Navvye/Olympiad-Problems-CSV huggingface dataset:
2005 South East Mathematical Olympiad
Let $n$ be positive integer, set $M = \{ 1, 2, \ldots, 2n \}$. Find the minimum positive integer $k$ such that for any subset $A$ (with $k$ elements) of set $M$, there exist four pairwise distinct elements in $A$ whose sum is $4n + 1$.
2005 South East Mathematical Olympiad
Find all positive integer solutions $(a, b, c)$ to the function $a^{2} + b^{2} + c^{2} = 2005$, where $a \leq b \leq c$.
2005 South East Mathematical Olympiad
Line $l$ tangents unit circle $S$ in point $P$. Point $A$ and circle $S$ are on the same side of $l$, and the distance from $A$ to $l$ is $h$ ($h > 2$). Two tangents of circle $S$ are drawn from $A$, and intersect line $l$ at points $B$ and $C$ respectively.



Find the value of $PB \cdot PC$.
2005 South East Mathematical Olympiad
Let $P(A)$ be the arithmetic-means of all elements of set $A = \{ a_1, a_2, \ldots, a_n \}$, namely $P(A) = \frac{1}{n} \sum^{n}_{i=1}a_i$. We denote $B$ "balanced subset" of $A$, if $B$ is a non-empty subset of $A$ and $P(B) = P(A)$. Let set $M = \{ 1, 2, 3, 4, 5, 6, 7, 8, 9 \}$.



Find the number of all "balanced subset" of $M$.
2005 South East Mathematical Olympiad
(1) Find the possible number of roots for the equation $|x + 1| + |x + 2| + |x + 3| = a$, where $x \in R$ and $a$ is parameter.



(2) Let $\{ a_1, a_2, \ldots, a_n \}$ be an arithmetic progression, $n \in \mathbb{N}$, and satisfy the condition

$$ \sum^{n}_{i=1}|a_i| = \sum^{n}_{i=1}|a_{i} + 1| = \sum^{n}_{i=1}|a_{i} - 2| = 507. $$

Find the maximum value of $n$.
2005 South East Mathematical Olympiad
Let $0 < \alpha, \beta, \gamma < \frac{\pi}{2}$ and $\sin^{3} \alpha + \sin^{3} \beta + \sin^3 \gamma = 1$. Prove that

$$ \tan^{2} \alpha + \tan^{2} \beta + \tan^{2} \gamma \geq \frac{3 \sqrt{3}}{2} . $$
2004 South East Mathematical Olympiad
Let real numbers a, b, c satisfy $a^2+2b^2+3c^2= \frac{3}{2}$, prove that $3^{-a}+9^{-b}+27^{-c}\ge1$.
2004 South East Mathematical Olympiad
In $\triangle$ABC, points D, M lie on side BC and AB respectively, point P lies on segment AD. Line DM intersects segments BP, AC (extended part), PC (extended part) at E, F and N respectively. Show that if DE=DF, then DM=DN.
2004 South East Mathematical Olympiad
(1) Determine if there exists an infinite sequence $\{a_n\}$ with positive integer terms, such that $a^2_{n+1}\ge 2a_na_{n+2}$ for any positive integer $n$.

(2) Determine if there exists an infinite sequence $\{a_n\}$ with positive irrational terms, such that  $a^2_{n+1}\ge 2a_na_{n+2}$ for any positive integer $n$.
2004 South East Mathematical Olympiad
Given a positive integer $n (n>2004)$, we put 1, 2, 3, …,$n^2$ into squares of an $n\times n$ chessboard with one number in a square. A square is called a “good square” if the square satisfies following conditions:

1) There are at least 2004 squares that are in the same row with the square such that any number within these 2004 squares is less than the number within the square.

2) There are at least 2004 squares that are in the same column with the square such that any number within these 2004 squares is less than the number within the square.

Find the maximum value of the number of the “good square”.
2004 South East Mathematical Olympiad
For $\theta\in[0, \dfrac{\pi}{2}]$, the following inequality $\sqrt{2}(2a+3)\cos(\theta-\dfrac{\pi}{4})+\dfrac{6}{\sin\theta+\cos\theta}-2\sin2\theta<3a+6$ is always true.

Determine the range of $a$.
2004 South East Mathematical Olympiad
ABC is an isosceles triangle with AB=AC. Point D lies on side BC. Point F is inside $\triangle$ABC and lies on the circumcircle of triangle ADC. The circumcircle of triangle BDF intersects side AB at point E. Prove that $CD\cdot EF+DF\cdot AE=BD\cdot AF$.
2004 South East Mathematical Olympiad
A tournament is held among $n$ teams, following such rules:

a) every team plays all others once at home and once away.(i.e. double round-robin schedule)

b) each team may participate in several away games in a week(from Sunday to Saturday).

c) there is no away game arrangement for a team, if it has a home game in the same week.

If the tournament finishes in 4 weeks, determine the maximum value of $n$.
2004 South East Mathematical Olympiad
Determine the number of ordered quadruples $(x, y, z, u)$ of integers, such that

$$\dfrac{x-y}{x+y}+\dfrac{y-z}{y+z}+\dfrac{z-u}{z+u}>0 \textrm{ and } 1\le x,y,z,u\le 10.$$
2021 Spain Mathematical Olympiad
Vertices $A, B, C$ of a equilateral triangle of side $1$ are in the surface of a sphere with radius $1$ and center $O$. Let $D$ be the orthogonal projection of $A$ on the plane $\alpha$ determined by points $B, C, O$. Let $N$ be one of the intersections of the line perpendicular to $\alpha$ passing through $O$ with the sphere. Find the angle $\angle DNO$.
2021 Spain Mathematical Olympiad
Given a positive integer $n$, we define $\lambda (n)$ as the number of positive integer solutions of $x^2-y^2=n$. We say that $n$ is olympic if $\lambda (n) = 2021$. Which is the smallest olympic positive integer? Which is the smallest olympic positive odd integer?
2021 Spain Mathematical Olympiad
We have $2021$ colors and $2021$ chips of each color. We place the $2021^2$ chips in a row. We say that a chip $F$ is bad if there is an odd number of chips that have a different color to $F$ both to the left and to the right of $F$.



(a) Determine the minimum possible number of bad chips.



(b) If we impose the additional condition that each chip must have at least one adjacent chip of the same color, determine the minimum possible number of bad chips.
2021 Spain Mathematical Olympiad
Let $a,b,c,d$ real numbers such that:



$$

a+b+c+d=0 \text{ and } a^2+b^2+c^2+d^2 = 12

$$

Find the minimum and maximum possible values for $abcd$, and determine for which values of $a,b,c,d$ the minimum and maximum are attained.
2021 Spain Mathematical Olympiad
We have $2n$ lights in two rows, numbered from $1$ to $n$ in each row. Some (or none) of the lights are on and the others are off, we call that a "state". Two states are distinct if there is a light which is on in one of them and off in the other. We say that a state is good if there is the same number of lights turned on in the first row and in the second row.



Prove that the total number of good states divided by the total number of states is:



$$

\frac{3 \cdot 5 \cdot 7 \cdots (2n-1)}{2^n n!}

$$
2021 Spain Mathematical Olympiad
Let $ABC$ be a triangle with $AB \neq AC$, let $I$ be its incenter, $\gamma$ its inscribed circle and $D$ the midpoint of $BC$. The tangent to $\gamma$ from $D$ different to $BC$ touches $\gamma$ in $E$. Prove that $AE$ and $DI$ are parallel.
2020 Spain Mathematical Olympiad
A polynomial $p(x)$ with real coefficients is said to be almeriense if it is of the form:



$$ 

p(x) = x^3+ax^2+bx+a

$$

And its three roots are positive real numbers in arithmetic progression. Find all almeriense polynomials such that $p\left(\frac{7}{4}\right) = 0$
2020 Spain Mathematical Olympiad
Consider the succession of integers $\{f(n)\}_{n=1}^{\infty}$ defined as:





$\bullet$ $f(1) = 1$.

$\bullet$ $f(n) = f(n/2)$ if $n$ is even.

$\bullet$ If $n > 1$ odd and $f(n-1)$ odd, then $f(n) = f(n-1)-1$.

$\bullet$ If $n > 1$ odd and $f(n-1)$ even, then $f(n) = f(n-1)+1$.



a) Compute $f(2^{2020}-1)$.



b) Prove that $\{f(n)\}_{n=1}^{\infty}$ is not periodical, that is, there do not exist positive integers $t$ and $n_0$ such that $f(n+t) = f(n)$ for all $n \geq n_0$.
2020 Spain Mathematical Olympiad
To each point of $\mathbb{Z}^3$ we assign one of $p$ colors.



Prove that there exists a rectangular parallelepiped with all its vertices in $\mathbb{Z}^3$ and of the same color.
2020 Spain Mathematical Olympiad
Ana and Benito play a game which consists of $2020$ turns. Initially, there are $2020$ cards on the table, numbered from $1$ to $2020$, and Ana possesses an extra card with number $0$. In the $k$-th turn, the player that doesn't possess card $k-1$ chooses whether to take the card with number $k$ or to give it to the other player. The number in each card indicates its value in points. At the end of the game whoever has most points wins. Determine whether one player has a winning strategy or whether both players can force a tie, and describe the strategy.
2020 Spain Mathematical Olympiad
In an acute-angled triangle $ABC$, let $M$ be the midpoint of $AB$ and $P$ the foot of the altitude to $BC$. Prove that if $AC+BC = \sqrt{2}AB$, then the circumcircle of triangle $BMP$ is tangent to $AC$.
2020 Spain Mathematical Olympiad
Let $S$ be a finite set of integers. We define $d_2(S)$ and $d_3(S)$ as:



$\bullet$ $d_2(S)$ is the number of elements $a \in S$ such that there exist $x, y \in \mathbb{Z}$ such that $x^2-y^2 = a$

$\bullet$ $d_3(S)$ is the number of elements $a \in S$ such that there exist $x, y \in \mathbb{Z}$ such that $x^3-y^3 = a$



(a) Let $m$ be an integer and $S = \{m, m+1, \ldots, m+2019\}$. Prove:



$$d_2(S) > \frac{13}{7} d_3(S)$$

(b) Let $S_n = \{1, 2, \ldots, n\}$ with $n$ a positive integer. Prove that there exists a $N$ so that for all $n > N$:



$$ d_2(S_n) > 4 \cdot d_3(S_n) $$
2019 Spain Mathematical Olympiad
An integer set T is orensan if there exist integers a<b<c, where a and c are part of T, but b is not part of T. Count the number of subsets T of {1,2,...,2019} which are orensan.
2019 Spain Mathematical Olympiad
Determine if there exists a finite set $S$ formed by positive prime numbers so that for each integer $n\geq2$, the number $2^2 + 3^2 +...+ n^2$ is a multiple of some element of $S$.
2019 Spain Mathematical Olympiad
The real numbers $a$, $b$ and $c$ verify that the polynomial $p(x)=x^4+ax^3+bx^2+ax+c$ has exactly three distinct real roots; these roots are equal to $\tan y$, $\tan 2y$ and  $\tan 3y$, for some real number $y$.

Find all possible values of $y$, $0\leq y < \pi$.
2019 Spain Mathematical Olympiad
Find all pairs of integers $(x,y)$ that satisfy the equation $3^4 2^3(x^2+y^2)=x^3y^3$
2019 Spain Mathematical Olympiad
We consider all pairs (x, y) of real numbers such that $0\leq x \leq y \leq 1$.Let $M (x,y)$ the maximum value of the set $$A=\{xy, 1-x-y+xy, x+y-2xy\}.$$Find the minimum value that $M(x,y)$ can take for all these pairs $(x,y)$.
2019 Spain Mathematical Olympiad
In the scalene triangle $ABC$, the bisector of angle A cuts side $BC$ at point $D$.

The tangent lines to the circumscribed circunferences of triangles $ABD$ and $ACD$ on point D, cut lines $AC$ and $AB$ on points $E$ and $F$ respectively. Let $G$ be the intersection point of lines $BE$ and $CF$.

Prove that angles $EDG$ and $ADF$ are equal.
2018 Spain Mathematical Olympiad
Find all positive integers $x$ such that $2x+1$ is a perfect square but none of the integers $2x+2, 2x+3, \ldots, 3x+2$ are perfect squares.
2018 Spain Mathematical Olympiad
Let $n$ be a positive integer. $2n+1$ tokens are in a row, each being black or white. A token is said to be balanced if the number of white tokens on its left plus the number of black tokens on its right is $n$. Determine whether the number of balanced tokens is even or odd.
2018 Spain Mathematical Olympiad
Let $ABC$ be an acute-angled triangle with circumcenter $O$ and let $M$ be a point on $AB$. The circumcircle of $AMO$ intersects $AC$ a second time on $K$ and the circumcircle of $BOM$ intersects $BC$ a second time on $N$.



Prove that $\left[MNK\right] \geq \frac{\left[ABC\right]}{4}$ and determine the equality case.
2018 Spain Mathematical Olympiad
Points on a spherical surface with radius $4$ are colored in $4$ different colors. Prove that there exist two points with the same color such that the distance between them is either $4\sqrt{3}$ or $2\sqrt{6}$.



(Distance is Euclidean, that is, the length of the straight segment between the points)
2018 Spain Mathematical Olympiad
Let $a, b$ be coprime positive integers. A positive integer $n$ is said to be weak if there do not exist any nonnegative integers $x, y$ such that $ax+by=n$. Prove that if $n$ is a  weak integer and $n < \frac{ab}{6}$, then there exists an integer $k \geq 2$ such that $kn$ is  weak.
2018 Spain Mathematical Olympiad
Find all functions such that $ f: \mathbb{R}^+ \rightarrow \mathbb{R}^+$ and $ f(x+f(y))=yf(xy+1)$ for every $ x,y\in \mathbb{R}^+$.
2017 Spain Mathematical Olympiad
Find the amount of different values given by the following expression:



$\frac{n^2-2}{n^2-n+2}$



where $ n \in \{1,2,3,..,100\}$
2017 Spain Mathematical Olympiad
A midpoint plotter is an instrument which draws the exact mid point of two point previously drawn. Starting off two points $1$ unit of distance apart and using only the midpoint plotter, you have to get two point which are strictly at a distance between $\frac{1}{2017}$ and $\frac{1}{2016}$ units, drawing the minimum amount of points. ¿Which is the minimum number of times you will need to use the midpoint plotter and what strategy should you follow to achieve it?
2017 Spain Mathematical Olympiad
Let $p$ be an odd prime and $S_{q} = \frac{1}{2*3*4} + \frac{1}{5*6*7} + ... + \frac{1}{q(q+1)(q+2)}$, where $q = \frac{3p-5}{2}$.

We write $\frac{1}{2}-2S_{q}$ in the form $\frac{m}{n}$, where $m$ and $n$ are integers. Prove that $m \equiv n (mod p)$
2017 Spain Mathematical Olympiad
You are given a row made by $2018$ squares, numbered consecutively from $0$ to $2017$. Initially, there is a coin in the square $0$. Two players $A$ and $B$ play alternatively, starting with $A$, on the following way: In his turn, each player can either make his coin advance $53$ squares or make the coin go back $2$ squares. On each move the coin can never go to a number less than $0$ or greater than $2017$. The player who puts the coin on the square $2017$ wins. ¿Who is the one with the wining strategy and how should he play to win?
2017 Spain Mathematical Olympiad
Let $a,b,c$ be positive real numbers so that $a+b+c = \frac{1}{\sqrt{3}}$. Find the maximum value of

$$27abc+a\sqrt{a^2+2bc}+b\sqrt{b^2+2ca}+c\sqrt{c^2+2ab}.$$
2017 Spain Mathematical Olympiad
In the triangle $ABC$, the respective mid points of the sides $BC$, $AB$ and $AC$ are $D$, $E$ and $F$. Let $M$ be the point where the internal bisector of the angle $\angle ADB$ intersects the side $AB$, and $N$ the point where the internal bisector of the angle $\angle ADC$ intersects the side $AC$. Also, let $O$ be the intersection point of $AD$ and $MN$, $P$ the intersection point of $AB$ and $FO$, and $R$ the intersection point of $AC$ and $EO$. Prove that $PR=AD$.
2016 Spain Mathematical Olympiad
Two real number sequences are guiven, one arithmetic $\left(a_n\right)_{n\in \mathbb {N}}$ and another geometric sequence $\left(g_n\right)_{n\in \mathbb {N}}$ none of them constant. Those sequences verifies $a_1=g_1\neq 0$, $a_2=g_2$ and $a_{10}=g_3$. Find with proof that, for every positive integer $p$, there is a positive integer $m$, such that $g_p=a_m$.
2016 Spain Mathematical Olympiad
Given a positive prime number $p$. Prove that there exist a positive integer $\alpha$ such that $p|\alpha(\alpha-1)+3$, if and only if there exist a positive integer $\beta$ such that $p|\beta(\beta-1)+25$.
2016 Spain Mathematical Olympiad
In the circumscircle of a triangle $ABC$, let $A_1$ be the point diametrically opposed to the vertex $A$. Let $A'$ the intersection point of $AA'$ and $BC$. The perpendicular to the line $AA'$ from $A'$ meets the sides $AB$ and $AC$ at $M$ and $N$, respectively. Prove that the points $A,M,A_1$ and $N$ lie on a circle which has the center on the height from $A$ of the triangle $ABC$.
2016 Spain Mathematical Olympiad
Let $m$ be a positive integer and $a$ and $b$ be distinct positive integers strictly greater than $m^2$ and strictly less than $m^2+m$. Find all integers $d$ such that $m^2 < d < m^2+m$ and $d$ divides $ab$.
2016 Spain Mathematical Olympiad
From all possible permutations from $(a_1,a_2,...,a_n)$ from the set $\{1,2,..,n\}$, $n\geq 1$, consider the sets that satisfies the $2(a_1+a_2+...+a_m)$ is divisible by $m$, for every $m=1,2,...,n$. Find the total number of permutations.
2016 Spain Mathematical Olympiad
Let $n\geq 2$ an integer. Find the least value of $\gamma$ such that for any positive real numbers $x_1,x_2,...,x_n$ with $x_1+x_2+...+x_n=1$ and any real $y_1+y_2+...+y_n=1$ and $0\leq y_1,y_2,...,y_n\leq \frac{1}{2}$ the following inequality holds:

$$x_1x_2...x_n\leq \gamma \left(x_1y_1+x_2y_2+...+x_ny_n\right)$$
2015 Spain Mathematical Olympiad
On the graph of a polynomial with integer coefficients, two points are chosen with integer coordinates. Prove that if the distance between them is an integer, then the segment that connects them is parallel to the horizontal axis.
2015 Spain Mathematical Olympiad
In triangle $ABC$, let $A'$ is the symmetrical of $A$ with respect to the circumcenter $O$ of $ABC$.

Prove that:

a) The sum of the squares of the tangents segments drawn from $A$ and $A'$ to the incircle of $ABC$ equals $$4R^2-4Rr-2r^2$$ where $R$ and $r$ are the radii of the circumscribed and inscribed circles of $ABC$ respectively.

b) The circle with center $A'$ and radius $A'I$ intersects the circumcircle of $ABC$ in a point $L$ such that $$AL=\sqrt{ AB.AC}$$ where $I$ is the centre of the inscribed circle of $ABC$.
2015 Spain Mathematical Olympiad
On the board is written an integer $N \geq 2$. Two players $A$ and $B$ play in turn, starting with $A$. Each player in turn replaces the existing number by the result of performing one of two operations: subtract 1 and divide by 2, provided that a positive integer is obtained. The player who reaches the number 1 wins.

Determine the smallest even number $N$ requires you to play at least $2015$ times to win ($B$ shifts are not counted).
2015 Spain Mathematical Olympiad
All faces of a polyhedron are triangles. Each of the vertices of this polyhedron is assigned independently one of three colors : green, white or black. We say that a face is Extremadura if its three vertices are of different colors, one green, one white and one black. Is it true that regardless of how the vertices's color, the number of Extremadura faces of this polyhedron is always even?
2015 Spain Mathematical Olympiad
Let  $p$ and $n$ be a natural numbers such that $p$ is a prime and  $1+np$ is a perfect square.  Prove that the number $n+1$ is sum of $p$ perfect squares.
2015 Spain Mathematical Olympiad
Let $ABC$ be a triangle. $M$, and $N$ points on $BC$, such that $BM=CN$, with $M$ in the interior of $BN$. Let $P$ and $Q$ be points in $AN$ and $AM$ respectively such that $\angle PMC= \angle MAB$, and $\angle QNB= \angle NAC$. Prove that $ \angle QBC= \angle PCB$.
2014 Spain Mathematical Olympiad
Is it possible to place the numbers $0,1,2,\dots,9$ on a circle so that the sum of any three consecutive numbers is a) 13, b) 14, c) 15?
2014 Spain Mathematical Olympiad
Given the rational numbers $r$, $q$, and $n$, such that $\displaystyle\frac1{r+qn}+\frac1{q+rn}=\frac1{r+q}$, prove that $\displaystyle\sqrt{\frac{n-3}{n+1}}$ is a rational number.
2014 Spain Mathematical Olympiad
Let $B$ and $C$ be two fixed points on a circle centered at $O$ that are not diametrically opposed. Let $A$ be a variable point on the circle distinct from $B$ and $C$ and not belonging to the perpendicular bisector of $BC$. Let $H$ be the orthocenter of $\triangle ABC$, and $M$ and $N$ be the midpoints of the segments $BC$ and $AH$, respectively. The line $AM$ intersects the circle again at $D$, and finally, $NM$ and $OD$ intersect at $P$. Determine the locus of points $P$ as $A$ moves around the circle.
2014 Spain Mathematical Olympiad
Let $(x_n)$ be a sequence of positive integers defined by $x_1=2$ and $x_{n+1}=2x_n^3+x_n$ for all integers $n\ge1$. Determine the largest power of $5$ that divides $x_{2014}^2+1$.
2014 Spain Mathematical Olympiad
Let $M$ be the set of all integers in the form of $a^2+13b^2$, where $a$ and $b$ are distinct itnegers.



i) Prove that the product of any two elements of $M$ is also an element of $M$.



ii) Determine, reasonably, if there exist infinite pairs of integers $(x,y)$ so that $x+y\not\in M$ but $x^{13}+y^{13}\in M$.
2014 Spain Mathematical Olympiad
$60$ points are on the interior of a unit circle (a circle with radius $1$). Show that there exists a point $V$ on the circumference of the circle such that the sum of the distances from $V$ to the $60$ points is less than or equal to $80$.
2013 Spain Mathematical Olympiad
Let $a,b,n$ positive integers with $a>b$ and $ab-1=n^2$. Prove that $a-b \geq \sqrt{4n-3}$ and study the cases where the equality holds.
2013 Spain Mathematical Olympiad
Find all the possible values of a positive integer $n$ for which the expression $S_n=x^n+y^n+z^n$ is constant for all real $x,y,z$ with $xyz=1$ and $x+y+z=0$.
2013 Spain Mathematical Olympiad
Let $k,n$ be positive integers with $n \geq k \geq 3$. We consider $n+1$ points on the real plane with none three of them on the same line. We colour any segment between the points with one of $k$ possibilities. We say that an angle is a "bicolour angle" iff its vertex is one of the $n+1$ points and the two segments that define it are of different colours. Show that there is always a way to colour the segments that makes more than $n \Big\lfloor{\frac{n}{k}}\Big\rfloor^2 \frac{k(k-1)}{2}$ bicolour angles.
2013 Spain Mathematical Olympiad
Are there infinitely many positive integers $n$ that can not be represented as $n = a^3+b^5+c^7+d^9+e^{11}$, where $a,b,c,d,e$ are positive integers? Explain why.
2013 Spain Mathematical Olympiad
Study if it there exist an strictly increasing sequence of integers $0=a_0<a_1<a_2<...$ satisfying the following conditions



$i)$ Any natural number can be written as the sum of two terms of the sequence (not necessarily distinct).



$ii)$For any positive integer $n$ we have $a_n > \frac{n^2}{16}$
2013 Spain Mathematical Olympiad
Let $ABCD$ a convex quadrilateral where:



$|AB|+|CD|=\sqrt{2} |AC|$ and $|BC|+|DA|=\sqrt{2} |BD|$



What form does the quadrilateral have?
2012 Spain Mathematical Olympiad
Determine if the number $\lambda_n=\sqrt{3n^2+2n+2}$ is irrational for all non-negative integers $n$.
2012 Spain Mathematical Olympiad
Find all functions $f:\mathbb{R}\to\mathbb{R}$ such that

$$(x-2)f(y)+f(y+2f(x))=f(x+yf(x))$$

for all $x,y\in\mathbb{R}$.
2012 Spain Mathematical Olympiad
Let $x$ and $n$ be integers such that $1\le x\le n$. We have $x+1$ separate boxes and $n-x$ identical balls. Define $f(n,x)$ as the number of ways that the $n-x$ balls can be distributed into the $x+1$ boxes. Let $p$ be a prime number. Find the integers $n$ greater than $1$ such that the prime number $p$ is a divisor of $f(n,x)$ for all $x\in\{1,2,\ldots ,n-1\}$.
2012 Spain Mathematical Olympiad
Find all positive integers $n$ and $k$ such that $(n+1)^n=2n^k+3n+1$.
2012 Spain Mathematical Olympiad
A sequence $(a_n)_{n\ge 1}$ of integers is defined by the recurrence

$$a_1=1,\ a_2=5,\ a_n=\frac{a_{n-1}^2+4}{a_{n-2}}\ \text{for}\ n\ge 2.$$

Prove that all terms of the sequence are integers and find an explicit formula for $a_n$.
2012 Spain Mathematical Olympiad
Let $ABC$ be an acute-angled triangle. Let $\omega$ be the inscribed circle with centre $I$, $\Omega$ be the circumscribed circle with centre $O$ and $M$ be the midpoint of the altitude $AH$ where $H$ lies on $BC$. The circle $\omega$ be tangent to the side $BC$ at the point $D$. The line $MD$ cuts $\omega$ at a second point $P$ and the perpendicular from $I$ to $MD$ cuts $BC$ at $N$. The lines $NR$ and $NS$ are tangent to the circle $\Omega$ at $R$ and $S$ respectively. Prove that the points $R,P,D$ and $S$ lie on the same circle.
2011 Spain Mathematical Olympiad
Each pair of vertices of a regular $67$-gon is joined by a line segment. Suppose $n$ of these segments are selected, and each of them is painted one of ten available colors. Find the minimum possible value of $n$ for which, regardless of which $n$ segments were selected and how they were painted, there will always be a vertex of the polygon that belongs to seven segments of the same color.
2011 Spain Mathematical Olympiad
Let $a$, $b$, $c$ be positive real numbers. Prove that $$ \frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}+\sqrt{\frac{ab+bc+ca}{a^2+b^2+c^2}}\ge\frac52$$ and determine when equality holds.