geometry problems from El Número de Oro Certamen , a Teacher's Olympiad in Argentina
with aops links[Profesores de Enseñanza Media]
collected inside aops here
1997- 2019
1999, 2006 - 2008 missing
The three bisectors of a triangle are of length less than or equal to $1$. Show that the area of the triangle is less than or equal to $1 /\sqrt3$.
Let $P$ be an interior point of a regular hexagon. $P$ is joined with each vertex of the hexagon, thus determining 6 triangles, which we alternately color red and blue. Prove that the sum of the areas of the $3$ red triangles coincides with the sum of the areas of the $3$ blue triangles.
In the sides $AD$ and $CD$ of the parallelogram $ABCD$ area $1$, points $N$ and $M$ are determined, respectively, so that $AN / AD = 1/3$ and $DM / DC = 1/4$. Find the areas of the four regions into which the segments $NC$ and $MB$ divide the parallelogram.
Let four points of the plane be given. Analyze in which cases there is a circle equidistant from all four.
What is the minimum number of circles of radius $1$ needed to cover a circle whose radius is the golden number $
\phi =\frac{1+\sqrt5}{2}$?
1999 missing
The following figure is made up of $3$ semicircles. Among the lines that pass through $O$, how many divide their perimeter into two equal parts? How many divide the area into two equal parts?
What is the minimum area that a square can have in the plane that has exactly $3$ integer coordinate points inside it?
Through the center of gravity of a triangle, a line $L$ is drawn that leaves two vertices of the triangle in one of the open half-planes that it determines. Show that the sum of the distances from these two vertices to $L$ is equal to the distance from the other vertex to $L$.
2001.10 (figure missing)
According to the situations described in the plane and in space by the following graphs, for what values of a, b, and c, in each case, does the diagonal line joining vertices A and B also pass through C?
Given a convex quadrilateral, find a point $O$ on its interior such that the segments that join $O$ with the midpoints of its sides decompose it into $4$ figures of equal area. .
Given in the plane $3$ parallel lines $L_i$ ($1\le i\le 3$), construct with a ruler and compass an equilateral triangle $A_1A_2A_3$ such that $A_i\in L_i \forall i$.
If three spheres have a point in common $P$, but no line through $P$ is tangent to all three, show that they have another point in common.
Find the dimensions of a rectangular parallelepiped of volume $1$ whose main diagonal measures $2$, also knowing that its lateral surface is twice the sum of its $3$ edges (dimensions).
Let $ABC$ be an equilateral triangle with side $1$. By one parallel to side $BC$ the triangle is divided into two figures of equal area. Calculate the distance between the two parallels.
Let us consider in $R^2$ an orthogonal coordinate system . Let $A$ and $B$ be two fixed points (different from the origin $O$) located on the axes $Ox$ and $Oy$ respectively and let $C$ be a variable point on the axis $Oz$. Find the locus determined by the center of gravity of the triangles $ABC$ as $C$ travels along the axis $Oz$.
Determine the volume of a regular tetrahedron knowing that the area of the section of the same determined by a plane perpendicular to the base that passes through a lateral edge of the tetrahedron is $4\sqrt2$ .
2003.10 (figure missing / incorrect)
Consider the cross of Lorraine formed by 13 squares of side 1 as indicated in the figure. Determine point Q on segment AB such that line PQ divides the cross into two surfaces of equal area.
Given any convex quadrilateral, show that with a ruler and compass a square of equal area can be constructed, indicating the construction steps .
Let $ABCDV$ be a pyramid whose base is the parallelogram $ABCD$ and $E, F$ points on the edges $VA$ and $VB$ respectively so that the line they determine is not parallel to $AB$. Determine a point $P$ on the interior of the face $VCD$ so that the intersection of the tetrahedron with the plane determined by $E, F$, and $P$ is a pentagon.
Making a $45^o$ turn centered in the center of a square with side a, a star with $8$ vertices is obtained. Show that it can be decomposed into $8$ pieces with which a new square can be formed. Calculate your side.
Indicate how to construct with a ruler and compass the quadrilateral $ABCD$ of which the length of the four sides and the segment that joins the midpoints of the opposite sides $DA$ and $BC$ are known.
Given $7$ points on a circle of radius $r$ such that the distance between any two of them is greater than or equal to $r$. What are the possible distributions of the points?
Let $P$ be a rectangular parallelepiped with sides $a, b$, and $c$ of lengths natural numbers such that $c$ measures equal to the diagonal of the rectangle with sides $a$ and $b$. Prove that the volume of $P$ is a multiple of $60$.
$ABCDE$ is a right pyramid with a square base of side $a$ and height $h$ . What is the ratio $a:h$ so that the lateral area of the pyramid (the sum of the areas of the four triangular faces) is equal to the lateral area of the cube with side $h$?
2006-2008 missing
If both asymptotes of a hyperbola each pass through two points of integer coordinates, show that on the hyperbola there are a finite number of points with integer coordinates.
Let $T$ be a triangle in the plane. If $T'$ is the image of $T$ by a symmetry with respect to a line, show that with only two cuts in $T$, it is possible to obtain pieces that, through rotations and translations in the plane, allow to cover $T'$ .
On a sphere of radius $1$ there are $m$ points. The minimum distance between two of these points is $\delta$. Prove that $m <\left(\frac{4}{\delta}\right) ^2$
Calculate the area of a triangle in space knowing that its orthogonal projections on the $xy, yz$ and $zx$ planes have areas $1/2, 1$ and $3/2$ respectively.
On each side of a triangle $ABC$, build three equally oriented right isosceles triangles towards the outside of the triangle, $ACB',BA'C$ and $CB'A$, right at $A', B'$ and $C'$ respectively. Show that $AA',BB'$ and $CC'$ are concurrent.
From a golden rectangle remove a square $C_1$, from the resulting rectangle a square $C_2$, is removed from the resulting rectangle a square $C_3$ is removed and from the resulting rectangle a square $C_4$ is removed as shown in the figure.
If this process were repeated indefinitely on the resulting rectangles, find the location of a point that would not be removed with any of the squares removed.
Let $ABCD$ be a convex quadrilateral and $E$ be the midpoint of $AB$. Show that it is possible to draw a line that passes through $E$ and divides the quadrilateral into two regions of equal area, indicating a construction with a ruler and compass to draw said line.
Show that in any triangle an ellipse tangent to the sides of the triangle can be inscribed at the respective midpoints.
If a set of points in space is such that every $5$ of them belong to a sphere of radius $r$, show that all the points belong to a sphere of radius $r$. Show that there exists a set of $5$ points in space such that each subset of $4$ points is on a sphere of radius $r$, but the $5$ points do not belong to any sphere of radius $r$.
Given a tetrahedron $T$ of volume $V$, let the points be: $O$ the center of gravity of $T$, $P$ the center of a face $C$ of $T$, $Q$ the center of an edge $a$ of $C$ and $R$ a vertex of the edge $a$. Find the volume of the tetrahedron $OPQR$.
Show that it is possible to make a partition into 4 congruent figures in the attached figure.
Given the regular pentagon ABCDE constructed as indicated in the figure from side 1, calculate the measure of FG and KL,
What is the maximum area that a parallelogram inscribed in a triangle can have? (The vertices of the parallelogram belong to the sides of the triangle). How many inscribed rectangles of maximum area can there be?
Let the tetrahedron $ABCD$ be in space. Can it be projected orthogonally on a plane $P$ in a parallelogram $A'B'C'D' $?
Given a triangle $ABC$, we denote $P$ a point in its interior and $D, E, F$ the feet of the perpendiculars of $P$ to the lines $BC, CA$ and $AB$ respectively. Find all points $P$ for which the sum is minimum: $\frac{BC}{PD}+\frac{CA}{PE}+\frac{AB}{PF}$
Let $ABCD$ be a right trapezoid circumscribed to a circle of radius $96$ cm, whose minor base is $168$ cm. Determine the perimeter and area of the trapezoid.
In figure, the triangles $ABC$ and $PQR$ are equilateral and $R$ is the midpoint of side $AB$. Find the relationship between the lengths of consecutive sides, from greatest to least, of the intersecting pentagon of both triangles.
Let $C$ be a circle of radius $1$. Characterize the locus of the midpoints of the chords of $C$ of length $3/2$.
Points $A, B,$ and $C$ are the vertices of a right triangle. Point $A$ is one end of a diameter $d$ of a sphere. Point $B$, on the diameter $d$, is $b$ cm from $A$. Point $C$, on the sphere, is $c$ cm from $B$. Determine the possible volumes of the sphere.
Given a triangle $ABC$, is it possible to circumscribe an equilateral triangle around it? In this case, indicate how to do it.
Given an acute angle of vertex $A$ and a point $X$ in its interior that belongs to the bisector of the angle, determine a segment that passes through $X$ and cuts the sides of the angle at $B$ and $C$ so that triangle $ABC$ has a minimum area.
Given a pyramid whose base is a convex quadrilateral, prove that it is possible to cut it with a plane whose intersection with the pyramid is a parallelogram.
Given a triangle $ABC$ of area $20$ cm$^2$, determine points $D, E$, and $F$ on sides $AB, BC$, and $CA$, respectively, so that triangle $ABE$ and quadrilateral $DBEF$ have equal areas and it is equal to $12$ cm$^2$.
A right triangle $ABC$ can be covered by $2$ congruent circles. Indicate how to do it with circles of the smallest radius $r$ possible.
In a regular tetrahedron with edge $a$, $3$ planes are drawn. Each plane passes through a vertex of the base and through the midpoints of the lateral edges opposite that vertex. Find the volume of the region of the tetrahedron that lies above the three given planes.
Let $ABC$ be the right triangle with legs $AB = 4$ and $BC = 3$. Point $O$ denotes the intersection of the bisector of angle $C$ with side $AB$. By drawing the circle $G$ with center $O$ and radius $OB$, the points $P$ and $Q$ are obtained, the intersection of $G$ with the bisector drawn, where $P$ is the point furthest from $C$. Calculate the ratio $PC/PQ$.
Let $ABC$ be a triangle and $D$ be the circumscribed circcle. Points $P$ and $Q$ are respectively the intersection of the bisector of angle $A$ with side $BC$ and with $D$. Point $R$ is the intersection of side $AC$ with the circle circumscribed to triangle $ABP$. Compare the areas of the triangles $RBQ$ and $RBC$.
An isosceles trapezoid with a perimeter of $64$ cm is inscribable and circumscribable. If the radius of the circumscribed circle is $16$ cm, find the length of the sides of the trapezoid.
Let $ABCD$ be a square with side $1+ \phi,$ where $\phi =\frac{1 + \sqrt5}{2}$ is the Golden Number. Divide the square into pieces that rearranged allow to obtain a rectangle such that one of its sides measures $\phi$.
From a point $P$ in the interior of an equilateral triangle with a side $1$ cm, perpendicular segments are drawn to the sides of the triangle, which is decomposed into $3$ quadrilaterals. Find the sum of the perimeters of these quadrilaterals.
Let $P, Q, R, S$ respectively be the center of gravity of the triangles $ABC, BCD, CDA$ and $DAB$ of the convex quadrilateral $ABCD$. Show that the lines $PD, QA, RB$, and $SC$ are concurrent.
The area of the shaded pentagon is $1$ cm $^2$. Find the area of the regular pentagon.
Given the rectangle $ABCD$, from a point $P$ on the diagonal $BD$ are drawn parallel to its sides that intersect them at the points $E, F, G$ and $H$ belonging to the sides $AB, AD, BC$ and $CD $, respectively. Determine the point $P$ such that the relationship between the areas of the rectangles $AEPF$ and $PGCH$, in some order, is $5/4$.
Given an isosceles triangle with sides $a, b$ and $c, b = c$, which verifies $\frac{a}{b}=\phi =\frac{1 + \sqrt5}{2}$ . Determine the angles of the triangle.
Given the triangle $ABC$ with sides $a, b, c$, construct with a ruler and compass a triangle of equal perimeter and greater area.
2019 El Número de Oro p7 (figure missing)
In the figure, ABC is an isosceles triangle, AB = AC, right in A. The region BDCE is bounded by the arcs BEC and BDC determined respectively by the circumference of center A and radius AB and that of center O and radius OB, where O midpoint of BC.
Find the relationship between the areas of the region BDCE and that of the triangle ABC.
Prove that there exists a golden rectangle whose diagonals have length $\sqrt{11\phi+7}$ where $\phi = \frac{1 + \sqrt5}{2}$ is the Golden Number.
Clarification:
A golden rectangle is called a rectangle whose sides, of lengths $a, b$, with $a> b$, verify $ \frac{a}{b} = \frac{1 + \sqrt5}{2} = \phi$ .
No comments:
Post a Comment