Found problems: 52
2021 Spain Mathematical Olympiad, 5
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:
$$
\frac{3 \cdot 5 \cdot 7 \cdots (2n-1)}{2^n n!}
$$
2018 Spain Mathematical Olympiad, 2
Let $n$ be a positive integer. $2n+1$ tokens are in a row, each being black or white. A token is said to be [i]balanced[/i] 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 [i]balanced[/i] tokens is even or odd.
2024 Spain Mathematical Olympiad, 3
Let $ABC$ be a scalene triangle and $P$ be an interior point such that $\angle PBA=\angle PCA$. The lines $PB$ and $PC$ intersect the internal and external bisectors of $\angle BAC$ at $Q$ and $R$, respectively. Let $S$ be the point such that $CS$ is parallel to $AQ$ and $BS$ is parallel to $AR$. Prove that $Q$, $R$ and $S$ are colinear.
1965 Spain Mathematical Olympiad, 1
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.
2018 Spain Mathematical Olympiad, 3
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.
2017 Spain Mathematical Olympiad, 6
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 $PR=AD$.
2019 Junior Balkan Team Selection Tests - Romania, 4
Let $n$ be a positive integer. $2n+1$ tokens are in a row, each being black or white. A token is said to be [i]balanced[/i] 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 [i]balanced[/i] tokens is even or odd.
2025 Spain Mathematical Olympiad, 1
Determine the number of distinct values which appear in the sequence \[\left\lfloor\frac{2025}{1}\right\rfloor,\left\lfloor\frac{2025}{2}\right\rfloor,\left\lfloor\frac{2025}{3}\right\rfloor,\dots,\left\lfloor\frac{2025}{2024}\right\rfloor,\left\lfloor\frac{2025}{2025}\right\rfloor.\]
2024 Spain Mathematical Olympiad, 6
Let $a$, $b$ and $n$ be positive integers, satisfying that $bn$ divides $an-a+1$. Let $\alpha=a/b$. Prove that, when the numbers $\lfloor\alpha\rfloor,\lfloor2\alpha\rfloor,\dots,\lfloor(n-1)\alpha\rfloor$ are divided by $n$, the residues are $1,2,\dots,n-1$, in some order.
2024 Spain Mathematical Olympiad, 5
Given two points $p_1=(x_1, y_1)$ and $p_2=(x_2, y_2)$ on the plane, denote by $\mathcal{R}(p_1,p_2)$ the rectangle with sides parallel to the coordinate axes and with $p_1$ and $p_2$ as opposite corners, that is, \[\{(x,y)\in \mathbb{R}^2:\min\{x_1, x_2\}\leq x\leq \max\{x_1, x_2\},\min\{y_1, y_2\}\leq y\leq \max\{y_1, y_2\}\}.\] Find the largest value of $k$ for which the following statement is true: for all sets $\mathcal{S}\subset\mathbb{R}^2$ with $|\mathcal{S}|=2024$, there exist two points $p_1, p_2\in\mathcal{S}$ such that $|\mathcal{S}\cap\mathcal{R}(p_1, p_2)|\geq k$.
2023 Spain Mathematical Olympiad, 4
Let $x_1\leq x_2\leq x_3\leq x_4$ be real numbers. Prove that there exist polynomials of degree two $P(x)$ and $Q(x)$ with real coefficients such that $x_1$, $x_2$, $x_3$ and $x_4$ are the roots of $P(Q(x))$ if and only if $x_1+x_4=x_2+x_3$.
2022 Spain Mathematical Olympiad, 3
Let $ABC$ be a triangle, with $AB<AC$, and let $\Gamma$ be its circumcircle. Let $D$, $E$ and $F$ be the tangency points of the incircle with $BC$, $CA$ and $AB$ respectively. Let $R$ be the point in $EF$ such that $DR$ is an altitude in the triangle $DEF$, and let $S$ be the intersection of the external bisector of $\angle BAC$ with $\Gamma$. Prove that $AR$ and $SD$ intersect on $\Gamma$.
2021 Spain Mathematical Olympiad, 1
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 $\angle DNO$.
2020 Spain Mathematical Olympiad, 2
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 $n \geq n_0$.
2017 Spain Mathematical Olympiad, 1
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\}$
2025 Spain Mathematical Olympiad, 6
Let $\mathbb{R}_{\neq 0}$ be the set of nonzero real numbers. Find all functions $f:\mathbb{R}_{\neq 0}\rightarrow\mathbb{R}_{\neq 0}$ such that, for all $x,y\in\mathbb{R}_{\neq 0}$, \[(x-y)f(y^2)+f\left(xy\,f\left(\frac{x^2}{y}\right)\right)=f(y^2f(y)).\]
2020 Spain Mathematical Olympiad, 3
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.
2017 Spain Mathematical Olympiad, 2
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, 4
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?
2018 Spain Mathematical Olympiad, 1
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.
2008 Spain Mathematical Olympiad, 2
Let $a$ and $b$ be two real numbers, with $0<a,b<1$. Prove that
\[\sqrt{ab^2+a^2b}+\sqrt{(1-a)(1-b)^2+(1-a)^2(1-b)}<\sqrt{2}\]
2016 Spain Mathematical Olympiad, 6
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:
$$x_1x_2...x_n\leq \gamma \left(x_1y_1+x_2y_2+...+x_ny_n\right)$$
2016 Spain Mathematical Olympiad, 4
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 $ab$.
2025 Spain Mathematical Olympiad, 3
We write the decimal expressions of $\sqrt{2}$ and $\sqrt{3}$ as \[\sqrt{2}=1.a_1a_2a_3\dots\quad\quad\sqrt{3}=1.b_1b_2b_3\dots\] where each $a_i$ or $b_i$ is a digit between 0 and 9. Prove that there exist at least 1000 values of $i$ between $1$ and $10^{1000}$ such that $a_i\neq b_i$.
2022 Spain Mathematical Olympiad, 6
Find all triples $(x,y,z)$ of positive integers, with $z>1$, satisfying simultaneously that \[x\text{ divides }y+1,\quad y\text{ divides }z-1,\quad z\text{ divides }x^2+1.\]