
Denote $n$ as a natural number, and $m$ as the result of writing the digits of $n$ in reverse order. Determine, if they exist, the numbers of three digits which satisfy $2m + S = n$, $S$ being the sum of the digits of $n$.
2002 Spain Mathematical Olympiad
Consider $2002$ segments on a plane, such that their lengths are the same. Prove that there exists such a straight line $r$ such that the sum of the lengths of the projections of the $2002$ segments about $r$ is less than $\frac{2}{3}$.
2002 Spain Mathematical Olympiad
In a regular polygon $H$ of $6n+1$ sides ($n$ is a positive integer), we paint $r$ vertices red, and the rest blue. Demonstrate that the number of isosceles triangles that have three of their vertices of the same color does not depend on the way we distribute the colors on the vertices of $H$.
2001 Spain Mathematical Olympiad
Prove that the graph of the polynomial $P(x)$ is symmetric in respect to point $A(a,b)$ if and only if there exists a polynomial $Q(x)$ such that:

$P(x) = b + (x-a)Q((x-a)^2)).$
2001 Spain Mathematical Olympiad
Let $P$ be a point on the interior of triangle $ABC$, such that the triangle $ABP$ satisfies $AP = BP$. On each of the other sides of $ABC$, build triangles $BQC$ and $CRA$ exteriorly, both similar to triangle $ABP$ satisfying: $$BQ = QC$$and $$CR = RA.$$Prove that the point $P,Q,C,$ and $R$ are collinear or are the vertices of a parallelogram.
2001 Spain Mathematical Olympiad
You have five segments of lengths $a_1, a_2, a_3, a_4,$ and $a_5$ such that it is possible to form a triangle with any three of them. Demonstrate that at least one of those triangles has angles that are all acute.
2001 Spain Mathematical Olympiad
The integers between $1$ and $9$ inclusive are distributed in the units of a $3$ x $3$ table. You sum six numbers of three digits: three that are read in the rows from left to right, and three that are read in the columns from top to bottom. Is there any such distribution for which the value of this sum is equal to $2001$?
2001 Spain Mathematical Olympiad
A quadrilateral $ABCD$ is inscribed in a circle of radius 1 whose diameter is $AB$. If the quadrilateral $ABCD$ has an incircle, prove that $CD \leq 2 \sqrt{5} - 2$.
2001 Spain Mathematical Olympiad
Define the function $f: \mathbb{N} \rightarrow \mathbb{N}$ which satisfies, for any $s, n \in \mathbb{N}$, the following conditions:

$f(1) = f(2^s)$ and if $n < 2^s$, then $f(2^s + n) = f(n) + 1.$

Calculate the maximum value of $f(n)$ when $n \leq 2001$ and find the smallest natural number $n$ such that $f(n) = 2001.$
2000 Spain Mathematical Olympiad
Consider the polynomials

$$P(x) = x^4 + ax^3 + bx^2 + cx + 1  \quad \text{and} \quad Q(x) = x^4 + cx^3 + bx^2 + ax + 1.$$

Find the conditions on the parameters $a, b, $c with $a\neq c$ for which $P(x)$ and $Q(x)$ have two common roots and, in such cases, solve the equations $P(x) = 0$ and $Q(x) = 0.$
2000 Spain Mathematical Olympiad
The figure shows a network of roads bounding $12$ blocks. Person $P$ goes from $A$ to $B,$ and person $Q$ goes from $B$ to $A,$ each going by a shortest path (along roads). The persons start simultaneously and go at the same constant speed. At each point with two possible directions to take, both have the same probability. Find the probability that the persons meet.



[asy]

import graph; size(150); real lsf = 0.5; pen dp = linewidth(0.7) + fontsize(10); defaultpen(dp); pen ds = black; 

draw((0,3)--(4,3),linewidth(1.2pt)); draw((4,3)--(4,0),linewidth(1.2pt)); draw((4,0)--(0,0),linewidth(1.2pt)); draw((0,0)--(0,3),linewidth(1.2pt)); draw((1,3)--(1,0),linewidth(1.2pt)); draw((2,3)--(2,0),linewidth(1.2pt)); draw((3,3)--(3,0),linewidth(1.2pt)); draw((0,1)--(4,1),linewidth(1.2pt)); draw((4,2)--(0,2),linewidth(1.2pt));

dot((0,0),ds); label("$A$", (-0.3,-0.36),NE*lsf); dot((4,3),ds); label("$B$", (4.16,3.1),NE*lsf); clip((-4.3,-10.94)--(-4.3,6.3)--(16.18,6.3)--(16.18,-10.94)--cycle);

[/asy]
2000 Spain Mathematical Olympiad
Two circles $C_1$ and $C_2$ with the respective radii $r_1$ and $r_2$ intersect in $A$ and $B.$ A variable line $r$ through $B$ meets $C_1$ and $C_2$ again at $P_r$ and $Q_r$ respectively. Prove that there exists a point $M,$ depending only on $C_1$ and $C_2,$ such that the perpendicular bisector of each segment $P_rQ_r$ passes through $M.$
2000 Spain Mathematical Olympiad
Find the largest integer $N$ satisfying the following two conditions:



(i) $\left[ \frac N3 \right]$ consists of three equal digits;



(ii) $\left[ \frac N3 \right] = 1 + 2 + 3 +\cdots + n$ for some positive integer $n.$
2000 Spain Mathematical Olympiad
Four points are given inside or on the boundary of a unit square. Prove that at least two of these points are on a mutual distance at most $1.$
2000 Spain Mathematical Olympiad
Show that there is no function $f : \mathbb N \to \mathbb N$ satisfying $f(f(n)) = n + 1$ for each positive integer $n.$
1999 Spain Mathematical Olympiad
The lines $t$ and $ t'$, tangent to the parabola $y = x^2$ at points $A$ and $B$ respectively, intersect at point $C$. The median of triangle $ABC$ from $C$ has length $m$. Find the area of $\triangle ABC$ in terms of $m$.
1999 Spain Mathematical Olympiad
Prove that there exists a sequence of positive integers $a_1,a_2,a_3, ...$  such that $a_1^2+a_2^2+...+a_n^2$ is a perfect square for all positive integers $n$.
1999 Spain Mathematical Olympiad
A one player game is played on the triangular board shown on the picture. A token is placed on each circle. Each

token is white on one side and black on the other. Initially, the token at one vertex of the triangle has the black side up,

while the others have the white sides up. A move consists of removing a token with the black side up and turning over the adjacent tokens (two tokens are adjacent if they are joined by a segment). Is it possible to remove all the tokens by a sequence of moves?
1999 Spain Mathematical Olympiad
A box contains $900$ cards, labeled from $100$ to $999$. Cards are removed one at a time without replacement. What is the smallest number of cards that must be removed to guarantee that the labels of at least three removed cards have equal sums of digits?
1999 Spain Mathematical Olympiad
The distances from the centroid $G$ of a triangle $ABC$ to its sides $a,b,c$ are denoted $g_a,g_b,g_c$ respectively. Let $r$ be the inradius of the triangle. Prove that:

a) $g_a,g_b,g_c \ge  \frac{2}{3}r$

b) $g_a+g_b+g_c \ge 3r$
1999 Spain Mathematical Olympiad
A plane is divided into $N$ regions by three families of parallel lines. No three lines pass through the same point. What is the smallest number of lines needed so that $N > 1999$?
1998 Spain Mathematical Olympiad
A unit square $ABCD$ with centre $O$ is rotated about $O$ by an angle $\alpha$. Compute the common area of the two squares.
1998 Spain Mathematical Olympiad
Find all four-digit numbers which are equal to the cube of the sum of their digits.
1998 Spain Mathematical Olympiad
Let $ABC$ be a triangle. Points $D$ and $E$ are taken on the line $BC$ such that $AD$ and $AE$ are parallel to the respective tangents to the circumcircle at $C$ and $B$. Prove that

$$\frac{BE}{CD}=\left(\frac{AB}{AC}\right)^2 $$
1998 Spain Mathematical Olympiad
Find the tangents of the angles of a triangle knowing that they are positive integers.
1998 Spain Mathematical Olympiad
Find all strictly increasing functions $f:\mathbb{N}\rightarrow\mathbb{N}$ that satisfy

$$f(n+f(n))=2f(n)\quad\text{for all}\ n\in\mathbb{N} $$
1998 Spain Mathematical Olympiad
Determine the values of $n$ for which an $n\times n$ square can be tiled with pieces of the type  //cdn.artofproblemsolving.com/images/061df8bfebee5139e0d9134ce7493bf2b65fe615.jpg.
1997 Spain Mathematical Olympiad
Compute the sum of the squares of the first $100$ terms of an arithmetic progression, given that their sum is $-1$ and that the sum of those among them having an even index is $1$.
1997 Spain Mathematical Olympiad
A square of side $5$ is divided into $25$ unit squares. Let $A$ be the set of the $16$ interior points of the initial square which are vertices of the unit squares. What is the largest number of points of $A$ no three of which form an isosceles right triangle?
1997 Spain Mathematical Olympiad
For each parabola $y = x^2+ px+q$ intersecting the coordinate axes in three distinct points, consider the circle passing through these points. Prove that all these circles pass through a single point, and find this point.
1997 Spain Mathematical Olympiad
Let $p$ be a prime number. Find all integers $k$ for which $\sqrt{k^2 -pk}$ is a positive integer.
1997 Spain Mathematical Olympiad
Prove that in every convex quadrilateral of area $1$, the sum of the lengths of the sides and diagonals is not smaller than $2(2+\sqrt2)$.