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Baltic Way 1990 - 2022 165p

geometry problems from Baltic Way Mathematical Team Contest
with aops links in the names


collected inside aops here

1990 - 2022
 

Baltic Way 1990.6
Let ABCD be a quadrangle, AD= BC, ÐA+ÐB = 120o 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.

Baltic Way 1990.7
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.

Baltic Way 1990.8
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 P1 and P2 are perpendicular.

Baltic Way 1990.9
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?

Baltic Way 1990.10
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?

Baltic Way 1991.16
Let two circles C1 and C2 (with radii r1 and r2) touch each other externally, and let l be their common tangent. A third circle C3 (with radius r3 < min(r1, r2)) is externally tangent to the two given circles and tangent to the line l. Prove that $\frac{1}{\sqrt{{{r}_{3}}}}=\frac{1}{\sqrt{{{r}_{1}}}}+\frac{1}{\sqrt{{{r}_{2}}}}$

Baltic Way 1991.17
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?

Baltic Way 1991.18
Is it possible to put two tetrahedra of volume 1/2 without intersection into a sphere with radius 1?

Baltic Way 1991.19
Let’s expand a little bit three circles, touching each other externally, so that three pairs of intersection points appear. Denote by A1, B1, C1 the three so obtained “external” points and by A2, B2, C2 the corresponding “internal” points. Prove the equality
                                       |A1B2| · |B1C2| · |C1A2| = |A1C2| · |C1B2| ·|B1A2|

Baltic Way 1991.20
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 $OC$.

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?

Baltic Way 1992.18
Show that in a non-obtuse triangle the perimenter of the triangle is always greater than two times the diameter of the circumcircle.

Baltic Way 1992.19
Let C be a circle in plane. Let C1 and C2 be nonintersecting circles touching C internally at points A and B respectively. Let t be a common tangent of C1 and C2 touching them at points D and E respectively, such that both C1 and C2 are on the same side of t. Let F be the point of intersection of AD and BE. Show that F lies on C.

Baltic Way 1992.20
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.

Baltic Way 1993.16
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.

Baltic Way 1993.17
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?

Baltic Way 1993.18
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.

Baltic Way 1993.19
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.

Baltic Way 1993.20
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.

Baltic Way 1994.11
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΄ = SN2.

Baltic Way 1994.12
The inscribed circle of the triangle A1A2A3 touches the sides A2A3, A3A1, A1A2 at points S1, S2, S3, respectively. Let O1, O2, O3 be the centres of the inscribed circles of triangles A1S2S3, A2S3S1, A3S1S2, respectively. Prove that the straight lines O1S1, O2S2, O3S3 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
\[a\left(\frac{1}{\beta}+\frac{1}{\gamma}\right)+b\left(\frac{1}{\gamma}+\frac{1}{\alpha}\right)+c\left(\frac{1}{\alpha}+\frac{1}{\beta}\right)\ge2\left(\frac{a}{\alpha}+\frac{b}{\beta}+\frac{c}{\gamma}\right)\]

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

Baltic Way 1995.16
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 $\alpha$.

Baltic Way 1995.18
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.

Baltic Way 1995.19
The following construction is used for training astronauts:
A circle C2 of radius 2R rolls along the inside of another, fixed circle C1 of radius nR, where n is an integer greater than 2. The astronaut is fastened to a third circle C3 of radius R which rolls along the inside of circle C2 in such a way that the touching point of the circles C2 and C3 remains at maximum distance from the touching point of the circles C1 and C2 at all times.
How many revolutions (relative to the ground) does the astronaut perform together with the circle C3 while the circle C2 completes one full lap around the inside of circle C1?

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
Baltic Way 1996.1
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.

Baltic Way 1996.2
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.

Baltic Way 1996.3
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.

Baltic Way 1996.4
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.

Baltic Way 1996.5
Let ABCD be a cyclic convex quadrilateral and let ra, rb, rc, rd be the radii of the circles inscribed i9n the triangles BCD, ACD, ABD, ABC, respectively. Prove that ra + rc = rb + rd.

Baltic Way 1997.11
On two parallel lines, the distinct points A1, A2, A3,… respectively B1, B2, B3,  … are marked in such a way that |AiAi+1| = 1 and |BiBi+1| = 2 for i = 1, 2, …(see figure). Provided that ÐA1A2B1= α, find the infinite sum ÐA1B1A2 + ÐA2B2A3 + Ð A3B3A4 + … .
Baltic Way 1997.12
Two circles C1 and C2 intersect in P and Q. A line through P intersects C1 and C2 again in A and B, respectively, and X is the midpoint of AB. The line through Q and X intersects C1 and C2 again in Y and Z, respectively. Prove that X is the midpoint of Y Z.

Baltic Way 1997.13
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 $\cot^2B\ge \cot A\cdot\cot C$.

Baltic Way 1997.15
In the acute triangle ABC, the bisectors of ÐA, ÐB and Ð C intersect the circumcircle again in A1, B1 and C1, respectively. Let M be the point of intersection of AB and B1C1, 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?

Baltic Way 1998.12
In a triangle ABC, ÐBAC = 90o. Point D lies on the side BC and satisfies ÐBDA = 2ÐBAD.
Prove that $\frac{1}{AD}=\frac{1}{2}\left( \frac{1}{BD}+\frac{1}{CD} \right)$.

Baltic Way 1998.13
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.

Baltic Way 1998.14
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.

Baltic Way 1998.15
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 = 90o.

Baltic Way 1999.11
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.

Baltic Way 1999.12
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.

Baltic Way 1999.13
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.

Baltic Way 1999.14
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.

Baltic Way 1999.15
Let ABC be a triangle with ÐC = 60o 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.

Baltic Way 2000.1
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.

Baltic Way 2000.2
Given an isosceles triangle ABC with ÐA = 90o. 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.

Baltic Way 2000.3
Given a triangle ABC with ÐA = 90o 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.

Baltic Way 2000.4
Given a triangle ABC with ÐA = 120o. 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 $PQ \ge \frac{\sqrt{3}}{2} (AB + AC)$ .

Baltic Way 2000.5
Let ABC be a triangle such that $\frac{BC}{AB-\text{ }BC}=\frac{AB+BC}{AC}$. Determine the ratio ÐA : Ð C.

Baltic Way 2001.6
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.

Baltic Way 2001.7
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.

Baltic Way 2001.8
Let ABCD be a convex quadrilateral, and let N be the midpoint of BC. Suppose further that ÐAND = 135o. Prove that $AB + CD + \frac{1}{\sqrt{2}} BC \ge AD$.

Baltic Way 2001.9
Given a rhombus ABCD, find the locus of the points P lying inside the rhombus and satisfying
ÐAPD + ÐBPC = 180o.

Baltic Way 2001.10
In a triangle ABC, the bisector of ÐBAC meets the side BC at the point D. Knowing that
BD· CD = AD2 and ÐADB = 45o, 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.

Baltic Way 2002.12
 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.

Baltic Way 2002.13
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.

Baltic Way 2002.14
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 = 90o .

Baltic Way 2002.15
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?

Baltic Way 2003.12
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.

Baltic Way 2003.13
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.

Baltic Way 2003.14
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.

Baltic Way 2003.15
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 ?

Baltic Way 2004.16
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.

Baltic Way 2004.17
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 ≤ x2 + y2 + z2 + u2  ≤ 50.

Baltic Way 2004.18
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 $\frac{1}{AX}+\frac{1}{XY}\ge \frac{4}{BC}$ .

Baltic Way 2004.19
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.

Baltic Way 2004.20
Three circular arcs w1,w2,w3 with common endpoints A and B are on the same side of the line AB, w2 lies between w1 and w3. Two rays emanating from B intersect these arcs at M1,M2,M3 and K1,K2,K3, respectively. Prove that $\frac{M_1M_2 }{ M_2M_3} = \frac{K_1K_2}{ K_2K_3}$.
.
Baltic Way 2005.11
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.

Baltic Way 2005.12
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 $5\times 3$?

Baltic Way 2005.14
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.

Baltic Way 2005.15
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.

Baltic Way 2006.11
The altitudes of a triangle are 12, 15, and 20. What is the area of the triangle?

Baltic Way 2006.12
Let ABC be a triangle, let B1 be the midpoint of the side AB and C1 the midpoint of the side AC. Let P be the point of intersection, other than A, of the circumscribed circles around the triangles ABC1 and AB1C. Let P1 be the point of intersection, other than A, of the line AP with the circumscribed circle around the triangle AB1C1. Prove that 2AP = 3AP1.

Baltic Way 2006.13
In a triangle ABC, points D, E lie on sides AB, AC respectively. The lines BE and CD intersect at F. Prove that if BC2 = 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.

Baltic Way 2006.15
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 .

Baltic Way 2007.11
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.

Baltic Way 2007.12
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 = 90o.

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 $t_1$.

Baltic Way 2007.14
In a convex quadrilateral ABCD we have ÐADC = 90o. 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.

Baltic Way 2007.15
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.

Baltic Way 2008.16
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.

Baltic Way 2008.17
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.

Baltic Way 2008.18
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.

Baltic Way 2008.20
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.

Baltic Way 2009.11
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.

Baltic Way 2009.12
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.

Baltic Way 2009.13
The point H is the orthocentre of a triangle ABC, and the segments AD, BE, CF are its altitudes. The points I1, I2, I3 are the incentres of the triangles EHF, FHD, DHE, respectively. Prove that the lines AI1, BI2, CI3 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 $A_1,A_2 ,\ldots ,A_n$?

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 $S_1+\ldots +S_m\ge 4$

Baltic Way 2010.11
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.

Baltic Way 2010.12
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.

Baltic Way 2010.13
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.

Baltic Way 2010.14
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.

Baltic Way 2010.15
The points M and N are chosen on the angle bisector AL of a triangle ABC such that ÐABM = ÐACN = 23o. X is a point inside the triangle such that BX = CX and ÐBXC = 2ÐBML. Find ÐMXN.

Baltic Way 2011.11
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.

Baltic Way 2011.12
Let P be a point inside a square ABCD such that PA : PB : PC is 1 : 2 : 3. Determine the angle ÐBPA.

Baltic Way 2011.13
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 .

Baltic Way 2011.14
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.

Baltic Way 2011.15
Let ABCD be a convex quadrilateral such that ÐADB = ÐBDC. Suppose that a point E on the side AD satisfies the equality AE · ED + BE2 = CD · AE. Show that ÐEBA = ÐDCB.

Baltic Way 2012.11
Let ABC be a triangle with ÐA = 60o. The point T lies inside the triangle in such a way that ÐATB=ÐBTC=ÐCTA=120o . Let M be the midpoint of BC. Prove that TA + TB + TC = 2AM.

Baltic Way 2012.12
Let P0, P1, . . . , P8 = P0 be successive points on a circle and Q be a point inside the polygon P0P1 … P7 such that ÐPi−1QPi = 45o 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.

Baltic Way 2012.13
Let ABC be an acute triangle, and let H be its orthocentre. Denote by HA, HB and HC the second intersection of the circumcircle with the altitudes from A, B and C respectively. Prove that the area of △HAHBHC does not exceed the area of △ABC.

Baltic Way 2012.14
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.

Baltic Way 2012.15
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.

Baltic Way 2013.11
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 = AD2 .

Baltic Way 2013.12
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.

Baltic Way 2013.13
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.

Baltic Way 2013.14
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.

Baltic Way 2013.15
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.

Baltic Way 2014.11
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.

Baltic Way 2014.12
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.

Baltic Way 2014.13
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.

Baltic Way 2014.14
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.

Baltic Way 2014.15
The sum of the angles A and C of a convex quadrilateral ABCD is less than 180ο. Prove that           AB · CD + AD · BC < AC (AB + AD).

Baltic Way 2015.11
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.

Baltic Way 2015.12
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.

Baltic Way 2015.13
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.

Baltic Way 2015.14
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.

Baltic Way 2015.15
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 = 90o.

Baltic Way 2016.16
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.

Baltic Way 2016.17
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.

Baltic Way 2016.18
Let ABCD be a parallelogram such that ÐBAD = 60o. Let K and L be the midpoints of BC and CD, respectively. Assuming that ABKL is a cyclic quadrilateral, find ÐABD.

Baltic Way 2016.19
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.

Baltic Way 2016.20
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 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.

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.

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.

Let $ABC$ be a triangle with $AB > AC$. The internal angle bisector of $\angle BAC$ intersects the side $BC$ at $D$. The circles with diameters $BD$ and $CD$ intersect the circumcircle of $\triangle ABC$ a second time at $P \not= B$ and $Q \not= C$, respectively. The lines $PQ$ and $BC$ intersect at $X$. Prove that $AX$ is tangent to the circumcircle of $\triangle ABC$.

Let $ABC$ be a triangle with circumcircle $\omega$. The internal angle bisectors of $\angle ABC$ and $\angle ACB$ intersect $\omega$ at $X\neq B$ and $Y\neq C$, respectively. Let $K$ be a point on $CX$ such that $\angle KAC = 90^\circ$. Similarly, let $L$ be a point on $BY$ such that $\angle LAB = 90^\circ$. Let $S$ be the midpoint of arc $CAB$ of $\omega$. Prove that $SK=SL$.

Let $ABC$ be an acute triangle with circumcircle $\omega$. Let $\ell$ be the tangent line to $\omega$ at $A$. Let $X$ and $Y$ be the projections of $B$ onto lines $\ell$ and $AC$, respectively. Let $H$ be the orthocenter of $BXY$. Let $CH$ intersect $\ell$ at $D$. Prove that $BA$ bisects angle $CBD$.

An acute triangle $ABC$ is given and let $H$ be its orthocenter. Let $\omega$ be the circle through $B$, $C$ and $H$, and let $\Gamma$ be the circle with diameter $AH$. Let $X\neq H$ be the other intersection point of $\omega$ and $\Gamma$, and let $\gamma$ be the reflection of $\Gamma$ over $AX$. Suppose $\gamma$ and $\omega$ intersect again at $Y\neq X$, and line $AH$ and $\omega$ intersect again at $Z \neq H$. Show that the circle through $A,Y,Z$ passes through the midpoint of segment $BC$.

On a plane, Bob chooses 3 points $A_0$, $B_0$, $C_0$ (not necessarily distinct) such that $A_0B_0+B_0C_0+C_0A_0=1$. Then he chooses points $A_1$, $B_1$, $C_1$ (not necessarily distinct) in such a way that $A_1B_1=A_0B_0$ and $B_1C_1=B_0C_0$. Next he chooses points $A_2$, $B_2$, $C_2$ as a permutation of points $A_1$, $B_1$, $C_1$. Finally, Bob chooses points $A_3$, $B_3$, $C_3$ (not necessarily distinct) in such a way that $A_3B_3=A_2B_2$ and $B_3C_3=B_2C_2$. What are the smallest and the greatest possible values of $A_3B_3+B_3C_3+C_3A_3$ Bob can obtain?
A point $P$ lies inside a triangle $ABC$. The points $K$ and $L$ are the projections of $P$ onto $AB$ and $AC$, respectively. The point $M$ lies on the line $BC$ so that $KM = LM$, and the point $P'$ is symmetric to $P$ with respect to $M$. Prove that $\angle BAP = \angle P'AC$.

Let $I$ be the incentre of a triangle $ABC$. Let $F$ and $G$ be the projections of $A$ onto the lines $BI$ and $CI$, respectively. Rays $AF$ and $AG$ intersect the circumcircles of the triangles $CFI$ and $BGI$ for the second time at points $K$ and $L$, respectively. Prove that the line $AI$ bisects the segment $KL$.

Let $D$ be the foot of the $A$-altitude of an acute triangle $ABC$. The internal bisector of the angle $DAC$ intersects $BC$ at $K$. Let $L$ be the projection of $K$ onto $AC$. Let $M$ be the intersection point of $BL$ and $AD$. Let $P$ be the intersection point of $MC$ and $DL$. Prove that $PK \perp AB$.

Let $ABC$ be a triangle with circumcircle $\Gamma$ and circumcentre $O$. Denote by $M$ the midpoint of $BC$. The point $D$ is the reflection of $A$ over $BC$, and the point $E$ is the intersection of $\Gamma$ and the ray $MD$. Let $S$ be the circumcentre of the triangle $ADE$. Prove that the points $A$, $E$, $M$, $O$, and $S$ lie on the same circle.

For which positive integers $n\geq4$ does there exist a convex $n$-gon with side lengths $1, 2, \dots, n$ (in some order) and with all of its sides tangent to the same circle?

Let $ABC$ be a triangle with circumcircle $\Gamma$ and circumcentre $O$. The circle with centre on the line $AB$ and passing through the points $A$ and $O$ intersects $\Gamma$ again in $D$. Similarly, the circle with centre on the line $AC$ and passing through the points $A$ and $O$ intersects $\Gamma$ again in $E$. Prove that $BD$ is parallel with $CE$.

An acute-angled triangle $ABC$ has altitudes $AD, BE$ and $CF$. Let $Q$ be an interior point of the segment $AD$, and let the circumcircles of the triangles $QDF$ and $QDE$ meet the line $BC$ again at points $X$ and $Y$ , respectively. Prove that $BX = CY$ .

Let $ABCD$ be a cyclic quadrilateral with $AB < BC$ and $AD < DC$. Let $E$ and $F$ be points on the sides $BC$ and $CD$, respectively, such that $AB = BE$ and $AD = DF$. Let further M denote the midpoint of the segment $EF$. Prove that $\angle BMD = 90^o$.

Let $\Gamma$ denote the circumcircle and $O$ the circumcentre of the acute-angled triangle $ABC$, and let $M$ be the midpoint of the segment $BC$. Let $T$ be the second intersection point of $\Gamma$ and the line $AM$, and $D$ the second intersection point of $\Gamma$ and the altitude from $A$. Let further $X$ be the intersection point of the lines $DT$ and $BC$. Let $P$ be the circumcentre of the triangle $XDM$. Prove that the circumcircle of the triangle $OPD$ passes through the midpoint of $XD$.

Let $\Omega$ be a circle, and $B, C$ are two fixed points on $\Omega$. Given a third point $A$ on $\Omega$, let $X$ and $Y$ denote the feet of the altitudes from $B$ and $C$, respectively, in the triangle $ABC$. Prove that there exists a fixed circle $\Gamma$ such that $XY$ is tangent to $\Gamma$ regardless of the choice of the point $A$.

source: www.math.olympiaadid.ut.ee/

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