geometry problems from Vietnamese Mathematical Olympiads (VMO)

with aops links in the names

with aops links in the names

1962 - 2019

Let $ ABCD$ is a tetrahedron. Denote by $ A'$, $ B'$ the feet of the perpendiculars from $ A$ and $ B$, respectively to the opposite faces. Show that $ AA'$ and $ BB'$ intersect if and only if $ AB$ is perpendicular to $ CD$. Do they intersect if $ AC = AD = BC = BD$?

Let be given a tetrahedron $ ABCD$ such that triangle $ BCD$ equilateral and $ AB = AC = AD$. The height is $ h$ and the angle between two planes $ ABC$ and $ BCD$ is $ \alpha$. The point $ X$ is taken on $ AB$ such that the plane $ XCD$ is perpendicular to $ AB$. Find the volume of the tetrahedron $XBCD$.

The tetrahedron $SABC$ has the perpendicular faces $SBC$ and $ABC$. The three angles at $S$ are all $60^o$ and $SB = SC = 1$. Find the volume of the tetrahedron.

The triangle ABC has half-perimeter $p$. Find the length of the side $a$ and the area $S$ in terms of $\angle A, \angle B$ and $p$. In particular, find $ S$ if $ p \approx 23.6$, $ \angle A \approx 52^{\circ}42'$, $ \angle B \approx 46^{\circ}16'$.

Let $P$ be a plane and two points $A \in (P),O \notin (P)$. For each line in $(P)$ through $A$, let $H$ be the foot of the perpendicular from $O$ to the line. Find the locus $(c)$ of $H$.
Denote by $(C)$ the oblique cone with peak $O$ and base $(c)$. Prove that all planes, either parallel to $(P)$ or perpendicular to $OA$, intersect $(C)$ by circles.
Consider the two symmetric faces of $(C)$ that intersect $(C)$ by the angles $\alpha$ and $\beta$ respectively. Find a relation between $\alpha$ and $\beta$.

At a time $t = 0$, a navy ship is at a point $O$, while an enemy ship is at a point $A$ cruising with speed $v$ perpendicular to $OA = a$. The speed and direction of the enemy ship do not change. The strategy of the navy ship is to travel with constant speed $u$ at a angle $0 < \phi < \pi /2$ to the line OA.

i) Let $\phi$ be chosen. What is the minimum distance between the two ships? Under what conditions will the distance vanish?

ii) If the distance does not vanish, what is the choice of $\phi$ to minimize the distance? What are directions of the two ships when their distance is minimum?

1965 VMO problem 2

$AB$ and $CD$ are two fixed parallel chords of the circle $S$. $M$ is a variable point on the circle. $Q$ is the intersection of the lines $MD$ and $AB$. $X$ is the circumcenter of the triangle $MCQ$.

Find the locus of $X$.

What happens to $X$ as $M$ tends to

(i) $D$,

(ii) $C$?

Find a point $E$ outside the plane of $S$ such that the circumcenter of the tetrahedron $MCQE$ has the same locus as $X$.

1966 VMO problem 2

$a, b$ are two fixed lines through $O$. Variable lines $x, y$ are parallel. $x$ intersects a at $A$ and $b$ at $C$, $y$ intersects $a$ at $B$ and $b$at $D$. The lines $AD$ and $BC$ meet at $M$. The line through $M$ parallel to $x$ meets $a$ at $L$ and $b$ at $N$. What can you say about $L, M, N$? Find the locus $M$.

1967 VMO problem 3

i) $ABCD$ is a rhombus. A tangent to the inscribed circle meets $AB, DA, BC, CD$ at $M, N, P, Q$ respectively. Find a relationship between $BM$ and $DN$.

ii) $ABCD$ is a rhombus and $P$ a point inside. The circles through $P$ with centers $A, B, C, D$ meet the four sides $AB, BC, CD, DA$ in eight points. Find a property of the resulting octagon. Use it to construct a regular octagon.

iii) Rotate the figure about the line $AC$ to form a solid. State a similar result.

1968 VMO problem 2

Let $(I, r)$ be a circle centered at $I$ of radius $r, x$ and $y$ be two parallel lines on the plane with a distance $h$ apart. A variable triangle $ABC$ with $A$ on $x, B$ and $C$ on $y$ has $(I, r)$ as its incircle.

i) Given $(I, r), \alpha$ and x, y, construct a triangle $ABC$ so that $\angle A = \alpha$.

ii) Calculate angles $\angle B$ and $\angle C$ in terms of $h, r$ and $\alpha$.

iii) If the incircle touches the side $BC$ at $D$, find a relation between $DB$ and $DC$.

1969 VMO problem 4

Two circles centers $O$ and $O'$, radii $R$ and $R'$, meet at two points. A variable line $L$ meets the circles at $A, C, B, D$ in that order and $\frac{AC}{AD} = \frac{CB}{BD}$. The perpendiculars from $O$ and $O'$ to $L$ have feet $H$ and $H'$.

Find the locus of $H$ and $H'$.

If $OO'^2 < R^2 + R'^2$, find a point $P$ on $L$ such that $PO + PO'$ has the smallest possible value.

Show that this value does not depend on the position of $L$.

Comment on the case $OO'^2 > R^2 + R'^2$.

1972 VMO problem 3

$ABC$ is a triangle. $U$ is a point on the line $BC$. $I$ is the midpoint of $BC$. The line through $C$ parallel to $AI$ meets the line $AU$ at $E$. The line through $E$ parallel to $BC$ meets the line $AB$ at $F$. The line through $E$ parallel to $AB$ meets the line $BC$ at $H$. The line through $H$ parallel to $AU$ meets the line $AB$ at $K$. The lines $HK$ and $FG$ meet at $T. V$ is the point on the line $AU$ such that $A$ is the midpoint of $UV$. Show that $V, T$ and $I$ are collinear.

1972 VMO problem 4

Let $ABCD$ be a regular tetrahedron with side $a$. Take $E,E'$ on the edge $AB, F, F'$ on the edge $AC$ and $G,G'$ on the edge AD so that $AE =a/6,AE' = 5a/6,AF= a/4,AF'= 3a/4,AG = a/3,AG'= 2a/3$. Compute the volume of $EFGE'F'G'$ in term of $a$ and find the angles between the lines $AB,AC,AD$ and the plane $EFG$.

i) Let $\phi$ be chosen. What is the minimum distance between the two ships? Under what conditions will the distance vanish?

ii) If the distance does not vanish, what is the choice of $\phi$ to minimize the distance? What are directions of the two ships when their distance is minimum?

1965 VMO problem 2

$AB$ and $CD$ are two fixed parallel chords of the circle $S$. $M$ is a variable point on the circle. $Q$ is the intersection of the lines $MD$ and $AB$. $X$ is the circumcenter of the triangle $MCQ$.

Find the locus of $X$.

What happens to $X$ as $M$ tends to

(i) $D$,

(ii) $C$?

Find a point $E$ outside the plane of $S$ such that the circumcenter of the tetrahedron $MCQE$ has the same locus as $X$.

1966 VMO problem 2

$a, b$ are two fixed lines through $O$. Variable lines $x, y$ are parallel. $x$ intersects a at $A$ and $b$ at $C$, $y$ intersects $a$ at $B$ and $b$at $D$. The lines $AD$ and $BC$ meet at $M$. The line through $M$ parallel to $x$ meets $a$ at $L$ and $b$ at $N$. What can you say about $L, M, N$? Find the locus $M$.

1967 VMO problem 3

i) $ABCD$ is a rhombus. A tangent to the inscribed circle meets $AB, DA, BC, CD$ at $M, N, P, Q$ respectively. Find a relationship between $BM$ and $DN$.

ii) $ABCD$ is a rhombus and $P$ a point inside. The circles through $P$ with centers $A, B, C, D$ meet the four sides $AB, BC, CD, DA$ in eight points. Find a property of the resulting octagon. Use it to construct a regular octagon.

iii) Rotate the figure about the line $AC$ to form a solid. State a similar result.

Let $(I, r)$ be a circle centered at $I$ of radius $r, x$ and $y$ be two parallel lines on the plane with a distance $h$ apart. A variable triangle $ABC$ with $A$ on $x, B$ and $C$ on $y$ has $(I, r)$ as its incircle.

i) Given $(I, r), \alpha$ and x, y, construct a triangle $ABC$ so that $\angle A = \alpha$.

ii) Calculate angles $\angle B$ and $\angle C$ in terms of $h, r$ and $\alpha$.

iii) If the incircle touches the side $BC$ at $D$, find a relation between $DB$ and $DC$.

1969 VMO problem 4

Two circles centers $O$ and $O'$, radii $R$ and $R'$, meet at two points. A variable line $L$ meets the circles at $A, C, B, D$ in that order and $\frac{AC}{AD} = \frac{CB}{BD}$. The perpendiculars from $O$ and $O'$ to $L$ have feet $H$ and $H'$.

Find the locus of $H$ and $H'$.

If $OO'^2 < R^2 + R'^2$, find a point $P$ on $L$ such that $PO + PO'$ has the smallest possible value.

Show that this value does not depend on the position of $L$.

Comment on the case $OO'^2 > R^2 + R'^2$.

$AB$ and $CD$ are perpendicular diameters of a circle. $L$ is the tangent to the circle at $A$. $M$ is a variable point on the minor arc $AC$. The ray $BM, DM$ meet the line $L$ at $P$ and $Q$ respectively. Show that $AP\cdot AQ = AB\cdot PQ$.

Show how to construct the point $M$ which gives$ BQ$ parallel to $DP$.

If the lines $OP$ and $BQ$ meet at $N$ find the locus of $N$.

The lines $BP$ and $BQ$ meet the tangent at $D$ at $P'$ and $Q'$ respectively. Find the relation between $P'$ and $Q$'.

The lines $D$P and $DQ$ meet the line $BC$ at $P"$ and $Q"$ respectively. Find the relation between $P"$ and $Q"$.

A plane $(P)$ passes through a vertex $A$ of a cube $ABCDEFGH$ and the three edges $AB,AD,AE$ make equal angles with $(P)$.

i) Compute the cosine of that common angle and find the perpendicular projection of the cube onto the plane.

ii) Find some relationships between $(P)$ and lines passing through two vertices of the cube and planes passing through three vertices of the cube.

1971 VMO problem 2

$ABCDA'B'C'D'$ is a cube (with $ABCD$ and $A'B'C'D'$ faces, and $AA', BB', CC', DD'$ edges). $L$ is a line which intersects or is parallel to the lines $AA', BC$ and $DB'$. $L$ meets the line $BC$ at $M$ (which may be the point at infinity). Let $m = |BM|$. The plane $MAA'$ meets the line $B'C'$ at $E$. Show that $|B'E| = m$. The plane $MDB'$ meets the line $A'D'$ at $F$.

Show that $|D'F| = m$.

Hence or otherwise show how to construct the point $P$ at the intersection of $L$ and the plane $A'B'C'D'$.

Find the distance between $P$ and the line $A'B'$ and the distance between $P$ and the line $A'D'$ in terms of $m$.

Find a relation between these two distances that does not depend on $m$.

Find the locus of $M$.

Let $S$ be the envelope of the line $L$ as $M$ varies. Find the intersection of $S$ with the faces of the cube.

1971 VMO problem 2

$ABCDA'B'C'D'$ is a cube (with $ABCD$ and $A'B'C'D'$ faces, and $AA', BB', CC', DD'$ edges). $L$ is a line which intersects or is parallel to the lines $AA', BC$ and $DB'$. $L$ meets the line $BC$ at $M$ (which may be the point at infinity). Let $m = |BM|$. The plane $MAA'$ meets the line $B'C'$ at $E$. Show that $|B'E| = m$. The plane $MDB'$ meets the line $A'D'$ at $F$.

Show that $|D'F| = m$.

Hence or otherwise show how to construct the point $P$ at the intersection of $L$ and the plane $A'B'C'D'$.

Find the distance between $P$ and the line $A'B'$ and the distance between $P$ and the line $A'D'$ in terms of $m$.

Find a relation between these two distances that does not depend on $m$.

Find the locus of $M$.

Let $S$ be the envelope of the line $L$ as $M$ varies. Find the intersection of $S$ with the faces of the cube.

1972 VMO problem 3

$ABC$ is a triangle. $U$ is a point on the line $BC$. $I$ is the midpoint of $BC$. The line through $C$ parallel to $AI$ meets the line $AU$ at $E$. The line through $E$ parallel to $BC$ meets the line $AB$ at $F$. The line through $E$ parallel to $AB$ meets the line $BC$ at $H$. The line through $H$ parallel to $AU$ meets the line $AB$ at $K$. The lines $HK$ and $FG$ meet at $T. V$ is the point on the line $AU$ such that $A$ is the midpoint of $UV$. Show that $V, T$ and $I$ are collinear.

1972 VMO problem 4

Let $ABCD$ be a regular tetrahedron with side $a$. Take $E,E'$ on the edge $AB, F, F'$ on the edge $AC$ and $G,G'$ on the edge AD so that $AE =a/6,AE' = 5a/6,AF= a/4,AF'= 3a/4,AG = a/3,AG'= 2a/3$. Compute the volume of $EFGE'F'G'$ in term of $a$ and find the angles between the lines $AB,AC,AD$ and the plane $EFG$.

In 1973 it did not take place.

Let $ABC$ be a triangle with $A = 90^o, AH$ the altitude, $P,Q$ the feet of the perpendiculars from $H$ to $AB,AC$ respectively. Let $M$ be a variable point on the line $PQ$. The line through $M$ perpendicular to $MH$ meets the lines $AB,AC$ at $R, S$ respectively.

i) Prove that circumcircle of $ARS$ always passes the fixed point $H$.

ii) Let $M_1$ be another position of $M$ with corresponding points $R_1, S_1$. Prove that the ratio $RR_1/SS_1$ is constant.

iii) The point $K$ is symmetric to $H$ with respect to $M$. The line through $K$ perpendicular to the line $PQ$ meets the line $RS$ at $D$. Prove that$ \angle BHR = \angle DHR, \angle DHS = \angle CHS$.

$C$ is a cube side $1$. The $12$ lines containing the sides of the cube meet at plane $p$ in $12$ points. What can you say about the $12$ points?

Let $ABCD$ be a tetrahedron with $BA \perp AC,DB \perp (BAC)$. Denote by $O$ the midpoint of $AB$, and $K$ the foot of the perpendicular from $O$ to $DC$. Suppose that $AC = BD$. Prove that $\frac{V_{KOAC}}{V_{KOBD}}=\frac{AC}{BD}$ if and only if $2AC \cdot BD = AB^2$.

In the space given a fixed line $\ell$ and a fixed point $A \notin \ell$, a variable line $d$ passes through $A$. Denote by $MN$ the common perpendicular between $d$ and $\ell$ ($M \in d,N \in \ell$). Find the locus of $M$ and the locus of the midpoint $I$ of $MN$.

1976 VMO problem 3

$P$ is a point inside the triangle $ABC$. The perpendicular distances from $P$ to the three sides have product $p$. Show that $p \le \frac{ 8 S^3}{27abc}$, where $S =$ area $ABC$ and $a, b, c$ are the sides. Prove a similar result for a tetrahedron.

$P$ is a point inside the triangle $ABC$. The perpendicular distances from $P$ to the three sides have product $p$. Show that $p \le \frac{ 8 S^3}{27abc}$, where $S =$ area $ABC$ and $a, b, c$ are the sides. Prove a similar result for a tetrahedron.

$L, L'$ are two skew lines in space and $p$ is a plane not containing either line. $M$ is a variable line parallel to $p$ which meets $L$ at $X$ and $L'$ at $Y$. Find the position of $M$ which minimises the distance $XY$. $L''$ is another fixed line. Find the line $M$ which is also perpendicular to $L''$ .

1978 VMO problem 3

The triangle $ABC$ has angle $A = 30^o$ and $AB = \frac{3}{4} AC$. Find the point $P$ inside the triangle which minimizes $5 PA + 4 PB + 3 PC$.

Let $P$ be a point inside a triangle $A_1A_2A_3$. For $i = 1, 2, 3$, line $PA_i$ intersects the side opposite to $A_i$ at $B_i$. Let $C_i$ and $D_i$ be the midpoints of $A_iB_i$ and $PB_i$, respectively. Prove that the areas of the triangles $C_1C_2C_3$ and $D_1D_2D_3$ are equal.

1980 VMO problem 4

Prove that for any tetrahedron in space, it is possible to find two perpendicular planes such that ratio between the projections of the tetrahedron on the two planes lies in the interval $[\frac{1}{\sqrt{2}}, \sqrt{2}].$

1981 VMO problem 3

2019 VMO problem

Let $ABC$ be triangle with $H$ is the orthocenter and $I$ is incenter. Denote $A_{1}, A_{2}, B_{1}, B_{2}, C_{1}, C_{2}$ be the points on the rays $AB, AC, BC, CA, CB$, respectively such that $$AA_{1} = AA_{2} = BC, BB_{1} = BB_{2} = CA, CC_{1} = CC_{2} = AB.$$Suppose that $B_{1}B_{2}$ cuts $C_{1}C_{2}$ at $A'$, $C_{1}C_{2}$ cuts $A_{1}A_{2}$ at $B'$ and $A_{1}A_{2}$ cuts $B_{1}B_{2}$ at $C'$.

a) Prove that area of triangle $A'B'C'$ is smaller than or equal to the area of triangle $ABC$.

b) Let $J$ be circumcenter of triangle $A'B'C'$. $AJ$ cuts $BC$ at $R$, $BJ$ cuts $CA$ at $S$ and $CJ$ cuts $AB$ at $T$. Suppose that $(AST), (BTR), (CRS)$ intersect at $K$. Prove that if triangle $ABC$ is not isosceles then $HIJK$ is a parallelogram.

2019 VMO problem

Let $ABC$ be an acute, nonisosceles triangle with inscribe in a circle $(O)$ and has orthocenter $H$. Denote $M,N,P$ as the midpoints of sides $BC,CA,AB$ and $D,E,F$ as the feet of the altitudes from vertices $A,B,C$ of triangle $ABC$. Let $K$ as the reflection of $H$ through $BC$. Two lines $DE,MP$ meet at $X$; two lines $DF,MN$ meet at $Y$.

a) The line $XY$ cut the minor arc $BC$ of $(O)$ at $Z$. Prove that $K,Z,E,F$ are concyclic.

b) Two lines $KE,KF$ cuts $(O)$ second time at $S,T$. Prove that $BS,CT,XY$ are concurrent.

The triangle $ABC$ has angle $A = 30^o$ and $AB = \frac{3}{4} AC$. Find the point $P$ inside the triangle which minimizes $5 PA + 4 PB + 3 PC$.

1978 VMO problem 6

Given a rectangular parallelepiped $ABCDA'B'C'D'$ with the bases $ABCD, A'B'C'D'$, the edges $AA',BB', CC',DD'$ and $AB = a,AD = b,AA' = c$. Show that there exists a triangle with the sides equal to the distances from $A,A',D$ to the diagonal $BD'$ of the parallelepiped. Denote those distances by $m_1,m_2,m_3$. Find the relationship between $a, b, c,m_1,m_2,m_3$.

Given a rectangular parallelepiped $ABCDA'B'C'D'$ with the bases $ABCD, A'B'C'D'$, the edges $AA',BB', CC',DD'$ and $AB = a,AD = b,AA' = c$. Show that there exists a triangle with the sides equal to the distances from $A,A',D$ to the diagonal $BD'$ of the parallelepiped. Denote those distances by $m_1,m_2,m_3$. Find the relationship between $a, b, c,m_1,m_2,m_3$.

1979 VMO problem 3

Let $ABC$ be a triangle with sides that are not equal. Find point $X$ on $BC$ such that $\frac{area \triangle ABX}{area \triangle ACX} = \frac{perimeter \triangle ABX}{perimeter \triangle ACX}$.

1980 VMO problem 3Let $ABC$ be a triangle with sides that are not equal. Find point $X$ on $BC$ such that $\frac{area \triangle ABX}{area \triangle ACX} = \frac{perimeter \triangle ABX}{perimeter \triangle ACX}$.

1979 VMO problem 6

$ABCD$ is a rectangle with $BC / AB = \sqrt2$. $ABEF$ is a congruent rectangle in a different plane. Find the angle $DAF$ such that the lines $CA$ and $BF$ are perpendicular. In this configuration, find two points on the line $CA$ and two points on the line $BF$ so that the four points form a regular tetrahedron.

$ABCD$ is a rectangle with $BC / AB = \sqrt2$. $ABEF$ is a congruent rectangle in a different plane. Find the angle $DAF$ such that the lines $CA$ and $BF$ are perpendicular. In this configuration, find two points on the line $CA$ and two points on the line $BF$ so that the four points form a regular tetrahedron.

Let $P$ be a point inside a triangle $A_1A_2A_3$. For $i = 1, 2, 3$, line $PA_i$ intersects the side opposite to $A_i$ at $B_i$. Let $C_i$ and $D_i$ be the midpoints of $A_iB_i$ and $PB_i$, respectively. Prove that the areas of the triangles $C_1C_2C_3$ and $D_1D_2D_3$ are equal.

1980 VMO problem 4

Prove that for any tetrahedron in space, it is possible to find two perpendicular planes such that ratio between the projections of the tetrahedron on the two planes lies in the interval $[\frac{1}{\sqrt{2}}, \sqrt{2}].$

1981 VMO problem 3

A plane $\rho$ and two points $M, N$ outside it are given. Determine the point $A$ on $\rho$ for which $\frac{AM}{AN}$ is minimal.

1981 VMO problem 6

Two circles $k_1$ and $k_2$ with centers $O_1$ and $O_2$ respectively touch externally at $A$. Let $M$ be a point inside $k_2$ and outside the line $O_1O_2$. Find a line $d$ through $M$ which intersects $k_1$ and $k_2$ again at $B$ and $C$ respectively so that the circumcircle of $\Delta ABC$ is tangent to $O_1O_2$.

1982 VMO problem 3

Let be given a triangle $ABC$. Equilateral triangles $BCA_1$ and $BCA_2$ are drawn so that $A$ and $A_1$ are on one side of $BC$, whereas $A_2$ is on the other side. Points $B_1,B_2,C_1,C_2$ are analogously defined. Prove that $ S_{ABC} + S_{A_1B_1C_1} = S_{A_2B_2C_2}.$

Two circles $k_1$ and $k_2$ with centers $O_1$ and $O_2$ respectively touch externally at $A$. Let $M$ be a point inside $k_2$ and outside the line $O_1O_2$. Find a line $d$ through $M$ which intersects $k_1$ and $k_2$ again at $B$ and $C$ respectively so that the circumcircle of $\Delta ABC$ is tangent to $O_1O_2$.

1982 VMO problem 3

Let be given a triangle $ABC$. Equilateral triangles $BCA_1$ and $BCA_2$ are drawn so that $A$ and $A_1$ are on one side of $BC$, whereas $A_2$ is on the other side. Points $B_1,B_2,C_1,C_2$ are analogously defined. Prove that $ S_{ABC} + S_{A_1B_1C_1} = S_{A_2B_2C_2}.$

1982 VMO problem 6

Let $ABCDA'B'C'D'$ be a cube (where $ABCD$ and $A'B'C'D'$ are faces and $AA',BB',CC',DD'$ are edges). Consider the four lines $AA', BC, D'C'$ and the line joining the midpoints of $BB'$ and $DD'$. Show that there is no line which cuts all the four lines.

1983 VMO problem 3

A triangle $ABC$ and a positive number $k$ are given. Find the locus of a point $M$ inside the triangle such that the projections of $M$ on the sides of $\Delta ABC$ form a triangle of area $k$.

Let $ABCDA'B'C'D'$ be a cube (where $ABCD$ and $A'B'C'D'$ are faces and $AA',BB',CC',DD'$ are edges). Consider the four lines $AA', BC, D'C'$ and the line joining the midpoints of $BB'$ and $DD'$. Show that there is no line which cuts all the four lines.

1983 VMO problem 3

A triangle $ABC$ and a positive number $k$ are given. Find the locus of a point $M$ inside the triangle such that the projections of $M$ on the sides of $\Delta ABC$ form a triangle of area $k$.

1983 VMO problem 6

Let be given a tetrahedron whose any two opposite edges are equal. A plane varies so that its intersection with the tetrahedron is a quadrilateral. Find the positions of the plane for which the perimeter of this quadrilateral is minimum, and find the locus of the centroid for those quadrilaterals with the minimum perimeter.

Let be given a tetrahedron whose any two opposite edges are equal. A plane varies so that its intersection with the tetrahedron is a quadrilateral. Find the positions of the plane for which the perimeter of this quadrilateral is minimum, and find the locus of the centroid for those quadrilaterals with the minimum perimeter.

1984 VMO problem 3

A square $ABCD$ of side length $2a$ is given on a plane $\Pi$ . Let $S$ be a point on the ray $Ax$ perpendicular to $\Pi$ such that $AS = 2a.$

a) Let $M \in BC$ and $N \in CD$ be two variable points.

i. Find the positions of $M,N$ such that $BM + DN \ge \frac{3}{2}$, planes $SAM$ and $SMN$ are perpendicular and $BM \cdot DN$ is minimum.

ii. Find $M$ and $N$ such that $\angle MAN = 45^{\circ}$ and the volume of $SAMN$ attains an extremum value. Find these values.

b) Let $Q$ be a point such that $\angle AQB = \angle AQD = 90^{\circ}$. The line $DQ$ intersects the plane $\pi$ through $AB$ perpendicular to $\Pi$ at $Q'$.

i. Find the locus of $Q'$.

ii. Let $K$ be the locus of points $Q$ and let $CQ$ meet $K$ again at $R$. Let $DR$ meets $\Pi$ at $R'$. Prove that $sin^2 \angle Q'DB + sin^2 \angle R'DB$ is independent of $Q$.

A square $ABCD$ of side length $2a$ is given on a plane $\Pi$ . Let $S$ be a point on the ray $Ax$ perpendicular to $\Pi$ such that $AS = 2a.$

a) Let $M \in BC$ and $N \in CD$ be two variable points.

i. Find the positions of $M,N$ such that $BM + DN \ge \frac{3}{2}$, planes $SAM$ and $SMN$ are perpendicular and $BM \cdot DN$ is minimum.

ii. Find $M$ and $N$ such that $\angle MAN = 45^{\circ}$ and the volume of $SAMN$ attains an extremum value. Find these values.

b) Let $Q$ be a point such that $\angle AQB = \angle AQD = 90^{\circ}$. The line $DQ$ intersects the plane $\pi$ through $AB$ perpendicular to $\Pi$ at $Q'$.

i. Find the locus of $Q'$.

ii. Let $K$ be the locus of points $Q$ and let $CQ$ meet $K$ again at $R$. Let $DR$ meets $\Pi$ at $R'$. Prove that $sin^2 \angle Q'DB + sin^2 \angle R'DB$ is independent of $Q$.

1984 VMO problem 6

Consider a trihedral angle $Sxyz$ with $\angle xSz = \alpha , \angle xSy = \beta$ and $\angle ySz =\gamma$. Let $A,B,C$ denote the dihedral angles at edges $y, z, x$ respectively.

a) Prove that $\frac{\sin\alpha}{\sin A}=\frac{\sin\beta}{\sin B}=\frac{\sin\gamma}{\sin C}$

b) Show that $\alpha + \beta = 180^{\circ}$ if and only if $\angle A + \angle B = 180^{\circ}.$

c) Assume that $\alpha=\beta =\gamma = 90^{\circ}$. Let $O \in Sz$ be a fixed point such that $SO = a$ and let $M,N$ be variable points on $x, y$ respectively. Prove that $\angle SOM +\angle SON +\angle MON$ is constant and find the locus of the incenter of $OSMN$.

Consider a trihedral angle $Sxyz$ with $\angle xSz = \alpha , \angle xSy = \beta$ and $\angle ySz =\gamma$. Let $A,B,C$ denote the dihedral angles at edges $y, z, x$ respectively.

a) Prove that $\frac{\sin\alpha}{\sin A}=\frac{\sin\beta}{\sin B}=\frac{\sin\gamma}{\sin C}$

b) Show that $\alpha + \beta = 180^{\circ}$ if and only if $\angle A + \angle B = 180^{\circ}.$

c) Assume that $\alpha=\beta =\gamma = 90^{\circ}$. Let $O \in Sz$ be a fixed point such that $SO = a$ and let $M,N$ be variable points on $x, y$ respectively. Prove that $\angle SOM +\angle SON +\angle MON$ is constant and find the locus of the incenter of $OSMN$.

1985 VMO problem 3

A parallelepiped with the side lengths $ a$, $ b$, $ c$ is cut by a plane through its intersection of diagonals which is perpendicular to one of these diagonals. Calculate the area of the intersection of the plane and the parallelepiped.

A parallelepiped with the side lengths $ a$, $ b$, $ c$ is cut by a plane through its intersection of diagonals which is perpendicular to one of these diagonals. Calculate the area of the intersection of the plane and the parallelepiped.

1985 VMO problem 6

A triangular pyramid $ O.ABC$ with base $ ABC$ has the property that the lengths of the altitudes from $ A$, $ B$ and $ C$ are not less than $ \frac{OB+OC}{2}$, $ \frac{OC +OA}{2}$ and $ \frac{OA +OB}{2}$, respectively. Given that the area of $ ABC$ is $ S$, calculate the volume of the pyramid.

A triangular pyramid $ O.ABC$ with base $ ABC$ has the property that the lengths of the altitudes from $ A$, $ B$ and $ C$ are not less than $ \frac{OB+OC}{2}$, $ \frac{OC +OA}{2}$ and $ \frac{OA +OB}{2}$, respectively. Given that the area of $ ABC$ is $ S$, calculate the volume of the pyramid.

1986 VMO problem 2

Let $ R$, $ r$ be respectively the circumradius and inradius of a regular $ 1986$-gonal pyramid. Prove that \[ \frac{R}{r}\ge 1 + \frac{1}{\cos\frac{\pi}{1986}}\] and find the total area of the surface of the pyramid when the equality occurs.

Let $ R$, $ r$ be respectively the circumradius and inradius of a regular $ 1986$-gonal pyramid. Prove that \[ \frac{R}{r}\ge 1 + \frac{1}{\cos\frac{\pi}{1986}}\] and find the total area of the surface of the pyramid when the equality occurs.

1986 VMO problem 4

Let $ ABCD$ be a square of side $ 2a$. An equilateral triangle $ AMB$ is constructed in the plane through $ AB$ perpendicular to the plane of the square. A point $ S$ moves on $ AB$ such that $ SB =x$. Let $ P$ be the projection of $ M$ on $ SC$ and $ E$, $ O$ be the midpoints of $ AB$ and $ CM$ respectively.

a) Find the locus of $ P$ as $ S$ moves on $ AB$.

b) Find the maximum and minimum lengths of $ SO$

Let $ ABCD$ be a square of side $ 2a$. An equilateral triangle $ AMB$ is constructed in the plane through $ AB$ perpendicular to the plane of the square. A point $ S$ moves on $ AB$ such that $ SB =x$. Let $ P$ be the projection of $ M$ on $ SC$ and $ E$, $ O$ be the midpoints of $ AB$ and $ CM$ respectively.

a) Find the locus of $ P$ as $ S$ moves on $ AB$.

b) Find the maximum and minimum lengths of $ SO$

Prove that among any five distinct rays $ Ox$, $ Oy$, $ Oz$, $ Ot$, $ Or$ in space there exist two which form an angle less than or equal to $ 90^{\circ}$.

1988 VMO problem 6

Let $ a$, $ b$, $ c$ be three pairwise skew lines in space. Prove that they have a common perpendicular if and only if $ S_a \circ S_b \circ S_c$ is a reflection in a line, where $ S_x$ denotes the reflection in line $ x$.

Let $ a$, $ b$, $ c$ be three pairwise skew lines in space. Prove that they have a common perpendicular if and only if $ S_a \circ S_b \circ S_c$ is a reflection in a line, where $ S_x$ denotes the reflection in line $ x$.

1989 VMO problem 3

A square $ ABCD$ of side length $ 2$ is given on a plane. The segment $ AB$ is moved continuously towards $ CD$ until $ A$ and $ C$ coincide with $ C$ and $ D$, respectively. Let $ S$ be the area of the region formed by the segment $ AB$ while moving. Prove that $ AB$ can be moved in such a way that $ S <\frac{5\pi}{6}$.

A square $ ABCD$ of side length $ 2$ is given on a plane. The segment $ AB$ is moved continuously towards $ CD$ until $ A$ and $ C$ coincide with $ C$ and $ D$, respectively. Let $ S$ be the area of the region formed by the segment $ AB$ while moving. Prove that $ AB$ can be moved in such a way that $ S <\frac{5\pi}{6}$.

1989 VMO problem 6

Let be given a parallelepiped $ ABCD.A'B'C'D'$. Show that if a line $ \Delta$ intersects three of the lines $ AB'$, $ BC'$, $ CD'$, $ DA'$, then it intersects also the fourth line.

Let be given a parallelepiped $ ABCD.A'B'C'D'$. Show that if a line $ \Delta$ intersects three of the lines $ AB'$, $ BC'$, $ CD'$, $ DA'$, then it intersects also the fourth line.

1990 VMO problem 3

A tetrahedron is to be cut by three planes which form a parallelepiped whose three faces and all vertices lie on the surface of the tetrahedron.

a) Can this be done so that the volume of the parallelepiped is at least $ \frac{9}{40}$ of the volume of the tetrahedron?

b) Determine the common point of the three planes if the volume of the parallelepiped is $ \frac{11}{50}$ of the volume of the tetrahedron.

A tetrahedron is to be cut by three planes which form a parallelepiped whose three faces and all vertices lie on the surface of the tetrahedron.

a) Can this be done so that the volume of the parallelepiped is at least $ \frac{9}{40}$ of the volume of the tetrahedron?

b) Determine the common point of the three planes if the volume of the parallelepiped is $ \frac{11}{50}$ of the volume of the tetrahedron.

1990 VMO problem 4

A triangle $ ABC$ is given in the plane. Let $ M$ be a point inside the triangle and $ A'$, $ B'$, $ C'$ be its projections on the sides $ BC$, $ CA$, $ AB$, respectively. Find the locus of $ M$ for which $ MA \cdot MA' = MB \cdot MB' = MC \cdot MC'$

A triangle $ ABC$ is given in the plane. Let $ M$ be a point inside the triangle and $ A'$, $ B'$, $ C'$ be its projections on the sides $ BC$, $ CA$, $ AB$, respectively. Find the locus of $ M$ for which $ MA \cdot MA' = MB \cdot MB' = MC \cdot MC'$

1991 VMO problem 3

Three mutually perpendicular rays $O_x,O_y,O_z$ and three points $A,B,C$ on $O_x,O_y,O_z$, respectively. A variable sphere є through $A, B,C$ meets $O_x,O_y,O_z$ again at $A', B',C'$, respectively. Let $M$ and $M'$ be the centroids of triangles $ABC$ and $A'B'C'$. Find the locus of the midpoint of $MM'$.

Three mutually perpendicular rays $O_x,O_y,O_z$ and three points $A,B,C$ on $O_x,O_y,O_z$, respectively. A variable sphere є through $A, B,C$ meets $O_x,O_y,O_z$ again at $A', B',C'$, respectively. Let $M$ and $M'$ be the centroids of triangles $ABC$ and $A'B'C'$. Find the locus of the midpoint of $MM'$.

1991 VMO problem 5

Let $G$ be centroid and $R$ the circunradius of a triangle $ABC$. The extensions of $GA,GB,GC$ meet the circuncircle again at $D,E,F$. Prove that:

$ \frac{3}{R}\leq\frac{1}{GD}+\frac{1}{GE}+\frac{1}{GF}\leq\sqrt{3}\left(\frac{1}{AB}+\frac{1}{BC}+\frac{1}{CA}\right) $

Let $G$ be centroid and $R$ the circunradius of a triangle $ABC$. The extensions of $GA,GB,GC$ meet the circuncircle again at $D,E,F$. Prove that:

$ \frac{3}{R}\leq\frac{1}{GD}+\frac{1}{GE}+\frac{1}{GF}\leq\sqrt{3}\left(\frac{1}{AB}+\frac{1}{BC}+\frac{1}{CA}\right) $

1992 VMO problem 1

Let $ABCD$ be a tetrahedron satisfying

i) $\widehat{ACD}+\widehat{BCD}=180^{0}$, and

ii) $\widehat{BAC}+\widehat{CAD}+\widehat{DAB}=\widehat{ABC}+\widehat{CBD}+\widehat{DBA}=180^{0}$.

Find value of $[ABC]+[BCD]+[CDA]+[DAB]$ if we know $AC+CB=k$ and $\widehat{ACB}=\alpha$

Let $ABCD$ be a tetrahedron satisfying

i) $\widehat{ACD}+\widehat{BCD}=180^{0}$, and

ii) $\widehat{BAC}+\widehat{CAD}+\widehat{DAB}=\widehat{ABC}+\widehat{CBD}+\widehat{DBA}=180^{0}$.

Find value of $[ABC]+[BCD]+[CDA]+[DAB]$ if we know $AC+CB=k$ and $\widehat{ACB}=\alpha$

1992 VMO problem 5

Let $H$ be a rectangle with angle between two diagonal $\leq 45^{0}$. Rotation $H$ around the its center with angle $0^{0}\leq x\leq 360^{0}$ we have rectangle $H_{x}$. Find $x$ such that $[H\cap H_{x}]$ minimum, where $[S]$ is area of $S$.

Let $H$ be a rectangle with angle between two diagonal $\leq 45^{0}$. Rotation $H$ around the its center with angle $0^{0}\leq x\leq 360^{0}$ we have rectangle $H_{x}$. Find $x$ such that $[H\cap H_{x}]$ minimum, where $[S]$ is area of $S$.

1993 VMO problem 2

$ABCD$ is a quadrilateral such that $AB$ is not parallel to $CD$, and $BC$ is not parallel to $AD$. Variable points $P, Q, R, S$ are taken on $AB, BC, CD, DA$ respectively so that $PQRS$ is a parallelogram. Find the locus of its center.

$ABCD$ is a quadrilateral such that $AB$ is not parallel to $CD$, and $BC$ is not parallel to $AD$. Variable points $P, Q, R, S$ are taken on $AB, BC, CD, DA$ respectively so that $PQRS$ is a parallelogram. Find the locus of its center.

1993 VMO problem 4

The tetrahedron $ABCD$ has its vertices on the fixed sphere $S$. Prove that $AB^{2}+AC^{2}+AD^{2}-BC^{2}-BD^{2}-CD^{2}$ is minimum iff $AB\perp AC,AC\perp AD,AD\perp AB$.

The tetrahedron $ABCD$ has its vertices on the fixed sphere $S$. Prove that $AB^{2}+AC^{2}+AD^{2}-BC^{2}-BD^{2}-CD^{2}$ is minimum iff $AB\perp AC,AC\perp AD,AD\perp AB$.

1994 VMO problem 2

$ABC$ is a triangle. Reflect each vertex in the opposite side to get the triangle $A'B'C'$. Find a necessary and sufficient condition on $ABC$ for $A'B'C'$ to be equilateral.

$ABC$ is a triangle. Reflect each vertex in the opposite side to get the triangle $A'B'C'$. Find a necessary and sufficient condition on $ABC$ for $A'B'C'$ to be equilateral.

1994 VMO problem 5

$S$ is a sphere center $O. G$ and $G'$ are two perpendicular great circles on $S$. Take $A, B, C$ on $G$ and $D$ on $G'$ such that the altitudes of the tetrahedron $ABCD$ intersect at a point. Find the locus of the intersection.

$S$ is a sphere center $O. G$ and $G'$ are two perpendicular great circles on $S$. Take $A, B, C$ on $G$ and $D$ on $G'$ such that the altitudes of the tetrahedron $ABCD$ intersect at a point. Find the locus of the intersection.

1995 VMO problem 3

Let a non-equilateral triangle $ ABC$ and $ AD,BE,CF$ are its altitudes. On the rays $ AD,BE,CF,$ respectively, let $ A',B',C'$ such that $ \frac {AA'}{AD} = \frac {BB'}{BE} = \frac {CC'}{CF} = k$. Find all values of $ k$ such that $ \triangle A'B'C'\sim\triangle ABC$ for any non-triangle $ ABC.$

Let a non-equilateral triangle $ ABC$ and $ AD,BE,CF$ are its altitudes. On the rays $ AD,BE,CF,$ respectively, let $ A',B',C'$ such that $ \frac {AA'}{AD} = \frac {BB'}{BE} = \frac {CC'}{CF} = k$. Find all values of $ k$ such that $ \triangle A'B'C'\sim\triangle ABC$ for any non-triangle $ ABC.$

1995 VMO problem 4

Let a tetrahedron $ ABCD$ and $ A',B',C',D'$ be the circumcenters of triangles $ BCD,CDA,DAB,ABC$ respectively. Denote planes $ (P_A),(P_B),(P_C),(P_D)$ be the planes which pass through $ A,B,C,D$ and perpendicular to $ C'D',D'A',A'B',B'C'$ respectively. Prove that these planes have a common point called $ I.$ If $ P$ is the center of the circumsphere of the tetrahedron, must this tetrahedron be regular?

Let a tetrahedron $ ABCD$ and $ A',B',C',D'$ be the circumcenters of triangles $ BCD,CDA,DAB,ABC$ respectively. Denote planes $ (P_A),(P_B),(P_C),(P_D)$ be the planes which pass through $ A,B,C,D$ and perpendicular to $ C'D',D'A',A'B',B'C'$ respectively. Prove that these planes have a common point called $ I.$ If $ P$ is the center of the circumsphere of the tetrahedron, must this tetrahedron be regular?

1996 VMO problem 2

Given a trihedral angle Sxyz. A plane (P) not through S cuts Sx,Sy,Sz respectively at A,B,C. On the plane (P), outside triangle ABC, construct triangles DAB,EBC,FCA which are confruent to the triangles SAB,SBC,SCA respectively. Let (T) be the sphere lying inside Sxyz, but not inside the tetrahedron SABC, toucheing the planes containing the faces of SABC. Prove that (T) touches the plane (P) at the circumcenter of triangle DEF.

Given a trihedral angle Sxyz. A plane (P) not through S cuts Sx,Sy,Sz respectively at A,B,C. On the plane (P), outside triangle ABC, construct triangles DAB,EBC,FCA which are confruent to the triangles SAB,SBC,SCA respectively. Let (T) be the sphere lying inside Sxyz, but not inside the tetrahedron SABC, toucheing the planes containing the faces of SABC. Prove that (T) touches the plane (P) at the circumcenter of triangle DEF.

1996 VMO problem 5

The triangle ABC has BC=1 and $ \angle BAC = a$. Find the shortest distance between its incenter and its centroid. Denote this shortest distance by $ f(a)$. When a varies in the interval $ (\frac {\pi}{3},\pi)$, find the maximum value of $ f(a)$.

The triangle ABC has BC=1 and $ \angle BAC = a$. Find the shortest distance between its incenter and its centroid. Denote this shortest distance by $ f(a)$. When a varies in the interval $ (\frac {\pi}{3},\pi)$, find the maximum value of $ f(a)$.

1997 VMO problem 1

Given a circle (O,R). A point P lies inside the circle, OP=d, d<R. We consider quadrilaterals ABCD, inscribed in (O), such that AC is perp to BD at point P. Evaluate the maximum and minimum values of the perimeter of ABCD in terms of R and d.

Given a circle (O,R). A point P lies inside the circle, OP=d, d<R. We consider quadrilaterals ABCD, inscribed in (O), such that AC is perp to BD at point P. Evaluate the maximum and minimum values of the perimeter of ABCD in terms of R and d.

Let be given a tetrahedron whose circumcenter is $O$. Draw diameters $AA_{1},BB_{1},CC_{1},DD_{1}$ of the circumsphere of $ABCD$. Let $A_{0},B_{0},C_{0},D_{0}$ be the centroids of triangle $BCD,CDA,DAB,ABC$. Prove that $A_{0}A_{1},B_{0}B_{1},C_{0}C_{1},D_{0}D_{1}$ are concurrent at a point, say, $F$. Prove that the line through $F$ and a midpoint of a side of $ABCD$ is perpendicular to the opposite side.

Let a triangle ABC and A',B',C' be the midpoints of the arcs BC,CA,AB respectively of its circumcircle. A'B',A'C' meets BC at $ A_1,A_2$ respectively. Pairs of point $ (B_1,B_2),(C_1,C_2)$ are similarly defined. Prove that $ A_1A_2 = B_1B_2 = C_1C_2$ if and only if triangle ABC is equilateral.

1999 VMO problem 5

]$ OA, OB, OC, OD$ are 4 rays in space such that the angle between any two is the same. Show that for a variable ray $ OX,$ the sum of the cosines of the angles $ XOA, XOB, XOC, XOD$ is constant and the sum of the squares of the cosines is also constant.

]$ OA, OB, OC, OD$ are 4 rays in space such that the angle between any two is the same. Show that for a variable ray $ OX,$ the sum of the cosines of the angles $ XOA, XOB, XOC, XOD$ is constant and the sum of the squares of the cosines is also constant.

Two circles $ (O_1)$ and $ (O_2)$ with respective centers $ O_1$, $ O_2$ are given on a plane. Let $ M_1$, $ M_2$ be points on $ (O_1)$, $ (O_2)$ respectively, and let the lines $ O_1M_1$ and $ O_2M_2$ meet at $ Q$. Starting simultaneously from these positions, the points $ M_1$ and $ M_2$ move clockwise on their own circles with the same angular velocity.

a) Determine the locus of the midpoint of $ M_1M_2$.

b) Prove that the circumcircle of $ \triangle M_1QM_2$ passes through a fixed point.

a) Determine the locus of the midpoint of $ M_1M_2$.

b) Prove that the circumcircle of $ \triangle M_1QM_2$ passes through a fixed point.

2001 VMO problem 1

A circle center $O$ meets a circle center $O'$ at $A$ and $B.$ The line $TT'$ touches the first circle at $T$ and the second at $T'$. The perpendiculars from $T$ and $T'$ meet the line $OO'$ at $S$ and $S'$. The ray $AS$ meets the first circle again at $R$, and the ray $AS'$ meets the second circle again at $R'$. Show that $R, B$ and $R'$ are collinear.

A circle center $O$ meets a circle center $O'$ at $A$ and $B.$ The line $TT'$ touches the first circle at $T$ and the second at $T'$. The perpendiculars from $T$ and $T'$ meet the line $OO'$ at $S$ and $S'$. The ray $AS$ meets the first circle again at $R$, and the ray $AS'$ meets the second circle again at $R'$. Show that $R, B$ and $R'$ are collinear.

An isosceles triangle $ ABC$ with $ AB = AC$ is given on the plane. A variable circle $ (O)$ with center $ O$ on the line $ BC$ passes through $ A$ and does not touch either of the lines $ AB$ and $ AC$. Let $ M$ and $ N$ be the second points of intersection of $ (O)$ with lines $ AB$ and $ AC$, respectively. Find the locus of the orthocenter of triangle $ AMN$.

2003 VMO problem 2

The circles $ C_{1}$ and $ C_{2}$ touch externally at $ M$ and the radius of $ C_{2}$ is larger than that of $ C_{1}$. $ A$ is any point on $ C_{2}$ which does not lie on the line joining the centers of the circles. $ B$ and $ C$ are points on $ C_{1}$ such that $ AB$ and $ AC$ are tangent to $ C_{1}$. The lines $ BM$, $ CM$ intersect $ C_{2}$ again at $ E$, $ F$ respectively. $ D$ is the intersection of the tangent at $ A$ and the line $ EF$. Show that the locus of $ D$ as $ A$ varies is a straight line.

The circles $ C_{1}$ and $ C_{2}$ touch externally at $ M$ and the radius of $ C_{2}$ is larger than that of $ C_{1}$. $ A$ is any point on $ C_{2}$ which does not lie on the line joining the centers of the circles. $ B$ and $ C$ are points on $ C_{1}$ such that $ AB$ and $ AC$ are tangent to $ C_{1}$. The lines $ BM$, $ CM$ intersect $ C_{2}$ again at $ E$, $ F$ respectively. $ D$ is the intersection of the tangent at $ A$ and the line $ EF$. Show that the locus of $ D$ as $ A$ varies is a straight line.

In a triangle $ ABC$, the bisector of $ \angle ACB$ cuts the side $ AB$ at $ D$. An arbitrary circle $ (O)$ passing through $ C$ and $ D$ meets the lines $ BC$ and $ AC$ at $ M$ and $ N$ (different from $ C$), respectively.

a) Prove that there is a circle $ (S)$ touching $ DM$ at $ M$ and $ DN$ at $ N$.

b) If circle $ (S)$ intersects the lines $ BC$ and $ CA$ again at $ P$ and $ Q$ respectively, prove that the lengths of the segments $ MP$ and $ NQ$ are constant as $ (O)$ varies.

a) Prove that there is a circle $ (S)$ touching $ DM$ at $ M$ and $ DN$ at $ N$.

b) If circle $ (S)$ intersects the lines $ BC$ and $ CA$ again at $ P$ and $ Q$ respectively, prove that the lengths of the segments $ MP$ and $ NQ$ are constant as $ (O)$ varies.

2005 VMO problem 2

Let $(O)$ be a fixed circle with the radius $R$. Let $A$ and $B$ be fixed points in $(O)$ such that $A,B,O$ are not collinear. Consider a variable point $C$ lying on $(O)$ ($C\neq A,B$). Construct two circles $(O_1),(O_2)$ passing through $A,B$ and tangent to $BC,AC$ at $C$, respectively. The circle $(O_1)$ intersects the circle $(O_2)$ in $D$ ($D\neq C$). Prove that:

a) $ CD\leq R $

b) The line $CD$ passes through a point independent of $C$ (i.e. there exists a fixed point on the line $CD$ when $C$ lies on $(O)$)

Let $(O)$ be a fixed circle with the radius $R$. Let $A$ and $B$ be fixed points in $(O)$ such that $A,B,O$ are not collinear. Consider a variable point $C$ lying on $(O)$ ($C\neq A,B$). Construct two circles $(O_1),(O_2)$ passing through $A,B$ and tangent to $BC,AC$ at $C$, respectively. The circle $(O_1)$ intersects the circle $(O_2)$ in $D$ ($D\neq C$). Prove that:

a) $ CD\leq R $

b) The line $CD$ passes through a point independent of $C$ (i.e. there exists a fixed point on the line $CD$ when $C$ lies on $(O)$)

Let $ABCD$ be a convex quadrilateral. Take an arbitrary point $M$ on the line $AB$, and let $N$ be the point of intersection of the circumcircles of triangles $MAC$ and $MBC$ (different from $M$). Prove that:

a) The point $N$ lies on a fixed circle;

b) The line $MN$ passes though a fixed point.

a) The point $N$ lies on a fixed circle;

b) The line $MN$ passes though a fixed point.

Let B,C be fixed points and A be roving point. Let H, G be orthecentre and centroid of triagle ABC. Known midpoint of HG lies on BC, find locus of A.

2007 VMO problem 6

Let ABCD be trapezium that is inscribed in circle (O) with larger edge BC. P is a point lying outer segment BC. PA cut (O) at N (that means PA isn't tangent of (O)), the circle with diameter PD intersect (O) at E, DE meet BC at N. Prove that MN always pass through a fixed point.

Let ABCD be trapezium that is inscribed in circle (O) with larger edge BC. P is a point lying outer segment BC. PA cut (O) at N (that means PA isn't tangent of (O)), the circle with diameter PD intersect (O) at E, DE meet BC at N. Prove that MN always pass through a fixed point.

2008 VMO problem 2

Given a triangle with acute angle $ \angle BEC,$ let $ E$ be the midpoint of $ AB.$ Point $ M$ is chosen on the ray of $ EC$ (or "opposite ray of CE") such that $ \angle BME = \angle ECA.$ Denote by $ \theta$ the measure of angle $ \angle BEC.$ Evaluate $ \frac {MC}{AB}$ in terms of $ \theta.$

Given a triangle with acute angle $ \angle BEC,$ let $ E$ be the midpoint of $ AB.$ Point $ M$ is chosen on the ray of $ EC$ (or "opposite ray of CE") such that $ \angle BME = \angle ECA.$ Denote by $ \theta$ the measure of angle $ \angle BEC.$ Evaluate $ \frac {MC}{AB}$ in terms of $ \theta.$

2008 VMO problem 7

Let $ AD$ is centroid of $ ABC$ triangle. Let $ (d)$ is the perpendicular line with $ AD$. Let $ M$ is a point on $ (d)$. Let $ E, F$ are midpoints of $ MB, MC$ respectively. The line through point $ E$ and perpendicular with $ (d)$ meet $ AB$ at $ P$. The line through point $ F$ and perpendicular with $ (d)$ meet $ AC$ at $ Q$. Let $ (d')$ is a line through point $ M$ and perpendicular with $ PQ$. Prove $ (d')$ always pass a fixed point.

Let $ AD$ is centroid of $ ABC$ triangle. Let $ (d)$ is the perpendicular line with $ AD$. Let $ M$ is a point on $ (d)$. Let $ E, F$ are midpoints of $ MB, MC$ respectively. The line through point $ E$ and perpendicular with $ (d)$ meet $ AB$ at $ P$. The line through point $ F$ and perpendicular with $ (d)$ meet $ AC$ at $ Q$. Let $ (d')$ is a line through point $ M$ and perpendicular with $ PQ$. Prove $ (d')$ always pass a fixed point.

2009 VMO problem 3

Let $ A$, $ B$ be two fixed points and $ C$ is a variable point on the plane such that $ \angle ACB = \alpha$ (constant) ($ 0^{\circ}\le \alpha\le 180^{\circ}$). Let $ D$, $ E$, $ F$ be the projections of the incenter $ I$ of triangle $ ABC$ to its sides $ \color{red}{AB}$, $ \color{red}{BC}$, $ \color{red}{CA}$ , respectively. Denoted by $ M$, $ N$ the intersections of $ AI$, $ BI$ with $ EF$, respectively. Prove that the length of the segment $ MN$ is constant and the circumcircle of triangle $ DMN$ always passes through a fixed point.

Let $ A$, $ B$ be two fixed points and $ C$ is a variable point on the plane such that $ \angle ACB = \alpha$ (constant) ($ 0^{\circ}\le \alpha\le 180^{\circ}$). Let $ D$, $ E$, $ F$ be the projections of the incenter $ I$ of triangle $ ABC$ to its sides $ \color{red}{AB}$, $ \color{red}{BC}$, $ \color{red}{CA}$ , respectively. Denoted by $ M$, $ N$ the intersections of $ AI$, $ BI$ with $ EF$, respectively. Prove that the length of the segment $ MN$ is constant and the circumcircle of triangle $ DMN$ always passes through a fixed point.

In plane,let a circle $(O)$ and two fixed points $B,C$ lies in $(O)$ such that $BC$ not is the diameter.Consider a point $A$ varies in $(O)$ such that $A\neq B,C$ and $AB\neq AC$.Call $D$ and $E$ respective is intersect of $BC$ and internal and external bisector of $\widehat{BAC}$,$I$ is midpoint of $DE$.The line that pass through orthocenter of $\triangle ABC$ and perpendicular with $AI$ intersects $AD,AE$ respective at $M,N$.

a) Prove that $MN$ pass through a fixed point

b) Determint the place of $A$ such that $S_{AMN}$ has maxium value

a) Prove that $MN$ pass through a fixed point

b) Determint the place of $A$ such that $S_{AMN}$ has maxium value

Let $AB$ be a diameter of a circle $(O)$ and let $P$ be any point on the tangent drawn at $B$ to $(O).$ Define $AP\cap (O)=C\neq A,$ and let $D$ be the point diametrically opposite to $C.$ If $DP$ meets $(O)$ second time in $E,$ then,

a) Prove that $AE, BC, PO$ concur at $M.$

b) If $R$ is the radius of $(O),$ find $P$ such that the area of $\triangle AMB$ is maximum, and calculate the area in terms of $R.$

Let $\triangle ABC$ be a triangle such that $\angle C$ and $\angle B$ are acute. Let $D$ be a variable point on $BC$ such that $D\neq B, C$ and $AD$ is not perpendicular to $BC.$ Let $d$ be the line passing through $D$ and perpendicular to $BC.$ Assume $d \cap AB= E, d \cap AC =F.$ If $M, N, P$ are the incentres of $\triangle AEF, \triangle BDE,\triangle CDF.$ Prove that $A, M, N, P$ are concyclic if and only if $d$ passes through the incentre of $\triangle ABC.$

Let $ABCD$ be a cyclic quadrilateral with circumcentre $O,$ and the pair of opposite sides not parallel with each other. Let $M=AB\cap CD$ and $N=AD\cap BC.$ Denote, by $P,Q,S,T;$ the intersection of the internal angle bisectors of $\angle MAN$ and $\angle MBN;$ $\angle MBN$ and $\angle MCN;$ $\angle MDN$ and $\angle MAN;$ $\angle MCN$ and $\angle MDN.$ Suppose that the four points $P,Q,S,T$ are distinct.

a) Show that the four points $P,Q,S,T$ are concyclic. Find the centre of this circle, and denote it as $I.$

b) Let $E=AC\cap BD.$ Prove that $E,O,I$ are collinear.

Let $ABC$ be a triangle such that $ABC$ isn't a isosceles triangle. $(I)$ is incircle of triangle touches $BC,CA,AB$ at $D,E,F$ respectively. The line through $E$ perpendicular to $BI$ cuts $(I)$ again at $K$. The line through $F$ perpendicular to $CI$ cuts $(I)$ again at $L$.$J$ is midpoint of $KL$.

a) Prove that $D,I,J$ collinear.

b) $B,C$ are fixed points,$A$ is moved point such that $\frac{AB}{AC}=k$ with $k$ is constant.$IE,IF$ cut $(I)$ again at $M,N$ respectively.$MN$ cuts $IB,IC$ at $P,Q$ respectively. Prove that bisector perpendicular of $PQ$ through a fixed point.

Let $ABC$ be a cute triangle.$(O)$ is circumcircle of $\triangle ABC$.$D$ is on arc $BC$ not containing $A$.Line $\triangle$ moved through $H$($H$ is orthocenter of $\triangle ABC$ cuts circumcircle of $\triangle ABH$,circumcircle $\triangle ACH$ again at $M,N$ respectively.

a) Find $\triangle$ satisfy $S_{AMN}$ max

b) $d_{1},d_{2}$ are the line through $M$ perpendicular to $DB$,the line through $N$ perpendicular to $DC$ respectively.

$d_{1}$ cuts $d_{2}$ at $P$.Prove that $P$ move on a fixed circle.

Let $ABC$ be an acute triangle, $(O)$ be the circumcircle, and $AB<AC.$ Let $I$ be the midpoint of arc $BC$ (not containing $A$). $K$ lies on $AC,$ $K\ne C$ such that $IK=IC.$ $BK$ intersects $(O)$ at the second point $D,$ $D\ne B$ and intersects $AI$ at $E.$ $DI$ intersects $AC$ at $F.$

a) Prove that $EF=\frac{BC}{2}.$

b) $M$ lies on $DI$ such that $CM$ is parallel to $AD.$ $KM$ intersects $BC$ at $N.$ The circumcircle of triangle $BKN$ intersects $(O)$ at the second point $P.$ Prove that $PK$ passes through the midpoint of segment $AD.$

Given a circle $(O)$ and two fixed points $B,C$ on $(O),$ and an arbitrary point $A$ on $(O)$ such that the triangle $ABC$ is acute. $M$ lies on ray $AB,$ $N$ lies on ray $AC$ such that $MA=MC$ and $NA=NB.$ Let $P$ be the intersection of $(AMN)$ and $(ABC),$ $P\ne A.$ $MN$ intersects $BC$ at $Q.$

a) Prove that $A,P,Q$ are collinear.

b) $D$ is the midpoint of $BC.$ Let $K$ be the intersection of $(M,MA)$ and $(N,NA),$ $K\ne A.$ $d$ is the line passing through $A$ and perpendicular to $AK.$ $E$ is the intersection of $d$ and $BC.$ $(ADE)$ intersects $(O)$ at $F,$ $F\ne A.$ Prove that $AF$ passes through a fixed point.

Given a circumcircle $(O)$ and two fixed points $B,C$ on $(O)$. $BC$ is not the diameter of $(O)$. A point $A$ varies on $(O)$ such that $ABC$ is an acute triangle. $E,F$ is the foot of the altitude from $B,C$ respectively of $ABC$. $(I)$ is a variable circumcircle going through $E$ and $F$ with center $I$.

a) Assume that $(I)$ touches $BC$ at $D$. Probe that $\frac{DB}{DC}=\sqrt{\frac{\cot B}{\cot C}}$.

b) Assume $(I)$ intersects $BC$ at $M$ and $N$. Let $H$ be the orthocenter and $P,Q$ be the intersections of $(I)$ and $(HBC)$. The circumcircle $(K)$ going through $P,Q$ and touches $(O)$ at $T$ ($T$ is on the same side with $A$ wrt $PQ$). Prove that the interior angle bisector of $\angle{MTN}$ passes through a fixed point.

Let $ABC$ be an acute triange with $B,C$ fixed. Let $D$ be the midpoint of $BC$ and $E,F$ be the foot of the perpendiculars from $D$ to $AB,AC$, respectively.

a) Let $O$ be the circumcenter of triangle $ABC$ and $M=EF\cap AO, N=EF\cap BC$. Prove that the circumcircle of triangle $AMN$ passes through a fixed point;

b) Assume that tangents of the circumcircle of triangle $AEF$ at $E,F$ are intersecting at $T$. Prove that $T$ is on a fixed line.

Given a triangle $ABC$ inscribed by circumcircle $(O)$. The angles at $B,C$ are acute angle. Let $M$ on the arc $BC$ that doesn't contain $A$ such that $AM$ is not perpendicular to $BC$. $AM$ meets the perpendicular bisector of $BC$ at $T$. The circumcircle $(AOT)$ meets $(O)$ at $N$ ($N\ne A$).

a) Prove that $\angle{BAM}=\angle{CAN}$.

b) Let $I$ be the incenter and $G$ be the foor of the angle bisector of $\angle{BAC}$. $AI,MI,NI$ intersect $(O)$ at $D,E,F$ respectively. Let ${P}=DF\cap AM, {Q}=DE\cap AN$. The circle passes through $P$ and touches $AD$ at $I$ meets $DF$ at $H$ ($H\ne D$).The circle passes through $Q$ and touches $AD$ at $I$ meets $DE$ at $K$ ($K\ne D$). Prove that the circumcircle $(GHK)$ touches $BC$.

Given an acute, non isoceles triangle $ABC$ and $(O)$ be its circumcircle, $H$ its orthocenter and $E, F$ are the feet of the altitudes from $B$ and $C$, respectively. $AH$ intersects $(O)$ at $D$ ($D\ne A$).

a) Let $I$ be the midpoint of $AH$, $EI$ meets $BD$ at $M$ and $FI$ meets $CD$ at $N$. Prove that $MN\perp OH$.

b) The lines $DE$, $DF$ intersect $(O)$ at $P,Q$ respectively ($P\ne D,Q\ne D$). $(AEF)$ meets $(O)$ and $AO$ at $R,S$ respectively ($R\ne A, S\ne A$). Prove that $BP,CQ,RS$ are concurrent.

Given an acute triangle $ABC$ and $(O)$ be its circumcircle. Let $G$ be the point on arc $BC$ that doesn't contain $O$ of the circumcircle $(I)$ of triangle $OBC$. The circumcircle of $ABG$ intersects $AC$ at $E$ and circumcircle of $ACG$ intersects $AB$ at $F$ ($E\ne A, F\ne A$).

a) Let $K$ be the intersection of $BE$ and $CF$. Prove that $AK,BC,OG$ are concurrent.

b) Let $D$ be a point on arc $BOC$ (arc $BC$ containing $O$) of $(I)$. $GB$ meets $CD$ at $M$ , $GC$ meets $BD$ at $N$. Assume that $MN$ intersects $(O)$ at $P$ nad $Q$. Prove that when $G$ moves on the arc $BC$ that doesn't contain $O$ of $(I)$, the circumcircle $(GPQ)$ always passes through two fixed points.

We have a scalene acute triangle $ABC$ (triangle with no two equal sides) and a point $D$ on side $BC$. Pick a point $E$ on side $AB$ and a point $F$ on side $AC$ such that $\angle DEB=\angle DFC$. Lines $DF,\, DE$ intersect $AB,\, AC$ at points $M,\, N$, respectively. Denote $(I_1),\, (I_2)$ by the circumcircles of triangles $DEM,\, DFN$ in that order. The circle $(J_1)$ touches $(I_1)$ internally at $D$ and touches $AB$ at $K$, circle $(J_2)$ touches $(I_2)$ internally at $D$ and touches $AC$ at $H$. $P$ is the intersection of $(I_1),\, (I_2)$ different from $D$. $Q$ is the intersection of $(J_1),\, (J_2)$ different from $D$.

a) Prove that all points $D,\, P,\, Q$ lie on the same line.

b) The circumcircles of triangles $AEF,\, AHK$ intersect at $A,\, G$. $(AEF)$ also cut $AQ$ at $A,\, L$. Prove that the tangent at $D$ of $(DQG)$ cuts $EF$ at a point on $(DLG)$.

Acute scalene triangle $ABC$ has $G$ as its centroid and $O$ as its circumcenter. Let $H_a,\, H_b,\, H_c$ be the projections of $A,\, B,\, C$ on respective opposite sides and $D,\, E,\, F$ be the midpoints of $BC,\, CA,\, AB$ in that order. $\overrightarrow{GH_a},\, \overrightarrow{GH_b},\, \overrightarrow{GH_c}$ intersect $(O)$ at $X,\,Y,\,Z$ respectively.

a) Prove that the circle $(XCE)$ pass through the midpoint of $BH_a$

b) Let $M,\, N,\, P$ be the midpoints of $AX,\, BY,\, CZ$ respectively. Prove that $\overleftrightarrow{DM},\, \overleftrightarrow{EN},\,\overleftrightarrow{FP}$ are concurrent.

2019 VMO problem

Let $ABC$ be triangle with $H$ is the orthocenter and $I$ is incenter. Denote $A_{1}, A_{2}, B_{1}, B_{2}, C_{1}, C_{2}$ be the points on the rays $AB, AC, BC, CA, CB$, respectively such that $$AA_{1} = AA_{2} = BC, BB_{1} = BB_{2} = CA, CC_{1} = CC_{2} = AB.$$Suppose that $B_{1}B_{2}$ cuts $C_{1}C_{2}$ at $A'$, $C_{1}C_{2}$ cuts $A_{1}A_{2}$ at $B'$ and $A_{1}A_{2}$ cuts $B_{1}B_{2}$ at $C'$.

a) Prove that area of triangle $A'B'C'$ is smaller than or equal to the area of triangle $ABC$.

b) Let $J$ be circumcenter of triangle $A'B'C'$. $AJ$ cuts $BC$ at $R$, $BJ$ cuts $CA$ at $S$ and $CJ$ cuts $AB$ at $T$. Suppose that $(AST), (BTR), (CRS)$ intersect at $K$. Prove that if triangle $ABC$ is not isosceles then $HIJK$ is a parallelogram.

2019 VMO problem

Let $ABC$ be an acute, nonisosceles triangle with inscribe in a circle $(O)$ and has orthocenter $H$. Denote $M,N,P$ as the midpoints of sides $BC,CA,AB$ and $D,E,F$ as the feet of the altitudes from vertices $A,B,C$ of triangle $ABC$. Let $K$ as the reflection of $H$ through $BC$. Two lines $DE,MP$ meet at $X$; two lines $DF,MN$ meet at $Y$.

a) The line $XY$ cut the minor arc $BC$ of $(O)$ at $Z$. Prove that $K,Z,E,F$ are concyclic.

b) Two lines $KE,KF$ cuts $(O)$ second time at $S,T$. Prove that $BS,CT,XY$ are concurrent.

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