Tournament of Towns 2001-10 (ToT) 69p (UC)

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

[O = Easier, A = Harder]

                                         2001-2010
                            (missing all 2004, 2006 fall , 2008 spring, 2009 spring) 

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.

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 problems are missing from aops

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 $\angle KNA = \angle LNA$.

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 fall problems are missing from aops

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 spring problems are missing from aops

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 spring problems are missing from aops

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 $KB = KC$.

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 P
2011 ToT Spring Junior Α P
2011 ToT Spring Senior O P
2011 ToT Spring Senior O P

2011 ToT Fall Junior O P
2011 ToT Fall Junior Α P
2011 ToT Fall Senior O P
2011 ToT Fall Senior O P

2012

2012 ToT Spring Junior O P
2012 ToT Spring Junior Α P
2012 ToT Spring Senior O P
2012 ToT Spring Senior O P

2012 ToT Fall Junior O P
2012 ToT Fall Junior Α P
2012 ToT Fall Senior O P
2012 ToT Fall Senior O P

2013

2013 ToT Spring Junior O P
2013 ToT Spring Junior Α P
2013 ToT Spring Senior O P
2013 ToT Spring Senior O P

2013 ToT Fall Junior O P
2013 ToT Fall Junior Α P
2013 ToT Fall Senior O P
2013 ToT Fall Senior O P

2014

2014 ToT Spring Junior O P
2014 ToT Spring Junior Α P
2014 ToT Spring Senior O P
2014 ToT Spring Senior O P

2014 ToT Fall Junior O P
2014 ToT Fall Junior Α P
2014 ToT Fall Senior O P
2014 ToT Fall Senior O P

2015

2015 ToT Spring Junior O P
2015 ToT Spring Junior Α P
2015 ToT Spring Senior O P
2015 ToT Spring Senior O P

2015 ToT Fall Junior O P
2015 ToT Fall Junior Α P
2015 ToT Fall Senior O P
2015 ToT Fall Senior O P

2016

2016 ToT Spring Junior O P
2016 ToT Spring Junior Α P
2016 ToT Spring Senior O P
2016 ToT Spring Senior O P

2016 ToT Fall Junior O P
2016 ToT Fall Junior Α P
2016 ToT Fall Senior O P
2016 ToT Fall Senior O P

2017

2017 ToT Spring Junior O P
2017 ToT Spring Junior Α P
2017 ToT Spring Senior O P
2017 ToT Spring Senior O P

2017 ToT Fall Junior O P
2017 ToT Fall Junior Α P
2017 ToT Fall Senior O P
2017 ToT Fall Senior O P


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

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