geometry problems from COMATEQ , a junior contest from Puerto Rico with aops links in the names
2017 - 2021
collected inside aops here
A blue square has a perimeter of $8$ cm. A red square has an area that is $9$ times the area of the blue square. What is the radius of the circle that passes through the vertices of the red square?
Each of the tips of a hexagram, or regular six point star, is an equilateral triangle. One can construct another hexagram, using the midpoint of the internal base of these triangles as vertices. If we repeat this process once again, we obtain a sequence of three hexagrams as shown in the figure.
What is the sum of the areas of the tips of these hexagrams, if we know that the most external equilateral triangles have sides of length $1$?
In the figure $ABCD$ is a rectangle where $AB=2BC$, $FC=3$ cm, $F$ is the midpoint of $DC$, $I$ is the midpoint of $AB$ and $GE$ is perpendicular to $AC$. Find the perimeter of the quadrilateral $AIFG$.
Consider the equilateral triangle and the circle shown in the figure. Both are tangent at the point $P$. Find the quotient between the circumference and the perimeter of the triangle.
Let $ABCD$ be a rectangle of area $s$. $E$ is a point of segment $AB$ so that $\overline{AE} =\frac1 3 \overline{AB}$, and $F$ is the intersection point of the extensions of segments $DE$ and $CB$. Find the area of triangle $EBF$ in terms of $s$.
Consider $ABCDE$ a regular pentagon with side $1$. Diagonals $AD$ and $EB$ intersect at a point $P$. Find the perimeter of quadrilateral $PBCD$.
Consider the following circle and regular hexagon (six congruent sides) as it is shown in the picture.
The segment $AB$ is a side of the hexagon and the diameter of the circle. Let $a$ and $b$ the areas of the circle and the hexagon respectively. What is the value of $a/b$ ?
Let $ABC$ be a triangle with an area of $4$ cm$^2$. Assume that $AC$ is divided in four equal parts, with segments parallel to $AB$ . Determine the area in cm$^2$ of the shaded region, if the dotted lines are parallel to $BC$ .
Juan drew a rectangle such that when subtractiing $6$ units to the base and adding $4$ units to the height it becomes a square with the same area as the rectangle. What is the area of the initial rectangle?
Determine the value of the angle marked with $\alpha$ in the following figure, knowing that $ABCDEFGHIJ$ is a regular decagon.
Suppose a regular hexagon and a square share a side as in the figure.
Let $A$ be the area of the hexagon and $B$ the area of the square. Calculate he quotient $A/B$.
Andrea is designing the logo of her new venture, to do this she begins by drawing a circumference as shown below. If she now wants to inscribe a square on this circumference, what area will this square have?
The rectangle $ABCD$ has area $120$ cm$^2$. The point E satisfies that $AD=2ED$ and $F$ is a point on $BC$. If the areas of the triangles $EGD$ and $ABH$ are known to be $12$ cm$^2$ and $10$ cm$^2$ respectively, what is the value of the area in square centimeters of quadrilateral $EHFG$?
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