geometry problems from Argentinian Training Lists for Cono Sur, Rioplatense, IberoAmerican with aops links in the names
collected inside aops: here
Cono Sur: 1993, 2013-14, 2018
IberoAmerican: 1994-99, 2019
Rioplatense: 2019
Cono Sur 2013 , 2014 , 2018
In triangle $ABC$, let $D, M$ be interior points of sides $BC$ and $AD$ , respectively. Lines $BM$ and $AC$ intersect at $E$ , lines $CM$ and $AB$ intersect at $F$ , and lines $EF$ and $AD$ intersect at $N$ . Prove that$$2\frac{AN}{DN}=\frac{AM}{DM}.$$
Let $I $ be the incenter of triangle $ABC$ . The inscribed circle is tangent to sides $BC$ , $CA$ and $AB$ and points $A',$ $B'$ and $C'$ respectively. Let $M$ be the midpoint of the segment $A'B'$. Let $P$ be the projection of $I$ on $AA'$. The straight lines $MP$ and $AC$ meet at $N$ . Prove that the lines $A'N$ and $B'C'$ are parallel.
Let $ABC$ be an acute triangle. We consider the points $M , N$ in the interior of the side $BC$ , $Q$ in the interior of the side $AB$ and $P$ in the interior of the side $AC$, such that $MNPQ$ is a rectangle. Prove that if the center of the rectangle $MNPQ$ is also the center of the triangle $ABC$ , then $AB=AC=3AP$.
Let $ABCD$ be a square and $E$ be a point on the side $AB$ . Lines $DE$ , $BC$ meet at $F,$ and lines $CE$ , $AF$ meet at $G$. Prove that the lines $BG$ and $DF$ are perpendicular.
Let $ABCD$ be a square and $P$ be an interior point such that $PA=1$ , $PB=\sqrt2$ and $PC=\sqrt3$.
a) Find the length of $PD$
b) Find the measure of the angle $\angle APB$
Let $ABC$ be a triangle with right angle $A$ . We consider the points $D$ in the interior of the side $AC$ and $E$ in the interior of the side $BD$ , so that $\angle ABC=\angle ECD=\angle CED$ . Provethat $BE=2AD$.
Let $ABCD$ be a quadrilateral inscribed in a circle with center $O$ . Lines $AC$ , $BD$ meet at $P$ and lines $AB$ , $CD$ meet at $Q$ . Let $R$ be the second intersection point of the circumscribed circles of the triangles $ABP$ and $CDP$.
a) Prove that $P , Q$ and $R$ are collinear.
b) Let $U$ and $V$ be the circumcenters of the triangles $ABP$ and $CDP$ , respectively. Prove that the points $U , R , O , V$ are concyclic.
In the triangle with acute angle $ABC$, with $AB\ne AC$. Let $D$ be the foot of the bisector of the angle $\angle A$. Let $E,F$ be the feet of the altitudes from $ B$ and $C$, respectively. The circles circumscribed by the triangles $DBF$ and $DCE$ meet for second time at $M$. Demonstrate that $ME=MF$.
Let $H$ be the orthocenter of the acute triangle $ABC$ and $P$ a point on the circumscribed circle of triangle $ABC$. Show that the Simson Line of $P$ divides the segment $PH$ in half.
The incircle of triangle $ABC$ is tangent to $AB$ and $AC$ at $D$ and $E$ respectively. Let $X$ be the point on the line $DE$ such that $DE$ is tangent to the circle that passes through $B, C$ and $X$. Prove that the angle $\angle BXC$ is obtuse.
Let $ABCD$ be a cyclic quadrilateral and $\omega_1$, $\omega_2$ the circles inscribed in the triangles $ABC$ and $BCD$. Show that the common exterior tangent to $\omega_1$ and $\omega_2$, other than $BC$, is parallel to $AD$.
We consider a circle with center $O$ and radius $ r$ and a line $\ell$ that does not pass through $O$. A cricket jumps between the points of the line and those of the circle, with jumps of length $ r$. Show that there are at most $8$ points that the cricket can reach.
Let $A$ be a point on the semicircle of diameter $BC$, and $X$ an arbitrary point inside the triangle $ABC$. Line $BX$ intersects semicircle a second time at $K$, and intersects segment $AC$ at $F$. Line$ CX$ intersects semicircle for second time at $L$, and intersects segment $AB$ at $E$. Prove that the circumscribed circles of the triangles $AKF$ and $AEL$ are tangent.
Let $D$ be the midpoint of side $BC$ of triangle $ABC$ with $AB \ne AC$ and $E$ the foot of the altitude $BC$. If $P$ is the intersection point of the perpendicular bisector of segment $DE$ with the perpendicular from $D$ on the bisector of $\angle BAC$, prove that $P$ lies on the Euler circle of triangle $ABC$.
Let $ABC$ be a triangle and let $Z$ be a point inside it that does not belong to the perimeter of $ABC$. Let $D$ be the intersection of the line $AZ$ with $BC$. Let $X$ be a point in $AB$ such that $XD$ bisects segment $BZ$. Similarly let $Y$ be a point in $AC$ such that $YD$ bisects segment $CZ$. Show that $Z$ lies in the interior of triangle $AXY$.
Let $ABC$ be a triangle with right angle at $A$. The altitude from $A$ intersects $BC$ at $H$ and $M$ is the midpoint of hypotenuse $BC$. On the legs and outside the triangle, the equilateral triangles $BAP$ and $ACQ$ are constructed. If $N$ is the intersection point of the lines $AM$ and $PQ$, show that the angles $\angle NHP$ and $\angle AHQ$ are equal.
We have the right triangle $ABC$ with $\angle ABC = 90^o$ and we choose points $D$ and $E$ in the segments $AB$ and $BC$ respectively such that $BD = BE$. Let $G$ be the point such that $DG\perp AB$ and $EG\perp BC$ and let $F$ be the intersection of lines $AE$ and $CD$. If $M$ is the midpoint of $AC$, show that FG and $BM$ intersect at a point on the circumscribed circle of $ABC$.
Danielle practices billiards at a circular table. He chooses two points $P_0$ and $P_1$ on the perimeter of the table, places the ball at point $P_0$ and hits it towards point $P_1$. By doing this, he realizes that the ball bounces $n$ times at different points $P_1, P_2,..., P_n$ where $P_n = P_0$. Show that the points $P_0, P_1,...,P_{n - 1}$ are the vertices of a regular polygon.
Let $ABCD$ be a convex quadrilateral with $AB\ge CD$. On the segment $AB$, points $E$ and $F$ are chosen and on the segment $CD$ points $G$ and $H$ are chosen such that $AE = BF = CG = DH <\frac{AB}{2}$. Let $P$, $Q$ and $R$ be the midpoints of $EG$, $FH$ and $CD$ respectively. It is known that $PR$ is parallel to $AD$ and $QR$ is parallel to $BC$.
a) Show that $ABCD$ is a trapezoid.
b) Let $d$ be the difference of the lengths of the parallel sides. Show that $2 PQ \le d$.
Let $P$ be an interior point of the acute triangle $ABC$. Show that if the symmetrics of $P$ with respect to the sides of the triangle belong to the circumcircle of the triangle, then $P$ is the orthocenter of $ABC$.
Let $ABC$ be a nondegenerate triangle and $I$ the point of intersection of its angle bisectors. $D$ is the midpoint of the segment $CI$ . If it is known that $\frac{AD}{DB} =\frac{BC}{AC}$ , show that $AC = BC$.
Two different circles $\omega_1$ and $\omega_2$ such that neither contains the other intersect at $A$ and $B$. Let $C$ be a point at $\omega_2$ ($C \ne A$ and $C \ne B$). Lines $CA$ and $CB$ intersect $\omega_1$ again at $F$ and $G$ respectively. The tangents to $\omega_1$ through $F$ and $G$ intersect at $D$. The lines $AG$ and $BF$ intersect at $E$. Prove that $C$, $D$ and $E$ are collinear.
Let $ABC$ be an acute triangle, $\Gamma$ the circle circumscribed to triangle $ABC$ and $D$ a point on segment $BC$. Let $M$ be the midpoint of $AD$, $E$ the foot of the perpendicular from $D$ on $AB$ and $F$ the intersection of the line $DE$ with the small arc $BC$ of $\Gamma$. If it is known that $\angle DAE = \angle AFE$, show that the lines $EM$ and $CF$ and the tangent line through $A$ to $\Gamma$ concur.
Let $ABCDEF$ be a convex hexagon with non-parallel sides and tangent to a circle $\Gamma$ at the midpoints $P, Q, R$ of the sides $AB$, $CD$, $EF$ respectively. $\Gamma$ is tangent to $BC$, $DE$ and $FA$ at points $X, Y, Z$ respectively. Line $AB$ intersects lines $EF$ and $CD$ at points $M$ and $N$ respectively. Lines $MZ$ and $NX$ intersect at point $K$. Let $r$ be the line joining the center of $\Gamma$ and $K$. Show that the seven lines $AD$, $PY$, $BE$, $RX$, $r$, $CF$ and $QZ$ have a point in common.
Given an acute triangle $ABC$ we construct exterior triangles $ABD$ and $ACE$ such that $\angle ADB =\angle AEC = 90^o$ and $\angle BAD = \angle CAE$. Let $A_1\in BC$, $B_1\in AC$, $C_1\in AB$ be the feet of the heights of triangle $ABC$, and let $K$ and $L$ be the midpoints of $BC_1$ and $CB_1$ respectively. Show that the circumcenters of the triangles $AKL$, $A_1B_1C_1$ and $DEA_1$ are collinear.
The inscribed circumference of triangle $ABC$ touches sides $BC$, $CA$, $AB$ at points $D, E, F$ respectively. In the segments $EF$, $FD$, $DE$ we consider the points $M, N, P$ respectively such that $BM + MC$, $CN + NA$, $AP + PB$ are minimal.
$\bullet$ Prove that the lines $AM$, $BN$, $CP$ are concurrent.
$\bullet$ Prove that $DM$, $EN$, $FP$ are altitudes of the triangle $DEF$.
Let $I$ be the incenter of the scalene triangle $ABC$, with $AB <AC$, and let $I'$ be the symmetric of $I$ wrt $BC$. The bisector $AI$ intersects the side $BC$ at $D$ and the circumscribed circle of $ABC$ at $E$. The line $EI'$ intersects the circumscribed circle of $ABC$ at $F$. Show that $\frac{AI}{IE} = \frac{ID}{DE}$ and $IA = IF$.
Rioplatense L2,L3 2019
Let $ABCD$ be a cyclic quadrilateral with $AB = AD$. Two points $M$ and $N$ interior of the segments $CD$ and $BC$ respectively are such that $DM + BN = MN$. Show that the circumcenter of triangle $AMN$ lies on segment $AC$.
Let $C_1$ and $B_1$ be points on the sides $AB$ and $AC$ of triangle $ABC$ respectively. Segments $BB_1$ and $CC_1$ intersect at $X$ and segments $B_1C_1$ and $AX$ intersect at $A_1$. The circumscribed circles of triangles $BXC_1$ and $CXB_1$ intersect side $BC$ at $D$ and $F$ respectively. Lines $B_1D$ and $C_1E$ intersect at $F$. Show that the lines $A_1F$, $B_1E$, $C_1D$ are parallel or concurrent.
Let $ABCD$ be a convex quadrilateral. Point $A_1$ belongs to the edge of $ABCD$ and is such that segment $AA_1$ divides quadrilateral $ABCD$ into two parts of equal area. In the same way, points $B_1$, $C_1$ and $D_1$ are defined. The lengths of segments $AA_1$, $BB_1$, $CC_1$, and $DD_1$ are known to be less than or equal to $1$. Show that $area \, (ABCD) <\frac23$.
A point $X$ is located in a right isosceles triangle $ABC$ with $\angle B = 90^o$. Prove the inequality$$ AX + BX+\sqrt2 CX \ge \sqrt5 AB$$and find all points $X$ for which equality applies.
IberoAmerican 1994-1999, 2019
Let $ABC$ be a triangle such that $\angle A <\angle A <90^o <\angle B$. Let $ P$ be the intersection of the external bisector of $\angle A$ with the line $BC$ and $Q$ the intersection of the external bisector of $\angle B$ with the line $AC$. Knowing that $AP = BQ = AB$, calculate $\angle A$.
Let $a, b$, and $c$ be the lengths of the sides of a triangle and let $p$ the semiperimeter and $ r$ the inradius of the triangle. Prove that
$$\frac{1}{(p-a)^2}+\frac{1}{(p - b)^2}+\frac{1}{(p-c)^2}\ge \frac{1}{r^2}$$
If $A$, $B$, $C$ and $D$ are four different points such that every circle passing through $A$ and $B$ has nonempty intersection with every circle passing through $C$ and $D$, show that the four points are collinear or concyclic.
Let $ABCDEF$ be a hexagon inscribed in a circle of radius $r$. Prove that if $AB = CD = EF = r$, then the midpoints of $BC$, $DE$ and $FA$ are vertices of an equilateral triangle.
Let $P$ be a convex polygon contained in a square of side $ 1$. Prove that the sum of the squares of the sides of $P$ is less than or equal to $4$.
Construct a triangle $ABC$ if the points $M, N$ and $ P$ in $AB, AC$ and $BC$ respectively , are known and verify the property$$\frac{MA}{MB} = \frac{PB}{PC} = \frac{NC}{CA} = k$$where $k$ is a fixed number between $0$ and $ 1$.
Find all right triangles with integer sides such that their area is twice their perimeter.
Let $A, B, C$ be three points of the plane with the two integer coordinates and let $R$ be the radius of the circle that contains $A$, $B$ and $C$. If $a, b$ and $c$ are the lengths of the sides of triangle $ABC$, prove that $abc \ge 2R$.
Let $ABCD$ be an cyclic quadrilateral, $M$ the orthocenter of triangle $ABC$, and $N$ the orthocenter of triangle $ABD$. Show that $MNDC$ is a parallelogram.
Let $P_1P_2P_3P_4P_5$ be a non-intersecting flat pentagon, totally contained between the line $ r$ that passes through $P_1$ and $P_5$ and the line $s$, parallel to $ r$ passing through $P_3$. Let $a> 0$. Prove that it is possible to choose points $P_6$ and $P_7$ such that $P_6P_2 = a$, so that it is possible to pave the plane with mosaics congruent to the heptagon $P_1P_2P_3P_4P_5P_6P_7$.
Consider the triangle $ABC$ with $AB <AC$.
a) Show that there are points $M, N$, such that $B \in MC$, $C \in NB$ and the inradii of the triangles $AMB$, $ABC$, and $ACN$ are equal.
b) If $M, N$, are as in (a), show that $MB <NC$ and $\angle MAB> \angle NAC$.
Show that if an octahedron has all its faces triangular and these faces are parallel in pairs, then the faces are equal in pairs.
Let $ABC$ be a triangle and $A', B', C'$ points on the sides $BC$, $CA$ and $AB$ respectively. Prove that if the lines $AA'$, $BB'$ and $CC'$ have a point in common $ P$, with $P \ne A$ and $P \ne A '$, then
$$\frac{B'C}{B'A} + \frac{C'B }{C'A} \ge 4 \frac{PA'}{PA}.$$Find the locus of the points $P$ for which the equality holds.
Prove that a triangle with prime sidelengths and integer area doesn't exist.
In right triangle $ABC$, with $\angle C = 90^o$, let $D$ be a point on side $AB$ and $M$ be the midpoint of $CD$. If $\angle AMD = \angle BMD$, show that$$\frac{\angle ACD}{\angle BCD}=\frac{\angle CDA}{\angle CDB}.$$
Let $O$ be the point of intersection of diagonals $AC$ and $BD$ of quadrilateral $ABCD$. Prove that the orthocenters of the four triangles $OAB$, $OBC$, $OCD$, $ODA$ are vertices of a parallelogram that is similar to the one determined by the centroids of the same triangles.
Let $AB$ and $AC$ be tangent to the circle $\Gamma$ at $B$ and $C$ respectively. Let $D$ be a point on the extension of $AB$, beyond $B$, and let $P$ be the second intersection point of $\Gamma$ with the circle circumscribed to the triangle $ACD$. Let $Q$ be the foot of the perpendicular from $B$ to $CD$. Prove that $\angle DPQ =2 \angle ADC$.
Let $ABC$ be a triangle and $M$ be a point. We consider the circles of diameters $AM$, $BM$, $CM$ and the circle that contains these three and is tangent to all three. Show that the radius $P$ of this large circle satisfies $P\ge 2r$, where $ r$ is the inradius of $ABC$.
You want to decorate a rectangular box with a ribbon that passes through its faces and forms various angles with the edges. If possible, the points where the tape crosses the edges are chosen so that the length of the closed path is minimal (local). This ensures that the ribbon can be adjusted and tied without slipping. Decide if there is a minimum path that crosses all $6$ faces exactly once.
Quadrilateral $ABCD$ is inscribed in a circle with center $O$. $AD$ and $BC$ intersect at $ P$. Let $ L$ and $M$ be the midpoints of $AD$ and $BC$ respectively, and let $Q$ and$ R$ be the feet of the perpendiculars from $O$ and $ P$ on $LM$. Prove that $LQ = RM$.
Let $a, b, c$ be the sides and $\alpha, \beta,\gamma$ and the angles (in radians) of a triangle of semiperimeter $s$. Prove that
$$\frac{bc}{\alpha(s-a)}+ \frac{ca}{\beta(s-b)}+\frac{ab}{\gamma(s-c)} \ge \frac{12s}{\pi}$$
Let $E$ and $F$ be points on sides $BC$ and $AD$, respectively, of quadrilateral $ABCD$. Let $ P$ be the intersection of $AE$ with $BF$ and $Q$ the intersection of $CF$ with $DE$. Show that $E$ and $F$ divide $BC$ and $AD$ (or $BC$ and $DA$) in the same ratio if and only if$$\frac{area\,\, (FPQ)}{area \,\,(EDA)}= \frac{area \,\, (EQP) }{area\,\, (FBC)}$$
In a triangle $ABC$, let $S$ be the midpoint of the median corresponding to vertex $A$ and $Q$ the intersection point of $BS$ with the side $AC$ . Show that $BS = 3QS$.
Let $A_1A_2... A_n $ be an $n$-sided polygon with centroid $G$, inscribed in a circle. The lines $A_1G$, $A_2G$,$ ...$, $A_nG$ intersect again the circle at $B_1$, $B_2$, $...,$ $B_n$, respectively. Prove that
$$\frac{A_1G}{GB_1}+\frac{A_2G}{GB_2}+...+\frac{A_nG}{GB_n}=n$$
Let $ABC$ be a right triangle with the right angle at $C$ and with legs $BC = a$ and $CA = b$. Let $D$ and $E$ be points on sides $AC$ and $AB$ respectively, so that $BD$ is the bisector of angle $B$ and $DE$ is parallel to $BC$. Prove that the area of the triangle $ADE$, $(ADE)$, satisfies the inequality$$(ADE)< \frac{a^4+b^4}{2(a+b)^2}$$
Let $ABC$ be a triangle, $G$ be the centroid, and $P$ be a variable point in the interior of $ABC$. Let $D, E, F$ be points on the sides $BC, CA, AB$ respectively, such that $PD\parallel AG$, $PE\parallel BG$, $PF\parallel CG$. Prove that the sum of areas $(PAF) + (PBD) + (PCE)$ is constant.
In a plane consider the equilateral triangles $ABC$ and $A'BC$. Let $D$ be a variable point on side $AC$. Line $A'D$ intersects line $AB$ at $E$. Line $BD$ intersects line $CE$ at $P$. Show that $P$ lies on the circle circumscribed to triangle $ABC$.
Triangle $ABC$ is right at $C$.
a) Show that the three ellipses that have foci at two vertices of the triangle and pass through the third have a point in common.
b) Show that the principal vertices of the ellipses in part (a), that is, the points at which each ellipse intersects the line joining its foci, form two collinear point themes.
Given a regular tetrahedron of height $1$, find the volume that will remain after removing, in a first step, the inscribed sphere; then, in a second step, the following spheres relative to each of the $4$ vertices, externally tangent to the inscribed sphere; and in the subsequent steps, following the process without interruption, the following $4$ spheres relative to the vertices, each tangent externally to the sphere, relative to the same vertex, created in the previous step.
Clarification. Given a sphere tangent to the internal faces adjacent to a vertex of a polyhedron we will call the following sphere relative to the vertex to the sphere tangent externally to the first sphere and tangent internally to the same faces of the polyhedron.
The triangle $ABC$ is inscribed in the circle $\Gamma$. Let $AA_1$, $BB_1$, $CC_1$ be the bisectors of the angles $\angle A, \angle B, \angle C$ with $A_1$, $B_1$, $C_1$ in $\Gamma$. Prove that the perimeter of the triangle is equal to
$$AA_1 \cos \left(\frac{\angle A}{2}\right)+BB_1 \cos \left(\frac{\angle B}{2}\right)+CC_1 \cos \left(\frac{\angle C}{2}\right)$$
Let $M$ be a variable point on side $BC$ of triangle $ABC$. A line through $M$ intersects lines $AB$ and $AC$ at $K$ and $L$ respectively, such that $M$ is the midpoint of $KL$. Point $K'$ is such that $ALKK'$ is a parallelogram. Determine the locus of $K'$ as $M$ moves in segment $BC$.
Let $ABCD$ be a square and $E, F$ be points on the sides $BC, CD$, respectively, such that $CE\ne CF$. Lines $AE$ and $AF$ intersect $BD$ at $P$ and $Q$, respectively. It is assumed that $BP \cdot CE = DQ\cdot CF$. Prove that the five points $C, E, F, P$ and $Q$ lie on a circle.
Let $P$ be an interior point of the equilateral triangle $ABC$ such that $PB\ne PC$. Let $D$ be the intersection point of $BP$ with $AC$ and $E$ be the intersection point of $PB$ with $AD$ and $E$ be the intersection point of $CP$ with $AB$. Assuming that $\frac{PB}{PC}= \frac{AD}{AE}$, find measure of the angle $\angle BPC$ .
Let $ABC$ be an acute triangle of orthocenter $H$ and circumcenter $O$. The perpendicular bisector of segment $AH$ intersects $AB$ at $ P$ and $AC$ at $Q$. Prove that $\angle AOP = \angle AOQ$.
Let $ABC$ be a triangle with $\angle A> 90^o$ and $AD$, $BE$, $CF$ its altitudes . Let $E', F'$ be the feet of the perpendiculars from $E$ and $F$ on $ BC$. Assuming $2E'F'= 2AD + BC$, find $\angle A$.
An isosceles triangle has integer sides; furthermore, the length of the base and the length of the other two sides are coprime. If the angle bisectors have rational lengths, show that the length of the equal sides is an odd perfect square.
Let $ABC$ be a non-equilateral triangle of circumcenter $O$ and incenter $I$. Let $D$ be the foot of the altitude drawn from $A$. If the circumradius $R$ is equal to the radius $r_a$ of the exscribed circle corresponding to $BC$, show that $O, I$, and $D$ are collinear.
Let $a, b$ and $c$ be the lengths of the sides of a scalene triangle $\Delta$ and let $l =\frac{b + c}{2}$, $m = \frac{c + a}{2}$ and $n =\frac{a + b}{2}$. Prove that $l, m$ and $n$ are lengths of the sides of a scalene triangle whose area is greater than that of triangle $\Delta$.
Let $ABCD$ be a trapezoid with the property that the triangles $ABC$, $ACD$, $ABD$ and $BCD$ have the same inradius. Prove that $A, B, C$ and $D$ are the vertices of a rectangle.
A parallelogram is inscribed in a regular hexagon in such a way that the centers of symmetry of both figures coincide. Prove that the area of the parallelogram is less than or equal to $2/3$ of the area of the hexagon.
Let $ABC$ be a non-obtuse triangle. For each point $P$ in segment $BC$ let $Q$ in segment $AC$ and $R$ in segment $AB$, such that triangle $PQR$ has minimum perimeter. Prove that if all the lines $QR$ are concurrent (when $P$ moves along $\overline{BC}$) then the angle $\angle BAC$ is right.
Let $P$ be a point in the interior of a circle, other than the center. Three lines are drawn through $P$ at angles of $60^o$, so that none of them is diameter; these divide the circle into $6$ regions. If we shade $3$ of the $6$ sectors thus determined alternately around $P$, we will have two regions within the circle: one shaded and the other unshaded. Prove that of these two regions, the one with the greatest area is the one that contains the center of the circle.
A circle $\Gamma$ of radius $R$ is tangent to a line $t$ at a point $C$. Another circle $\omega$ of radius $r <R$ is tangent to $t$ at a point $B$ and intersects $\Gamma$ at points $A$ and $A'$, $A$ being closest to $t$. What is the locus of the centers of the circles circumscribed to the triangles $ABC$ when the circle $\omega$ rolls on the line $t$?
In a convex polygon $P$, some diagonals have been drawn that do not intersect inside $P$. Show that there are at least two vertices of P$$ such that neither of them is an endpoint of any of the diagonals drawn.
On the sides $AB$ and $BC$ of the square $ABCD$, the points $E$ and $F$ were chosen, respectively, such that $BE = BF$. Let $BN$ be the altitude of triangle $BCE$. Prove that $DNF$ is a right angle.
The centers of the spheres circumscribed and inscribed to a given tetrahedron coincide. Prove that the faces are congruent triangles.
Let $P$ be a point in the interior of a tetrahedron $ABCD$. Show that the sum of the angles $APB$, $APC$, $APD$, $BPC$, $BPD$, $CPD$ is greater than $540^o$
Let $A_1A_2A_3$ be a triangle ,points $B_1$, $B_2$, $B_3$ be on the sides $A_2A_3$,$ A_3A_1$, $A_1A_2$, respectively (which do not coincide with any vertex). Show that the perpendicular bisectors of the three segments $A_1B_1$ are never concurrent
Given four points in space $A, B, C, D$. Let $M$ and $N$ be the midpoints of $AC$ and $BD$ respectively. Prove that
$$AB^2 + BC^2 + CD^2 + DA^2 = AC^2 + BD^2 + 4MN^2.$$
Through the ends $A$ and $B$ of a diameter $AB$ of a given circle, the tangents $\ell$ and $m$ are drawn, parallel to each other. Let $C$ be a point on $\ell$ other than $A$ and let $q_1$, $q2$ be two rays with origin at $C$. Suppose that $q_i$ ($i = 1, 2$) intersects the circle at points $D_i$, $E_i$ (with $D_1$ between $C$ and $E_i$). Rays $AD_1$, $AD_2$, $AE_1$, $AE_2$ intersect m at points $M_1$, $M_2$, $N_1$, $N_2$, respectively. Prove that $M_1M_2 = N_1N_2$.
In triangle $ABC$, let $ r$ be the inradius and let $r_A$ be the radius of the circle tangent to $AB$, to $AC$ and externally tangent to the circle inscribed at $ABC$, $r_B$ and $r_C$ are defined similarly. Prove that
$$r_A + r_B + r_C\ge r$$and the equality holds if and only if the triangle is equilateral.
In a tetrahedron $ABCD$ the foot of the height drawn from each vertex coincides with the incenter of the opposite face. Prove that the tetrahedron is regular.
In a sphere of radius $ 1$, consider a maximum circle $H$. On one of the hemispheres determined by $H$ consider three circles of equal radii, tangent to each other, in pairs, and tangent to $H$. Determine the radius of another circle, located on the same hemisphere and that is tangent to the three circles.
A lattice triangle has the following property: "the product of the lengths of two of its sides is a prime number and its area is a prime number". Find the area and determine all lattice triangles that have the above property.
Consider a square$ ABCD$ and the arbitrary points $P$ and $Q$ on the sides $AB$ and $BC$, respectively. Lines $AQ$ and $PC$ intersect $PD$ and $QD$ at points X and Y. Show that lines $AY$, $CX$, and $BD$ are concurrent.
Let $A_1A_2A_3$ be a non-right triangle. The points $O_1$, $O_2$, $O_3$ are the centers of the circles $\Gamma_1$, $\Gamma_2$, $\Gamma_3$ tangent in pairs (internally or externally); the circle $\Gamma_1$ passes through $A_2$ and $A_3$, the circle $\Gamma_2$ passes through $A_3$ and $A_1$, the circle $\Gamma_3$ passes through $A_1$ and $A_2$. Knowing that the triangles $A_1A_2A_3$ and $O_1O_2O_3$ are similar, determine their angles.
Consider a semicircle of diameter $AB = 2R$ and a chord $CD$ of fixed length $c$. Let $E$ be the intersection of $AC$ with $BD$ and $F$ the intersection of $AD$ with $BC$. Show that the segment $EF$ has constant length and is perpendicular to $AB$. Express $EF$ in terms of $R$ and $c$.
A regular heptagon $A_1A_2... A_7$ is inscribed in circle $C$. Point $P$ is taken on the shorter arc $A_7A_1$.
Prove that $PA_1+PA_3+PA_5+PA_7 = PA_2+PA_4+PA_6$.
We consider a circle $\Gamma$, an exterior point $A$ and a line tangent to $\Gamma$ through $A$, where $T$ is the point of tangency. A variable line $r$ is drawn through $A$ that cuts the circle $\Gamma$ at $P$ and $B$ in such a way that $P$ belongs to segment $AB$. For each $ r$ we take, in the same half plane as $T$, the point $C$ on the perpendicular bisector of $AB$ such that the distance from $C$ to $r$ is equal to $AB$.
a) Determine the locus of $C$ as line $ r$ varies.
b) Prove that the area of the triangle $APC$ is constant as line $ r$ varies.
Prove that in a convex quadrilateral of area $1$ the sum of all the sides and all the diagonals is greater than or equal to $4 +\sqrt8.$
Given a convex polygon, prove that we can find a point $Q$ inside the polygon and three vertices $A_1$, $A_2$, $A_3$ (not necessarily consecutive) such that each ray $A_iQ$ ($i = 1, 2,3$) forms an acute angle with each one of the sides that have a vertex at $A_i$.
Let $P$ be an interior point of the tetrahedron $ABCD$ and $S_A,S_B,S_C,S_D$ the center of gravity of the tetrahedra $PBCD$, $PCDA$, $PDAB$, $PABC$. Prove that the volume of the tetrahedron $S_AS_BS_CS_D$ is equal to $\frac{1}{64}$ of the volume of tetrahedron $ABCD$.
Let $M$ be the set of all tetrahedra whose inscribed and circumscribed spheres are concentric. We denote with $ r$ and $R$ the radii of said spheres. Determine the range of values $\frac{R}{r}$ for tetrahedra of $M$.
Through a point $P$ inside a sphere, draw three chords perpendicular in pairs. Prove that the sum of the squares of their lengths does not depend on their directions.
Let $ABCD$ be a parallelogram. Circle $S_1$ passes through vertex $C$ and is tangent to sides $BA$ and $AD$ at points $P_1$ and $Q_1$ respectively. Circle $S_2$ passes through vertex $B$ and is tangent to sides $DC$ and $AD$ at points $P_2$ and $Q_2$ respectively. Let $d_1$ and $d_2$ be the distances from $C$ and $B$ to the lines $P_1Q_1$ and $P_2Q_2$ respectively. Find all possible values of the ratio $d_1: d_2$.
Let $ABC$ be a triangle, $I$ its incenter, $\Gamma$ its circumscribed circle, and $M$ the midpoint of side $BC$. Suppose that there are two different points $P, Q$ on the line $IM$ such that $PB = PI$ and $QC = QI$. Let $G$ be the point of intersection of the lines $PB$ and $QC$. Prove that the line $IG$ and the perpendicular bisector of segment $BC$ intersect at a point on $\Gamma$.
$ABC$ is a triangle with incenter $I$. We construct points $P$ and $Q$ such that $AB$ is the bisector of $\angle IAP$, $AC$ is the bisector of $\angle QAI$ and $\angle PBI + \angle IAB = \angle QCI + \angle IAC = 90^o$. The lines $IP$, $IQ$ intersect $AB$, $AC$ at $K, L$ respectively. The circumscribed circles of the triangles $APK$ and $AQL$ intersect at $A$ and $R$. Show that $R$ lies on the line joining the midpoints of $BC$ and $KL$.
Let $A_1H_1$, $A_2H_2$ and $A_3H_3$ be the altitudes and $A_1L_1$, $A_2L_2$ and $A_3L_3$ the angle bisectors of an acute triangle $A_1A_2A_3$. Prove that $area \, (L_1L_2L_3) \ge area \, (H_1H_2H_3)$.
Let $ABC$ be an acute triangle with $AB \ne AC$ and $H$ its orthocenter. Let $D$ and $E$ be the intersections of $BH$ and $CH$ with $AC$ and $AB$ respectively, and $P$ the foot of the perpendicular from $A$ on $DE$. The circumcircle of $BPC$ intersects $DE$ at a point $Q \ne P$. Show that lines $AP$ and $QH$ intersect at a point on the circumcircle of $ABC$.
In triangle $ABC$, the inscribed circle $\omega$ is tangent to sides $BC$, $CA$ , $AB$ at points $D, E, F$ respectively. Let $M$ and $N$ be the midpoints of $DE$ and $DF$ respectively. Points $D', E', F'$ are considered on the line $MN$ such that $D'E = D'F$, $BE' || DF$ and $CF' || DE$. Show that the lines $DD'$, $EE'$ and $FF'$ are concurrent.
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