geometry problems from the old International Mathematics Tournament of Towns (started in 1980)
with aops links in the names
1980 - 1986 complete
1987-2000 under construction
with aops links in the names
1980 - 1986 complete
1987-2000 under construction
J-S stand for Junior / Senior
O - A stand for O - A level
[O = Easier / Training, A = Harder ]
[O = Easier / Training, A = Harder ]
We are given convex quadrilateral $ABCD$. Each of its sides is divided into $N$ line segments of equal length. The points of division of side $AB$ are connected with the points of division of side $CD$ by straight lines (which we call the first set of straight lines), and the points of division of side BC are connected with the points of division of side $DA$ by straight lines (which we call the second set of straight lines) as shown in the diagram, which illustrates the case $N = 4$.
This forms $N^2$ smaller quadrilaterals. From these we choose $N$ quadrilaterals in such a way that any two are at least divided by one line from the first set and one line from the second set. Prove that the sum of the areas of these chosen quadrilaterals is equal to the area of $ABCD$ divided by $N$.
(A Andjans, Riga)
009 1981 TOT Spring J3
$ABCD$ is a convex quadrilateral inscribed in a circle with centre $O$, and with mutually perpendicular diagonals. Prove that the broken line $AOC$ divides the quadrilateral into two parts of equal area.
(V Varvarkin)
012 1981 TOT Spring S1
We will say that two pyramids touch each other by faces if they have no common interior points and if the intersection of a face of one of them with a face of the other is either a triangle or a polygon. Is it possible to place $8$ tetrahedra in such a way that every two of them touch each other by faces?
(A Andjans, Riga)
016 1982 TOT Spring J2
The lengths of all sides and both diagonals of a quadrilateral are less than $1$ metre.
Prove that it may be placed in a circle of radius $0.9$ metres.
028 1982 TOT Autumn S2
Does there exist a polyhedron (not necessarily convex) which could have the following complete list of edges?
$AB, AC, BC, BD, CD, DE, EF, EG, FG, FH, GH, AH$.
030 1982 TOT Autumn S4
(a) $K_1,K_2,..., K_n$ are the feet of the perpendiculars from an arbitrary point $M$ inside a given regular $n$-gon to its sides (or sides produced). Prove that the sum $\overrightarrow{MK_1} + \overrightarrow{MK_2} + ... + \overrightarrow{MK_n}$ equals $\frac{n}{2}\overrightarrow{MO}$, where $O$ is the centre of the $n$-gon.
(b) Prove that the sum of the vectors whose origin is an arbitrary point $M$ inside a given regular tetrahedron and whose endpoints are the feet of the perpendiculars from $M$ to the faces of the tetrahedron equals $\frac43 \overrightarrow{MO}$, where $O$ is the centre of the tetrahedron.
(VV Prasolov, Moscow)
040 1983 TOT Spring S-O2 S-A2
On sides $AB, BC$ and $CA$ of triangle $ABC$ are located points $P, M$ and $K$, respectively, so that $AM, BK$ and $CP$ intersect in one point and the sum of the vectors $\overrightarrow{AM}, \overrightarrow{BK}$ and $\overrightarrow{CP}$ equals $ \overrightarrow{0}$. Prove that $K, M$ and $P$ are midpoints of the sides of triangle $ABC$ on which they are located.
044 1983 TOT Autumn J1
Inside square $ABCD$ consider a point $M$. Prove that the points of intersection of the medians of triangles $ABM, BCM, CDM$ and $DAM$ form a square.
(V Prasolov)
049 1983 TOT Autumn S1
On sides $CB$ and $CD$ of square $ABCD$ are chosen points $M$ and $K$ so that the perimeter of triangle $CMK$ equals double the side of the square. Find angle $MAK$.
054 1984 TOT Spring J-O2
In the convex pentagon $ABCDE, AE = AD, AB = AC$, and angle $CAD$ equals the sum of angles $AEB$ and $ABE$. Prove that segment $CD$ is double the length of median $AM$ of triangle $ABE$.
059 1984 TOT Spring J-A4
Show how to cut an isosceles right triangle into a number of triangles similar to it in such a way that every two of these triangles is of different size.
(AV Savkin)
061 1984 TOT Spring S-O2 S-A2
Six altitudes are constructed from the three vertices of the base of a tetrahedron to the opposite sides of the three lateral faces. Prove that all three straight lines joining two base points of the altitudes in each lateral face are parallel to a certain plane.
(IF Sharygin, Moscow)
067 1984 TOT Autumn Training J1
In triangle $ABC$ the bisector of the angle at $B$ meets $AC$ at $D$ and the bisector of the angle at $C$ meets $AB$ at $E$. These bisectors intersect at $O$ and the lengths of $OD$ and $OE$ are equal. Prove that either $\angle BAC = 60^o$ or triangle $ABC$ is isosceles.
070 1984 TOT Autumn Training J4
Inside a rectangle is inscribed a quadrilateral, which has a vertex on each side of the rectangle. Prove that the perimeter of the inscribed quadrilateral is not smaller than double the length of a diagonal of the rectangle.
(V. V . Proizvolov , Moscow)
075 1984 TOT Autumn Training S1
In convex hexagon $ABCDEF, AB$ is parallel to $CF, CD$ is parallel to $BE$ and $EF$ is parallel to $AD$. Prove that the areas of triangles $ACE$ and $BDF$ are equal .
076 1984 TOT Autumn Training S3
In $\vartriangle ABC, \angle ABC = \angle ACB = 40^o$ . $BD$ bisects $\angle ABC$ , with $D$ located on $AC$. Prove that $BD + DA = BC$.
080 1985 TOT Spring Training J1
A median , a bisector and an altitude of a certain triangle intersect at an inner point $O$ . The segment of the bisector from the vertex to $O$ is equal to the segment of the altitude from the vertex to $O$ . Prove that the triangle is equilateral .
088 1985 TOT Spring J4
A square is divided into $5$ rectangles in such a way that its $4$ vertices belong to $4$ of the rectangles , whose areas are equal , and the fifth rectangle has no points in common with the side of the square (see diagram) . Prove that the fifth rectangle is a square.
090 1985 TOT Spring Training S1
In quadrilateral ABCD it is given that $AB = BC = 1, \angle ABC = 100^o$ , and $\angle CDA = 130^o$ . Find the length of $BD$.
093 1985 TOT Spring S1
Prove that the area of a unit cube's projection on any plane equals the length of its projection on the perpendicular to this plane.
Prove that each student would obtain the same result .
We are given a convex quadrilateral and point $M$ inside it . The perimeter of the quadrilateral has length $L$ while the lengths of the diagonals are $D_1$ and $D_2$. Prove that the sum of the distances from $M$ to the vertices of the quadrilateral are not greater than $L + D_1 + D_2$ .
(V. Prasolov)
(a) The point $O$ lies inside the convex polygon $A_1A_2A_3...A_n$ . Consider all the angles $A_iOA_j$ where $i, j$ are distinct natural numbers from $1$ to $n$ . Prove that at least $n- 1$ of these angles are not acute .
(b) Same problem for a convex polyhedron with $n$ vertices.
(V. Boltyanskiy, Moscow)
In triangle $ABC, AH$ is an altitude ($H$ is on $BC$) and $BE$ is a bisector ($E$ is on $AC$) . We are given that angle $BEA$ equals $45^o$ .Prove that angle $EHC$ equals $45^o$ .
(I. Sharygin , Moscow)
This forms $N^2$ smaller quadrilaterals. From these we choose $N$ quadrilaterals in such a way that any two are at least divided by one line from the first set and one line from the second set. Prove that the sum of the areas of these chosen quadrilaterals is equal to the area of $ABCD$ divided by $N$.
(A Andjans, Riga)
009 1981 TOT Spring J3
$ABCD$ is a convex quadrilateral inscribed in a circle with centre $O$, and with mutually perpendicular diagonals. Prove that the broken line $AOC$ divides the quadrilateral into two parts of equal area.
(V Varvarkin)
012 1981 TOT Spring S1
We will say that two pyramids touch each other by faces if they have no common interior points and if the intersection of a face of one of them with a face of the other is either a triangle or a polygon. Is it possible to place $8$ tetrahedra in such a way that every two of them touch each other by faces?
(A Andjans, Riga)
016 1982 TOT Spring J2
The lengths of all sides and both diagonals of a quadrilateral are less than $1$ metre.
Prove that it may be placed in a circle of radius $0.9$ metres.
028 1982 TOT Autumn S2
Does there exist a polyhedron (not necessarily convex) which could have the following complete list of edges?
$AB, AC, BC, BD, CD, DE, EF, EG, FG, FH, GH, AH$.
030 1982 TOT Autumn S4
(a) $K_1,K_2,..., K_n$ are the feet of the perpendiculars from an arbitrary point $M$ inside a given regular $n$-gon to its sides (or sides produced). Prove that the sum $\overrightarrow{MK_1} + \overrightarrow{MK_2} + ... + \overrightarrow{MK_n}$ equals $\frac{n}{2}\overrightarrow{MO}$, where $O$ is the centre of the $n$-gon.
(b) Prove that the sum of the vectors whose origin is an arbitrary point $M$ inside a given regular tetrahedron and whose endpoints are the feet of the perpendiculars from $M$ to the faces of the tetrahedron equals $\frac43 \overrightarrow{MO}$, where $O$ is the centre of the tetrahedron.
(VV Prasolov, Moscow)
040 1983 TOT Spring S-O2 S-A2
On sides $AB, BC$ and $CA$ of triangle $ABC$ are located points $P, M$ and $K$, respectively, so that $AM, BK$ and $CP$ intersect in one point and the sum of the vectors $\overrightarrow{AM}, \overrightarrow{BK}$ and $\overrightarrow{CP}$ equals $ \overrightarrow{0}$. Prove that $K, M$ and $P$ are midpoints of the sides of triangle $ABC$ on which they are located.
044 1983 TOT Autumn J1
Inside square $ABCD$ consider a point $M$. Prove that the points of intersection of the medians of triangles $ABM, BCM, CDM$ and $DAM$ form a square.
(V Prasolov)
049 1983 TOT Autumn S1
On sides $CB$ and $CD$ of square $ABCD$ are chosen points $M$ and $K$ so that the perimeter of triangle $CMK$ equals double the side of the square. Find angle $MAK$.
054 1984 TOT Spring J-O2
In the convex pentagon $ABCDE, AE = AD, AB = AC$, and angle $CAD$ equals the sum of angles $AEB$ and $ABE$. Prove that segment $CD$ is double the length of median $AM$ of triangle $ABE$.
059 1984 TOT Spring J-A4
Show how to cut an isosceles right triangle into a number of triangles similar to it in such a way that every two of these triangles is of different size.
(AV Savkin)
061 1984 TOT Spring S-O2 S-A2
Six altitudes are constructed from the three vertices of the base of a tetrahedron to the opposite sides of the three lateral faces. Prove that all three straight lines joining two base points of the altitudes in each lateral face are parallel to a certain plane.
(IF Sharygin, Moscow)
067 1984 TOT Autumn Training J1
In triangle $ABC$ the bisector of the angle at $B$ meets $AC$ at $D$ and the bisector of the angle at $C$ meets $AB$ at $E$. These bisectors intersect at $O$ and the lengths of $OD$ and $OE$ are equal. Prove that either $\angle BAC = 60^o$ or triangle $ABC$ is isosceles.
070 1984 TOT Autumn Training J4
Inside a rectangle is inscribed a quadrilateral, which has a vertex on each side of the rectangle. Prove that the perimeter of the inscribed quadrilateral is not smaller than double the length of a diagonal of the rectangle.
(V. V . Proizvolov , Moscow)
075 1984 TOT Autumn Training S1
In convex hexagon $ABCDEF, AB$ is parallel to $CF, CD$ is parallel to $BE$ and $EF$ is parallel to $AD$. Prove that the areas of triangles $ACE$ and $BDF$ are equal .
076 1984 TOT Autumn Training S3
In $\vartriangle ABC, \angle ABC = \angle ACB = 40^o$ . $BD$ bisects $\angle ABC$ , with $D$ located on $AC$. Prove that $BD + DA = BC$.
080 1985 TOT Spring Training J1
A median , a bisector and an altitude of a certain triangle intersect at an inner point $O$ . The segment of the bisector from the vertex to $O$ is equal to the segment of the altitude from the vertex to $O$ . Prove that the triangle is equilateral .
088 1985 TOT Spring J4
A square is divided into $5$ rectangles in such a way that its $4$ vertices belong to $4$ of the rectangles , whose areas are equal , and the fifth rectangle has no points in common with the side of the square (see diagram) . Prove that the fifth rectangle is a square.
In quadrilateral ABCD it is given that $AB = BC = 1, \angle ABC = 100^o$ , and $\angle CDA = 130^o$ . Find the length of $BD$.
093 1985 TOT Spring S1
Prove that the area of a unit cube's projection on any plane equals the length of its projection on the perpendicular to this plane.
094 1985 TOT Spring S2
The radius $OM$ of a circle rotates uniformly at a rate of $360/n$ degrees per second , where $n$ is a positive integer . The initial radius is $OM_0$. After $1$ second the radius is $OM_1$ , after two more seconds (i.e. after three seconds altogether) the radius is $OM_2$ , after $3$ more seconds (after $6$ seconds altogether) the radius is $OM_3$, ..., after $n - 1$ more seconds its position is $OM_{n-1}$. For which values of $n$ do the points $M_0, M_1 , ..., M_{n-1}$ divide the circle into $n$ equal arcs?
(a) Is it true that the powers of $2$ are such values?
(b) Does there exist such a value which is not a power of $2$?
(V. V. Proizvolov , Moscow)
A teacher gives each student in the class the following task in their exercise book .
"Take two concentric circles of radius $1$ and $10$ . To the smaller circle produce three tangents whose intersections $A, B$ and $C$ lie in the larger circle . Measure the area $S$ of triangle $ABC$, and areas $S_1 , S_2$ and $S_3$ , the three sector-like regions with vertices at $A, B$ and $C$ (as in the diagram). Find the value of $S_1 +S_2 +S_3 -S$."Prove that each student would obtain the same result .
(V. Prasolov)
(b) Same problem for a convex polyhedron with $n$ vertices.
(V. Boltyanskiy, Moscow)
In triangle $ABC, AH$ is an altitude ($H$ is on $BC$) and $BE$ is a bisector ($E$ is on $AC$) . We are given that angle $BEA$ equals $45^o$ .Prove that angle $EHC$ equals $45^o$ .
(I. Sharygin , Moscow)
Through vertices $A$ and $B$ of triangle $ABC$ are constructed two lines which divide the triangle into four regions (three triangles and one quadrilateral). It is known that three of them have equal area. Prove that one of these three regions is the quadrilateral .
(G . Galperin , A . Savin, Moscow)
110 1986 TOT Spring J3
We are given the square $ABCD$. On sides $AB$ and $CD$ we are given points$ K$ and $L$ respectively, and on segment $KL$ we are given point $M$ . Prove that the second point of intersection (i.e. the one other than $M$) of the points of intersection of circles circumscribed about triangles $AKM$ and $MLC$ lies on the diagonal $AC$.
(V . N . Dubrovskiy)
115 1986 TOT Spring S3
Vectors coincide with the edges of an arbitrary tetrahedron (possibly non-regular). Is it possible for the sum of these six vectors to equal the zero vector?
(Problem from Leningrad)
117 1986 TOT Spring S5
Square $ABCD$ and circle $O$ intersect in eight points, forming four curvilinear triangles, $AEF , BGH , CIJ$ and $DKL$ ($EF , GH, IJ$ and $KL$ are arcs of the circle) . Prove that
(a) The sum of lengths of $EF$ and $IJ$ equals the sum of the lengths of $GH$ and $KL$.
(b) The sum of the perimeters of curvilinear triangles $AEF$ and $CIJ$ equals the sum of the perimeters of the curvilinear triangles $BGH$ and $DKL$.
( V . V . Proizvolov , Moscow)
123 1986 TOT Autumn J5
Find the locus of the orthocentres (i.e. the point where three altitudes meet) of the triangles inscribed in a given circle .
(A. Andjans, Riga)
126 1986 TOT Autumn S1
We are given trapezoid $ABCD$ and point $M$ on the intersection of its diagonals. The parallel sides are $AD$ and $BC$ and it is known that $AB$ is perpendicular to $AD$ and that the trapezoid can have an inscribed circle. If the radius of this inscribed circle is $R$ find the area of triangle $DCM$ .
133 1987 TOT Spring Training J2
In an acute angled triangle the bases of the altitudes are joined to form a new triangle. In this new triangle it is known that two sides are parallel to sides of the original triangle . Prove that the third side is also parallel to one of the sides of the original triangle .
139 1987 TOT Spring Main J4
Angle $A$ of the acute-angled triangle $ABC$ equals $60^o$ . Prove that the bisector of one of the angles formed by the altitudes drawn from $B$ and $C$, passes through the circumcircle 's centre.
(V . Pogrebnyak , year 12 student , Vinnitsa)
144 1987 TOT Spring Training S2
In $3$ dimensional space we are given a parallelogram $ABCD$ and plane $M$. The distances from vertices $A, B$ and $C$ to plane $M$ are $a, b$ and $c$ respectively. Find the distance $d$ from vertex $D$ to the plane $M$ .
145 1987 TOT Spring Main S2
to be continued(G . Galperin , A . Savin, Moscow)
110 1986 TOT Spring J3
We are given the square $ABCD$. On sides $AB$ and $CD$ we are given points$ K$ and $L$ respectively, and on segment $KL$ we are given point $M$ . Prove that the second point of intersection (i.e. the one other than $M$) of the points of intersection of circles circumscribed about triangles $AKM$ and $MLC$ lies on the diagonal $AC$.
(V . N . Dubrovskiy)
115 1986 TOT Spring S3
Vectors coincide with the edges of an arbitrary tetrahedron (possibly non-regular). Is it possible for the sum of these six vectors to equal the zero vector?
(Problem from Leningrad)
The bisector of angle $BAD$ in the parallelogram $ABCD$ intersects the lines $BC$ and $CD$ at the points $K$ and $L$ respectively. It is known that $ABCD$ is not a rhombus. Prove that the centre of the circle passing through the points $C, K$ and $L$ lies on the circle passing through the points $B, C$ and $D$.
Square $ABCD$ and circle $O$ intersect in eight points, forming four curvilinear triangles, $AEF , BGH , CIJ$ and $DKL$ ($EF , GH, IJ$ and $KL$ are arcs of the circle) . Prove that
(a) The sum of lengths of $EF$ and $IJ$ equals the sum of the lengths of $GH$ and $KL$.
(b) The sum of the perimeters of curvilinear triangles $AEF$ and $CIJ$ equals the sum of the perimeters of the curvilinear triangles $BGH$ and $DKL$.
( V . V . Proizvolov , Moscow)
123 1986 TOT Autumn J5
Find the locus of the orthocentres (i.e. the point where three altitudes meet) of the triangles inscribed in a given circle .
(A. Andjans, Riga)
126 1986 TOT Autumn S1
We are given trapezoid $ABCD$ and point $M$ on the intersection of its diagonals. The parallel sides are $AD$ and $BC$ and it is known that $AB$ is perpendicular to $AD$ and that the trapezoid can have an inscribed circle. If the radius of this inscribed circle is $R$ find the area of triangle $DCM$ .
133 1987 TOT Spring Training J2
In an acute angled triangle the bases of the altitudes are joined to form a new triangle. In this new triangle it is known that two sides are parallel to sides of the original triangle . Prove that the third side is also parallel to one of the sides of the original triangle .
139 1987 TOT Spring Main J4
Angle $A$ of the acute-angled triangle $ABC$ equals $60^o$ . Prove that the bisector of one of the angles formed by the altitudes drawn from $B$ and $C$, passes through the circumcircle 's centre.
(V . Pogrebnyak , year 12 student , Vinnitsa)
144 1987 TOT Spring Training S2
In $3$ dimensional space we are given a parallelogram $ABCD$ and plane $M$. The distances from vertices $A, B$ and $C$ to plane $M$ are $a, b$ and $c$ respectively. Find the distance $d$ from vertex $D$ to the plane $M$ .
Α disk of radius $1$ is covered by seven identical disks. Prove that their radii are not less than $\frac12$ .
148 1987 TOT Spring Main S5
Perpendiculars are drawn from an interior point $M$ of the equilateral triangle $ABC$ to its sides , intersecting them at points $D, E$ and $F$ . Find the locus of all points $M$ such that $DEF$ is a right triangle .
(J . Tabov , Sofia)
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