geometry problems from International Olympiad Tuymaada

with aops links in the names

Point $M$ lies inside triangle $ABC$. Prove that for any other point $N$ lying inside the triangle $ABC$, at least one of the following three inequalities is fulfilled: $AN>AM, BN>BM, CN>CM$.

Let a convex polyhedron be given with volume $V$ and full surface $S$.

Prove that inside a polyhedron it is possible to arrange a ball of radius $\frac{V}{S}$.

In three houses $A,B$ and $C$, forming a right triangle with the legs $AC=30$ and $CB=40$, live three beetles $a,b$ and $c$, capable of moving at speeds of $2, 3$ and $4$, respectively. Suppose that you simultaneously release these bugs from point $M$ and mark the time after which beetles reach their homes. Find on the plane such a point $M$, where is the last time to reach the house a bug would be minimal.

Prove that in space there is a sphere containing exactly $1994$ points with integer coordinates.

Give a geometric proof of the statement that the fold line on a sheet of paper is straight.

Given a circle of radius $r= 1995$. Show that around it you can describe exactly $16$ primitive Pythagorean triangles. The primitive Pythagorean triangle is a right-angled triangle, the lengths of the sides of which are expressed by coprime integers.

Inside the triangle $ABC$ a point $M$ is given . Find the points $P,Q$ and $R$ lying on the sides $AB,BC$ and $AC$ respectively and such so that the sum $MP+PQ+QR+RM$ is the smallest.

Given a segment of length $7\sqrt3$ .

Is it possible to use only compass to construct a segment of length $\sqrt7$?

Given a tetrahedron $ABCD$, in which $AB=CD= 13 , AC=BD=14$ and $AD=BC=15$.

Show that the centers of the inscribed sphere and sphere around it coincide, and find the radii of these spheres.

Using only angle with angle $\frac{\pi}{7}$ and a ruler, constuct angle $\frac{\pi}{14}$

Find a right triangle that can be cut into $365$ equal triangles.

The segment of length $\ell$ with the ends on the border of a triangle divides the area of that triangle in half. Prove that $\ell >r\sqrt2$, where $r$ is the radius of the inscribed circle of the triangle.

Given the tetrahedron $ABCD$, whose opposite edges are equal, that is, $AB=CD, AC=BD$ and $BC=AD$. Prove that exist exactly $6$ planes intersecting the triangular angles of the tetrahedron and dividing the total surface and volume of this tetrahedron in half.

A right triangle is inscribed in parabola $y=x^2$. Prove that it's hypotenuse is not less than $2$.

Given the pyramid $ABCD$. Let $O$ be the middle of edge $AC$. Given that $DO$ is the height of the pyramid, $AB=BC=2DO$ and the angle $ABC$ is right. Cut this pyramid into $8$ equal and similar to it pyramids.

Tuymaada 1999 Seniors 5

Tuymaada 2000 Juniors 3 / Seniors 2

with aops links in the names

1994 - 2019

Prove that inside a polyhedron it is possible to arrange a ball of radius $\frac{V}{S}$.

Inside the triangle $ABC$ a point $M$ is given . Find the points $P,Q$ and $R$ lying on the sides $AB,BC$ and $AC$ respectively and such so that the sum $MP+PQ+QR+RM$ is the smallest.

Is it possible to use only compass to construct a segment of length $\sqrt7$?

Show that the centers of the inscribed sphere and sphere around it coincide, and find the radii of these spheres.

Tuymaada 1999 Seniors 5

In the triangle
ABC we have ÐABC = 100

^{ o}, ÐACB = 65^{o}, M ä AB, N ä AC, and ÐMCB = 55^{o}, ÐNBC = 80^{o}. Find ÐNMC.
(St.Petersburg folklore)

Let $O$ be the center of the circle described around the the triangle $ABC$. The centers of the circles described around the squares $OAB,OBC,OCA$ lie at the vertices of a regular triangle. Prove that the triangle $ABC$ is right.

A tangent

*l*to the circle inscribed in a rhombus meets its sides AB and BC at points E and F respectively. Prove that the product AE · CF is independent of the choice of*l*.
Tuymaada 2001 Juniors 3

Let ABC be an acute isosceles triangle ($AB=BC$) inscribed in a circle with center $O$ . The line through the midpoint of the chord $AB$ and point $O$ intersects the line $AC$ at $L$ and the circle at the point $P$. Let the bisector of angle $BAC$ intersects the circle at point $K$. Lines $AB$ and $PK$ intersect at point $D$. Prove that the points $L,B,D$ and $P$ lie on the same circle.

Let ABC be an acute isosceles triangle ($AB=BC$) inscribed in a circle with center $O$ . The line through the midpoint of the chord $AB$ and point $O$ intersects the line $AC$ at $L$ and the circle at the point $P$. Let the bisector of angle $BAC$ intersects the circle at point $K$. Lines $AB$ and $PK$ intersect at point $D$. Prove that the points $L,B,D$ and $P$ lie on the same circle.

On the side $AB$ of an isosceles triangle $AB$ ($AC=BC$) lie points $P$ and $Q$ such that $\angle PCQ \le \frac{1}{2} \angle ACB$. Prove that $PQ \le \frac{1}{2} AB$.

ABCD is a convex
quadrilateral, half-lines DA and CB meet at point Q, half-lines BA and CD meet
at point P. It is known that ÐAQB = ÐAPD. The bisector of angle ÐAQB meets the sides AB and CD of
the quadrilateral at points X and Y , respectively, the bisector of angle ÐAPD meets the sides AD and BC at
points Z and T, respectively. The circumcircles of triangles ZQT and XPY meet
at point K inside the quadrilateral. Prove that K lies on the diagonal AC.

( S. Berlov)

Tuymaada 2002 Juniors 2

Points on the sides $ BC $, $ CA $ and $ AB $ of the triangle $ ABC $ are respectively $ A_1 $, $ B_1 $ and $ C_1 $ such that $ AC_1: C_1B = BA_1: A_1C = CB_1: B_1A = 2: 1 $. Prove that if triangle $ A_1B_1C_1 $ is equilateral, then triangle $ ABC $ is also equilateral.

Tuymaada 2002 Juniors 8

The circle with the center of $ O $ touches the sides of the corner with the vertex $ A $ at the points of $ K $ and $ M $. The tangent to the circle intersects the segments $ AK $ and $ AM $ at points $ B $ and $ C $ respectively, and the line $ KM $ intersects the segments $ OB $ and $ OC $ at the points $ D $ and $ E $. Prove that the area of the triangle $ ODE $ is equal to a quarter of the area of a triangle $ BOC $ if and only if the angle $ A $ is $ 60^\circ $.

Tuymaada 2002 Seniors 3

Points on the sides $ BC $, $ CA $ and $ AB $ of the triangle $ ABC $ are respectively $ A_1 $, $ B_1 $ and $ C_1 $ such that $ AC_1: C_1B = BA_1: A_1C = CB_1: B_1A = 2: 1 $. Prove that if triangle $ A_1B_1C_1 $ is equilateral, then triangle $ ABC $ is also equilateral.

Tuymaada 2002 Juniors 8

The circle with the center of $ O $ touches the sides of the corner with the vertex $ A $ at the points of $ K $ and $ M $. The tangent to the circle intersects the segments $ AK $ and $ AM $ at points $ B $ and $ C $ respectively, and the line $ KM $ intersects the segments $ OB $ and $ OC $ at the points $ D $ and $ E $. Prove that the area of the triangle $ ODE $ is equal to a quarter of the area of a triangle $ BOC $ if and only if the angle $ A $ is $ 60^\circ $.

Tuymaada 2002 Seniors 3

A circle having
common centre with the circumcircle of triangle ABC meets the sides of the triangle
at six points forming convex hexagon A

_{1}A_{2}B_{1}B_{2}C_{1}C_{2}(A_{1}and A_{2}lie on BC, B_{1}and B_{2}lie on AC, C_{1}and C_{2}lie on AB). If A_{1}B_{1}is parallel to the bisector of angle B, prove that A_{2}C_{2}is parallel to the bisector of angle C.
(S. Berlov)

The points D and
E on the circumcircle of an acute triangle ABC are such that AD = AE = BC. Let
H be the common point of the altitudes of triangle ABC. It is known that AH

^{2}= BH^{2}+ CH^{2}. Prove that H lies on the segment DE.
(D. Shiryaev)

Tuymaada 2003 Juniors 3

In the acute triangle $ ABC $, the point $ I $ is the center of the inscribed the circle, the point $ O $ is the center of the circumscribed circle and the point $ I_a $ is the center the excircle tangent to the side $ BC $ and the extensions of the sides $ AB $ and $ AC $. Point $ A'$ is symmetric to vertex $ A $ with respect to the line $ BC $. Prove that $ \angle IOI_a = \angle IA'I_a $.

Tuymaada 2003 Juniors 7

Through the point $ K $ lying outside the circle $ \omega $, the tangents are drawn $ KB $ and $ KD $ to this circle ($ B $ and $ D $ are tangency points) and a line intersecting a circle at points $ A $ and $ C $. The bisector of angle $ ABC $ intersects the segment $ AC $ at the point $ E $ and circle $ \omega $ at $ F $. Prove that $ \angle FDE = 90^\circ $.

Tuymaada 2003 Seniors 2

In the acute triangle $ ABC $, the point $ I $ is the center of the inscribed the circle, the point $ O $ is the center of the circumscribed circle and the point $ I_a $ is the center the excircle tangent to the side $ BC $ and the extensions of the sides $ AB $ and $ AC $. Point $ A'$ is symmetric to vertex $ A $ with respect to the line $ BC $. Prove that $ \angle IOI_a = \angle IA'I_a $.

Tuymaada 2003 Juniors 7

Through the point $ K $ lying outside the circle $ \omega $, the tangents are drawn $ KB $ and $ KD $ to this circle ($ B $ and $ D $ are tangency points) and a line intersecting a circle at points $ A $ and $ C $. The bisector of angle $ ABC $ intersects the segment $ AC $ at the point $ E $ and circle $ \omega $ at $ F $. Prove that $ \angle FDE = 90^\circ $.

Tuymaada 2003 Seniors 2

In a
quadrilateral ABCD sides AB and CD are equal, ÐA
= 150

^{o}, ÐB = 44^{o}, ÐC = 72^{o}. Perpendicular bisector of the segment AD meets the side BC at point P. Find ÐAPD.
(F. Bakharev)

In a convex
quadrilateral ABCD we have AB · CD = BC· DA and 2ÐA
+ ÐC = 180

^{o}. Point P lies on the circumcircle of triangle ABD and is the midpoint of the arc BD not containing A. It is known that the point P lies inside the quadrilateral ABCD. Prove that ÐBCA = ÐDCP.
(S. Berlov)

Tuymaada 2004 Juniors 3

Point $ O $ is the center of the circumscribed circle of an acute triangle $ Abc $. A certain circle passes through the points $ B $ and $ C $ and intersects sides $ AB $ and $ AC $ of a triangle. On its arc lying inside the triangle, points $ D $ and $ E $ are chosen so that the segments $ BD $ and $ CE $ pass through the point $ O $. Perpendicular $ DD_1 $ to $ AB $ side and perpendicular $ EE_1 $ to $ AC $ side intersect at $ M $. Prove that the points $ A $, $ M $ and $ O $ lie on the same straight line.

Point $ O $ is the center of the circumscribed circle of an acute triangle $ Abc $. A certain circle passes through the points $ B $ and $ C $ and intersects sides $ AB $ and $ AC $ of a triangle. On its arc lying inside the triangle, points $ D $ and $ E $ are chosen so that the segments $ BD $ and $ CE $ pass through the point $ O $. Perpendicular $ DD_1 $ to $ AB $ side and perpendicular $ EE_1 $ to $ AC $ side intersect at $ M $. Prove that the points $ A $, $ M $ and $ O $ lie on the same straight line.

The incircle of triangle ABC touches its sides AB and BC at points P and Q. The line PQ meets the circumcircle of triangle ABC at points X and Y. Find ÐXBY if ÐABC = 90

^{ o}.
(A. Smirnov)

Tuymaada 2004 Seniors 3
An acute
triangle ABC is inscribed in a circle of radius 1 with centre O, all the angles
of ABC are greater than 45

^{o}. B_{1}is the foot of perpendicular from B to CO, B_{2}is the foot of perpendicular from B1 to AC. Similarly, C_{1}is the foot of perpendicular from C to BO, C_{2}is the foot of perpendicular from C_{1}to AB. The lines B_{1}B2 and C_{1}C_{2}intersect at A_{3}. The points B_{3}and C_{3}are defined in the same way. Find the circumradius of triangle A_{3}B_{3}C_{3}.
(F.Bakharev & F.Petrov)

Tuymaada 2005 Juniors 2

Points $ X $ and $ Y $ are the midpoints of the sides $ AB $ and $ AC $ of the triangle $ ABC $, $ I $ is the center of its inscribed circle, $ K $ is the point of tangency of the inscribed circles with side $ BC $. The external angle bisector at the vertex $ B $ intersects the line $ XY $ at the point $ P $, and the external angle bisector at the vertex of $ C $ intersects $ XY $ at $ Q $. Prove that the area of the quadrilateral $ PKQI $ is equal to half the area of the triangle $ ABC $.

Points $ X $ and $ Y $ are the midpoints of the sides $ AB $ and $ AC $ of the triangle $ ABC $, $ I $ is the center of its inscribed circle, $ K $ is the point of tangency of the inscribed circles with side $ BC $. The external angle bisector at the vertex $ B $ intersects the line $ XY $ at the point $ P $, and the external angle bisector at the vertex of $ C $ intersects $ XY $ at $ Q $. Prove that the area of the quadrilateral $ PKQI $ is equal to half the area of the triangle $ ABC $.

Tuymaada 2005 Juniors 7

The point $ I $ is the center of the inscribed circle of the triangle $ ABC $. The points $ B_1 $ and $ C_1 $ are the midpoints of the sides $ AC $ and $ AB $, respectively. It is known that $ \angle BIC_1 + \angle CIB_1 = 180^\circ $. Prove the equality $ AB + AC = 3BC $

Tuymaada 2005 Seniors 4

The point $ I $ is the center of the inscribed circle of the triangle $ ABC $. The points $ B_1 $ and $ C_1 $ are the midpoints of the sides $ AC $ and $ AB $, respectively. It is known that $ \angle BIC_1 + \angle CIB_1 = 180^\circ $. Prove the equality $ AB + AC = 3BC $

Tuymaada 2005 Seniors 4

In a triangle
ABC, let A

_{1}, B_{1}, C_{1}be the points where the excircles touch the sides BC, CA and AB respectively. Prove that AA_{1}, BB_{1}and CC_{1}are the sidelenghts of a triangle.
(L.
Emelyanov)

Let I be the
incentre of triangle ABC. A circle containing the points B and C meets the segments
BI and CI at points P and Q respectively. It is known that BP · CQ = PI · QI. Prove
that the circumcircle of the triangle PQI is tangent to the circumcircle of
ABC.

(S. Berlov)

Tuymaada 2006 Juniors 1

On the equal $ AC $ and $ BC $ of an isosceles right triangle $ ABC $ , points $ D $ and $ E $ are marked respectively, so that $ CD = CE $. Perpendiculars on the straight line $ AE $, passing through the points $ C $ and $ D $, intersect the side $ AB $ at the points $ P $ and $ Q $.Prove that $ BP = PQ $.

Tuymaada 2006 Juniors 7

The median $ BM $ of a triangle $ ABC $ intersects the circumscribed circle at point $ K $. The circumcircle of the triangle $ KMC $ intersects the segment $ BC $ at point $ P $, and the circumcircle of $ AMK $ intersects the extension of $ BA $ at $ Q $. Prove that $ PQ> AC $.

Tuymaada 2006 Seniors 3

On the equal $ AC $ and $ BC $ of an isosceles right triangle $ ABC $ , points $ D $ and $ E $ are marked respectively, so that $ CD = CE $. Perpendiculars on the straight line $ AE $, passing through the points $ C $ and $ D $, intersect the side $ AB $ at the points $ P $ and $ Q $.Prove that $ BP = PQ $.

Tuymaada 2006 Juniors 7

The median $ BM $ of a triangle $ ABC $ intersects the circumscribed circle at point $ K $. The circumcircle of the triangle $ KMC $ intersects the segment $ BC $ at point $ P $, and the circumcircle of $ AMK $ intersects the extension of $ BA $ at $ Q $. Prove that $ PQ> AC $.

Tuymaada 2006 Seniors 3

A line d is
given in the plane. Let B äd and A another
point, not on d, and such that AB is not perpendicular on d. Let ω
be a variable circle touching d at B and letting A outside, and X and Y the
points on ω such that AX and AY are tangent to the circle. Prove
that the line XY passes through a fixed point.

(F. Bakharev)

Let ABC be a
triangle, G it`s centroid, H it`s orthocenter, and M the midpoint of the arc AC
(not containing B). It is known that MG = R, where R is the radius of the
circumcircle. Prove that BG≥ BH.

(F. Bakharev)

Tuymaada 2007 Juniors 4

An acute-angle non-isosceles triangle $ ABC $ is given. The point $ H $ is its orthocenter, the points $ O $ and $ I $ are the centers of its circumscribed and inscribed circles, respectively. The circumcircle of the triangle $ OIH $ passes through the vertex $ A $. Prove that one of the angles of the triangle is $ 60^\circ $.

An acute-angle non-isosceles triangle $ ABC $ is given. The point $ H $ is its orthocenter, the points $ O $ and $ I $ are the centers of its circumscribed and inscribed circles, respectively. The circumcircle of the triangle $ OIH $ passes through the vertex $ A $. Prove that one of the angles of the triangle is $ 60^\circ $.

On the $ AB $ side of the triangle $ ABC $, points $ X $ and $ Y $ are chosen, on the side of $ AC $ is a point of $ Z $, and on the side of $ BC $ is a point of $ T $. Wherein $ XZ \parallel BC $, $ YT \parallel AC $. Line $ TZ $ intersects the circumscribed circle of triangle $ ABC $ at points $ D $ and $ E $. Prove that points $ X $, $ Y $, $ D $ and $ E $ lie on the same circle.

Tuymaada 2007 Seniors 3
AA

_{1}, BB_{1}, CC_{1}are altitudes of an acute triangle ABC. A circle passing through A_{1}and B_{1}touches the arc AB of its circumcircle at C_{2}. The points A_{2}, B_{2}are defined similarly. Prove that the lines AA_{2}, BB_{2}, CC_{2}are concurrent.
(R. Sakhipov)

Point D is
chosen on the side AB of triangle ABC. Point L inside the triangle ABC is such that
BD = LD and ÐLAB = ÐLCA = ÐDCB. It is known that ÐALD + ÐABC = 180

^{o}. Prove that ÐBLC = 90^{o}.
(R. Sakhipov)

Point I

_{1}is the reflection of incentre I of triangle ABC across the side BC. The circumcircle of BCI_{1}intersects the line II_{1}again at point P. It is known that P lies outside the incircle of the triangle ABC. Two tangents drawn from P to the latter circle touch it at points X and Y . Prove that the line XY contains a medial line of the triangle ABC.
(L.
Emelyanov)

Let ABCD be an
isosceles trapezoid with AD k BC. Its diagonals AC and BD intersect at point M.
Points X and Y on the segment AB are such that AX = AM, BY = BM. Let Z be the
midpoint of XY and N is the point of intersection of the segments XD and Y C. Prove
that the line ZN is parallel to the bases of the trapezoid.

(A. Akopyan &A. Myakishev)

A convex hexagon
is given. Let s be the sum of the lengths of the three segments connecting the
midpoints of its opposite sides. Prove that there is a point in the hexagon
such that the sum of its distances to the lines containing the sides of the
hexagon does not exceed s.

(N. Sedrakyan)

In a cyclic
quadrilateral ABCD the sides AB and AD are equal, CD > AB + BC. Prove that ÐABC > 120

^{o}.
(from olympiad materials)

M is the
midpoint of base BC in a trapezoid ABCD. A point P is chosen on the base AD.
The line PM meets the line CD at a point Q such that C lies between Q and D. The
perpendicular to the bases drawn through P meets the line BQ at K. Prove that ÐQBC = ÐKDA.

(S. Berlov)

On the side AB
of a cyclic quadrilateral ABCD there is a point X such that diagonal BD bisects CX and diagonal AC bisects DX. What
is the minimum possible value of AB / CD?

(S. Berlov)

A triangle ABC is
given. Let B1 be the reflection of B across the line AC, C1 the reflection of C
across the line AB, and O1 the reflection of the circumcentre of ABC across the
line BC. Prove that the circumcentre of AB1C1 lies on the line AO1.

(A. Akopyan)

Let ABC be an
acute triangle, H its orthocentre, D a point on the side [BC], and P a point such
that ADPH is a parallelogram. Show that ÐBPC
> ÐBAC.

(S. Berlov)

Let ABC be a
triangle, I its incenter, ω its incircle, P
a point such that PI $\perp$ BC and PA // BC, Qä(AB), R ä (AC) such that
QR // BC and QR tangent to ω. Show that ÐQPB = ÐCPR.

(V. Smykalov)

In acute
triangle ABC, let H denote its orthocenter and let D be a point on side BC. Let
P be the point so that ADPH is a parallelogram. Prove that ÐDCP < ÐBHP.

(S. Berlov)

In a cyclic
quadrilateral ABCD, the extensions of sides AB and CD meet at point P, and the
extensions of sides AD and BC meet at point Q. Prove that the distance between
the orthocenters of triangles APD and AQB is equal to the distance between the
orthocenters of triangles CQD and BPC.

(L. Emelyanov)

An excircle of
triangle ABC touches the side AB at P and the extensions of sides AC and BC at
Q and R, respectively. Prove that if the midpoint of PQ lies on the
circumcircle of ABC, then the midpoint of PR also lies on that circumcircle.

(S. Berlov)

A circle passing
through the vertices A and B of a cyclic quadrilateral ABCD intersects diagonals
AC and BD at E and F, respectively. The lines AF and BC meet at a point P, and
the lines BE and AD meet at a point Q. Prove that PQ is parallel to CD.

(A. Akopyan)

Circles ω

_{1}and ω_{2}intersect at points A and B, and M is the midpoint of AB. Points S_{1}and S_{2}lie on the line AB (but not between A and B). The tangents drawn from S_{1}to ω_{1 }touch it at X_{1}and Y_{1}, and the tangents drawn from S_{2}to ω_{2}touch it at X_{2}and Y_{2}. Prove that if the line X_{1}X2 passes through M, then line Y_{1}Y_{2}also passes through M.
(A. Akopyan)

In a convex
hexagon AC΄BA΄CB΄,
every two opposite sides are equal. Let A

_{1}denote the point of intersection of BC with the perpendicular bisector of AA΄. Define B_{1}and C_{1}similarly. Prove that A_{1}, B_{1}, and C_{1}are collinear.
(A. Akopyan)

A rectangle ABCD
is given. Segment DK is equal to BD and lies on the half-line DC. M is the
midpoint of BK. Prove that AM is the angle bisector of ÐBAC.

(S. Berlov)

A circle is
contained in a quadrilateral with successive sides of lengths 3, 6, 5 and 8.
Prove that the length of its radius is less than 3.

(K. Kokhas)

Point P is taken
in the interior of the triangle ABC, so that ÐPAB
= ÐPCB =1/4 (ÐA
+ ÐC). Let L be the
foot of the angle bisector of ÐB. The line PL
meets the circumcircle of APC at point Q. Prove that QB is the angle bisector
of ÐAQC.

(S. Berlov)

Quadrilateral
ABCD is both cyclic and circumscribed. Its incircle touches its sides AB and CD
at points X and Y , respectively. The perpendiculars to AB and CD drawn at A
and D, respectively, meet at point U, those drawn at X and Y meet at point V ,
and finally, those drawn at B and C meet at point W. Prove that points U, V and
W are collinear.

(A. Golovanov)

ABCDEF is a
convex hexagon, such that in it AC // DF, BD // AE and CE // BF. Prove that

AB

^{2}+ CD^{2}+ EF^{2}= BC^{2}+ DE^{2}+ AF^{2}.
(N.
Sedrakyan)

The point A1 on
the perimeter of a convex quadrilateral ABCD is such that the line AA

_{1}divides the quadrilateral into two parts of equal area. The points B_{1}, C_{1}, D_{1}are defined similarly. Prove that the area of the quadrilateral A_{1}B_{1}C_{1}D_{1}is greater than a quarter of the area of ABCD.
(L.
Emelyanov)

Points X and Y
inside the rhombus ABCD are such that Y is inside the convex quadrilateral BXDC
and 2ÐXBY = 2ÐXDY = ÐABC. Prove that the lines AX and
CY are parallel.

(S. Berlov)

Points A

_{1}, A_{2}, A_{3}, A_{4}are the vertices of a regular tetrahedron of edge length 1. The points B_{1}and B_{2}lie inside the figure bounded by the plane A_{1}A_{2}A_{3 }and the spheres of radius 1 and centres A_{1}, A_{2}, A_{3}. Prove that B_{1}B_{2}< max{B_{1}A_{1},B_{1}A_{2},B_{1}A_{3},B_{1}A_{4}}.
(A. Kupavsky)

The points K and
L on the side BC of a triangle △ABC are such that ÐBAK = ÐCAL = 90

^{o}. rove that the midpoint of the altitude drawn from A, the midpoint of KL and the circumcentre of △ABC are collinear.
(A.
Akopyan, S. Boev, P. Kozhevnikov)

Radius of the
circle ω

_{A}with centre at vertex A of a triangle △ABC is equal to the radius of the excircle tangent to BC. The circles ω_{B}and ω_{C}are defined similarly. Prove that if two of these circles are tangent then every two of them are tangent to each other.
(L.
Emelyanov)

A parallelogram
ABCD is given. The excircle of triangle △ABC touches the sides AB at L and the
extension of BC at K. The line DK meets the diagonal AC at point X, the line BX
meets the median CC1 of triangle △ABC at Y . Prove that the line Y L, median
BB1 of triangle △ABC and its bisector CC′ have a common point.

(A.
Golovanov)

D is midpoint of
AC for △ABC. Bisectors of ÐACB, ÐABD are perpendicular. Find max
value for ÐBAC.

(S. Berlov)

CL is bisector
of ÐC of ABC and
intersect circumcircle at K. I - incenter of ABC. IL = LK. Prove, that CI = IK.

(D.
Shiryaev)

In △ABC
points M,O are midpoint of AB and circumcenter. It is true, that OM = R - r.
Bisector of external ÐA intersect BC
at D and bisector of external ÐC intersect AB
at E. Find possible values of ÐCED

(D.
Shiryaev)

The point D on
the altitude AA1 of an acute triangle ABC is such that \BDC = 90◦; H is the
orthocentre of ABC. A circle with diameter AH is constructed. Prove that the
tangent drawn from B to this circle is equal to BD.

(L.
Emelyanov)

The numbers a,
b, c, d satisfy 0 < a ≤ b ≤ d ≤ c and a + c = b + d. Prove that for every
internal point P of a segment with length a this segment is a side of a
circumscribed quadrilateral with consecutive sides a, b, c, d, such that its
incircle contains P.

(L.
Emelyanov)

Altitudes AA

_{1}, BB_{1}, CC_{1}of an acute triangle ABC meet at H. A_{0}, B_{0}, C_{0}are the midpoints of BC, CA, AB respectively. Points A_{2}, B_{2}, C_{2}on the segments AH, BH, HC_{1}respectively are such that ÐA_{0}B_{2}A_{2}= ÐB_{0}C_{2}B_{2}= ÐC_{0}A_{2}C_{2}= 90^{o}. Prove that the lines AC_{2}, BA_{2}, CB_{2}are concurrent.
(A.Pastor)

The diagonals AC
and BD of a cyclic quadrilateral ABCD are perpendicular and meet at point P.
The point Q on the segment PC is such that AP=QC. Prove that the perimeter of
the triangle BQD is at least 2AC.

(A.
Kuznetsov)

BL is the
bisector of an isosceles triangle ABC. A point D is chosen
on the base BC and a point E is chosen on the lateral
side AB so that AE=1/2 AL=CD . Prove that LE=LD.

(A.
Kuznetsov)

A
point E lies on the extension of the side AD of the rectangle ABCD over D. The
ray EC meets the circumcircle ω
of ABE at the point F ≠ E. The rays DC and AF meet at P. H is the foot of the
perpendicular drawn from C to the line

*l*going through E and parallel to AF. Prove that the line PH is tangent to ω.
(A. Kuznetsov)

Two
points A and B are given in the plane. A point X is called their

*preposterous midpoint*if there is a Cartesian coordinate system in the plane such that the coordinates of A and B in this system are non-negative, the abscissa of X is the geometric mean of the abscissae of A and B, and the ordinate of X is the geometric mean of the ordinates of A and B. Find the locus of all the*preposterous midpoints*of A and B.
(K. Tyschuk)

A circle touches the side $AB$ of the triangle $ABC$ at $A$, touches the side $BC$ at $P$ and intersects the side $AC$ at $Q$. The line symmetrical to $PQ$ with respect to $AC$ meets the line $AP$ at $X$. Prove that $PC=CX$.

Additional information for Junior League:

Show that this point lies belongs to $\omega$, the circumcircle of $OAC$

(S. Berlov)

Quadrilateral $ABCD$ with perpendicular diagonals is inscribed in a circle with centre $O$. The tangents to this circle at $A$ and $C$ together with line $BD$ form the triangle $\Delta$. Prove that the circumcircles of $BOD$ and $\Delta$ are tangent.Additional information for Junior League:

Show that this point lies belongs to $\omega$, the circumcircle of $OAC$

(A. Kuznetsov)

A point $P$ on the side $AB$ of a triangle $ABC$ and points $S$ and $T$ on the sides $AC$ and $BC$ are such that $AP=AS$ and $BP=BT$. The circumcircle of $PST$ meets the sides $AB$ and $BC$ again at $Q$ and $R$, respectively. The lines $PS$ and $QR$ meet at $L$. Prove that the line $CL$ bisects the segment $PQ$.
(A. Antropov)

A triangle $ABC$ with $AB < AC$ is inscribed in a circle $\omega$. Circles $\gamma_1$ and $\gamma_2$ touch the lines $AB$ and $AC$, and their centres lie on the circumference of $\omega$. Prove that $C$ lies on a common external tangent to $\gamma_1$ and $\gamma_2$.

(A. Kuznetsov)

A circle $\omega$ touches the sides $A$B and $BC$ of a triangle $ABC$ and intersects its side $AC$ at $K$. It is known that the tangent to $\omega$ at $K$ is symmetrical to the line $AC$ with respect to the line $BK$. What can be the difference $AK -CK$ if $AB = 9$ and $BC = 11$?

(S. Berlov)

A trapezoid $ABCD$ with $BC // AD$ is given. The points $B'$ and $C'$ are symmetrical to $B$ and $C$ with respect to $CD$ and $AB$, respectively. Prove that the midpoint of the segment joining the circumcentres of $ABC'$ and $B'CD$ is equidistant from $A$ and $D$.

(A. Kuznetsov)

In $\triangle ABC$ $\angle B$ is obtuse and $AB \ne BC$. Let $O$ is the circumcenter and $\omega$ is the circumcircle of this triangle. $N$ is the midpoint of arc $ABC$. The circumcircle of $\triangle BON$ intersects $AC$ on points $X$ and $Y$. Let $BX \cap \omega = P \ne B$ and $BY \cap \omega = Q \ne B$. Prove that $P, Q$ and reflection of $N$ with respect to line $AC$ are collinear.

(A. Kuznetsov)

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