geometry problems from Baltic Way Mathematical Team Contest

with aops links in the names

with aops links in the names

[the missing solution file in English is available only in Finnish below]

Baltic Way all 2009 Finish in pdf with solutions

Baltic Way all problems 2002-06 EN in pdf with solutions for A4 print

Baltic Way all problems 2002-06 EN in pdf with solutions for A5 print

from latex

Consider two points $A(x_1, y_1)$ and $B(x_2, y_2)$ on the graph of the function $y = \frac{1}{x}$ such that $0 < x_1 < x_2$ and $AB = 2 \cdot OA$, where $O = (0, 0)$. Let $C$ be the midpoint of the segment $AB$. Prove that the angle between the $x$-axis and the ray $OA$ is equal to three times the angle between the $x$-axis and the ray $OC$.

Baltic Way
1997.15

Let $n$ be a positive integer. Consider $n$ points in the plane such that no three of them are collinear and no two of the distances between them are equal. One by one, we connect each point to the two points nearest to it by line segments (if there are already other line segments drawn to this point, we do not erase these). Prove that there is no point from which line segments will be drawn to more than $11$ points.

A unit square is cut into $m$ quadrilaterals $Q_1,\ldots ,Q_m$. For each $i=1,\ldots ,m$ let $S_i$ be the sum of the squares of the four sides of $Q_i$. Prove that $S_1+\ldots +S_m\ge 4$

The altitudes $BB_1$ and $CC_1$ of an acute triangle $ABC$ intersect in point $H$. Let $B_2$ and $C_2$ be points on the segments $BH$ and $CH$, respectively, such that $BB_2=B_1H$ and $CC_2=C_1H$. The circumcircle of the triangle $B_2HC_2$ intersects the circumcircle of the triangle $ABC$ in points $D$ and $E$. Prove that the triangle $DEH$ is right-angled.

Let $ABC$ be a triangle with $AB = AC$. Let $M$ be the midpoint of $BC$. Let the circles with diameters $AC$ and $BM$ intersect at points $M$ and $P$. Let $MP$ intersect $AB$ at $Q$. Let $R$ be a point on $AP$ such that $QR \parallel BP$. Prove that $CP$ bisects $\angle RCB$.

Let $ABC$ be a triangle and $H$ its orthocenter. Let $D$ be a point lying on the segment $AC$ and let $E$ be the point on the line $BC$ such that $BC\perp DE$. Prove that $EH\perp BD$ if and only if $BD$ bisects $AE$.

Let $ABCDEF$ be a convex hexagon in which $AB=AF$, $BC=CD$, $DE=EF$ and $\angle ABC = \angle EFA = 90^{\circ}$. Prove that $AD\perp CE$.

Let $ABC$ be a triangle with $\angle ABC = 90^{\circ}$, and let $H$ be the foot of the altitude from $B$. The points $M$ and $N$ are the midpoints of the segments $AH$ and $CH$, respectively. Let $P$ and $Q$ be the second points of intersection of the circumcircle of the triangle $ABC$ with the lines $BM$ and $BN$, respectively. The segments $AQ$ and $CP$ intersect at the point $R$. Prove that the line $BR$ passes through the midpoint of the segment $MN$.

Let $n \geq 4$, and consider a (not necessarily convex) polygon $P_1P_2 ... P_n$ in the plane. Suppose that, for each $P_k$, there is a unique vertex $Q_k\ne P_k$ among $P_1,..., P_n$ that lies closest to it. The polygon is then said to be hostile if $Q_k\ne P_{k\pm 1}$ for all $k$ (where $P_0 = P_n$, $P_{n+1} = P_1$).

(a) Prove that no hostile polygon is convex.

(b) Find all $n \geq 4$ for which there exists a hostile $n$-gon.

source: www.math.olympiaadid.ut.ee/

Baltic Way all 2009 Finish in pdf with solutions

Baltic Way all problems 2002-06 EN in pdf with solutions for A4 print

Baltic Way all problems 2002-06 EN in pdf with solutions for A5 print

from latex

1990 - 2019

Let ABCD be a
quadrangle, AD= BC, ÐA+ÐB = 120

^{o}and let P be a point exterior to the quadrangle such that P and A lie at opposite sides of the line DC and the triangle DPC is equilateral. Prove that the triangle APB is also equilateral.
The midpoint of
each side of a convex pentagon is connected by a segment with the intersection
point of the medians of the triangle formed by the remaining three vertices of
the pentagon. Prove that all five such segments intersect at one point.

Let P be a point
on the circumcircle of a triangle ABC. It is known that the base points of the
perpendiculars drawn from P onto the lines AB, BC and CA lie on one straight
line (called a Simson line). Prove that the Simson lines of two diametrically
opposite points P

_{1}and P_{2}are perpendicular.
Two equal
triangles are inscribed into an ellipse. Are they necessarily symmetrical with
respect either to the axes or to the centre of the ellipse?

A segment AB of
unit length is marked on the straight line t. The segment is then moved on the
plane so that it remains parallel to t at all times, the traces of the points A
and B do not intersect and finally the segment returns onto t. How far can the
point A now be from its initial position?

Let two circles
C

_{1}and C_{2 }(with radii r_{1}and r_{2}) touch each other externally, and let*l*be their common tangent. A third circle C_{3}(with radius r_{3}< min(r_{1}, r_{2})) is externally tangent to the two given circles and tangent to the line*l*. Prove that $\frac{1}{\sqrt{{{r}_{3}}}}=\frac{1}{\sqrt{{{r}_{1}}}}+\frac{1}{\sqrt{{{r}_{2}}}}$
Let the
coordinate planes have the reflection property. A beam falls onto one of them.
How does the final direction of the beam after reflecting from all three
coordinate planes depend on its initial direction?

Is it possible
to put two tetrahedra of volume 1/2 without intersection into a sphere with
radius 1?

Let’s expand a
little bit three circles, touching each other externally, so that three pairs
of intersection points appear. Denote by A

_{1}, B_{1}, C_{1}the three so obtained “external” points and by A_{2}, B_{2}, C_{2}the corresponding “internal” points. Prove the equality
|A

_{1}B_{2}| · |B_{1}C_{2}| · |C_{1}A_{2}| = |A_{1}C_{2}| · |C_{1}B_{2}| ·|B_{1}A_{2}|
All faces of a convex polyhedron are parallelograms. Can the polyhedron have exactly 1992 faces?

Baltic Way
1992.17
Quadrangle ABCD
is inscribed in a circle with radius 1 in such a way that the diagonal AC is a
diameter of the circle, while the other diagonal BD is as long as AB. The
diagonals intesect at P. It is known that the length of PC is 2 / 5. How long
is the side CD?

Show that in a
non-obtuse triangle the perimenter of the triangle is always greater than two
times the diameter of the circumcircle.

Let C be a
circle in plane. Let C

_{1}and C_{2}be nonintersecting circles touching C internally at points A and B respectively. Let t be a common tangent of C_{1}and C_{2}touching them at points D and E respectively, such that both C_{1}and C_{2}are on the same side of t. Let F be the point of intersection of AD and BE. Show that F lies on C.
Let a ≤b ≤c be
the sides of a right triangle, and let 2p be its perimeter. Show that

p(p - c) - (p -
a)(p - b) = S , where S is the area of the triangle.

Two circles,
both with the same radius r, are placed in the plane without intersecting each
other. A line in the plane intersects the first circle at the points A, B and
the other at points C, D, so that AB =BC= CD = 14 cm. Another line intersects
the circles at E, F, respectively G, H so that EF = FG = GH = 6 cm. Find the
radius r.

Let's consider
three pairwise non-parallel straight constant lines in the plane. Three points
are moving along these lines with different nonzero velocities, one on each
line (we consider the movement to have taken place for infinite time and
continue infinitely in the future). Is it possible to determine these straight
lines, the velocities of each moving point and their positions at some "zero"
moment in such a way that the points never were, are or will be collinear?

In the triangle
ABC, AB = 15, BC = 12, AC = 13. Let the median AM and bisector BK intersect at
point O, where MäBC, KäAC. Let OL ⊥ AB, L äAB. Prove that ÐOLK = ÐOLM.

A convex
quadrangle ABCD is inscribed in a circle with center O. The angles AOB, BOC,
COD and DOA, taken in some order, are of the same size as the angles of the
quadrangle ABCD. Prove that ABCD is a square.

Let Q be a unit
cube. We say that a tetrahedron is good if all its edges are equal and all of
its vertices lie on the boundary of Q. Find all possible volumes of good
tetrahedra.

Let NS and EW be
two perpendicular diameters of a circle C. A line

*l*touches C at point S. Let A and B be two points on C, symmetric with respect to the diameter EW. Denote the intersection points of*l*with the lines NA and NB by A΄ and B΄, respectively. Show that SA΄ · SB΄ = SN^{2}.
The inscribed
circle of the triangle A

_{1}A_{2}A_{3}touches the sides A_{2}A_{3}, A_{3}A_{1}, A_{1}A_{2}at points S_{1}, S_{2}, S_{3}, respectively. Let O_{1}, O_{2}, O_{3}be the centres of the inscribed circles of triangles A_{1}S_{2}S_{3}, A_{2}S_{3}S_{1}, A_{3}S_{1}S_{2}, respectively. Prove that the straight lines O_{1}S_{1}, O_{2}S_{2}, O3S_{3}intersect at one point.
Find the smallest number $a$ such that a square of side $a$ can contain five disks of radius $1$, so that no two of the disks have a common interior point.

Baltic Way 1994.14

Let $\alpha,\beta,\gamma$ be the angles of a triangle opposite to its sides with lengths $a,b,c$ respectively. Prove the inequality

\[a\left(\frac{1}{\beta}+\frac{1}{\gamma}\right)+b\left(\frac{1}{\gamma}+\frac{1}{\alpha}\right)+c\left(\frac{1}{\alpha}+\frac{1}{\beta}\right)\ge2\left(\frac{a}{\alpha}+\frac{b}{\beta}+\frac{c}{\gamma}\right)\]

Let $\alpha,\beta,\gamma$ be the angles of a triangle opposite to its sides with lengths $a,b,c$ respectively. Prove the inequality

\[a\left(\frac{1}{\beta}+\frac{1}{\gamma}\right)+b\left(\frac{1}{\gamma}+\frac{1}{\alpha}\right)+c\left(\frac{1}{\alpha}+\frac{1}{\beta}\right)\ge2\left(\frac{a}{\alpha}+\frac{b}{\beta}+\frac{c}{\gamma}\right)\]

Does there exist a triangle such that the lengths of all its sides and altitudes are integers and its perimeter is equal to $1995$?

In the triangle
ABC, let

*l*be the bisector of the external angle at C. The line through the midpoint O of AB parallel to*l*meets AC at E. Determine CE, if AC = 7 and CB= 4.
Prove that there exists a number $\alpha$ such that for any triangle $ABC$ the inequality

\[ \max(h_A,h_B,h_C)\le \alpha\cdot\min(m_A,m_B,m_C)\]

where $h_A,h_B,h_C$ denote the lengths of the altitudes and $m_A,m_B,m_C$ denote the lengths of the medians. Find the smallest possible value of $\alpha$.

\[ \max(h_A,h_B,h_C)\le \alpha\cdot\min(m_A,m_B,m_C)\]

where $h_A,h_B,h_C$ denote the lengths of the altitudes and $m_A,m_B,m_C$ denote the lengths of the medians. Find the smallest possible value of $\alpha$.

Let M be the
midpoint of the side AC of a triangle ABC and let H be the footpoint of the
altitude from B. Let P and Q be orthogonal projections of A and C on the
bisector of the angle B. Prove that the four points H, P, M and Q lie on the
same circle.

The following
construction is used for training astronauts:

A circle C

_{2}of radius 2R rolls along the inside of another, fixed circle C_{1}of radius nR, where n is an integer greater than 2. The astronaut is fastened to a third circle C_{3}of radius R which rolls along the inside of circle C_{2}in such a way that the touching point of the circles C_{2}and C_{3}remains at maximum distance from the touching point of the circles C_{1}and C_{2}at all times.
How many
revolutions (relative to the ground) does the astronaut perform together with
the circle C

_{3}while the circle C_{2}completes one full lap around the inside of circle C_{1}?
All the vertices of a convex pentagon are on lattice points. Prove that the area of the pentagon is at least $\frac{5}{2}$.

Bogdan Enescu

Let
α be the angle between two lines
containing the diagonals of a regular 1996-gon, and let β≠ 0 be another such angle. Prove that α / β is a rational
number.

In the figure
below, you see three half-circles. The circle C is tangent to two of the
half-circles and to the line PQ perpendicular to the diameter AB. The area of
the shaded region is 39π, and the area of the circle C is
9π.
Find the length of the diameter AB.

Let ABCD be a
unit square and let P and Q be points in the plane such that Q is the
circumcentre of triangle BPC and D be the circumcentre of triangle PQA. Find
all possible values of the length of segment PQ.

ABCD is a
trapezium (AD // BC). P is the point on the line AB such that 6 CPD is maximal.
Q is the point on the line CD such that ÐBQA
is maximal. Given that P lies on the segment AB, prove that Ð CPD = Ð BQA.

Let ABCD be a
cyclic convex quadrilateral and let r

_{a}, r_{b}, r_{c}, r_{d}be the radii of the circles inscribed i9n the triangles BCD, ACD, ABD, ABC, respectively. Prove that r_{a}+ r_{c}= r_{b}+ r_{d}.
On
two parallel lines, the distinct points A

_{1}, A_{2}, A_{3},… respectively B_{1}, B_{2}, B_{3}, … are marked in such a way that |A_{i}A_{i+1}| = 1 and |B_{i}B_{i+1}| = 2 for i = 1, 2, …(see figure). Provided that ÐA_{1}A_{2}B_{1}= α, find the infinite sum ÐA_{1}B_{1}A_{2}+ ÐA_{2}B_{2}A_{3}+ Ð A_{3}B_{3}A_{4}+ … .
Two circles C

_{1}and C_{2 }intersect in P and Q. A line through P intersects C_{1}and C_{2}again in A and B, respectively, and X is the midpoint of AB. The line through Q and X intersects C_{1}and C_{2}again in Y and Z, respectively. Prove that X is the midpoint of Y Z.
Five distinct
points A, B, C, D and E lie on a line with AB = BC = CD = DE. The point F lies
outside the line. Let G be the circumcentre of the triangle ADF and H the
circumcentre of the triangle BEF. Show that the lines GH and FC are
perpendicular.

Baltic Way 1997.14

In the triangle $ABC$, $AC^2$ is the arithmetic mean of $BC^2$ and $AB^2$. Show that $\cot^2B\ge \cot A\cdot\cot C$.

In the triangle $ABC$, $AC^2$ is the arithmetic mean of $BC^2$ and $AB^2$. Show that $\cot^2B\ge \cot A\cdot\cot C$.

In the acute
triangle ABC, the bisectors of ÐA, ÐB and Ð C intersect the circumcircle
again in A

_{1}, B_{1}and C_{1}, respectively. Let M be the point of intersection of AB and B_{1}C_{1}, and let N be the point of intersection of BC and A1B1. Prove that MN passes through the incentre of triangle ABC.
If $a,b,c$ be the lengths of the sides of a triangle. Let $R$ denote its circumradius. Prove that

\[ R\ge \frac{a^2+b^2}{2\sqrt{2a^2+2b^2-c^2}}\]

When does equality hold?

\[ R\ge \frac{a^2+b^2}{2\sqrt{2a^2+2b^2-c^2}}\]

When does equality hold?

In
a triangle ABC, ÐBAC = 90

^{o}. Point D lies on the side BC and satisfies ÐBDA = 2ÐBAD.
Prove that $\frac{1}{AD}=\frac{1}{2}\left(
\frac{1}{BD}+\frac{1}{CD} \right)$.

In a convex
pentagon ABCDE, the sides AE and BC are parallel and ÐADE = ÐBDC. The diagonals AC and BE
intersect in P. Prove that ÐEAD =ÐBDP and ÐCBD = ÐADP.

Given triangle
ABC with AB< AC. The line passing through B and parallel to AC meets the external
angle bisector of ÐBAC at D. The
line passing through C and parallel to AB meets this bisector at E. Point F
lies on the side AC and satisfies the equality FC = AB. Prove that DF = FE.

Given acute
triangle ABC. Point D is the foot of the perpendicular from A to BC. Point E
lies on the segment AD and satisfies the equation $\frac{AE }{ED} = \frac{CD }{DB}$.
Point F is the foot of the perpendicular from D to BE. Prove that ÐAFC = 90

^{o}.
Prove that for
any four points in the plane, no three of which are collinear, there exists a
circle such that such that three of the four points are on the circumference
and the fourth point is either on the circumference or inside the circle.

In a triangle
ABC it is given that 2AB = AC + BC. Prove that the incentre of ABC, the circumcenter
of ABC, and the midpoints of AC and BC are concyclic.

The bisectors of
the angles A and B of the triangle ABC meet the sides BC and CA at the points D
and E, respectively. Assuming that AE + BD = AB, determine the angle C.

Let ABC be an
isosceles triangle with AB = AC. Points D and E lie on the sides AB and AC,

respectively.
The line passing through B and parallel to AC meets the line DE at F. The line

passing through
C and parallel to AB meets the line DE at G. Prove that

$\frac{ [DBCG]
}{ [FBCE] } = \frac{AD}{AE}$ where [PQRS] denotes the area of the
quadrilateral PQRS.

Let ABC be a
triangle with ÐC = 60

^{o}and AC < BC. The point D lies on the side BC and
satisfies BD =
AC. The side AC is extended to the point E where AC = CE. Prove that AB = DE.

Let K be a point
inside the triangle ABC. Let M and N be points such that M and K are on
opposite sides of the line AB, and N and K are on opposite sides of the line
BC. Assume that ÐMAB = ÐMBA = ÐNBC = ÐNCB = ÐKAC = ÐKCA. Show that MBNK is a parallelogram.

Given an
isosceles triangle ABC with ÐA = 90

^{o}. Let M be the midpoint of AB. The line passing through A and perpendicular to CM intersects the side BC at P. Prove that ÐAMC =ÐBMP.
Given a triangle
ABC with ÐA = 90

^{o}and AB ≠ AC. The points D, E, F lie on the sides BC, CA, AB, respectively, in such a way that AFDE is a square. Prove that the line BC, the line FE and the line tangent at the point A to the circumcircle of the triangle ABC intersect in one point.
Given a triangle
ABC with ÐA = 120

^{o}. The points K and L lie on the sides AB and AC, respectively. Let BKP and CLQ be equilateral triangles constructed outside the triangle ABC. Prove that $PQ \ge \frac{\sqrt{3}}{2} (AB + AC)$ .
Let ABC be a
triangle such that $\frac{BC}{AB-\text{ }BC}=\frac{AB+BC}{AC}$. Determine the
ratio ÐA : Ð C.

The points A,B,C,D,E
lie on the circle c in this order and satisfy AB // EC and AC // ED. The line
tangent to the circle c at E meets the line AB at P. The lines BD and EC meet
at Q. Prove that AC = PQ.

Given a
parallelogram ABCD. A circle passing through A meets the line segments AB, AC and
AD at inner points M, K, N, respectively. Prove that AB ·AM + AD·AN = AK · AC.

Let ABCD be a
convex quadrilateral, and let N be the midpoint of BC. Suppose further that ÐAND = 135

^{o}. Prove that $AB + CD + \frac{1}{\sqrt{2}} BC \ge AD$.
Given a rhombus
ABCD, find the locus of the points P lying inside the rhombus and satisfying

ÐAPD + ÐBPC = 180

^{o}.
In a triangle
ABC, the bisector of ÐBAC meets the
side BC at the point D. Knowing that

BD· CD = AD

^{2}and ÐADB = 45^{o}, determine the angles of triangle ABC.Let $n$ be a positive integer. Consider $n$ points in the plane such that no three of them are collinear and no two of the distances between them are equal. One by one, we connect each point to the two points nearest to it by line segments (if there are already other line segments drawn to this point, we do not erase these). Prove that there is no point from which line segments will be drawn to more than $11$ points.

A set S of four distinct points is given in
the plane. It is known that for any point XäS the remaining points
can be denoted by Y , Z and W so that XY = XZ + XW. Prove that all the four
points lie on a line.

Let ABC be an
acute triangle with ÐBAC > ÐBCA, and let D be a point on side
AC such that AB = BD. Furthermore, let F be a point on the circumcircle of
triangle ABC such that line FD is perpendicular to side BC and points F , B lie
on different sides of line AC. Prove that line FB is perpendicular to side AC.

Let L, M and N
be points on sides AC, AB and BC of triangle ABC, respectively, such that BL is
the bisector of angle ABC and segments AN , BL and CM have a common point.
Prove that if ÐALB = ÐMNB then ÐLNM = 90

^{o}.
A spider and a fly
are sitting on a cube. The fly wants to maximize the shortest path to the
spider along the surface of the cube. Is it necessarily best for the fly to be
at the point opposite to the spider? ("Opposite" means "symmetric
with respect to the center of the cube".)

Is it possible to select $1000$ points in the plane so that $6000$ pairwise distances between them are equal?

Let ABCD be a
square. Let M be an inner point on side BC and N be an inner point on side CD
with ÐMAN = 45

^{ o}. Prove that the circumcenter of AMN lies on AC.
Let ABCD be a
rectangle and BC = 2 AB. Let E be the midpoint of BC and P an arbitrary inner
point of AD. Let F and G be the feet of perpendiculars drawn correspondingly
from A to BP and from D to CP . Prove that the points E, F , P , G are
concyclic.

Let ABC be an
arbitrary triangle and AMB, BNC, CKA regular triangles outward of ABC. Through
the midpoint of MN a perpendicular to AC is constructed, similarly through
midpoints of NK respectively. KM perpendiculars to AB resp. BC are constructed.
Prove that these 3 perpendiculars intersect at the same point.

Let P be the
intersection point of the diagonals AC and BD in a cyclic quadrilateral. A
circle through P touches the side CD in the midpoint M of this side and
intersects the segments BD and AC in the points Q and R respectively. Let S be
a point on the segment BD such that BS = DQ. The parallel to AB through S
intersects AC at T . Prove that AT = RC.

A circle is divided into $13$ segments, numbered consecutively from $1$ to $13$. Five fleas called $A,B,C,D$ and $E$ are sitting in the segments $1,2,3,4$ and $5$. A flea is allowed to jump to an empty segment five positions away in either direction around the circle. Only one flea jumps at the same time, and two fleas cannot be in the same segment. After some jumps, the fleas are back in the segments $1,2,3,4,5$, but possibly in some other order than they started. Which orders are possible ?

Through a point
P exterior to a given circle pass a secant and a tangent to the circle. The
secant intersects the circle at A and B, and the tangent touches the circle at
C on the same side of the diameter through P as A and B. The projection of C on
the diameter is Q. Prove that QC bisects ÐAQB.

Consider a
rectangle with side lengths 3 and 4, and pick an arbitrary inner point on each
side. Let x, y, z and u denote the side lengths of the quadrilateral spanned by
these points. Prove that

25 ≤ x

^{2}+ y^{2}+ z^{2}+ u^{2 }≤ 50.
A ray emanating
from the vertex A of the triangle ABC intersects the side BC at X and the
circumcircle of ABC at Y . Prove that $\frac{1}{AX}+\frac{1}{XY}\ge
\frac{4}{BC}$ .

D is the midpoint
of the side BC of the given triangle ABC. M is a point on the side BC such that
ÐBAM = ÐDAC. L is the second intersection
point of the circumcircle of the triangle CAM with the side AB. K is the second
intersection point of the circumcircle of the triangle BAM with the side AC.
Prove that KL // BC.

Three circular
arcs w

_{1},w_{2},w_{3}with common endpoints A and B are on the same side of the line AB, w_{2}lies between w_{1}and w_{3}. Two rays emanating from B intersect these arcs at M_{1},M_{2},M_{3}and K_{1},K_{2},K_{3}, respectively. Prove that $\frac{M_1M_2 }{ M_2M_3} = \frac{K_1K_2}{ K_2K_3}$.
.

Let the points D
and E lie on the sides BC and AC, respectively, of the triangle ABC, satisfying
BD = AE. The line joining the circumcentres of the triangles ADC and BEC meets the
lines AC and BC at K and L, respectively. Prove that KC = LC.

Let ABCD be a
convex quadrilateral such that BC = AD. Let M and N be the midpoints

of AB and CD,
respectively. The lines AD and BC meet the line MN at P and Q, respectively.

Prove that CQ =
DP.

What the smallest number of circles of radius $\sqrt{2}$ that are needed to cover a rectangle

$(a)$ of size $6\times 3$?

$(b)$ of size $5\times 3$?

$(a)$ of size $6\times 3$?

$(b)$ of size $5\times 3$?

Let the medians
of the triangle ABC meet at M. Let D and E be different points on the

line BC such
that DC = CE = AB, and let P and Q be points on the segments BD and BE,

respectively,
such that 2BP = PD and 2BQ = QE. Determine ÐPMQ.

Let the lines e
and f be perpendicular and intersect each other at O. Let A and B lie on e and C
and D lie on f , such that all the five points A, B, C, D and O are distinct.
Let the lines b and d pass through B and D respectively, perpendicularly to AC,
let the lines a and c pass through A and C respectively, perpendicularly to BD.
Let a and b intersect at X and c and d intersect at Y. Prove that XY passes
through O.

The altitudes of
a triangle are 12, 15, and 20. What is the area of the triangle?

Let ABC be a
triangle, let B

_{1}be the midpoint of the side AB and C_{1}the midpoint of the side AC. Let P be the point of intersection, other than A, of the circumscribed circles around the triangles ABC_{1}and AB_{1}C. Let P1 be the point of intersection, other than A, of the line AP with the circumscribed circle around the triangle AB_{1}C_{1}. Prove that 2AP = 3AP_{1}.
In a triangle
ABC, points D, E lie on sides AB, AC respectively. The lines BE and CD intersect
at F. Prove that if BC

^{2}= BD · BA + CE · CA, then the points A, D, F, E lie on a circle.
There are $2006$ points marked on the surface of a sphere. Prove that the surface can be cut into $2006$ congruent pieces so that each piece contains exactly one of these points inside it.

Let the medians
of the triangle ABC intersect at point M. A line t through M intersects the
circumcircle of ABC at X and Y so that A and C lie on the same side of t. Prove
that

BX· BY = AX· AY
+ CX· CY .

In triangle ABC let AD, BE and CF be the altitudes. Let the points P, Q, R and S fulfill the following requirements:

(1) P is the circumcentre of triangle ABC.

(2) All the
segments PQ, QR and RS are equal to the circumradius of triangle ABC.

(3) The oriented
segment PQ has the same
direction as the oriented segment AD.
Similarly, QR

has the same
direction as BE, and RS has the same direction as CF.

Prove that S is the incentre of triangle ABC.

Let M be a point on the arc AB of the circumcircle of the
triangle ABC which does not
contain C. Suppose that the
projections of M onto the lines
AB and BC lie on the sides themselves, not on their extensions. Denote
these projections by X and Y , respectively. Let K and N be the midpoints of AC
and XY ,respectively.
Prove that ÐMNK =
90

^{o}.
Let $t_1,t_2,\ldots,t_k$ be different straight lines in space, where $k>1$. Prove that points $P_i$ on $t_i$, $i=1,\ldots,k$, exist such that $P_{i+1}$ is the projection of $P_i$ on $t_{i+1}$ for $i=1,\ldots,k-1$, and $P_1$ is the projection of $P_k$ on $t_1$.

In a convex
quadrilateral ABCD we have ÐADC = 90

^{o}. Let E and F be the projections of B onto the lines AD and AC, respectively. Assume that F lies between A and C, that A lies between D and E, and that the line EF passes through the midpoint of the segment BD. Prove that the quadrilateral ABCD is cyclic.
The incircle of
the triangle ABC touches the
side AC at the point D. Another circle passes through D and touches the rays BC and BA, the latter at the point A. Determine the ratio AD / DC.

Let ABCD be a parallelogram. The circle
with diameter AC intersects the
line BD at points P and Q. The perpendicular to the line AC passing through the point C intersects the lines AB
and AD at points X and Y , respectively. Prove that the points P , Q, X and Y lie on the same circle.

Assume that a, b, c and d are the sides of a quadrilateral
inscribed in a given circle. Prove that the product (ab +cd)(ac + bd)(+ bc) acquires its maximum when the
quadrilateral is a square.

Let AB be a diameter of a circle S, and let L be the tangent at A.
Furthermore, let c be a fixed,
positive real, and consider all pairs of points X and Y lying on
L, on opposite sides of A, such that AX · AY = c . The lines BX
and BY intersect S at points P and Q,
respectively. Show that all the lines PQ
pass through a common point.

In a circle of diameter $ 1$, some chords are drawn. The sum of their lengths is greater than $ 19$. Prove that there is a diameter intersecting at least $ 7$ chords.

Let M be a point on BC and N be a point on AB such
that AM and CN are angle bisectors of the
triangle ABC . Given that $\frac{
\angle BNM}{ \angle MNC} = \frac{ \angle
BMN}{ \angle NMA}$ prove that
the triangle ABC is isosceles.

Let M be the midpoint of the side AC of a triangle ABC, and let K be a point on the ray BA
beyond A. The line KM intersects the side BC at the point L. P is the point on the segment BM such that PM is
the bisector of the angle LPK.
The line ℓ passes through A and is parallel to BM. Prove that the projection of the
point M onto the line ℓ belongs to the line PK.

In a
quadrilateral ABCD we have AB // CD and AB = 2CD. A line ℓ is perpendicular
to CD and contains the point C. The circle with centre D and radius DA intersects the line ℓ
at points P and Q. Prove that AP⊥BQ.

The point H is the orthocentre of a triangle ABC, and the segments AD, BE, CF are its
altitudes. The points I

_{1}, I_{2}, I_{3}are the incentres of the triangles EHF, FHD, DHE, respectively. Prove that the lines AI_{1}, BI_{2}, CI_{3}intersect at a single point.
For which $n\ge 2$ is it possible to find $n$ pairwise non-similar triangles $A_1, A_2,\ldots , A_n$ such that each of them can be divided into $n$ pairwise non-similar triangles, each of them similar to one of $A_1,A_2 ,\ldots ,A_n$?

Let ABCD be a
square and let S be the point of intersection of its diagonals AC and BD. Two
circles k, k΄
go through A, C and B, D, respectively. Furthermore, k and k΄ intersect in exactly two different points P and Q.
Prove that S lies on PQ.

Let ABCD be a
convex quadrilateral with precisely one pair of parallel sides.

a) Show that the
lengths of its sides AB, BC, CD, DA (in this order) do not form an arithmetic
progression.

b) Show that
there is such a quadrilateral for which the lengths of its sides AB, BC, CD, DA
form an arithmetic progression after the order of the lengths is changed.

In an acute
triangle ABC, the segment CD is an altitude and H is the orthocentre. Given
that the circumcentre of the triangle lies on the line containing the bisector
of the angle DHB, determine all possible values of ÐCAB.

Assume that all
angles of a triangle ABC are acute. Let D and E be points on the sides AC and
BC of the triangle such that A,B,D, and E lie on the same circle. Further suppose
the circle through D,E, and C intersects the side AB in two points X and Y.
Show that the midpoint of XY is the foot of the altitude from C to AB.

The points M and
N are chosen on the angle bisector AL of a triangle ABC such that ÐABM = ÐACN = 23

^{o}. X is a point inside the triangle such that BX = CX and ÐBXC = 2ÐBML. Find ÐMXN.
Let AB and CD be
two diameters of the circle C. For an arbitrary point P on C, let R and S be
the feet of the perpendiculars from P to AB and CD, respectively. Show that the
length of RS is independent of the choice of P.

Let P be a point
inside a square ABCD such that PA : PB : PC is 1 : 2 : 3. Determine the angle ÐBPA.

Let E be an
interior point of the convex quadrilateral ABCD. Construct triangles ∆ABF, ∆BCG,
∆CDH and ∆DAI on the outside of the quadrilateral such that the similarities ∆ABF
~∆DCE, ∆BCG ~∆ADE, ∆CDH ~∆BAE and ∆DAI ~ ∆CBE hold. Let P,Q, R and S be the
projections of E on the lines AB, BC, CD and DA, respectively. Prove that if
the quadrilateral PQRS is cyclic, then EF · CD = EG · DA = EH · AB = EI · BC .

The incircle of
a triangle ABC touches the sides BC, CA, AB at D, E, F, respectively. Let G be
a point on the incircle such that FG is a diameter. The lines EG and FD
intersect at H. Prove that CH// AB.

Let ABCD be a
convex quadrilateral such that ÐADB = ÐBDC. Suppose that a point E on the
side AD satisfies the equality AE · ED + BE

^{2}= CD · AE. Show that ÐEBA = ÐDCB.
Let ABC be a triangle with ÐA = 60

^{o}. The point T lies inside the triangle in such a way that ÐATB=ÐBTC=ÐCTA=120^{o }. Let M be the midpoint of BC. Prove that TA + TB + TC = 2AM.
Let P

_{0}, P_{1}, . . . , P_{8}= P_{0}be successive points on a circle and Q be a point inside the polygon P_{0}P_{1}… P_{7}such that ÐP_{i−}_{1}QP_{i}= 45^{o}for i = 1, . . . , 8. Prove that the sum $\sum\limits_{i=1}^{8}{{{P}_{i-1}}{{P}_{i}}^{2}}$ is minimal if and only if Q is the centre of the circle.
Let ABC be an acute triangle, and let H be its orthocentre. Denote by H

_{A}, H_{B}and H_{C}the second intersection of the circumcircle with the altitudes from A, B and C respectively. Prove that the area of △H_{A}H_{B}H_{C}does not exceed the area of △ABC.
Given a triangle
ABC, let its incircle touch the
sides BC, CA, AB at D, E, F, respectively. Let G be
the midpoint of the segment DE.
Prove that ÐEFC =ÐGFD.

The circumcentre
O of a given cyclic
quadrilateral ABCD lies inside
the quadrilateral but not on the diagonal AC. The diagonals of the quadrilateral intersect at I. The circumcircle of the triangle AOI meets the sides AD and AB at points P and
Q, respectively, the
circumcircle of the triangle COI meets
the sides CB and CD at points R and S,
respectively. Prove that PQRS is
a parallelogram.

In an acute
triangle ABC with AC > AB, let D be the projection of A on BC, and let E and
F be the projections of D on AB and AC, respectively. Let G be the intersection
point of the lines AD and EF. Let H be the second intersection point of the line
AD and the circumcircle of triangle ABC. Prove that AG · AH = AD

^{2}.
A trapezoid ABCD
with bases AB and CD is such that the circumcircle of the triangle BCD
intersects the line AD in a point E, distinct from A and D. Prove that the
circumcircle of the triangle ABE is tangent to the line BC.

All faces of a
tetrahedron are right-angled triangles. It is known that three of its edges
have the same length s. Find the volume of the tetrahedron.

Circles β
and β
of the same radius intersect in two points, one of which is P. Denote by A and
B, respectively, the points diametrically opposite to P on each of α
and β.
A third circle of the same radius passes through P and intersects α
and β in the points X and Y , respectively. Show that the
line XY is parallel to the line AB.

Four circles in
a plane have a common center. Their radii form a strictly increasing arithmetic
progression. Prove that there is no square with each vertex lying on a different
circle.

Let Γ
be the circumcircle of an acute triangle ABC. The perpendicular to AB from C
meets AB at D and Γ again at E. The bisector of
angle C meets AB at F and Γ again at G. The
line GD meets Γ
again at H and the line HF meets Γ again at I.
Prove that AI = EB.

Triangle ABC is
given. Let M be the midpoint of the segment AB and T be the midpoint of the arc
BC not containing A of the circumcircle of ABC. The point K inside the triangle
ABC is such that MATK is an isosceles trapezoid with AT // MK. Show that AK =
KC.

Let ABCD be a
square inscribed in a circle ω and let P be a
point on the shorter arc AB of ω. Let CP Ç BD = R and DP Ç AC = S. Show that triangles ARB and DSR have
equal areas.

Let ABCD be a
convex quadrilateral such that the line BD bisects the angle ABC. The
circumcircle of triangle ABC intersects the sides AD and CD in the points P and
Q, respectively. The line through D and parallel to AC intersects the lines BC
and BA at the points R and S, respectively. Prove that the points P, Q, R and S
lie on a common circle.

The sum of the
angles A and C of a convex quadrilateral ABCD is less than 180

^{ο}. Prove that AB · CD + AD · BC < AC (AB + AD).
The diagonals of
the parallelogram ABCD intersect at E. The bisectors of ÐDAE and ÐEBC intersect at F. Assume that
ECFD is a parallelogram. Determine the ratio AB : AD.

A circle passes
through vertex B of the triangle ABC, intersects its sides AB and BC at points
K and L, respectively, and touches the side AC at its midpoint M. The point N
on the arc BL (which does not contain K) is such that ÐLKN = ÐACB. Find ÐBAC given that the triangle CKN
is equilateral.

Let D be the
footpoint of the altitude from B in the triangle ABC, where AB = 1. The
incentre of triangle BCD coincides with the centroid of triangle ABC. Find the
lengths of AC and BC.

In the
non-isosceles triangle ABC the altitude from A meets side BC in D. Let M be the
midpoint of BC and let N be the reflection of M in D. The circumcircle of the
triangle AMN intersects the side AB in P ≠A and the side AC in Q ≠A. Prove that
AN, BQ and CP are concurrent.

In triangle ABC,
the interior and exterior angle bisectors of ÐBAC
intersect the line BC in D and E, respectively. Let F be the second point of
intersection of the line AD with the circumcircle of the triangle ABC. Let O be
the circumcentre of the triangle ABC and let D΄ be the reflection
of D in O. Prove that ÐD΄FE = 90

^{o.}
In triangle ABC, the points D and E are the intersections of the angular bisectors from C and B with the sides AB and
AC, respectively. Points F and G on the extensions of AB
and AC beyond B and C, respectively, satisfy BF
= CG = BC. Prove that FG // DE.

Let ABCD be a convex quadrilateral with AB = AD. Let T be a
point on the diagonal AC such
that ÐABT +
ÐADT =
ÐBCD.
Prove that AT + AC ≥ AB + AD.

Let ABCD be a parallelogram such that ÐBAD = 60

^{o}. Let K and L be the midpoints of BC and CD, respectively. Assuming that ABKL is a cyclic quadrilateral, find ÐABD.
Consider
triangles in the plane where each vertex has integer coordinates. Such a
triangle can be legally transformed by
moving one vertex parallel to the opposite side to a different point with
integer coordinates. Show that if two triangles have the same area, then there exists
a series of legal transformations that transforms one to the other.

Let ABCD be a cyclic quadrilateral with AB and CD not parallel. Let M be
the midpoint of CD. Let P be a point inside ABCD such that PA = PB = CM. Prove
that AB, CD and the perpendicular bisector of MP are concurrent.

Let $H$ and $I$ be the orthocentre and incentre, respectively, of an acute angled triangle $ABC$. The circumcircle of the triangle $BCI$ intersects the segment $AB$ at the point $P$ different from $B$. Let $K$ be the projection of $H$ onto $AI$ and $Q$ the reflection of $P$ in $K$. Show that $B$, $H$ and $Q$ are collinear.

Line $\ell$ touches circle $S_1$ in the point $X$ and circle $S_2$ in the point $Y$. We draw a line $m$ which is parallel to $\ell$ and intersects $S_1$ in a point $P$ and $S_2$ in a point $Q$. Prove that the ratio $XP/YQ$ does not depend on the choice of $m$.

Let $ABC$ be a triangle in which $\angle ABC = 60^\circ$. Let $I$ and $O$ be the incentre and circumcentre of $ABC$, respectively. Let $M$ be the midpoint of the arc $BC$ of the circumcircle of $ABC$, which does not contain the point $A$. Determine $\angle BAC$ given that $MB = OI$.

Let $P$ be a point inside the acute angle $\angle BAC$. Suppose that $\angle ABP = \angle ACP = 90^\circ$. The points $D$ and $E$ are on the segments $BA$ and $CA$, respectively, such that $BD = BP$ and $CP = CE$. The points $F$ and $G$ are on the segments $AC$ and $AB$, respectively, such that $DF$ is perpendicular to $AB$ and $EG$ is perpendicular to $AC$. Show that $PF = PG$.

Let $n \ge 3$ be an integer. What is the largest possible number of interior angles greater than $180^\circ$ in an $n$-gon in the plane, given that the $n$-gon does not intersect itself and all its sides have the same length?

The points $A,B,C,D$ lie, in this order, on a circle $\omega$, where $AD$ is a diameter of $\omega$. Furthermore, $AB=BC=a$ and $CD=c$ for some relatively prime integers $a$ and $c$. Show that if the diameter $d$ of $\omega$ is also an integer, then either $d$ or $2d$ is a perfect square.

The bisector of the angle $A$ of a triangle $ABC$ intersects $BC$ in a point $D$ and intersects the circumcircle of the triangle $ABC$ in a point $E$. Let $K,L,M$ and $N$ be the midpoints of the segments $AB,BD,CD$ and $AC$, respectively. Let $P$ be the circumcenter of the triangle $EKL$, and $Q$ be the circumcenter of the triangle $EMN$. Prove that $\angle PEQ=\angle BAC$.

A quadrilateral $ABCD$ is circumscribed about a circle $\omega$. The intersection point of $\omega$ and the diagonal $AC$, closest to $A$, is $E$. The point $F$ is diametrally opposite to the point $E$ on the circle $\omega$. The tangent to $\omega$ at the point $F$ intersects lines $AB$ and $BC$ in points $A_1$ and $C_1$, and lines $AD$ and $CD$ in points $A_2$ and $C_2$, respectively. Prove that $A_1C_1=A_2C_2$.

Two circles in the plane do not intersect and do not lie inside each other. We choose diameters $A_1B_1$ and $A_2B_2$ of these circles such that the segments $A_1A_2$ and $B_1B_2'$ intersect. Let $A$ and $B$ be the midpoints of the segments $A_1A_2$ and $B_1B_2$, and $C$ be the intersection point of these segments. Prove that the orthocenter of the triangle $ABC$ belongs to a fixed line that does not depend on the choice of diameters.

(a) Prove that no hostile polygon is convex.

(b) Find all $n \geq 4$ for which there exists a hostile $n$-gon.

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