Found problems: 10
2002 Chile National Olympiad, 2
Determine all natural numbers $n$ for which it is possible to construct a rectangle of sides $15$ and $n$, with pieces congruent to:
[asy]
unitsize(0.6 cm);
draw((0,0)--(3,0));
draw((0,1)--(3,1));
draw((0,2)--(1,2));
draw((2,2)--(3,2));
draw((0,0)--(0,2));
draw((1,0)--(1,2));
draw((2,0)--(2,2));
draw((3,0)--(3,2));
draw((5,-0.5)--(6,-0.5));
draw((4,0.5)--(7,0.5));
draw((4,1.5)--(7,1.5));
draw((5,2.5)--(6,2.5));
draw((4,0.5)--(4,1.5));
draw((5,-0.5)--(5,2.5));
draw((6,-0.5)--(6,2.5));
draw((7,0.5)--(7,1.5));
[/asy]
The squares of the pieces have side $1$ and the pieces cannot overlap or leave free spaces
2014 Chile National Olympiad, 5
Prove that if a quadrilateral $ABCD$ can be cut into a finite number of parallelograms, then $ABCD$ is a parallelogram.
2022 Chile Junior Math Olympiad, 1
Find all real numbers $x, y, z$ that satisfy the following system
$$\sqrt{x^3 - y} = z - 1$$
$$\sqrt{y^3 - z} = x - 1$$
$$\sqrt{z^3 - x} = y - 1$$
2002 Chile National Olympiad, 4
All naturals from $1$ to $2002$ are placed in a row. Can the signs: $+$ and $-$ be placed between each consecutive pair of them so that the corresponding algebraic sum is $0$?
2022 Chile National Olympiad, 1
Find all real numbers $x, y, z$ that satisfy the following system
$$\sqrt{x^3 - y} = z - 1$$
$$\sqrt{y^3 - z} = x - 1$$
$$\sqrt{z^3 - x} = y - 1$$
2002 Chile National Olympiad, 7
A convex polygon of sides $\ell_1, \ell_2, ..., \ell_n$ is called [i]ordered [/i] if for all reordering $( \sigma (1), \sigma (2), ..., \sigma (n))$ of the set $(1, 2,..., n)$ there exists a point $P$ inside the polygon such that $d_{\sigma (1)} < _{\sigma (2)} <...< d_{\sigma (n)}$ , where $d_i$ represents the distance between $P$ and side $\ell_i$. Find all the convex ordered polygons.
2002 Chile National Olympiad, 1
A Metro ticket, which has six digits, is considered a "lucky number" if its six digits are different and their first three digits add up to the same as the last three (A number such as $026134$ is "lucky number"). Show that the sum of all the "lucky numbers" is divisible by $2002$.
2002 Chile National Olympiad, 6
Determine all three-digit numbers $N$ such that the average of the six numbers that can be formed by permutation of its three digits is equal to $N$.
2002 Chile National Olympiad, 3
Given the line $AB$, let $M$ be a point on it. Towards the same side of the plane and with bases $AM$ and $MB$, squares $AMCD$ and $MBEF$ are constructed. Let $N$ be the point (different from $M$) where the circumcircles circumscribed to both squares intersect and let $N_1$ be the point where the lines $BC$ and $AF$ intersect. Prove that the points $N$ and $N_1$ coincide. Prove that as the point $M$ moves on the line $AB$, the line $MN$ moves always passing through a fixed point.
2002 Chile National Olympiad, 5
Given a right triangle $T$, where the coordinates of its vertices are integers, let $E$ be the number of points of integer coordinates that belong to the edge of the triangle $T$, $I$ the number of points of integer coordinates that belong to the interior of the triangle $T$. Show that the area $A(T)$ of triangle $T$ is given by: $A(T) = \frac{E}{2}+I -1$.