Navvye/Olympiad-Problems-CSV huggingface dataset:
1993 Polish MO Finals
Denote $g(k)$ as the greatest odd divisor of $k$. Put $f(k) = \dfrac{k}{2} + \dfrac{k}{g(k)}$ for $k$ even, and $2^{(k+1)/2}$ for $k$ odd. Define the sequence $x_1, x_2, x_3, ...$ by $x_1 = 1$, $x_{n+1} = f(x_n)$. Find $n$ such that $x_n = 800$.
1993 Polish MO Finals
Let be given a convex polyhedron whose all faces are triangular. The vertices of the polyhedron are colored using three colors. Prove that the number of faces with vertices in all the three colors is even.
1993 Polish MO Finals
Find out whether it is possible to determine the volume of a tetrahedron knowing the areas of its faces and its circumradius.
1992 Polish MO Finals
Segments $AC$ and $BD$ meet at $P$, and $|PA| = |PD|$, $|PB| = |PC|$. $O$ is the circumcenter of the triangle $PAB$. Show that $OP$ and $CD$ are perpendicular.
1992 Polish MO Finals
Find all functions $f : \mathbb{Q}^{+} \rightarrow \mathbb{Q}^{+}$, where $\mathbb{Q}^{+}$ is the set of positive rationals, such that $f(x+1) = f(x) + 1$ and $f(x^3) = f(x)^3$ for all $x$.
1992 Polish MO Finals
Show that for real numbers $x_1, x_2, ... , x_n$ we have:

$$ \sum\limits_{i=1}^n \sum\limits_{j=1}^n \dfrac{x_ix_j}{i+j} \geq 0  $$

When do we have equality?
1992 Polish MO Finals
The functions $f_0, f_1, f_2, ...$ are defined on the reals by $f_0(x) = 8$ for all $x$, $f_{n+1}(x) = \sqrt{x^2 + 6f_n(x)}$. For all $n$ solve the equation $f_n(x) = 2x$.
1992 Polish MO Finals
The base of a regular pyramid is a regular $2n$-gon $A_1A_2...A_{2n}$. A sphere passing through the top vertex $S$ of the pyramid cuts the edge $SA_i$ at $B_i$ (for $i = 1, 2, ... , 2n$). Show that $\sum\limits_{i=1}^n SB_{2i-1} = \sum\limits_{i=1}^n SB_{2i}$.
1992 Polish MO Finals
Show that $(k^3)!$ is divisible by $(k!)^{k^2+k+1}$.
1991 Polish MO Finals
Prove or disprove that there exist two tetrahedra $T_1$ and $T_2$ such that:

(i) the volume of $T_1$ is greater than that of $T_2$;

(ii) the area of any face of $T_1$ does not exceed the area of any face of $T_2$.
1991 Polish MO Finals
Let $X$ be the set of all lattice points in the plane (points $(x, y)$ with $x, y \in \mathbb{Z}$). A path of length $n$ is a chain $(P_0, P_1, ... , P_n)$ of points in $X$ such that $P_{i-1}P_i = 1$ for $i = 1, ... , n$. Let $F(n)$ be the number of distinct paths beginning in $P_0=(0,0)$ and ending in any point $P_n$ on line $y = 0$. Prove that $F(n) = \binom{2n}{n}$
1991 Polish MO Finals
Define

$$ N=\sum\limits_{k=1}^{60}e_k k^{k^k} $$

where $e_k \in \{-1, 1\}$ for each $k$. Prove that $N$ cannot be the fifth power of an integer.
1991 Polish MO Finals
On the Cartesian plane consider the set $V$ of all vectors with integer coordinates. Determine all functions $f : V \rightarrow \mathbb{R}$ satisfying the conditions:

(i) $f(v) = 1$ for each of the four vectors $v \in V$ of unit length.

(ii) $f(v+w) = f(v)+f(w)$ for every two perpendicular vectors $v, w \in V$

(Zero vector is considered to be perpendicular to every vector).
1991 Polish MO Finals
Two noncongruent circles $k_1$ and $k_2$ are exterior to each other. Their common tangents intersect the line through their centers at points $A$ and $B$. Let $P$ be any point of $k_1$. Prove that there is a diameter of $k_2$ with one endpoint on line $PA$ and the other on $PB$.
1991 Polish MO Finals
If $x, y, z$ are real numbers satisfying $x^2 +y^2 +z^2 = 2$, prove the inequality

$$ x + y + z \leq 2 + xyz $$

When does equality occur?
1990 Polish MO Finals
Find all functions $f : \mathbb{R} \longrightarrow \mathbb{R}$ that satisfy

$$ (x - y)f(x + y) - (x + y)f(x - y) = 4xy(x^2 - y^2)  $$
1990 Polish MO Finals
Let $x_1, x_2, . . . , x_n$ be positive numbers. Prove that



$$ \sum\limits_{i=1}^n \dfrac{x_i ^2}{x_i ^2+x_{i+1}x_{i+2}} \leq n-1 $$



Where $x_{n+1}=x_1$ and $x_{n+2}=x_2$.
1990 Polish MO Finals
In a tournament, every two of the $n$ players played exactly one match with each other (no

draws). Prove that it is possible either

(i) to partition the league in two groups $A$ and $B$ such that everybody in $A$ defeated everybody in $B$; or

(ii) to arrange all the players in a chain $x_1, x_2, . . . , x_n, x_1$ in such a way that each player defeated his successor.
1990 Polish MO Finals
A triangle whose all sides have length not smaller than $1$ is inscribed in a square of side length $1$. Prove that the center of the square lies inside the triangle or on its boundary.
1990 Polish MO Finals
Suppose that $(a_n)$ is a sequence of positive integers such that $\lim\limits_{n\to \infty} \dfrac{n}{a_n}=0$

Prove that there exists $k$ such that there are at least $1990$ perfect squares between $a_1 + a_2 + ... + a_k$ and $a_1 + a_2 + ... + a_{k+1}$.
1990 Polish MO Finals
Prove that for all integers $n > 2$,

$$ 3| \sum\limits_{i=0}^{[n/3]} (-1)^i C _n ^{3i} $$
1989 Polish MO Finals
An even number of politicians are sitting at a round table. After a break, they come back and sit down again in arbitrary places. Show that there must be two people with the same number of people sitting between them as before the break..



Additional problem:

Solve the problem when the number of people is in a form $6k+3$.
1989 Polish MO Finals
$k_1, k_2, k_3$ are three circles. $k_2$ and $k_3$ touch externally at $P$, $k_3$ and $k_1$ touch externally at $Q$, and $k_1$ and $k_2$ touch externally at $R$. The line $PQ$ meets $k_1$ again at $S$, the line $PR$ meets $k_1$ again at $T$. The line $RS$ meets $k_2$ again at $U$, and the line $QT$ meets $k_3$ again at $V$. Show that $P, U, V$ are collinear.
1989 Polish MO Finals
The edges of a cube are labeled from $1$ to $12$. Show that there must exist at least eight triples $(i, j, k)$ with $1 \leq i < j < k \leq 12$ so that the edges $i, j, k$ are consecutive edges of a path. Also show that there exists labeling in which we cannot find nine such triples.
1989 Polish MO Finals
$n, k$ are positive integers. $A_0$ is the set $\{1, 2, ... , n\}$. $A_i$ is a randomly chosen subset of $A_{i-1}$ (with each subset having equal probability). Show that the expected number of elements of $A_k$ is $\dfrac{n}{2^k}$
1989 Polish MO Finals
Three circles of radius $a$ are drawn on the surface of a sphere of radius $r$. Each pair of circles touches externally and the three circles all lie in one hemisphere. Find the radius of a circle on the surface of the sphere which touches all three circles.
1989 Polish MO Finals
Show that for positive reals $a, b, c, d$ we have

$$ \left(\dfrac{ab + ac + ad + bc + bd + cd}{6} \right)^3 \geq \left(\dfrac{abc + abd + acd + bcd}{4}\right)^2 $$
1988 Polish MO Finals
The real numbers $x_1, x_2, ... , x_n$ belong to the interval $(0,1)$ and satisfy $x_1 + x_2 + ... + x_n = m + r$, where $m$ is an integer and $r \in [0,1)$. Show that $x_1 ^2 + x_2 ^2 + ... + x_n ^2 \leq m + r^2$.
1988 Polish MO Finals
For a permutation $P = (p_1, p_2, ... , p_n)$ of $(1, 2, ... , n)$ define $X(P)$ as the number of $j$ such that $p_i < p_j$ for every $i < j$. What is the expected value of $X(P)$ if each permutation is equally likely?
1988 Polish MO Finals
$W$ is a polygon which has a center of symmetry $S$ such that if $P$ belongs to $W$, then so does $P'$, where $S$ is the midpoint of $PP'$. Show that there is a parallelogram $V$ containing $W$ such that the midpoint of each side of $V$ lies on the border of $W$.
1988 Polish MO Finals
$d$ is a positive integer and $f : [0,d] \rightarrow \mathbb{R}$ is a continuous function with $f(0) = f(d)$. Show that there exists $x \in [0,d-1]$ such that $f(x) = f(x+1)$.
1988 Polish MO Finals
The sequence $a_1, a_2, a_3, ...$ is defined by $a_1 = a_2 = a_3 = 1$, $a_{n+3} = a_{n+2}a_{n+1} + a_n$. Show that for any positive integer $r$ we can find $s$ such that $a_s$ is a multiple of $r$.
1988 Polish MO Finals
Find the largest possible volume for a tetrahedron which lies inside a hemisphere of radius $1$.
1987 Polish MO Finals
There are $n \ge 2$ points in a square side $1$. Show that one can label the points $P_1, P_2, ... , P_n$ such that $\sum_{i=1}^n |P_{i-1} - P_i|^2 \le 4$, where we use cyclic subscripts, so that $P_0$ means $P_n$.
1987 Polish MO Finals
A regular $n$-gon is inscribed in a circle radius $1$. Let $X$ be the set of all arcs $PQ$, where $P, Q$ are distinct vertices of the $n$-gon. $5$ elements $L_1, L_2, ... , L_5$ of $X$ are chosen at random (so two or more of the $L_i$ can be the same). Show that the expected length of $L_1 \cap L_2 \cap L_3 \cap L_4 \cap L_5$ is independent of $n$.
1987 Polish MO Finals
$w(x)$ is a polynomial with integer coefficients. Let $p_n$ be the sum of the digits of the number $w(n)$. Show that some value must occur infinitely often in the sequence $p_1, p_2, p_3, ...$ .
1987 Polish MO Finals
Let $S$ be the set of all tetrahedra which satisfy:

(1) the base has area $1$,

(2) the total face area is $4$, and

(3) the angles between the base and the other three faces are all equal.

Find the element of $S$ which has the largest volume.
1987 Polish MO Finals
Find the smallest $n$ such that $n^2 -n+11$ is the product of four primes (not necessarily distinct).
1987 Polish MO Finals
A plane is tiled with regular hexagons of side $1$. $A$ is a fixed hexagon vertex.

Find the number of paths $P$ such that:

(1) one endpoint of $P$ is $A$,

(2) the other endpoint of $P$ is a hexagon vertex,

(3) $P$ lies along hexagon edges,

(4) $P$ has length $60$, and

(5) there is no shorter path along hexagon edges from $A$ to the other endpoint of $P$.
1986 Polish MO Finals
A square of side $1$ is covered with $m^2$ rectangles.

Show that there is a rectangle with perimeter at least $\frac{4}{m}$.
1986 Polish MO Finals
Find the maximum possible volume of a tetrahedron which has three faces with area $1$.
1986 Polish MO Finals
$p$ is a prime and $m$ is a non-negative integer $< p-1$.

Show that $ \sum_{j=1}^p j^m$ is divisible by $p$.
1986 Polish MO Finals
Find all $n$ such that there is a real polynomial $f(x)$ of degree $n$ such that $f(x) \ge f'(x)$ for all real $x$.
1986 Polish MO Finals
There is a chess tournament with $2n$ players ($n > 1$). There is at most one match between each pair of players. If it is not possible to find three players who all play each other, show that there are at most $n^2$ matches. Conversely, show that if there are at most $n^2$ matches, then it is possible to arrange them so that we cannot find three players who all play each other.
1986 Polish MO Finals
$ABC$ is a triangle. The feet of the perpendiculars from $B$ and $C$ to the angle bisector at $A$ are $K, L$ respectively. $N$ is the midpoint of $BC$, and $AM$ is an altitude. Show that $K,L,N,M$ are concyclic.
1985 Polish MO Finals
Find the largest $k$ such that for every positive integer $n$ we can find at least $k$ numbers in the set $\{n+1, n+2, ... , n+16\}$ which are coprime with $n(n+17)$.
1985 Polish MO Finals
Given a square side $1$ and $2n$ positive reals $a_1, b_1, ... , a_n, b_n$ each  $\le  1$ and satisfying  $\sum a_ib_i \ge  100$. Show that the square can be covered with rectangles $R_i$ with sides length $(a_i, b_i)$ parallel to the square sides.
1985 Polish MO Finals
The function $f : R \to R$ satisfies $f(3x) = 3f(x) - 4f(x)^3$ for all real $x$ and is continuous at $x = 0$. Show that $|f(x)| \le 1$ for all $x$.
1985 Polish MO Finals
$P$ is a point inside the triangle $ABC$ is a triangle. The distance of $P$ from the lines $BC, CA, AB$ is $d_a, d_b, d_c$ respectively. If $r$ is the inradius, show that $$\frac{2}{ \frac{1}{d_a} + \frac{1}{d_b} + \frac{1}{d_c}} < r < \frac{d_a + d_b + d_c}{2}$$
1985 Polish MO Finals
$p(x,y)$ is a polynomial such that $p(cos t, sin t) = 0$ for all real $t$.

Show that there is a polynomial $q(x,y)$ such that $p(x,y) = (x^2 + y^2 - 1) q(x,y)$.
1985 Polish MO Finals
There is a convex polyhedron with $k$ faces.

Show that if more than $k/2$ of the faces are such that no two have a common edge,

then the polyhedron cannot have an inscribed sphere.
1984 Polish MO Finals
Find the number of all real functions $f$ which map the sum of $n$ elements into the sum of their images, such that $f^{n-1}$ is a constant function and $f^{n-2}$ is not. Here $f^0(x) = x$ and $f^k = f \circ f^{k-1}$ for $k \ge 1$.
1984 Polish MO Finals
Let $n$ be a positive integer. For all $i, j \in \{1,2,...,n\}$ define $a_{j,i} = 1$ if $j = i$ and $a_{j,i} = 0$ otherwise. Also, for $i = n+1,...,2n$ and $j = 1,...,n$ define $a_{j,i} = -\frac{1}{n}$.

Prove that for any permutation $p$ of the set $\{1,2,...,2n\}$ the following inequality holds: $\sum_{j=1}^{n}\left|\sum_{k=1}^{n} a_{j,p}(k)\right| \ge \frac{n}{2}$
1984 Polish MO Finals
Let $W$ be a regular octahedron and $O$ be its center. In a plane $P$ containing $O$ circles $k_1(O,r_1)$ and $k_2(O,r_2)$ are chosen so that $k_1 \subset P\cap W  \subset k_2$. Prove that $\frac{r_1}{r_2}\le \frac{\sqrt3}{2}$
1984 Polish MO Finals
A coin is tossed $n$ times, and the outcome is written in the form ($a_1,a_2,...,a_n$), where $a_i = 1$ or $2$ depending on whether the result of the $i$-th toss is the head or the tail, respectively. Set $b_j = a_1 +a_2 +...+a_j$ for $j = 1,2,...,n$, and let $p(n)$ be the probability that the sequence $b_1,b_2,...,b_n$ contains the number $n$. Express $p(n)$ in terms of $p(n-1)$ and $p(n-2)$.
1984 Polish MO Finals
A regular hexagon of side $1$ is covered by six unit disks. Prove that none of the vertices of the hexagon is covered by two (or more) discs.
1984 Polish MO Finals
Cities $P_1,...,P_{1025}$ are connected to each other by airlines $A_1,...,A_{10}$ so that for any two distinct cities $P_k$ and $P_m$ there is an airline offering a direct flight between them. Prove that one of the airlines can offer a round trip with an odd number of flights.
1983 Polish MO Finals
On the plane are given a convex $n$-gon $P_1P_2....P_n$ and a point $Q$ inside it, not lying on any of its diagonals. Prove that if $n$ is even, then the number of triangles $P_iP_jP_k$ containing the point $Q$ is even.
1983 Polish MO Finals
Let be given an irrational number $a$ in the interval $(0,1)$ and a positive integer $N$.

Prove that there exist positive integers $p,q,r,s$ such that $\frac{p}{q} < a <\frac{r}{s}, \frac{r}{s} -\frac{p}{q}<\frac{1}{N}$, and $rq- ps = 1$.
1983 Polish MO Finals
Consider the following one-player game on an infinite chessboard. If two horizontally or vertically adjacent squares are occupied by a pawn each, and a square on the same line that is adjacent to one of them is empty, then it is allowed to remove the two pawns and place a pawn on the third (empty) square. Prove that if in the initial position all the pawns were forming a rectangle with the number of squares divisible by $3$, then it is not possible to end the game with only one pawn left on the board.
1983 Polish MO Finals
Prove that if natural numbers $a,b,c,d$ satisfy the equality $ab = cd$, then $\frac{gcd(a,c)gcd(a,d)}{gcd(a,b,c,d)}= a$
1983 Polish MO Finals
On the plane are given unit vectors $\overrightarrow{a_1},\overrightarrow{a_2},\overrightarrow{a_3}$. Show that one can choose numbers $c_1,c_2,c_3 \in \{-1,1\}$ such that the length of the vector $c_1\overrightarrow{a_1}+c_2\overrightarrow{a_2}+c_3\overrightarrow{a_3}$ is at least $2$.
1983 Polish MO Finals
Prove that if all dihedral angles of a tetrahedron are acute, then all its faces are acute-angled triangles.
1982 Polish MO Finals
Find a way of arranging $n$ girls and $n$ boys around a round table for which $d_n-c_n$ is maximum, where dn is the number of girls sitting between two boys and $c_n$ is the number of boys sitting between two girls.
1982 Polish MO Finals
In a cyclic quadrilateral $ABCD$ the line passing through the midpoint of $AB$ and the intersection point of the diagonals is perpendicular to $CD$. Prove that either the sides $AB$ and $CD$ are parallel or the diagonals are perpendicular
1982 Polish MO Finals
Find all pairs of positive numbers $(x,y)$ which satisfy the system of equations

$x^2 +y^2 = a^2 +b^2$

$x^3 +y^3 = a^3 +b^3$

where $a$ and $b$ are given positive numbers.
1982 Polish MO Finals
On a plane is given a finite set of points. Prove that the points can be covered by open squares $Q_1,Q_2,...,Q_n$ such that $1 \le\frac{N_j}{S_j} \le 4$ for $j = 1,...,n,$ where $N_j$ is the number of points from the set inside square $Q_j$ and $S_j$ is the area of $Q_j$.
1982 Polish MO Finals
Integers $x_0,x_1,...,x_{n-1}, x_n = x_0, x_{n+1} = x_1$ satisfy the inequality $(-1)^{x_k} x_{k-1}x_{k+1} >0$ for $k = 1,2,...,n$. Prove that the difference $\sum_{k=0}^{n-1}x_k -\sum_{k=0}^{n-1}|x_k|$ is divisible by $4$.
1982 Polish MO Finals
Prove that the sum of dihedral angles in an arbitrary tetrahedron is greater than $2\pi$
1977 Polish MO Finals
Let $ABCD$ be a tetrahedron with $\angle BAD = 60^{\cdot}$, $\angle BAC = 40^{\cdot}$, $\angle ABD = 80^{\cdot}$, $\angle ABC = 70^{\cdot}$. Prove that the lines $AB$ and $CD$ are perpendicular.
1977 Polish MO Finals
Let $s \geq 3$ be a given integer. A sequence $K_n$ of circles and a sequence $W_n$

of convex $s$-gons satisfy:

$$ K_n \supset W_n \supset K_{n+1}  $$ for all $n = 1, 2, ...$

Prove that the sequence of the radii of the circles $K_n$ converges to zero.
1977 Polish MO Finals
Consider the set $A = \{0, 1, 2, . . . , 2^{2n} - 1\}$. The function $f : A \rightarrow A$ is given by: $f(x_0 + 2x_1 + 2^2x_2 + ... + 2^{2n-1}x_{2n-1})=$$(1 - x_0) + 2x_1 + 2^2(1 - x_2) + 2^3x_3 + ... + 2^{2n-1}x_{2n-1}$

for every $0-1$ sequence $(x_0, x_1, . . . , x_{2n-1})$. Show that if $a_1, a_2, . . . , a_9$ are consecutive terms of an arithmetic progression, then the sequence $f(a_1), f(a_2), . . . , f(a_9)$ is not increasing.
1977 Polish MO Finals
A function $h : \mathbb{R} \rightarrow \mathbb{R}$ is differentiable and satisfies $h(ax) = bh(x)$ for all $x$, where $a$ and $b$ are given positive numbers and $0 \not = |a| \not = 1$. Suppose that $h'(0) \not = 0$ and the function $h'$ is continuous at $x = 0$. Prove that $a = b$ and that there is a real number $c$ such that $h(x) = cx$ for all $x$.
1977 Polish MO Finals
Show that for every convex polygon there is a circle passing through three consecutive vertices of the polygon and containing the entire polygon
1977 Polish MO Finals
Consider the polynomial $W(x) = (x - a)^kQ(x)$, where $a \neq 0$, $Q$ is a nonzero polynomial, and $k$ a natural number. Prove that $W$ has at least $k + 1$ nonzero coefficients.
2012 Portugal MO
A five-digit positive integer $abcde_{10}$ ($a\neq 0$) is said to be a range if its digits satisfy the inequalities $a<b>c<d>e$. For example, $37452$ is a range. How many ranges are there?
2012 Portugal MO
In triangle $[ABC]$, the bissector of the angle $\angle{BAC}$ intersects the side $[BC]$ at $D$. Suppose that $\overline{AD}=\overline{CD}$. Find the lengths $\overline{BC}$, $\overline{AC}$ and $\overline{AB}$ that minimize the perimeter of $[ABC]$, given that all the sides of the triangles $[ABC]$ and $[ADC]$ have integer lengths.
2012 Portugal MO
Helena and Luis are going to play a game with two bags with marbles. They play alternately and on each turn they can do one and only one of the following moves:



Take out a marble from one bag.

Take out a marble from each bag.

Take out a marble from one bag and then put it into the other bag.



The player who leaves both bags empty wins the game.



Before starting the game, Helena counted out the marbles of each bag and said to Luis: "You may start!", while she thought "I will certainly win...". What are the possible distributions of the marbles in the bags?
2012 Portugal MO
Find the number of positive integers $n$ such that $1\leq n\leq 1000$ and $n$ is divisible by $\lfloor \sqrt[3]{n} \rfloor$.
2012 Portugal MO
Let $[ABC]$ be a triangle. Points $D$, $E$, $F$ and $G$ are such $E$ and $F$ are on the lines $AC$ and $BC$, respectively, and $[ACFG]$ and $[BCED]$ are rhombus. Lines $AC$ and $BG$ meet at $H$; lines $BC$ and $AD$ meet at $I$; lines $AI$ and $BH$ meet at $J$. Prove that $[JICH]$ and $[ABJ]$ have equal area.
2012 Portugal MO
Isabel wants to partition the set $\mathbb{N}$ of the positive integers into $n$ disjoint sets $A_{1}, A_{2}, \ldots, A_{n}$. Suppose that for each $i$ with $1\leq i\leq n$, given any positive integers $r, s\in A_{i}$ with $r\neq s$, we have $r+s\in A_{i}$. If $|A_{j}|=1$ for some $j$, find the greatest positive integer that may belong to $A_{j}$.
2010 Portugal MO
There are several candles of the same size on the Chapel of Bones. On the first day a candle is lit for a hour. On the second day two candles are lit for a hour, on the third day three candles are lit for a hour, and successively, until the last day, when all the candles are lit for a hour. On the end of that day, all the candles were completely consumed. Find all the possibilities for the number of candles.
2010 Portugal MO
On a circumference, points $A$ and $B$ are on opposite arcs of diameter $CD$. Line segments $CE$ and $DF$ are perpendicular to $AB$ such that $A-E-F-B$ (i.e., $A$, $E$, $F$ and $B$ are collinear on this order). Knowing $AE=1$, find the length of $BF$.
2010 Portugal MO
On each day, more than half of the inhabitants of Évora eats sericaia as dessert. Show that there is a group of 10 inhabitants of Évora such that, on each of the last 2010 days, at least one of the inhabitants ate sericaia as dessert.
2010 Portugal MO
Giraldo wrote five distinct natural numbers on the vertices of a pentagon. And next he wrote on each side of the pentagon the least common multiple of the numbers written of the two vertices who were on that side and noticed that the five numbers written on the sides were equal. What is the smallest number Giraldo could have written on the sides?
2010 Portugal MO
Show that any triangle has two sides whose lengths $a$ and $b$ satisfy $\frac{\sqrt{5}-1}{2}<\frac{a}{b}<\frac{\sqrt{5}+1}{2}$.
2010 Portugal MO
Consider a square $(p-1)\times(p-1)$, where $p$ is a prime number, which is divided by squares $1\times 1$ whose sides are parallel to the initial square's sides. Show that it is possible to select $p$ vertices such that there are no three collinear vertices.
2009 Portugal MO
A circumference was divided in $n$ equal parts. On each of these parts one number from $1$ to $n$ was placed such that the distance between consecutive numbers is always the same. Numbers $11$, $4$ and $17$ were in consecutive positions. In how many parts was the circumference divided?
2009 Portugal MO
Points $N$ and $M$ are on the sides $CD$ and $BC$ of square $ABCD$, respectively. The perimeter of triangle $MCN$ is equal to the double of the length of the square's side. Find $\angle MAN$.
2009 Portugal MO
Duarte wants to draw a square whose side's length is $2009$ cm and which is divided in $2009\times2009$ squares whose side's length is $1$ cm and whose sides are parallel to the original square's one, without taking the pencil out of the paper. Starting on one of the vertex of the giant square, what is the length of the shortest line that allows him to make this drawing?
2009 Portugal MO
João calculated the product of the non zero digits of each integer from $1$ to $10^{2009}$ and then he summed these $10^{2009}$ products. Which number did he obtain?
2009 Portugal MO
Circumferences $C_1$ and $C_2$ have different radios and are externally tangent on point $T$. Consider points $A$ on $C_1$ and $B$ on $C_2$, both different from $T$, such that $\angle BTA=90^{\circ}$. What is the locus of the midpoints of line segments $AB$ constructed that way?
2009 Portugal MO
Two players play the following game on a circular board with 2009 houses. The two plays put, alternatively, on an empty house, one of three pieces, called explorer (E), trap (T) or stone (S). A treasure is a sequence of three consecutive filled houses such that the first one (on any direction) has an explorer and the middle one doesn't have a trap. For example, STE is not a treasure, while TEE is a treasure. The first player forming a treasure wins. Can any of the players guarantee the victory? And, in affirmative case, who?
2008 Portugal MO
What is the maximum number of triangles with vertices on the points of the fork/graph which is possible to construct?
2008 Portugal MO
Let $AEBC$ be a cyclic quadrilateral. Let $D$ be a point on the ray $AE$ which is outside the circumscribed circumference of $AEBC$. Suppose that $\angle CAB=\angle BAE$. Prove that $AB=BD$ if and only if $DE=AC$.