Given the equilateral triangle $ABC$, side $a$, and its circumscribed circumference, is considered the circle segment limited by the string $AB$ and the arc (of $120^{\circ}$) with the same ends. When cutting this circular segment with lines parallel to the side $BC$, a segment of points inside the mentioned circular segment is determined on each of them. Determine the maximum length of those rectilinear segments.
Given the equation $x^{2}+a x+1=0$, determine
a) the interval in which the real number $a$ should be kept so that the roots of that equation are imaginary.
b) The geometric place of the representative points of those roots in the usual graphical representation of complex numbers, when $a$ travels the previous interval.
It is considered a convex polygon of $n$ sides. All their diagonal lines are drawn and it is assumed that in no case do three of them concur at a point that is not a vertex, and that there are also no diagonals that are parallel. Under these conditions, you want to calculate:
a) The total number of intersection points of these diagonals, excluding vertices.
b) How many of these points are inside the polygon, and how many exteriors.
The performance tax of personal work is a function $f(x)$ of the total $x$ of annual remuneration (in pesetas). knowing that
a) $f(x)$ It is a continuous function.
b) the derivative $d f(x) / d x$ in the interval $0 \leq x\lt 60000$ is constant and equal to zero; in the interval $60000\lt x\lt P$ is constant and equal to 1; and to $x\gt P$ is constant and equal to 0.14 .
c) $f(0)=0$ y $f(140000)=14000$. Determine the value of the quantity $P$ of pesetas and graphically represent the function $y=f(x)$.
In a vertical plane we consider the points $A$ and $B$ located on a horizontal line, and the semicircumference of ends $A, B$ located in the lower half plane. A segment of length $a$, equal to the diameter of the semicircle, moves in such a way that it always contains the point $A$, and that one of its ends runs through the given semicircle. Determine the value of the cosine of the angle that that segment must form with the horizontal line, so that its midpoint is as low as possible.
Given a regular pentagon, its five diagonal segments are drawn. It is asked to determine the total number of triangles that appear constructed in the figure and classify this set of triangles in classes of equal triangles (directly or inversely) with each other.
It is considered a file with 1000 numbered tokens, ordered in its natural order. The following operation is applied to that file: the first file of the file is placed interspersed between the penultimate and the last of the file, and the second, at the end of all, leaving, therefore, in the first place the one previously occupied by the third. Observing the succession of positions occupied by each of the cards, show that after 1000 analogous operations, applied successively (each to the ordering resulting from the previous operation), the file returns to its natural order. Check that an analog result could not be obtained ($n$ operations for a file of $n$ tabs) if it were a file with an odd number $n$ of tabs.
Find the smallest value $x$ such that, given any point inside an equilateral triangle of side $1$, we can always choose two points on the sides of the triangle, collinear with the given point and a distance $x$ apart.
Given a set of $n$ different positive reals $\{a_{1}, a_{2}, \ldots , a_{n}\}$. Take all possible non-empty subsets and form their sums. Prove we get at least $\frac{n(n + 1)}{2}$ different sums.
Given an isosceles triangle, find the locus of the point $P$ inside the triangle such that the distance from $P$ to the base equals the geometric mean of the distances to the sides.
Given a triangle $ABC$. Let the line through $C$ parallel to the angle bisector of $B$ meet the angle bisector of $A$ at $D$, and let the line through $C$ parallel to the angle bisector of $A$ meet the angle bisector of $B$ at $E$. Prove that if $DE$ is parallel to $AB$, then $CA = CB$.
Can we label each vertex of a 45-gon with one of the digits $0, 1, \ldots, 9$ so that for each pair of distinct digits $i, j$ one of the 45 sides has vertices labeled $i, j$?
Given a circle $c$ and two fixed points $A$, $B$ on it. $M$ is another point on $c$, and $K$ is the midpoint of $BM$. $P$ is the foot of the perpendicular from $K$ to $AM$.
a. Prove that $KP$ passes through a fixed point (as $M$ varies).
We place labeled points on a circle as follows. At step 1, take two points at opposite ends of a diameter and label them both $1$. At step $n \gt 1$, place a point at the midpoint of each arc created at step $n - 1$ and label it with the sum of the labels at the two adjacent points. What is the total sum of the labels after step $n$?
For example, after step 4 we have: $1$, $4$, $3$, $5$, $2$, $5$, $3$, $4$, $1$, $4$, $3$, $5$, $2$, $5$, $4$, $3$, $4$.
a. A $6 \times 6$ board is tiled with $2 \times 1$ dominos. Prove that we can always divide the board into two rectangles each of which is tiled separately (with no domino crossing the dividing line).
8 players compete in a tournament. Everyone plays everyone else just once. The winner of a game gets 1, the loser 0, or each gets $\frac{1}{2}$ if the game is drawn. The final result is that everyone gets a different score and the player placing second gets the same as the total of the four bottom players. What was the result of the game between the player placing third and the player placing seventh?