Tournament of Towns 1998 - 2013,2020 (ToT) 162p

geometry problems from International Matehmatics Tournament of Towns
with aops links in the names

ToT older years 1980-2000 (uc)

Tournament of Towns Junior 2001-19 EN (solutions 2001-15)

Tournament of Towns Senior 2001-19 EN (solutions 2001-15)

[O = Easier, A = Harder]

                                         1998 -2013, 2020

[when a problem is proposed in both Junior and Senior, 

only Juniors is mentioned]

1998

1998 ToT Spring Junior O P3
$AB$ and $CD$ are segments lying on the two sides of an angle whose vertex is $O$. $A$ is between $O$ and $B$, and $C$ is between $O$ and $D$ . The line connecting the midpoints of the segments $AD$ and $BC$ intersects $AB$ at $M$ and $CD$ at $N$. Prove that $\frac{OM}{ON}=\frac{AB}{CD}$

(V Senderov)

1998 ToT Spring Junior O P5
Pinocchio claims that he can divide an isoceles triangle into three triangles, any two of which can be put together to form a new isosceles triangle. Is Pinocchio lying?

(A Shapovalov)

1998 ToT Spring Junior A P2
$ABCD$ is a parallelogram. A point $M$ is found on the side $AB$ or its extension such that $\angle MAD = \angle  AMO$ where $O$ is the point of intersection of the diagonals of the parallelogram. Prove that $MD = MG$.

(M Smurov)

1998 ToT Spring Senior O P1
Pinocchio claims that he can take some non-right-angled triangles , all of which are similar to one another and some of which may be congruent to one another, and put them together to form a rectangle. Is Pinocchio lying?

(A Fedotov)

1998 ToT Spring Senior O P5
A circle with center $O$ is inscribed in an angle. Let $A$ be the reflection of $O$ across one side of the angle. Tangents to the circle from $A$ intersect the other side of the angle at points $B$ and $C$. Prove that the circumcenter of triangle $ABC$ lies on the bisector of the original angle.

(I.Sharygin)

1998 ToT Spring Senior A P4
A point $M$ is found inside a convex quadrilateral $ABCD$ such that triangles $AMB$ and $CMD$ are isoceles ($AM = MB, CM = MD$) and $\angle AMB= \angle CMD = 120^o$ . Prove that there exists a point N such that triangles$BNC$ and $DNA$ are equilateral.

(I.Sharygin)

1998 ToT Autumn Junior O P3
In a triangle $ABC$ the points $A'$, $B'$ and $C'$ lie on the sides $BC$, $CA$ and $AB$, respectively. It is known that $\angle AC'B' = \angle B'A'C$, $\angle CB'A' = \angle A'C'B$ and $\angle BA'C' = \angle C'B'A$. Prove that $A'$, $B'$ and $C'$ are the midpoints of the corresponding sides.

(V Proizvolov)

1998 ToT Autumn Junior A P3
Segment $AB$ intersects two equal circles, is parallel to the line joining their centres, and all the points of intersection of the segment and the circles lie between $A$ and $B$. From the point $A$ tangents to the circle nearest to $A$ are drawn, and from the point $B$ tangents to the circle nearest to $B$ are also drawn. It turns out that the quadrilateral formed by the four tangents extended contains both circles. Prove that a circle can be drawn so that it touches all four sides of the quadrilateral.

 (P Kozhevnikov)

1998 ToT Autumn Senior A P5

The sum of the length, width, and height of a rectangular parallelepiped will be called its size. Can it happen that one rectangular parallelepiped contains another one of greater size?

(A Shen)

1999

1999 ToT Spring Junior O P2
$ABC$ is a right-angled triangle. A square $ABDE$ is constructed on the opposite side of the hypothenuse $AB$ from $C$. The bisector of $\angle C$ cuts $DE$ at $F$. If $AC = 1$ and $BC = 3$, compute $\frac{DF}{EF}$.

(A Blinkov)

1999 ToT Spring Junior A P2
Let $O$ be the intersection point of the diagonals of a parallelogram $ABCD$ . Prove that if the line $BC$ is tangent to the circle passing through the points $A, B$, and $O$, then the line $CD$ is tangent to the circle passing through the points $B, C$ and $O$.

(A Zaslavskiy)

1999 ToT Spring Junior A P5
The sides $AB$ and $AC$ are tangent at points $P$ and $Q$, respectively, to the incircle of a triangle $ABC. R$ and $S$ are the midpoints of the sides $AC$ and $BC$, respectively, and $T$ is the intersection point of the lines $PQ$ and $RS$. Prove that $T$ lies on the bisector of the angle $B$ of the triangle.

(M Evdokimov)

1999 ToT Spring Senior A P1

A convex polyhedron is floating in a sea. Can it happen that $90\%$ of its volume is below the water level, while more than half of its surface area is above the water level?

(A Shapovalov)

1999 ToT Spring Senior A P2

Let all vertices of a convex quadrilateral $ABCD$ lie on the circumference of a circle with centre $O$. Let $F$ be the second point of intersection of the circumcircles of the triangles $ABO$ and $CDO$. Prove that the circle passing through the points $A, F$ and $D$ also passes through the point of intersection of the segments $AC$ and $BD$.

(A Zaslavskiy)

1999 ToT Autumn Junior O P1
A right-angled triangle made of paper is folded along a straight line so that the vertex at the right angle coincides with one of the other vertices of the triangle and a quadrilateral is obtained .
(a) What is the ratio into which the diagonals of this quadrilateral divide each other?
(b) This quadrilateral is cut along its longest diagonal. Find the area of the smallest piece of paper thus obtained if the area of the original triangle is $1$ .

(A Shapovalov)

1999 ToT Autumn Senior O P1

The incentre of a triangle is joined by three segments to the three vertices of the triangle, thereby dividing it into three smaller triangles. If one of these three triangles is similar to the original triangle, find its angles.

(A Shapovalov)

1999 ToT Autumn Senior A P4
Points $K, L$ on sides $AC, CB$ respectively of a triangle $ABC$ are the points of contact of the excircles with the corresponding sides . Prove that the straight line through the midpoints of $KL$ and $AB$
(a) divides the perimeter of triangle $ABC$ in half,
(b) is parallel to the bisector of angle $ACB$.

( L Emelianov)

2000

2000 ToT Spring Junior O P2
In a quadrilateral $ABCD$ of area $1$, the parallel sides $BC$ and $AD$ are in the ratio $1 :2$ . $K$ is the midpoint of the diagonal $AC$ and $L$ is the point of intersection of the line $DK$ and the side $AB$. Determine the area of the quadrilateral $BCKL$ .

(M G Sonkin)

2000 ToT Spring Junior A P2
Two parallel sides of a quadrilateral have integer lengths. Prove that this quadrilateral can be cut into congruent triangles.

(A Shapovalov)

2000 ToT Spring Junior A P3
$A$ is a fixed point inside a given circle. Determine the locus of points $C$ such that $ABCD$ is a rectangle with $B$ and $D$ on the circumference of the given circle.

(M Panov)

2000 ToT Spring Senior O P1
The diagonals of a convex quadrilateral $ABCD$ meet at $P$. The sum of the areas of triangles $PAB$ and $PCD$ is equal to the sum of areas of triangles $PAD$ and $PCB$. Prove that $P$ is the midpoint of either $AC$ or $BD$.

(Folklore)

2000 ToT Spring Senior A P4
The chords $AC$ and $BD$ of a, circle with centre $O$ intersect at the point $K$. The circumcentres of triangles $AKB$ and $CKD$ are $M$ and $N$ respectively. Prove that $OM = KN$.

(A Zaslavsky )

2000 ToT Autumn Junior O P2

$ABCD$ is parallelogram, $M$ is the midpoint of side $CD$ and $H$ is the foot of the perpendicular from $B$ to line $AM$. Prove that $BCH$ is an isosceles triangle.

(M Volchkevich)

2000 ToT Autumn Junior A P2
In triangle $ABC, AB = AC$. A line is drawn through $A$ parallel to $BC$. Outside triangle $ABC$, a circle is drawn tangent to this line, to the line $BC$, to $AB$ and to the incircle of $ABC$. If the radius of this circle is $1$ , determine the inradius of $ABC$.

(RK Gordin)

2000 ToT Autumn Senior O P1
Triangle $ABC$ is inscribed in a circle. Chords $AM$ and $AN$ intersect side $BC$ at points $K$ and $L$ respectively. Prove that if a circle passes through all of the points $K, L, M$ and $N$, then $ABC$ is an isosceles triangle.

(V Zhgun)

2000 ToT Autumn Senior O P3
In each lateral face of a pentagonal prism at least one of the four angles is equal to $f$. Find all possible values of $f$.

(A Shapovalov)

2000 ToT Autumn Senior A P3
In a triangle $ABC, AB = c, BC = a, CA = b$, and $a < b < c$. Points $B'$ and $A'$ are chosen on the rays $BC$ and $AC$ respectively so that $BB'= AA'= c$. Points $C''$ and $B''$ are chosen on the rays $CA$ and $BA$ so that $CC'' = BB'' = a$. Find the ratio of the segment $A'B'$ to the segment $C'' B''$.

(R Zhenodarov)

2001

2001 ToT Spring Junior O P2
One of the midlines of a triangle is longer than one of its medians. Prove that the triangle has an obtuse angle.

2001 ToT Spring Junior A P3
Point $A$ lies inside an angle with vertex $M$. A ray issuing from point $A$ is reflected in one side of the angle at point $B$, then in the other side at point $C$ and then returns back to point $A$ (the ordinary rule of reflection holds). Prove that the center of the circle circumscribed about triangle $\triangle BCM$ lies on line $AM$.

2001 ToT Spring Junior A P6 , Senior A P3
Let $AH_A$, $BH_B$ and $CH_C$ be the altitudes of triangle $\triangle ABC$. Prove that the triangle whose vertices are the intersection points of the altitudes of triangles $\triangle AH_BH_C$, $\triangle BH_AH_C$ and $\triangle CH_AH_B$ is equal to triangle $\triangle H_AH_BH_C$.

2001 ToT Spring Senior O P3
Points $X$ and $Y$ are chosen on the sides $AB$ and $BC$ of the triangle $\triangle ABC$. The segments $AY$ and $CX$ intersect at the point $Z$. Given that $AY = YC$ and $AB = ZC$, prove that the points $B$, $X$, $Z$, and $Y$ lie on the same circle.

2001 ToT Fall Junior O P1
In the quadrilateral $ABCD$, $AD$ is parallel to $BC$.  $K$ is a point on $AB$. Draw the line through $A$ parallel to $KC$ and the line through $B$ parallel to $KD$. Prove that these two lines intersect at some point on $CD$.

2001 ToT Fall Senior O P1
An altitude of a pentagon is the perpendicular drop from a vertex to the opposite side. A median of a pentagon is the line joining a vertex to the midpoint of the opposite side. If the five altitudes and the five medians all have the same length,prove that the pentagon is regular.

2002

2002 ToT Spring Junior O P4
Quadrilateral $ABCD$ is circumscribed about a circle $\Gamma$ and $K,L,M,N$ are points of tangency of sides $AB,BC,CD,DA$ with $\Gamma$ respectively. Let $S\equiv KM\cap LN$. If quadrilateral $SKBL$ is cyclic then show that $SNDM$ is also cyclic.

2002 ToT Spring Junior A P3

Let $E$ and $F$ be the respective midpoints of $BC,CD$ of a convex quadrilateral $ABCD$. Segments $AE,AF,EF$ cut the quadrilateral into four triangles whose areas are four consecutive integers. Find the maximum possible area of $\Delta BAD$.

2002 ToT Spring Senior O P2
$\Delta ABC$ and its mirror reflection $\Delta A^{\prime}B^{\prime}C^{\prime}$ is arbitrarily placed on the plane. Prove the midpoints of $AA^{\prime},BB^{\prime},CC^{\prime}$ are collinear.

2002 ToT Spring Senior A P5
Let $AA_1,BB_1,CC_1$ be the altitudes of acute $\Delta ABC$. Let $O_a,O_b,O_c$ be the incentres of $\Delta AB_1C_1,\Delta BC_1A_1,\Delta CA_1B_1$ respectively. Also let $T_a,T_b,T_c$ be the points of tangency of the incircle of $\Delta ABC$ with $BC,CA,AB$ respectively. Prove that $T_aO_cT_bO_aT_cO_b$ is an equilateral hexagon.

2002 ToT Fall Junior O P5
An angle and a point $A$ inside it is given. Is it possible to draw through $A$ three straight lines so that on either side of the angle one of three points of intersection of these lines be the midpoint of two other points of intersection with that side?

2002 ToT Fall Junior A P4
Point $P$ is chosen in the plane of triangle $ABC$ such that $\angle{ABP}$ is congruent to $\angle{ACP}$ and $\angle{CBP}$ is congruent to $\angle{CAP}$. Show $P$ is the orthocentre.

2002 ToT Fall Senior A P5
Two circles $\Gamma_1,\Gamma_2$ intersect at $A,B$. Through $B$ a straight line $\ell$ is drawn and $\ell\cap \Gamma_1=K,\ell\cap\Gamma_2=M\;(K,M\neq B)$. We are given $\ell_1\parallel AM$ is tangent to $\Gamma_1$ at $Q$. $QA\cap \Gamma_2=R\;(\neq A)$ and further $\ell_2$ is tangent to $\Gamma_2$ at $R$. Prove that:

• $\ell_2\parallel AK$
• $\ell,\ell_1,\ell_2$ have a common point.

2003

2003 ToT Spring Junior O P3
Points $K$ and $L$ are chosen on the sides $AB$ and $BC$ of the isosceles $\triangle ABC$ ($AB = BC$) so that $AK +LC = KL$. A line parallel to $BC$ is drawn through midpoint $M$ of the segment $KL$, intersecting side $AC$ at point $N$. Find the value of $\angle KNL$.

2003 ToT Spring Junior A P2
Triangle $ABC$ is given. Prove that $\frac{R}{r} > \frac{a}{h}$, where $R$ is the radius of the circumscribed circle, $r$ is the radius of the inscribed circle, $a$ is the length of the longest side, $h$ is the length of the shortest altitude.

2003 ToT Spring Junior A P6
A trapezoid with bases $AD$ and $BC$ is circumscribed about a circle, $E$ is the intersection point of the diagonals. Prove that $\angle AED$ is not acute.

2003 ToT Spring Senior O P3
Point $M$ is chosen in triangle $ABC$ so that the radii of the circumcircles of triangles $AMC, BMC$, and $BMA$ are no smaller than the radius of the circumcircle of $ABC$. Prove that all four radii are equal.

2003 ToT Spring Senior A P1
A triangular pyramid $ABCD$ is given. Prove that $\frac Rr > \frac ah$, where $R$ is the radius of the circumscribed sphere, $r$ is the radius of the inscribed sphere, $a$ is the length of the longest edge, $h$ is the length of the shortest altitude (from a vertex to the opposite face).

2003 ToT Spring Senior A P4
A right triangle $ABC$ with hypotenuse $AB$ is inscribed in a circle. Let $K$ be the midpoint of the arc $BC$ not containing $A, N$ the midpoint of side $AC$, and $M$ a point of intersection of ray $KN$ with the circle. Let $E$ be a point of intersection of tangents to the circle at points $A$ and $C$. Prove that $\angle EMK = 90^\circ$

2003 ToT Fall Junior O P2
In $7$-gon $A_1A_2A_3A_4A_5A_6A_7$ diagonals $A_1A_3, A_2A_4, A_3A_5, A_4A_6, A_5A_7, A_6A_1$ and $A_7A_2$ are congruent to each other and diagonals $A_1A_4, A_2A_5, A_3A_6, A_4A_7, A_5A_1, A_6A_2$ and $A_7A_3$ are also congruent to each other. Is the polygon necessarily regular?

2003 ToT Fall Junior A P5
A point $O$ lies inside of the square $ABCD$. Prove that the difference between the sum of angles $OAB, OBC, OCD , ODA$ and $180^{\circ}$ does not exceed $45^{\circ}$.

2003 ToT Fall Senior O P4
Each side of $1 \times 1$ square is a hypothenuse of an exterior right triangle. Let $A, B, C, D$ be the vertices of the right angles and $O_1, O_2, O_3, O_4$ be the centers of the incircles of these triangles. Prove that
a) The area of quadrilateral $ABCD$ does not exceed $2$;
b) The area of quadrilateral $O_1O_2O_3O_4$ does not exceed $1$.

2003 ToT Fall Senior A P4
In a triangle $ABC$, let $H$ be the point of intersection of altitudes, $I$ the center of incircle, $O$ the center of circumcircle, $K$ the point where the incircle touches $BC$. Given that $IO$ is parallel to $BC$, prove that $AO$ is parallel to $HK$.

2003 ToT Fall Senior A P6
Let $O$ be the center of insphere of a tetrahedron $ABCD$. The sum of areas of faces $ABC$ and $ABD$ equals the sum of areas of faces $CDA$ and $CDB$. Prove that $O$ and midpoints of $BC, AD, AC$ and $BD$ belong to the same plane.

2004

2004 ToT Spring Junior O P1
In triangle ABC the bisector of angle A, the perpendicular to side AB from its midpoint, and the altitude from vertex B, intersect in the same point. Prove that the bisector of angle A, the perpendicular to side AC from its midpoint, and the altitude from vertex C also intersect in the same point.

2004 ToT Spring Junior Α P4
Two circles intersect in points A and B. Their common tangent nearer B touches the circles at points E and F, and intersects the extension of AB at the point M. The point K is chosen on the extention of AM so that KM = MA. The line KE intersects the circle containing E again at the point C. The line KF intersects the circle containing F again at the point D. Prove that the points A, C and D are collinear.

2004 ToT Spring Senior O P1
Segments AB, BC and CD of the broken line ABCD are equal and are tangent to a circle with centre at the point O. Prove that the point of contact of this circle with BC, the point O and the intersection point of AC and BD are collinear.

2004 ToT Spring Senior O P3
Perimeter of a convex quadrilateral is 2004 and one of its diagonals is 1001. Can another diagonal be 1? 2? 1001?

2004 ToT Spring Senior Α P3
The perpendicular projection of a triangular pyramid on some plane has the largest possible area. Prove that this plane is parallel to either a face or two opposite edges of the pyramid.

2004 ToT Spring Senior Α P5
The parabola $y = x^2$ intersects a circle at exactly two points A and B. If their tangents at A coincide, must their tangents at B also coincide?

2004 ToT Fall Junior O P4
A circle and a straight line with no common points are given. With compass and straightedge construct a square with two adjacent vertices on the circle and two other vertices on the line (it is known that such a square exists).

2004 ToT Fall Junior Α P2

An incircle of triangle ABC touches the sides BC, CA and AB at points A', B' and C' respectively. Is it necessarily true that triangle ABC is equilateral if AA' = BB' = CC'?

2004 ToT Fall Junior Α P5
Point K belongs to side BC of triangle ABC. Incircles of triangles ABK and ACK touch
BC at points M and N respectively. Prove that BM · CN > KM · KN.

2004 ToT Fall Senior O P1
Three circles pass through point X and A, B, C are their intersection points (other than X). Let A'  be the second point of intersection of straight line AX and the circle circumscribed around triangle BCX. Define simiarly points B' , C' . Prove that triangles ABC' , AB' C, and A' BC are similar.

2004 ToT Fall Senior Α P4
A circle with the center I is entirely inside of a circle with center O. Consider all possible chords AB of the larger circle which are tangent to the smaller one. Find the locus of the centers of the circles circumscribed about the triangle AIB.

2004 ToT Fall Senior Α P7
Let \angleAOB be obtained from \angle COD by rotation (ray AO transforms into ray CO). Let E and F be the points of intersection of the circles inscribed into these angles. Prove that \angle AOE = \angle DOF.

2005

2005 ToT Spring Junior O P4, Senior O P3
$M$ and $N$ are the midpoints of sides $BC$ and $AD$, respectively, of a square $ABCD$. $K$ is an arbitrary point on the extension of the diagonal $AC$ beyond $A$. The segment $KM$ intersects the side $AB$ at some point $L$. Prove that $\angle KNA = \angle LNA$.

2005 ToT Spring Junior A P2

The altitudes $AD$ and $BE$ of triangle $ABC$ meet at its orthocentre $H$. The midpoints of $AB$ and $CH$ are $X$ and $Y$, respectively. Prove that $XY$ is perpendicular to $DE$.

2005 ToT Spring Senior A P2
A circle $\omega_1$ with centre $O_1$ passes through the centre $O_2$ of a second circle $\omega_2$. The tangent lines to $\omega_2$ from a point $C$ on $\omega_1$ intersect $\omega_1$ again at points $A$ and $B$ respectively. Prove that $AB$ is perpendicular to $O_1O_2$.

2005 ToT Spring Senior A P5
Prove that if a regular icosahedron and a regular dodecahedron have a common circumsphere, then they have a common insphere.

2005 ToT Fall Junior O P1
In triangle $ABC$, points $M_1, M_2$ and $M_3$ are midpoints of sides $AB$, $BC$ and $AC$, respectively, while points $H_1, H_2$ and $H_3$ are bases of altitudes drawn from $C$, $A$ and $B$, respectively. Prove that one can construct a triangle from segments $H_1M_2, H_2M_3$ and $H_3M_1$.

2005 ToT Fall Junior A P2
The extensions of sides $AB$ and $CD$ of a convex quadrilateral $ABCD$ intersect at $K$. It is known that $AD = BC$. Let $M$ and $N$ be the midpoints of sides $AB$ and $CD$. Prove that the triangle $MNK$ is obtuse.

2005 ToT Fall Senior O P2
A segment of length $\sqrt2 + \sqrt3 + \sqrt5$ is drawn. Is it possible to draw a segment of unit length using a compass and a straightedge?

2005 ToT Fall Senior O P4
On all three sides of a right triangle $ABC$ external squares are constructed; their centers denoted by $D$, $E$, $F$. Show that the ratio of the area of triangle $DEF$ to the area of triangle $ABC$ is:
a) greater than $1$;
b)  at least $2$.

2005 ToT Fall Senior A P5
In triangle $ABC$ bisectors $AA_1, BB_1$ and $CC_1$ are drawn. Given $\angle A : \angle B : \angle C = 4 : 2 : 1$, prove that $A_1B_1 = A_1C_1$.

2006

2006 ToT Spring Junior O P1
Let $\angle A$ in a triangle $ABC$ be $60^\circ$. Let point $N$ be the intersection of $AC$ and perpendicular bisector to the side $AB$ while point $M$ be the intersection of $AB$ and perpendicular bisector to the side $AC$. Prove that $CB = MN$.

2006 ToT Spring Junior A P3
On sides $AB$ and $BC$ of an acute triangle $ABC$ two congruent rectangles $ABMN$ and $LBCK$ are constructed (outside of the triangle), so that $AB = LB$. Prove that straight lines $AL, CM$ and $NK$ intersect at the same point.

2006 ToT Spring Senior O P4
Quadrilateral $ABCD$ is a cyclic, $AB = AD$. Points $M$ and $N$ are chosen on sides $BC$ and $CD$ respectfully so that $\angle MAN =1/2 (\angle BAD)$. Prove that  $MN = BM + ND$.

2006 ToT Spring Senior A P4

In triangle $ABC$ let $X$ be some fixed point on bisector $AA'$ while point $B'$  be intersection of  $BX$ and $AC$ and point $C'$ be intersection of $CX$ and $AB$. Let point $P$ be intersection of segments $A'B'$ and $CC'$ while point $Q$  be intersection of segments $A'C'$ and $BB'$. Prove τhat $\angle PAC = \angle QAB$.

2006 ToT Fall Junior O P4
Given triangle ABC, BC is extended beyond B to the point D such that BD = BA. The bisectors of the exterior angles at vertices B and C intersect at the point M. Prove that quadrilateral ADMC is cyclic.

2006 ToT Fall Junior A P1
Two regular polygons, a 7-gon and a 17-gon are given. For each of them two circles are drawn, an inscribed circle and a circumscribed circle. It happened that rings containing the polygons have equal areas. Prove that sides of the polygons are equal.

2006 ToT Fall Junior A P4
A circle of radius R is inscribed into an acute triangle. Three tangents to the circle split the triangle into three right angle triangles and a hexagon that has perimeter Q. Find the sum of diameters of circles inscribed into the three right triangles.

2006 ToT Fall Senior O P2
The incircle of the quadrilateral ABCD touches AB, BC, CD and DA at E, F, G and H respectively. Prove that the line joining the incentres of triangles HAE and FCG is perpendicular to the line joining the incentres of triangles EBF and GDH.

2006 ToT Fall Senior O P5
Can a regular octahedron be inscribed in a cube in such a way that all vertices of the octahedron are on cube's edges?

2006 ToT Fall Senior A P2
Suppose ABC is an acute triangle. Points A1, B1 and C1 are chosen on sides BC, AC and AB
respectively so that the rays A1A, B1B and C1C are bisectors of triangle A1B1C1. Prove that
AA1, BB1 and CC1 are altitudes of triangle ABC.

2007

2007 ToT Spring Junior O P1
The sides of a convex pentagon are extended on both sides to form five triangles. If these triangles are congruent to one another, does it follow that the pentagon is regular?

2007 ToT Spring Junior A P2

$K, L, M$ and $N$ are points on sides $AB, BC, CD$ and $DA$, respectively, of the unit square $ABCD$ such that $KM$ is parallel to $BC$ and $LN$ is parallel to  $AB$. The perimeter of triangle $KLB$ is equal to $1$. What is the area of triangle $MND$?

2007 ToT Spring Junior A P6
In the quadrilateral $ABCD$, $AB = BC = CD$ and $\angle BMC = 90^\circ$, where $M$ is the midpoint of $AD$. Determine the acute angle between the lines $AC$ and $BD$.

2007 ToT Spring Senior O P3
$B$ is a point on the line which is tangent to a circle at the point $A$. The line segment $AB$ is rotated about the centre of the circle through some angle to the line segment $A'B'$. Prove that the line $AA'$ passes through the midpoint of $BB'$.

2007 ToT Spring Senior A P7
$T$ is a point on the plane of triangle $ABC$ such that $\angle ATB = \angle  BTC = \angle  CTA = 120^\circ$. Prove that the lines symmetric to $AT, BT$ and $CT$ with respect to $BC, CA$ and $AB$, respectively, are concurrent.

2007 ToT Fall Junior O P3
$D$ is the midpoint of the side $BC$ of triangle $ABC$. $E$ and $F$ are points on $CA$ and $AB$ respectively, such that $BE$ is perpendicular to $CA$ and $CF$ is perpendicular to $AB$. If $DEF$ is an equilateral triangle, does it follow that $ABC$ is also equilateral?

2007 ToT Fall Junior A P1
Let $ABCD$ be a rhombus. Let $K$ be a point on the line $CD$, other than $C$ or $D$, such that $AD = BK$. Let $P$ be the point of intersection of $BD$ with the perpendicular bisector of $BC$. Prove that $A, K$ and $P$ are collinear.

2007 ToT Fall Junior P P2
Let us call a triangle “almost right angle triangle” if one of its angles differs from $90^\circ$ by no more than $15^\circ$. Let us call a triangle “almost isosceles triangle” if two of its angles differs from each other by no more than $15^\circ$. Is it true that that any acute triangle is either “almost right angle triangle” or “almost isosceles triangle”?

2007 ToT Fall Junior P P3

A triangle with sides $a, b, c$ is folded along a line $\ell$ so that a vertex $C$ is on side $c$. Find the segments on which point $C$ divides $c$, given that the angles adjacent to $\ell$ are equal.

2007 ToT Fall Senior O P3
Give a construction by straight-edge and compass of a point $C$ on a line $\ell$ parallel to a segment $AB$, such that the product $AC \cdot BC$ is minimum.

2007 ToT Fall Senior A P2
Let $K, L, M$ and $N$ be the midpoints of the sides $AB, BC, CD$ and $DA$ of a cyclic quadrilateral $ABCD$. Let $P$ be the point of intersection of $AC$ and $BD$. Prove that the circumradii of triangles $PKL, PLM, PMN$ and $PNK$ are equal to one another.

2007 ToT Fall Senior P P4

Jim and Jane divide a triangular cake between themselves. Jim chooses any point in the cake and Jane makes a straight cut through this point and chooses the piece. Find the size of the piece that each of them can guarantee for himself/herself (both of them want to get as much as possible).

2008

2008 ToT Spring Junior O P1
In the convex hexagon ABCDEF, AB, BC and CD are respectively parallel to DE, EF and FA. If AB = DE, prove that BC = EF and CD = FA.

2008 ToT Spring Junior A P2
A line parallel to the side AC of triangle ABC cuts the side AB at K and the side BC at M. O is the point of intersection of AM and CK. If AK = AO and KM = MC, prove that AM = KB.

2008 ToT Spring Junior A P7
A convex quadrilateral ABCD has no parallel sides. The angles between the diagonal AC and the four sides are 55^o, 55^o, 19^o and 16^o in some order. Determine all possible values of the acute angle between AC and BD.

2008 ToT Spring Senior O P3
In triangle ABC, \angle A = 90^o. M is the midpoint of BC and H is the foot of the altitude from
A to BC. The line passing through M and perpendicular to AC meets the circumcircle of
triangle AMC again at P. If BP intersects AH at K, prove that AK = KH.

2008 ToT Spring Senior A P4
Each of Peter and Basil draws a convex quadrilateral with no parallel sides. The angles between a diagonal and the four sides of Peter's quadrilateral are \alpha, \alpha, \beta and \gamma  in some order. The angles between a diagonal and the four sides of Basil's quadrilateral are also \alpha, \alpha, \beta and \gamma   in some order. Prove that the acute angle between the diagonals of Peter's quadrilateral is equal to the acute angle between the diagonals of Basil's quadrilateral.

2008 ToT Spring Senior A P7

Each of three lines cuts chords of equal lengths in two given circles. The points of intersection of these lines form a triangle. Prove that its circumcircle passes through the midpoint of the segment joining the centres of the circles.

2008 ToT Fall Junior O P3
Acute triangle $A_1A_2A_3$ is inscribed in a circle of radius $2$. Prove that one can choose points $B_1, B_2, B_3$ on the arcs $A_1A_2, A_2A_3, A_3A_1$ respectively, such that the numerical value of the area of the hexagon $A_1B_1A_2B_2A_3B_3$ is equal to the numerical value of the perimeter of the triangle $A_1A_2A_3.$

2008 ToT Fall Junior A P3
In his triangle $ABC$ Serge made some measurements and informed Ilias about the lengths of median $AD$ and side $AC$. Based on these data Ilias proved the assertion: angle $CAB$ is obtuse, while angle $DAB$ is acute. Determine a ratio $AD/AC$ and prove Ilias' assertion (for any triangle with such a ratio).

2008 ToT Fall Junior A P6

Let $ABC$ be a non-isosceles triangle. Two isosceles triangles $AB'C$ with base $AC$ and $CA'B$ with base $BC$ are constructed outside of triangle $ABC$. Both triangles have the same base angle $\varphi$. Let $C_1$ be a point of intersection of the perpendicular from $C$ to $A'B'$ and the perpendicular bisector of the segment $AB$. Determine the value of $\angle AC_1B.$

2008 ToT Fall Senior O P3
A $30$-gon $A_1A_2\cdots A_{30}$ is inscribed in a circle of radius $2$. Prove that one can choose a point $B_k$ on the arc $A_kA_{k+1}$ for $1 \leq k \leq 29$ and a point $B_{30}$ on the arc $A_{30}A_1$, such that the numerical value of the area of the $60$-gon $A_1B_1A_2B_2 \dots A_{30}B_{30}$ is equal to the numerical value of the perimeter of the original $30$-gon.

2008 ToT Fall Senior A P4
Let $ABCD$ be a non-isosceles trapezoid. Define a point $A1$ as intersection of circumcircle of triangle $BCD$ and line $AC$. (Choose $A_1$ distinct from $C$). Points $B_1, C_1, D_1$ are defined in similar way. Prove that $A_1B_1C_1D_1$ is a trapezoid as well.

2009

2009 ToT Spring Junior O P5
In rhombus ABCD, angle A equals 120^ο. Points M and N are chosen on sides BC and CD so that angle NAM equals 30^ο. Prove that the circumcenter of triangle NAM lies on a diagonal of of the rhombus.

2009 ToT Spring Junior A P7
Angle C of an isosceles triangle ABC equals 120^ο. Each of two rays emitting from vertex C (inwards the triangle) meets AB at some point (P_i) reflects according to the rule the angle of incidence equals the angle of refkection" and meets lateral side of triangle ABC at point Q_i (i = 1,2). Given that angle between the rays equals 60^ο, prove that area of triangle P_1CP_2 equals the sum of areas of triangles AQ_1P_1 and BQ_2P_2 (AP_1 < AP_2).

2009 ToT Spring Senior O P5
Suppose that X is an arbitrary point inside a tetrahedron. Through each vertex of the tetrahedron, draw a straight line that is parallel to the line segment connecting X with the intersection point of the medians of the opposite face. Prove that these four lines meet at the same point.

2009 ToT Spring Senior A P4
Three planes dissect a parallelepiped into eight hexahedrons such that all of their faces are quadrilaterals (each plane intersects two corresponding pairs of opposite faces of the parallelepiped and does not intersect the remaining two faces). One of the hexahedrons has a circumscribed sphere. Prove that each of these hexahedrons has a circumscribed sphere.

2009 ToT Fall Junior A P4
Let $ABCD$ be a rhombus. $P$ is a point on side $BC$ and $Q$ is a point on side $CD$ such that $BP = CQ$. Prove that centroid of triangle $APQ$ lies on the segment $BD.$

2009 ToT Fall Senior O P2
$A, B, C, D, E$ and $F$ are points in space such that $AB$ is parallel to $DE$, $BC$ is parallel to $EF$, $CD$ is parallel to $FA$, but $AB \neq DE$. Prove that all six points lie in the same plane.

2009 ToT Fall Senior A P3
Every edge of a tetrahedron is tangent to a given sphere. Prove that the three line segments joining the points of tangency of the three pairs of opposite edges of the tetrahedron are concurrent.

2009 ToT Fall Senior A P5
Let $XY Z$ be a triangle. The convex hexagon $ABCDEF$ is such that $AB; CD$ and $EF$ are parallel and equal to $XY; Y Z$ and $ZX$, respectively. Prove that area of triangle with vertices at the midpoints of $BC; DE$ and $FA$ is no less than area of triangle $XY Z.$

2010

2010 ToT Spring Junior O P3
An angle is given in a plane. Using only a compass, one must find out
(a) if this angle is acute. Find the minimal number of circles one must draw to be sure.
(b) if this angle equals $31^{\circ}$.(One may draw as many circles as one needs).

2010 ToT Spring Junior A P2
Let $M$ be the midpoint of side $AC$ of the triangle $ABC$. Let $P$ be a point on the side $BC$. If $O$ is the point of intersection of $AP$ and $BM$ and $BO = BP$, determine the ratio $\frac{OM}{PC}$ .

2010 ToT Spring Senior O P5
A needle (a segment) lies on a plane. One can rotate it $45^{\circ}$ round any of its endpoints. Is it possible that after several rotations the needle returns to initial position with the endpoints interchanged?

2010 ToT Spring Senior A P6
Quadrilateral $ABCD$ is circumscribed around the circle with centre $I$. Let points $M$ and $N$ be the midpoints of sides $AB$ and $CD$ respectively and let $\frac{IM}{AB} = \frac{IN}{CD}$. Prove that $ABCD$ is either a trapezoid or a parallelogram.

2010 ToT Fall Junior O P2
In a quadrilateral $ABCD$ with an incircle, $AB = CD; BC < AD$ and $BC$ is parallel to $AD$. Prove that the bisector of $\angle C$ bisects the area of $ABCD$

2010 ToT Fall Junior A P1
A round coin may be used to construct a circle passing through one or two given points on the plane. Given a line on the plane, show how to use this coin to construct two points such that they dene a line perpendicular to the given line. Note that the coin may not be used to construct a circle tangent to the given line.

2010 ToT Fall Junior A P6

In acute triangle $ABC$, an arbitrary point $P$ is chosen on altitude $AH$. Points $E$ and $F$ are the midpoints of sides $CA$ and $AB$ respectively. The perpendiculars from $E$ to $CP$ and from $F$ to $BP$ meet at point $K$. Prove that $KB = KC$.

2010 ToT Fall Senior O P2

The diagonals of a convex quadrilateral $ABCD$ are perpendicular to each other and intersect at the point $O$. The sum of the inradii of triangles $AOB$ and $COD$ is equal to the sum of the inradii of triangles $BOC$ and $DOA$.

(a) Prove that $ABCD$ has an incircle.

(b) Prove that $ABCD$ is symmetric about one of its diagonals.

2010 ToT Fall Senior A P5
The quadrilateral $ABCD$ is inscribed in a circle with center $O$. The diagonals $AC$ and $BD$ do not pass through $O$. If the circumcentre of triangle $AOC$ lies on the line $BD$, prove that the circumcentre of triangle $BOD$ lies on the line $AC$.

2011

2011 ToT Spring Junior O P4
Each diagonal of a convex quadrilateral divides it into two isosceles triangles. The two diagonals of the same quadrilateral divide it into four isosceles triangles. Must this quadrilateral be a square?

2011 ToT Spring Junior Α P1
Does there exist a hexagon that can be divided into four congruent triangles by a straight cut?

2011 ToT Spring Junior Α P5
$AD$ and $BE$ are altitudes of an acute triangle $ABC$. From $D$, perpendiculars are dropped to $AB$ at $G$ and $AC$ at $K$. From $E$, perpendiculars are dropped to $AB$ at $F$ and $BC$ at $H$. Prove that $FG$ is parallel to $HK$ and $FK = GH$.

2011 ToT Spring Senior O P1
The faces of a convex polyhedron are similar triangles. Prove that this polyhedron has two pairs of congruent faces.

2011 ToT Spring Senior O P4
Four perpendiculars are drawn from four vertices of a convex pentagon to the opposite sides. If these four lines pass through the same point, prove that the perpendicular from the fifth vertex to the opposite side also passes through this point.

2011 ToT Spring Senior A P3
(a) Does there exist an innite triangular beam such that two of its cross-sections are similar but not congruent triangles?
(b) Does there exist an innite triangular beam such that two of its cross-sections are equilateral triangles of sides $1$ and $2$ respectively?

2011 ToT Spring Senior A P5
In the convex quadrilateral $ABCD, BC$ is parallel to $AD$. Two circular arcs $\omega_1$ and $\omega_3$ pass through $A$ and $B$ and are on the same side of $AB$. Two circular arcs $\omega_2$ and $\omega_4$ pass through $C$ and $D$ and are on the same side of $CD$. The measures of $\omega_1, \omega_2, \omega_3$ and $\omega_4$ are $\alpha, \beta,\beta$ and $\alpha$ respectively. If $\omega_1$ and $\omega_2$ are tangent to each other externally, prove that so are $\omega_3$ and $\omega_4$.

2011 ToT Fall Junior O P1
$P$ and $Q$ are points on the longest side $AB$ of triangle $ABC$ such that $AQ = AC$ and $BP = BC$. Prove that the circumcentre of triangle $CPQ$ coincides with the incentre of triangle $ABC$.

2011 ToT Fall Junior Α P2

On side $AB$ of triangle $ABC$ a point $P$ is taken such that $AP = 2PB$. It is known that $CP = 2PQ$ where $Q$ is the midpoint of $AC$. Prove that $ABC$ is a right triangle.

2011 ToT Fall Senior O P3
In a convex quadrilateral $ABCD, AB = 10, BC = 14, CD = 11$ and $DA = 5$. Determine the angle between its diagonals.

2011 ToT Fall Senior A P3
In triangle $ABC$, points $A_1,B_1,C_1$ are bases of altitudes from vertices $A,B,C$, and points $C_A,C_B$ are the projections of $C_1$ to $AC$ and $BC$ respectively. Prove that line $C_AC_B$ bisects the segments $C_1A_1$ and $C_1B_1$.

2012

2012 ToT Spring Junior O P3
In the parallelogram $ABCD$, the diagonal $AC$ touches the incircles of triangles $ABC$ and $ADC$ at $W$ and $Y$ respectively, and the diagonal $BD$ touches the incircles of triangles $BAD$ and $BCD$ at $X$ and $Z$ respectively. Prove that either $W,X, Y$ and $Z$ coincide, or $WXYZ$ is a rectangle.

2012 ToT Spring Junior Α P7
Let $AH$ be an altitude of an equilateral triangle $ABC$. Let $I$ be the incentre of triangle $ABH$, and let $L, K$ and $J$ be the incentres of triangles $ABI,BCI$ and $CAI$ respectively. Determine $\angle KJL$.

2012 ToT Spring Senior O P1
Each vertex of a convex polyhedron lies on exactly three edges, at least two of which have the same length. Prove that the polyhedron has three edges of the same length.

2012 ToT Spring Senior O P4
A quadrilateral $ABCD$ with no parallel sides is inscribed in a circle. Two circles, one passing through $A$ and $B$, and the other through $C$ and $D$, are tangent to each other at $X$. Prove that the locus of $X$ is a circle.

2012 ToT Spring Senior A P4

Alex marked one point on each of the six interior faces of a hollow unit cube. Then he connected by strings any two marked points on adjacent faces. Prove that the total length of these strings is at least $6\sqrt2$.

2012 ToT Spring Senior A P5
Let $\ell$ be a tangent to the incircle of triangle $ABC$. Let $\ell_a,\ell_b$ and $\ell_c$ be the respective images of $\ell$ under reflection across the exterior bisector of $\angle A,\angle  B$ and $\angle C$. Prove that the triangle formed by these lines is congruent to $ABC$.

2012 ToT Fall Junior O P4
A circle touches sides $AB, BC, CD$ of a parallelogram $ABCD$ at points $K, L, M$ respectively. Prove that the line $KL$ bisects the height of the parallelogram drawn from the vertex $C$ to $AB$.

2012 ToT Fall Junior Α P4
Given a triangle $ABC$. Suppose I is its incentre, and $X, Y, Z$ are the incentres of triangles $AIB, BIC$ and $AIC$ respectively. The incentre of triangle $XYZ$ coincides with $I$. Is it necessarily true that triangle $ABC$ is regular?

2012 ToT Fall Junior Α P6
(a) A point $A$ is marked inside a circle. Two perpendicular lines drawn through $A$ intersect the circle at four points. Prove that the centre of mass of these four points does not depend on the choice of the lines.
(b) A regular $2n$-gon ($n \ge 2$) with centre $A$ is drawn inside a circle (A does not necessarily coincide with the centre of the circle). The rays going from $A$ to the vertices of the $2n$-gon mark $2n$ points on the circle. Then the $2n$-gon is rotated about $A$. The rays going from $A$ to the new locations of vertices mark new $2n$ points on the circle. Let $O$ and $N$ be the centres of gravity of old and new points respectively. Prove that $O = N$

2012 ToT Fall Senior O P2
Given a convex polyhedron and a sphere intersecting each its edge at two points so that each edge is trisected (divided into three equal parts). Is it necessarily true that all faces of the polyhedron are
(a) congruent polygons?
(b) regular polygons?

2012 ToT Fall Senior A P4
In a triangle $ABC$ two points, $C_1$ and $A_1$ are marked on the sides $AB$ and $BC$ respectively (the points do not coincide with the vertices). Let $K$ be the midpoint of $A_1C_1$ and $I$ be the incentre of the triangle $ABC$. Given that the quadrilateral $A_1BC_1I$ is cyclic, prove that the angle $AKC$ is obtuse.

2012 ToT Fall Senior A P6
(a) A point $A$ is marked inside a sphere. Three perpendicular lines drawn through $A$ intersect the sphere at six points. Prove that the centre of gravity of these six points does not depend on the choice of such three lines.
(b) An icosahedron with the centre $A$ is placed inside a sphere (its centre does not necessarily coincide with the centre of the sphere). The rays going from $A$ to the vertices of the icosahedron mark $12$ points on the sphere. Then the icosahedron is rotated about its centre. New rays mark new $12$ points on the sphere. Let $O$ and $N$ be the centres of mass of old and new points respectively. Prove that $O = N$.

2013

2013 ToT Spring Junior O P1
There are six points on the plane such that one can split them into two triples each creating a triangle. Is it always possible to split these points into two triples creating two triangles with no common point (neither inside, nor on the boundary)?

2013 ToT Spring Junior O P3
In a quadrilateral $ABCD$, angle $B$ is equal to $150^o$, angle $C$ is right, and sides $AB$ and $CD$ are equal. Determine the angle between $BC$ and the line connecting the midpoints of sides $BC$ and $AD$.  

2013 ToT Spring Junior Α P5
A point in the plane is called a node if both its coordinates are integers. Consider a triangle with vertices at nodes containing exactly two nodes inside. Prove that the straight line connecting these nodes either passes through a vertex or is parallel to a side of the triangle.

2013 ToT Spring Junior Α P6

Let $ABC$ be a right-angled triangle, $I$ its incenter and $B_0, A_0$ points of tangency of the incircle with the legs $AC$ and $BC$ respectively. Let the perpendicular dropped to $AI$ from $A_0$ and the perpendicular dropped to $BI$ from $B_0$ meet at point $P$. Prove that the lines $CP$ and $AB$ are perpendicular.

2013 ToT Spring Senior O P2
Let $C$ be a right angle in triangle $ABC$. On legs $AC$ and$BC$ the squares $ACKL, BCMN$ are constructed outside of triangle. If $CE$ is an altitude of the triangle prove that $LEM$ is a right angle.

2013 ToT Spring Senior A P3
A point in the plane is called a node if both its coordinates are integers. Consider a triangle with vertices at nodes containing at least two nodes inside. Prove that there exists a pair of internal nodes such that a straight line connecting them either passes through a vertex or is parallel to side of the triangle.

2013 ToT Fall Junior O P4
Let $ABC$ be an isosceles triangle. Suppose that points $K$ and $L$ are chosen on lateral sides $AB$ and $AC$ respectively so that $AK = CL$ and $\angle ALK + \angle LKB = 60^o$. Prove that $KL = BC$.

2013 ToT Fall Junior Α P3
Assume that $C$ is a right angle of triangle $ABC$ and $N$ is a midpoint of the semicircle, constructed on $CB$ as on diameter externally. Prove that $AN$ divides the bisector of angle $C$ in half.

2013 ToT Fall Junior Α P5
A $101$-gon is inscribed in a circle. From each vertex of this polygon a perpendicular is dropped to the opposite side or its extension. Prove that at least one perpendicular drops to the side.

2013 ToT Fall Senior O P2
On the sides of triangle $ABC$, three similar triangles are constructed with triangle $YBA$ and triangle $ZAC$ in the exterior and triangle $XBC$ in the interior. (Above, the vertices of the triangles are ordered so that the similarities take vertices to corresponding vertices, for example, the similarity between triangle $YBA$ and triangle $ZAC$ takes $Y$ to $Z, B$ to $A$ and $A$ to $C$). Prove that $AYXZ$ is a parallelogram

2013 ToT Fall Senior A P2
Let $ABC$ be an equilateral triangle with centre $O$. A line through $C$ meets the circumcircle of triangle $AOB$ at points $D$ and $E$. Prove that points $A, O$ and the midpoints of segments $BD, BE$ are concyclic.


2020

2020 ToT Spring Junior O P3
Let $ABCD$ be a rhombus, let $APQC$ be a parallelogram such that the point $B$ lies inside it and the side $AP$ is equal to the side of the rhombus. Prove that $B$ is the orthocenter of the triangle $DPQ$.

Egor Bakaev

2020 ToT Spring Junior A P5
Let $ABCD$ be an inscribed trapezoid. The base $AB$ is $3$ times longer than $CD$. Tangents to the circumscribed circle at the points $A$ and $C$ intersect at the point $K$. Prove that the angle $KDA$ is a right angle.

Alexandr Yuran

2020 ToT Spring Senior O P5
Given are two circles which intersect at points $P$ and $Q$. Consider an arbitrary line $\ell$ through $Q$, let the second points of intersection of this line with the circles be $A$ and $B$ respectively. Let $C$ be the point of intersection of the tangents to the circles in those points. Let $D$ be the intersection of the line $AB$ and the bisector of the angle $CPQ$. Prove that all possible $D$ for any choice of $\ell$ lie on a single circle.

Alexey Zaslavsky

2020 ToT Spring Senior A P5

Let $ABCD$ be an inscribed quadrilateral. Let the circles with diameters $AB$ and $CD$ intersect at two points $X_1$ and $Y_1$, the circles with diameters $BC$ and $AD$ intersect at two points $X_2$ and $Y_2$, the circles with diameters $AC$ and $BD$ intersect at two points $X_3$ and $Y_3$. Prove that the lines $X_1Y_1, X_2Y_2$ and $X_3Y_3$ are concurrent.

Maxim Didin

Oral Rounds

2011 ToT Oral P3

Three pairwise intersecting rays are given. At some point in time not on every ray from its beginning a point begins to move with speed. It is known that these three points form a triangle at any time, and the center of the circumscribed circle of this the triangle also moves uniformly and on a straight line. Is it true, that all these triangles are similar to each other?

2016 ToT Oral P2

On plane there is fixed ray $s$ with vertex $A$ and a point $P$ not on the line which contains $s$. We choose a random point $K$ which lies on ray. Let $N$ be a point on a ray outside $AK$ such that $NK=1$. Let $M$ be a point such that $NM=1,M \in PK$ and $M!=K.$ Prove that all lines $NM$, provided by some point $K$, touch some fixed circle.

2016 ToT Oral P5

In convex hexagonal pyramid 11 edges are equal to 1.Find all possible values of 12th edge.

2020 ToT Oral (15 March) P2

At heights $AA_0, BB_0, CC_0$ of an acute-angled non-equilateral triangle $ABC$, points $A_1, B_1, C_1$ were marked, respectively, so that $AA_1 = BB_1 = CC_1 = R$, where $R$ is the radius of the circumscribed circle of triangle $ABC$. Prove that the center of the circumscribed circle of the triangle $A_1B_1C_1$ coincides with the center of the inscribed circle of triangle $ABC$.

E. Bakaev

sources:
www.math.toronto.edu/oz/turgor/
www.turgor.ru/en