geometry problems from Moldovan Mathematical Olympiads with aops links in the names
Olimpiada Republicană la Matematică
collected inside aops: VII-VIII plane,
2002, 2014 - 2020
Consider an angle $ \angle DEF$, and the fixed points $ B$ and $ C$ on the semiline $ (EF$ and the variable point $ A$ on $ (ED$. Determine the position of $ A$ on $ (ED$ such that the sum $ AB+AC$ is minimum.
In a triangle $ ABC$, the angle bisector at $ B$ intersects $ AC$ at $ D$ and the circumcircle again at $ E$. The circumcircle of the triangle $ DAE$ meets the segment $ AB$ again at $ F$. Prove that the triangles $ DBC$ and $ DBF$ are congruent.
Consider a circle $ \Gamma(O,R)$ and a point $ P$ found in the interior of this circle. Consider a chord $ AB$ of $ \Gamma$ that passes through $ P$. Suppose that the tangents to $ \Gamma$ at the points $ A$ and $ B$ intersect at $ Q$. Let $ M\in QA$ and $ N\in QB$ s.t. $ PM\perp QA$ and $ PN\perp QB$. Prove that the value of $ \frac {1}{PN} + \frac {1}{PM}$ doesn't depend of choosing the chord $ AB$.
In a triangle $ ABC$, the bisectors of the angles at $ B$ and $ C$ meet the opposite sides $ B_1$ and $ C_1$, respectively. Let $ T$ be the midpoint $ AB_1$. Lines $ BT$ and $ B_1C_1$ meet at $ E$ and lines $ AB$ and $ CE$ meet at $ L$. Prove that the lines $ TL$ and $ B_1C_1$ have a point in common.
Let $ ABCD$ be a convex quadrilateral and let $ N$ on side $ AD$ and $ M$ on side $ BC$ be points such that $ \dfrac{AN}{ND}=\dfrac{BM}{MC}$. The lines $ AM$ and $ BN$ intersect at $ P$, while the lines $ CN$ and $ DM$ intersect at $ Q$. Prove that if $ S_{ABP}+S_{CDQ}=S_{MNPQ}$, then either $ AD\parallel BC$ or $ N$ is the midpoint of $ DA$.
The circles $ C_1$ and $ C_2$ with centers $ O_1$ and $ O_2$ respectively are externally tangent. Their common tangent not intersecting the segment $ O_1O_2$ touches $ C_1$ at $ A$ and $ C_2$ at $ B$. Let $ C$ be the reflection of $ A$ in $ O_1O_2$ and $ P$ be the intersection of $ AC$ and $ O_1O_2$. Line $ BP$ meets $ C_2$ again at $ L$. Prove that line $ CL$ is tangent to the circle $ C_2$.
The sides $ AB$,$ BC$ and $ CA$ of the triangle $ ABC$ are tangent to the incircle of the triangle $ ABC$ with center $ I$ at the points $ C_1$,$ A_1$ and $ B_1$, respectively.Let $ B_2$ be the midpoint of the side $ AC$.Prove that the lines $ B_1I$, $ A_1C_1$ and $ BB_2$ are concurrent.
Let the triangle $ ADB_1$ s.t. $ m(\angle DAB_1)\ne 90^\circ$.On the sides of this triangle externally are constructed the squares $ ABCD$
and $ AB_1C_1D_1$ with centers $ O_1$ and $ O_2$, respectively.Prove that the circumcircles of the triangles $ BAB_1$, $ DAD_1$ and
$ O_1AO_2$ share a common point, that differs from $ A$.
Let $ P$ be a polyhedron whose all edges are congruent and tangent to a sphere. Suppose that one of the facesof $ P$ has an odd number of sides. Prove that all vertices of $ P$ lie on a single sphere.
The circumradius of a tetrahedron $ ABCD$ is $ R$, and the lenghts of the segments connecting the vertices $ A,B,C,D$ with the centroids of the opposite faces are equal to $ m_a,m_b,m_c$ and $ m_d$, respectively. Prove that: $ m_a+m_b+m_c+m_d\leq \dfrac{16}{3}R$
Let A,B,C be three collinear points and a circle T(A,r).
If M and N are two diametrical opposite variable points on T,
Find locus geometrical of the intersection BM and CN.
2003-2005 missing
2006 grades 8-9, 11-12 missing
A convex quadrilateral $ ABCD$ is inscribed in a circle. The tangents to the circle through $ A$ and $ C$ intersect at a point $ P$, such that this point $ P$ does not lie on $ BD$, and such that $ PA^{2}=PB\cdot PD$. Prove that the line $ BD$ passes through the midpoint of $ AC$.
Let a triangle $ABC$ satisfy $AC = BC$; in other words, let $ABC$ be an isosceles triangle with base $AB$. Let $P$ be a point inside the triangle $ABC$ such that $\angle PAB = \angle PBC$. Denote by $M$ the midpoint of the segment $AB$. Show that $\angle APM + \angle BPC = 180^{\circ}$.
Consider an octogon with equal angles and rational side lengths. Prove that it has a symmetry center.
2007 grades 8-10, 12 missing
$ABCDA_{1}B_{1}C_{1}D_{1}$ is a cube with side length $4a$. Points $E$ and $F$ are taken on $(AA_{1})$ and $(BB_{1})$ such that $AE=B_{1}F=a$. $G$ and $H$ are midpoints of $(A_{1}B_{1})$ and $(C_{1}D_{1})$, respectively.
Find the minimum value of the $CP+PQ$, where $P\in[GH]$ and $Q\in[EF]$.
Given a tetrahedron $VABC$ with edges $VA$, $VB$ and $VC$ perpendicular any two of them. The sum of the lengths of the tetrahedron's edges is $3p$. Find the maximal volume of $VABC$.
2006 grades 8, 10-12 missing
From the vertex $ A$ of the equilateral triangle $ ABC$ a line is drown that intercepts the segment $ [BC]$ in the point $ E$. The point $ M \in (AE$ is such that $ M$ external to $ ABC$, $ \angle AMB = 20 ^\circ$ and $ \angle AMC = 30 ^ \circ$. What is the measure of the angle $ \angle MAB$?
2009 grades 11-12 missing
On the lines $AB$ are located $2009$ different points that do not belong to the segment $[AB]$. Prove that the sum of the distances from point $A$ to these points is not equal to the sum of the distances from point $B$ to these points.
Triangle $ABC$ with $AB = 10$ cm ¸and $\angle C= 15^o$, is right at $B$. Point $D \in (AC)$ is the foot of the altitude taken from $B$. Find the distance from point $D$ to the line $AB$.
The circle $C_1$ of center $O$ and the circle $C_2$ intersect at points $A$ and $B$, so that point $O$ lies on circle $C_2$. The lines $d$ and $e$ are tangent at point $A$ to the circles $C_1$ and $C_2$ respectively. If the line $e$ intersects the circle $C_1$ at point $D$, prove that the lines $BD$ and $d$ are parallel.
Prove that a right triangle has an angle equal to $30^o$ if and only if the center of the circle inscribed in this triangle is located on the perpendicular bisector of the median taken from the vertex of the right angle of the triangle.
Let $ABC$ be an equilateral triangle. The points $M$ and $K$ are located in different half-planes with respect to line $BC$, so that the point $M \in (AB)$ ¸and the triangle $MKC$ is equilateral. Prove that the lines $AC$ and $BK$ are parallel.
A side of an arbitrary triangle has a length greater than $1$. Prove that the given triangle it can be cut into at least $2$ triangles, so that each of them has a side of length equal to $1$.
Let the triangle $ABC$ be with $| AB | > | AC |$. Point M is the midpoint of the side $[BC]$, and point $I$ is the center of the circle inscribed in the triangle ABC such that the relation $| AI | = | MI |$. Prove that points $A, B, M, I$ are located on the same circle.
Let the isosceles triangle $ABC$ with $| AB | = | AC |$. The point $M$ is the midpoint of the base $[BC]$, the point $N$ is the orthogonal projection of the point $M$ on the line $AC$, and the point $P$ is located on the segment $(MC)$ such that $| MP | = | P C | \sin^2 C$. Prove that the lines $AP$ and $BN$ are perpendicular.
2010 - 2013 missing
Let $a, b, c$ be the lengths of the sides of a triangle, and $r$ be the radius of the circle circumscribed triangle around it
Prove that $a \sqrt{bc} \le r (b+c)$.
The triangle $ABC$ is equilateral. On the ray $[AC$, beyond point $C$, point $D$ is considered and on the ray $[BC$, beyond point $C$, point $E$ is considered, such that $[BD]=[DE]$ . Prove that $[AD]=[CE]$.
Let $ABC$ be an acute triangle with $\angle ABC=67^o 30'$. The point $M$ , located on side $(BC)$, it is the foot of the altitude taken from the vertex $A$, such that $BC=\sqrt2 AM $. Prove that the triangle $ABC$ is isosceles.
Let the right triangle $ABC$ with $\angle BAC=90^o$ and $\angle BCA=15^o$. On the leg $(AC)$ and on hypotenuse $(BC)$, points $D$ and $E$ are taken respectively such that $\angle BDE= 90^o$ and $DE =2 AD$. Find the measure of the angle $ADB$.
The bisectors of the angles $A$ and $B$ of the triangle $ABC$ intersect the circumscribed circle of the triangle at points $D$, respectively $E$. Let $AD\cap BE = \{I\}$, $DE\cap BC = \{P\}$ ¸and $DE\cap AC = \{M\}$. Prove that the $IMCP$ quadrilateral is a rhombus.
In the triangle $ABC$ with $\angle A = 90^o$ we construct the height $AD$, $D \in (BC)$ .Denote by $E$ ,$F$ the feet of the perpendiculars, constructed from $D$ on the sides [AB], $[AC]$ respectively . Prove that $\sqrt5 \cdot BC\ge \sqrt2 (BF + CE)$.
Let $O$ be the center of the circle inscribed in a triangle $ABC$. The sides $AB$ and $BC$ are tangent to this circle at points $D$ and $E$, respectively. Let $F$ be the point of intersection of the lines $AO$ and $DE$. Prove that the lines $CF$ and $AF$ are perpendicular.
Let $P$ be an interior point of the midline of the triangle $ABC$, determined by the mispoints of $AB$ and $AC$. The lines $BP$ and $CP$ intersect the sides $AC$ and $AB$ at points$ D$ and $E$, respectively. Prove that the ratio of the distances from points $A$ and $P$ to the line $DE$ is equal to $2$.
By the point P, which is at distance $\alpha$ from the point of intersection of the diagonals of a parallelogram is drawn a plane that does not intersect the parallelogram. Prove that sum the distances from the vertices of the parallelogram to this plane is not greater than $4\alpha$ .
Four circles $S_1, S_2, S_3$ and $S_4$ have centers on the circle $S$. Circles $S_1$ and $S_2$ intersects at points $A_1$ and $B_1$ , circles $S_2$ and $S_3$ intersects at points $A_2$ and $B_2$, circles $S_3$ and $S_4$ intersects at points $A_3$ and $B_3$ , and circles $S_4$ and $S_1$ intersects at points $A_4$ and $B_4$. Points $A_1, A_2, A_3$ and $A_4$ lie on the circle S, and the points $B_1, B_2 , B_3$ and $B_4$ are different and lie inside the circle $S$. Prove that the quadrilateral $B_1B_2B_3B_4$ is rectangle.
The base of a quadrilateral pyramid is a convex quadrilateral, two sides of which are of length equal to $8$ cm, and others of lengths equal to $6$ cm. The height of the pyramid is $6$c m long and the side faces form with the plane of the base angles measuring $60^o$. Determine the volume of the pyramid.
Let be the right triangle $ABC$ ($\angle ACB=90^o$). Let $M$ and $N$ be interior points of the sides $AC$ and $BC$, respectively, and $L$ the point of intersection of the lines $AN$ and $BM$. Prove that point $C$ and the orthocenters of the triangles $AML$ and $BNL$ are collinear.
The measures of two angles of a triangle are equal to $20^o$ and $40^o$, and the difference in the lengths of the larger sides is equal to $18$ cm. Find the length of the angle bisector of the triangle from the vertice of the third angle.
A rectangular sheet has been folded so that the upper left vertice coincides with the midpoint of the bottom side (see drawing). Find the lengths of the sides if triangles $1$ and $2$ in the drawing are congruent and the area of the rectangle corresponding to the sheet is equal to $486$ cm$^2$.
Points $E$ and $F$ are located on the sides $BC$ and $CD$ of the square $ABCD$, respectively, so that $CF = BE$ and $BF \cup DE = \{G\}$. Prove that the lines $AG$ and$ EF$ are perpendicular.
In triangle $ABC$, point $D$ is the midpoint of the side BC. Line through the vertice $B$ and the midpoint of the segment $AD$ intersects the side $AC$ at point $E$. The line passing through point $C$, parallel to the line $AD$ intersects the extension of the side $BA$ at point $K$. Prove that points $D, E$ and $K$ are collinear.
Let $ABCD$ be an isosceles trapezoid ($AB\parallel CD$, $CD <AB$), where $AD = DC = BC$ and $\angle A = 40^o$. On the ray $(AD$ consider the point $E$ such that$ D\in (AE)$ and $DE = AC$. Prove that $EC\perp BD$.
Let $ABCD$ be a parallelogram with the sides $AB = CD = a$, $AD = BC = b$ and $\angle A = 30^o$. A line passing through $C$ intersects the lines $AB$ and $AD$ outside the parallelogram at the points $P$ and $Q$, respectively. Prove that the area of the parallelogram does not exceed $\frac{1}{8}(a \cdot AQ + b \cdot AP).$
Let $AB$ be the diameter of the circle $\omega$ with the center in $O$ and $OC$ a radius perpendicular to $AB$, and $M$ a point inside the segment $OC$ . Let $N$ be the second intersection point of the line $AM$ with $\omega$ , and $P$ the point of intersection of the tangents of $\omega$ taken at points $N$ and $B$. Prove that the points $M, O, P$ and $N$ are located on the same circle.
Let $X$ be a point in triangle $ABC$ where $AB >AC$. The line taken by $X$ parallel to the side $AB$ intersects the other two sides at points $P$ and $Q$. The line drawn through the $X$ parallel to the side $AC$ intersects the other two sides at points $R$ and $S$. Find the locus of the points X in the triangle $ABC$ for which $PQ = RS$.
The bisector of the angle $BCA$ of the triangle $AC$ intersects for the second time the circle circumscribed that triangle at the point $E$ and intersects the perpendicular bisectors of the segments $AC$ and $BC$ at the points $P$ and $Q$. The points $K$ and $L$ are the midpoints of the segments $AC$ and $BC$ respectively. Prove that the areas of the triangles $RPK$ and $RQL$ are equal.
Determine the locus of the points $M$ inside the convex quadrilateral $ABCD$ for which the areas of the quadrilaterals $MBCD$ and $MBAD$ are equal.
The base of the pyramid $VABC$ is the isosceles triangle $ABC$ with $[AB]=[ AC]$. The edge $VA$ forms with the planes of the faces $ABC$ and $VBC$ congruent angles measuring $45^o$ . Point $A$ and the midpoints of the edges $AB, AC, BC, VB$ and $VC$ lie on a sphere of radius $1$ cm . Determine the area of the face $VAC$.
In the pyramid $VABC$ the length of the edge AB is equal to $12$ cm, and the length of the edge $CD$ with $8$ cm. The distance between the lines $AB$ and $CD$ is equal to $6$ cm. Determine the measure of the angle formed by the lines $AB$ and $CD$, if it is known that the volume of the pyramid $VABC$ is equal with $48$ cm$^2$
On the side $[BC]$ of the equilateral triangle $ABC$, points $K$ and $L$ are taken so that $BK= KL=LC$. On the side $[AC]$, the point $M$ is taken so that $AM: MC=1: 2$ . Prove that $\angle AKM+ \angle ALM = 30^o$.
In the triangle $ABC$, the length of the median $[BM]$, $(M \in [AC])$ is equal to the length of the side $[AC]$ ($BM = AC$). On the lines $BA$ and $AC$, points $D$ and $E$ are taken, respectively, so that $D\notin [AB]$, $E\notin [AC]$, $AD= AB$ and $CE =CM$. Prove that lines $DM$ and $BE$ are perpendicular.
Let the triangle $ABC$ be with $\angle B=105^o$, $\angle C = 30^o$ . If $D \in (BC)$ so that $[AD]$ is the median of the triangle, to calculate the measure of the angle $DAC$.
Consider a point $P$ inside square $ABCD$ s.t. $\angle PAB= \angle PBA= 15 ^{\circ}$. Show that $\triangle PCD$ is equilateral
Two parallel lines $\ell$ and $m$ are given , and a point $A$ between them. An arbitrary line $s$ is taken, perpendicular on lines $\ell$ and $m$ , which intersect them at points $B$ and $C$, respectively. Determine the locus of points, which are the centers of the circles circumscribed to the triangles $ABC$.
Two circles are given , $C_1 (O_1)$ and $C_2 (O_2)$ , which are internally tangent at point $K$. Small circle $C_2 (O_2)$ pass through the center $O_1$ of the large circle $C_1 (O_1)$ . $MN$ is a chord of the big circle $C_1 (O_1)$ and also tangent to the small circle at point $C$. The chords $KM$ and $KN$ intersect the small circle at points $A$ and $B$, respectively. Segments $KC$ , $AB$ intersect at point $L$.
a) Prove that $CN: CM = LB: LA$.
b) Find the length of the segment $MN$, if $LB: LA= 2: 3$ and the radius of the small circle is equal to $\sqrt{23}$ .
Let $ABC$ be a triangle in which $a \le b \le c$ (where $a = BC, b =AC, c = AB$).
Prove the equivalence of the following statements:
1) there is a point $X$ inside the triangle $ABC$ with the property that the segments cut from the sides of the triangle on the line passing through $X$ parallel to these sides are congruent,
2) $bc < a(b + c)$.
A circle is inscribed in the triangle $ABC$. The sides of the triangle $BC,CA,AB$ are tangent to the circle at the points $A_1,B_1,C_1$ respectively. The segment $AA_1$ intersects for second time the inscribed circle at point $Q$. Line $\ell$ passes through the point $A$ and is parallel to the line $BC$. Lines $A_1B_1$ and $A_1C_1$ intersect line $\ell$ at points $R$ and $P$ respectively. Prove that $\angle PQR=\angle B_1QC_1$.
Let $M$ be the set of all tetrahedrons $ABCD$ which have four edges $|BC|= a$, $|AC|= b$, $|AB| =c$ and $|DB|= d$, such that $(a+b+c)\sqrt{d }=1$. .Find all tetrahedra, with the minimum product of the lengths of all edges, from the tetrahedra of set $M$ with maximum volume .
Let $a, b, c$ be the lengths of the sides of the triangle $ABC$ , and $\alpha, \beta, \gamma$ the measures of opposite angles respectively. Show that if $a+c=b ctg\frac{\beta}{2} $, then the triangle is right.
The point $K$ divides the height of a regular hexagonal pyramid into the ratio $2:1$ at the top. The area of the section of the pyramid with a plane, passing through K, parallel to a lateral face, is equal to $14$ cm$^2$. Determine the lateral area of the pyramid.
The base of the pyramid $VABC$ is the equilateral triangle $ABC$ with the side of $9$ cm, and the height of the pyramid is $6$ cm. Determine the minimum value of the radius of the sphere, which contains the vertices of the pyramid.
Let the triangle $ABC$ with $\angle BAC= 90^o$ , $\angle ACB=15^o$. On the side $BC$ take the point $D$, and on the side $AC$ take the point $E$, so that $\angle BAD = \angle EBC = 30^o$. Prove that $AB=DE$.
In the triangle $ABC$ with $\angle A= 90^o$ we build the bisector $CD$ of the angle $ACB$, $D\in (AB)$ and $BE \perp CD$, $E\in CD$ . Knowing $CD = 2 DE$ , find the measures of the acute angles of the triangle $ABC$.
In any triangle $ABC$, $D\in (BC)$ and $E\in (AB)$ such that $BC= 3CD$ and $AB = 2 AE$. $P$ is the midpoint of $[CE]$. show that the points $A, P, D$ are collinear.
Let $ABCD$ be a right trapezoid with $AB \parallel CD$ and $\angle B =75^o$. Point $H$ belongs to the line $BC$ so that $AH \perp BC$ and $CD \perp BH$. Find the area of the trapezoid $ABCD$, if $AD+ AH = 8$.
In the acute triangle $ABC$ , the median $AM$ is larger than the side $AB$. Prove that the triangle $ABC$ can be cut into three parts from which a rhombus can be formed.
Let $P$ be a point, located inside a triangle $ABC$, such that $\angle CAP= \angle CBP$. Let $D$ be the midpoint of the side $AB$, and $M$ and $N$ the projections of the point $P$ on the sides $BC$ and $AC$, respectively. Prove that $DM = DN$.
Two circles have a common chord $AB$ . Through point $B$ goes a straight line which intersects the circles at points $C$ and $D$, so that $B$ is between $C$ and $D$. The tangents to the circles, taken through points $C$ and $D$, intersect at a point $E$. Compare $AD \cdot AC$ with $AB \cdot AE$.
Inside a right trapezoid $ABCD$ , with right angles at the vertices $A$ and $B$ , there are two circles.One of them is tangent to the sides and the large base $AD$, and the other is tangent to the lateral sides , at the small base $BC$ and in the first circle.
a) The line passing through the centers of the circles intersects $AD$ at the point $P$. Prove that $\frac{AP}{PD} =\sin (\angle D)$
b) Determine the area of the trapezoid, if the radii of the circles are $\frac43$ and $\frac13$.
The base of an oblique prism is an equilateral triangle. The length of the height of the prism is equal with $5$ cm. The areas of the side faces are equal to $30$ cm$^2$ , $30$ cm$^2$ and $45$ cm$^2$ . Determine the length the side of the base triangle.
Let $ABCDA_1B_1C_1D_1$ be a cube with an edge of $1$ m. Determine the minimum value of the length the segment with the ends on $(AB_1)$ and $(BC_1)$ and forms an angle of $45^o$ with the plane of the face $ABCD$.
Consider the triangle $ABC$ with $BC=2AC$. Let $D \in (BC)$ such that $BD=3DC$. Find the value of the ratio $\frac{AB}{AD}$.
Let the angle $\angle XOY=30^o$. Let $M$ be a point located inside this angle and $A,B$ are the orthogonal projections of that point on its sides. Prove that $OM=2AB.$
Perimeter of a triangle $ABC$ is $8$ cm. Points $D$ and $E$ belong to sides $AB$ and $CA$ so that $DE\parallel BC$ and $DE$ is tangent to the incircle of triangle $ABC$. Find the maximum value of the length of the segment $DE$.
The point of intersection of the altitudes of an acute triangle is equidistant from the midpoints of it's sides. Prove that the triangle is equilateral.
In the triangle $ABC$ with $\angle A= 90^o$, we construct $AD \perp BC$ ($D \in (BC)$) and the angle bisector $AE$, ($E \in (BC)$). Denote with $L$ and $F$ the orthogonal projections of the point $E$ on the legs $[AB]$ and $[AC]$, respectively . Prove that the lines $AD, BF$ and $CL$ are concurrent.
Given the acute triangle $ABC$, inscribed in circle with center $O$. We construct the altitude $AD$, ($D \in BC$), and the angle bisector $AE$, ($E \in BC$), which intersects the circle at point $F$. Denote with $L$ the intersection of the line $AO$ with the circle. Prove that the lines $FD$ and $LE$ intersect on the circle, circumscribed around the triangle $ABC$.
It is considered a circle $C (O)$ and a chord $[MN]$ of this circle. In one of the arcs determined by the chord $[MN]$ two circles are inscribed $C_1(O_1)$ and $C_2(O_2)$ , $O_1 \ne O_2$ , which are tangent to circle $C (O)$ at points $A$ and $B$, and at the chord $[MN]$ in points $C$ and $D$, respectively. Prove that points $A, B, C$ and $D$ lie on a circle.
Let $ABC$ be an arbitrary triangle $A_1,B_1$ and $C_1$ be three points, $A_1 \in (BC)$, $B_1 \in (AC)$, $C_1 \in (AB)$ , such that $ \frac{BA_1}{A_1C}=\frac{CB_1}{B_1A}=\frac{AC_1}{C_1B}$ . Prove that the lengths of the segments Prove that exists a triangle with sidelengths the segments $[AA_1]$, $[ BB_1]$ and $[CC_1]$.
Inside a large circle are built three small congruent circles, so that each small circle is tangent to the large circle and the other two small circles. From an arbitrary point of view $M$, located on the large circle and different from the tangent points, one tangent is drawn to each small circle. Denote $\ell_1, \ell_2, \ell_3$ the lengths of segments of the tangents, taken from the point $M$ to the respective tangent points located on the small circles. Prove that one of these lengths is equal to the sum of the other two.
Given the cube $ABCDA_1B_1C_1D_1$. The point $K$ is an inner point of the edge $BB_1$ such that $\frac{BK}{KB_1}=m$ . Through the points $K$ and $C_1$ a plane $\alpha$ is drawn , parallel to the line $BD_1$ .
a) Denote $P$ the point of intersection of the plane with the line $A_1B_1$. Find the value of the ratio $\frac{A_1P}{PB_1}$.
b) The plane $\alpha$ divides the cube into $2$ parts. Find the ratio of the volume of these parts.
Let the pyramid $ABCD$, where $BC=a, CA=b, AB=c, DA=a_1, DB=b_1, DC=c_1$. Determine the measure of the acute angle formed by the lines of edges $AD$ and $BC$.
Let $VABCD$ be a regular quadrilateral pyramid, in which the length of the side of the base $ABCD$ is equal to $5$, and the length of the height of the pyramid is equal to $10\sqrt2$. The pyramid is cut by a plane passing through the vertex $A$ and is parallel to the diagonal $BD$ of the base, so that the line $AB$ forms with this plane an angle of $30^o$. Determine the area of the obtained section.
Let $ABC$ be an isosceles triangle with $AB=AC$ and $\angle BAC= 72^o$ . On the side $AB$ the points $D$ and $E$ are taken so that $\angle ACD = \angle DCE =\angle ECB$, and the point $F$ belongs to the side $(BC)$ , so that $EF$ is the bisector of the angle $BEC$. Prove that $AF \perp CE$.
Let the triangle $ABC$ be $AB=AC$ and $\angle B > 30^o$ . Inside the triangle we consider a point $M$ such that $\angle MBC = 30^o$ and $\angle MAB =\frac34 \angle BAC$. Determine $\angle AMC$.
Give the right triangle $ABC$ with $\angle A=90^o$ . The angle bisector of $ABC$ intersects the perpendicular bisector of the side $[AC]$ at point $D$, located outside the triangle $ABC$. Prove that $\vartriangle BDC$ is right.
The $M$ point is the midpoint of the base $[AC]$ of an isosceles triangle $ABC$. $[MH]$ is orthogonal to $[BC]$ side. Point $P$ is the midpoint of the segment $[MH]$. Prove that $[AH]$ is orthogonal to $[BP]$.
In the square $ABCD$, the points $E$ and $F$ belong to the sides $(AD)$ and $(DC)$, respectively. The diagonal $AC$ intersects $BE$ and $BF$ at points $H$ and $G$, respectively, If $\angle EBF = 45^o$, and $EG \cap HF = \{O\}$, prove that the lines $BO$ and $EF$ are perpendicular.
Let $a$ and $b$ be two parallel lines. The circle $\Omega$ is tangent to line $a$ at point $A$ and intersects the line $b$ at the distinct points $B$ and $C$. The point $T$ is located on the line $a$. $BT$ and $CT$ intersect again the circle $\Omega$ at the points $M$ and $N$, respectively . Show that the line $MN$ bisects the segment $[AT]$.
Let $ABC$ be an isosceles triangle with $AC = BC$ . Let $M$ be the midpoint of the side $AB$, $N$ be the foot of the perpendicular drawn from $M$ on the $AC$, and $P$ the midpoint of the segment $MN$. Prove that the lines $BN$ and $CP$ are perpendicular.
In the triangle $ABC$, the median $AM$ and the angle bisector $BN$ intersect at point $P$. Determine the measures of the angles of the triangle $ABC$, if the lines $MN$ and B$C$ are known to be perpendicular, and $BP: AN =3: 2$.
A sphere passes through all the vertices of a face of the cube and is tangent to all the edges of the face opposite of the cube. Find the ratio of the volume of the sphere to the volume of the cube.
A plane, which contains an edge, divides a regular tetrahedron into two bodies, the volumes which is reported as $3: 5$. Determine the measures of the angles at which the secant plane divides the dihedral angle of the tetrahedron.
In the regular quadrilateral pyramid $VABCD$ , the height has length $h$ and is also the diameter of a sphere and $\angle AVB =\phi$ . Determine the length of the curve obtained by the intersection of the sphere with the lateral surface of the pyramid.
Given the triangle $ABC$ with the altitides $ BE$ and $CF$ , $E \in (AC)$, $F \in (AB)$ . The point $P$ belongs to the segment $(BE)$ such that $BP=AC$ and the point $Q$ belongs to the extension of the segment $(CF)$, so that $F \in (CQ)$ and $CQ=AB$. Determine the measure of the angle $QAP$.
Let $ABCD$ be a square, and the point $E$ is the midpoint of the side $AD$. If $BD \cap CE =\{F\}$ , prove that $AF \perp BE$
The angle bisectors $AA_1$ and $CC_1$ are taken in the acute triangle $ABC$. Prove that if the lengths of the perpendiculars, constructed from the point $B$ on the lines $AA_1$ and $CC_1$ are equal, then the triangle $ABC$ is isosceles.
Find the measure of the angle $B$ of the triangle $ABC$, if it is known that the altitudes constructed from the vertices $A$ and $C$ intersects inside the triangle and one of them is divided by the point of intersection in equal segments, and the other in the ratio $2:1$, considering from the top.
Angle bisectors $BB_1$ and $CC_1$ of triangle ABC intersect at point $O$, $\angle BOC=120^o$. The circle, circumscribed around the triangle $BC_1O$, intersects the side $BC$ in point $D$. Prove that $AD \perp B_1C_1$.
Let $ABC$ be an acute the triangle with $AB > AC$ . The point $F$ , located on the side $(BC)$, it is the foot of the altitude drawn from the vertice $A$ , and $H$ is the orthocenter of the triangle $ABC$ . On the ray $(BC$ take a point $D$ such that $C \in (BD)$. The circle circumscribed around the triangle $DFH$ intersects the segment $(AD)$ for second time ar point $N$ such that the point $N$ also lies on the circle circumscribed around the triangle $ABC$. Prove that line $NH$ passes through the midpoint of the side $(BC)$.
Inside the isosceles triangle $ABC$ ($AC=BC$) with $\angle C= 80^o$, the point $P$ is located such that $\angle PAB=30^o$ and $\angle PBA=10^o$ . Determine the measure in degrees of the angle $\angle CPB$.
Find the maximum possible area of the quadrilateral whose sidelengths are equal to $1$ cm, $2\sqrt2$ cm, $3$ cm and $4$ cm.
Let two segments intersect in the tetrahedron that connect the ends of some edge with the centers of the circles inscribed in the faces opposite to these ends. Prove that the two line segments that connect the ends of an edge crossing the original edge to the centers of the circles inscribed in the remaining two opposite faces also intersect.
Let $ABC$ be a fixed equilateral triangle. For every arbitrary line $\ell $ that what goes through vertice $B$ consider the points $D_{\ell}$ and $E_{\ell}$ , which represents the foot of the perpendicular taken from the points $A$ and $C$, respectively on the line $\ell$. Determine the locus of the points $P_{\ell}$ , which form an equilateral triangle $P_{\ell}D_{\ell}E_{\ell}$.
The isosceles acute triangle $ABC$, $\angle B= \angle C= \alpha$, is the base of the prism $ABCA_1B_1C_1$. The lateral edge $A_1A$ is perpendicular to the edge $AC$, and $\angle A_1AB=\beta <90^o$. Determine the lateral area of the prism, if $ A_1A=BC=a$ .
Let the parallelepiped $ABCDA_1B_1C_1D_1$, where $\angle A_1AD=\angle A_1AB= \angle DAB= 60^o$, and $C_1A_1 = \sqrt7$ cm, $C_1B= \sqrt{13}$ cm, $C_1D = \sqrt{19}$ cm. Determine the distance from point $A$ to the plane $A_1BD$.
source: http://aee.edu.md/content/ordine
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