geometry problems from Belarusian Mathematical Olympiads
with aops links in the names
with aops links in the names
[category/ grade :A / XI , B / X , C / IX, D / VIII]
collected inside aops here
1995 -2019 grades VIII - XI
(missing XI: 1996, 2013, 2016)
Mark six points in a plane so that any three of them are vertices of a nondegenerate isosceles triangle.
Given a triangle $ABC$,let $K$ be the midpoint of $AB$ and $L$ a be point on $AC$ such that $AL=LC+CB$.Prove that ${\angle}KLB=90^\circ$ if and only if $AC=3CB$
Two circles touch in $M$, and lie inside a rectangle $ABCD$. One of them touches the sides $AB$ and $AD$, and the other one touches $AD,BC,CD$. The radius of the second circle is four times that of the first circle. Find the ratio in which the common tangent of the circles in $M$ divides $AB$ and $CD$.
In a star-shaped closed broken line $ABCDEA$, $AB$ meets $CD$ and $DE$ at $P$ and $Q$, $BC$ meets $DE$ and $EA$ at $R$ and $S$, and $CD$ meets $EA$ at $T$, respectively, and $AP = QB, BR = SC, CT = PD, DQ = RE$. Prove that $ES = TA$.
The rectangle $ABCD$ is partitioned into five rectangles $P_1,P_2,P_3,P_4, P5$. If $P_5$ is a square, and $P_1,P_2,P_3, P_4$ have the same area, prove that $ABCD$ is a square.
Let $M$ be a point on the semicircle with diameter $AB, K$ be a point on $AB$, and $P,Q$ be the circumcenters of triangles $AMK,MKB$. Prove that the points $M,K,P,Q$ lie on a circle.
A point $B$ inside a regular hexagon $A_1A_2...A_6$ is given, such that ${\angle}A_2A_1B={\angle}A_4A_3B=50^\circ$. Find ${\angle}A_1A_2B$
Let $AK,BL,CM$ be the altitudes of an acute triangle $ABC$. If $9\overrightarrow{AK} +4\overrightarrow{BL}+7\overrightarrow{CM}= \overrightarrow{0}$, prove that one of the angles of $\vartriangle ABC$ is equal to $60^o$
In a triangle $ABC$ with $\angle B = 3 \angle A$, let $M,N$ be chosen on side $CA$ so that $\angle CBM = \angle MBN = \angle NBA$. Suppose that $X$ is an arbitrary point on $BC, L$ the intersection of $AX$ and $BN$, and $K$ the intersection of $NX$ and $BM$. Prove that $KL$ and $AC$ are parallel.
The center $O_1$ of a circle $S_1$ lies on a circle $S_2$ with center $O_2$. The radius of $S_2$ is greater than that of $S_1$. Let $A$ be the intersection of $S_1$ and $O_1O_2$. Consider a circle $S$ centered at an arbitrary point $X$ on $S_2$ and passing through $A$, and let $Y \ne A$ be the intersection of $S$ and $S_2$. Prove that all lines $XY$ are concurrent as $X$ runs along $S_2$.
year 1996 missing
1997 Belarusian MO D 8.2
Points $D$ and $E$ are taken on side $CB$ of triangle $ABC$, with $D$ between $C$ and $E$,
such that $\angle BAE =\angle CAD$. If $AC < AB$, prove that $AC.AE < AB.AD$.
1997 Belarusian MO D 8.6
If ABCD is as convex quadrilateral with $\angle ADC = 30$ and $BD = AB+BC+CA$,
prove that $BD$ bisects $\angle ABC$.
1998 Belarusian MO C 9.5
Points $D$ and $E$ are taken on side $CB$ of triangle $ABC$, with $D$ between $C$ and $E$,
such that $\angle BAE =\angle CAD$. If $AC < AB$, prove that $AC.AE < AB.AD$.
1997 Belarusian MO D 8.6
If ABCD is as convex quadrilateral with $\angle ADC = 30$ and $BD = AB+BC+CA$,
prove that $BD$ bisects $\angle ABC$.
Points $D,M,N$ are chosen on the sides $AC,AB,BC$ of a triangle $ABC$ respectively, so that the intersection point $P$ of $AN$ and $CM$ lies on $BD$. Prove that $BD$ is a median of the triangle if and only if $AP : PN = CP : PM$.
We are given a mechanism that can perform the following operations:
$\bullet$ Joining any two points of a plane by a straight line;
$\bullet$ Constructing the reflection $X$ of a given point $P$ in a given line $\ell$.
Given a triangle A$BC$, using the given mechanism, construct
(a) its centroid,
(b) its circumcenter.
A pentagon $A_1A_2A_3A_4A_5$ is inscribed in a circle, $B$ being the intrersection point of $A_1A_4$ and $A_2A_5$. Given that $\angle A_4A_1A_3 = \angle A_5A_2A_4$ and $\angle A_2A_4A_1 = \angle A_3A_5A_2$, prove that $\angle A1A3B = \angle BA3A5$.
In a trapezoid $ABCD$ with $AB \parallel CD$ it holds that $\angle ADB+\angle DBC = 180^o$. Prove that $AB\cdot BC = AD \cdot DC$
Different points $A_1,A_2,A_3,A_4,A_5$ lie on a circle so that $A_1A_2 = A_2A_3 = A_3A_4 =A_4A_5$. Let $A_6$ be the diametrically opposite point to $A_2$, and $A_7$ be the intersection of $A_1A_5$ and $A_3A_6$. Prove that the lines $A_1A_6$ and $A_4A_7$ are perpendicular
A triangle $A_1B_1C_1$ is a parallel projection of a triangle $ABC$ in space. The parallel projections $A_1H_1$ and $C_1L_1$ of the altitude $AH$ and the bisector $CL$ of $\vartriangle ABC$ respectively are drawn. Using a ruler and compass, construct a parallel projection of :
(a) the orthocenter,
(b) the incenter of $\vartriangle ABC$.
Find the angle between the diagonals of the convex quadrilateral, if it is known that they are equal to each other, and also twice the segment, connecting the midpoints of some two opposite sides.
a) Prove that for any convex quadrilateral one of its midlines (i.e., straight lines connecting the midpoints of opposite sides) splits it into two parts, the area of each of which is not less than $3/8$ of the area of the entire quadrangle.
b) Will the statement of part a) remain true if the number $3/8$ gets replaced by a larger number?
On the sides $AB, BC$ and $AC$ of the triangle $ABC$, points $K, H$ and $T$ are marked, respectively, so that $AH \perp BC$, $\angle BCK = \angle ACK$ and $AT=TC$ . Find the perimeter of triangle $ABC$, if the lengths of the segments $KM = 2$ cm, $MN = 1$ cm and $NC = 3$ cm are known, where$ M$ is the intersection point of $AH$ and $CK$, and $N$ is the intersection point of $HT$ and $CK$.
Prove that if the convex octagon $ABCDEFGH$ can be split into six parallelograms in the manner shown in the figure, the four straight lines $AE, BF, CG$ and $DH$ intersect at one point.
Points $N,L$ on side AC and points $P, K$ on side $BC$ of triangle $ABC$ are such that $AK ,BL$ are bisectors, and $AP , BN$ are altitudes of the triangle. Let $O$ and $I$ be the centers of the inscribed and circumscribed circles of triangle $ABC$ respectively. Prove that $N, P$ and $I$ lie on one straight line if and only if $L, K$ and $O$ lie on the same straight line .
You are given a regular decagon $A_1A_2... A_{10}$. Let $A$ be the intersection point of lines $A_1A_4$ and $A_2A_5, B$ is the point of intersection of lines $A_1A_6$ and $A_2A_7$, and $C$ is the point of intersection of lines $A_1A_9$ and $A_2A_{10}$. Find the angles of $ABC$.
1998 Grade A 10. day 1 not available
Points $M$ and $N$ are marked on the straight line containing the side $AC$ of triangle $ABC$ so that $MA = AB$ and $NC = CB$ (the order of the points on the line: $M, A, C, N$). Prove that the center of the circle inscribed in triangle $ABC$ lies on the common chord of the circles circumscribed around triangles $MCB$ and $NAB$ .
Let $ABCDE$ be a pentagon with $AE = ED$, $AB + CD = BC$, and $\angle BAE +\angle CDE = 180^o$ . Prove that $\angle AED = 2\angle BEC$.
Let $ ABC$ be an isosceles right triangle and $M$ be the midpoint of its hypotenuse $AB$. Points $D$ and $E$ are taken on the legs $AC$ and $BC$ respectively such that $AD=2DC$ and $BE=2EC$. Lines $AE$ and $DM$ intersect at $F$. Show that $FC$ bisects the $\angle DFE$.
In an acute-angled triangle $ABC$, the circle with diameter $AB$ intersects $CA$ at $L$ and $CB$ at $N$. The segment $LN$ intersects the median $CM$ at $K$. Compute $CM$, given that $AB = 9$ and $CK =\frac35 CM$.
Let $AB$ and $CD$ be perpendicular diameters of a circle, and K be a point on the circle other than $A,B,C,D$. Let the lines $AK$ and $CD$ meet at $M$, and the lines $DK$ and $BC$ meet at $N$. Prove that $MN$ is parallel to $AB$.
A circle is inscribed in an isosceles trapezoid $ABCD$.The diagonal $AC$ intersects the circle at $K$ and $L$, in the order $A,K,L,C$. Find the value of ${\sqrt[4]{\frac{AL{\cdot}KC}{AK{\cdot}LC}}}$.
Let $P$ and $Q$ be points on the side $AB$ of the triangle $\triangle ABC$ (with $P$ between $A$ and $Q$) such that $\angle ACP = \angle PCQ = \angle QCB$,and let $AD$ be the angle bisector of $\angle BAC$. Line $AD$ meets lines $CP$ and $CQ$ at $M$ and $N$ respectively. Given that $PN = CD$ and $3\angle BAC = 2\angle BCA$, prove that triangles $\triangle CQD$ and $\triangle QNB$ have the same area.
A circle is inscribed in the trapezoid ABCD. Let K, L, M, N be the points of tangency of this circle with the diagonals AC and BD, respectively (K is between A and L, and M is between B and N). Given that $AK\cdot LC=16$ and $BM\cdot ND=\frac94$, find the radius of the circle.
1999 Belarusian MO A 11.7
Let $O$ be the center of circle $ W$. Two equal chords $AB$ and $CD $ of $ W $ intersect at $L $ such that $AL>LB $ and $DL>LC$. Let $M $ and $ N $ be points on $AL$ and $DL$ respectively such that $\angle ALC=2 \angle MON$. Prove that the chord of $W$ passing through $M $and $N$ is equal to $AB$ and $CD$.
2000 Belarusian MO B 10.6
The lateral sides and diagonals of a trapezoid intersect a line $l$, determining three equal segments on it. Must $l$ be parallel to the bases of the trapezoid?
2000 Belarusian MO A 11.6
A vertex of a tetrahedron is called perfect if the three edges at this vertex are sides of a certain triangle. How many perfect vertices can a tetrahedron have?
2001 Belarusian MO B 10.8
2001 Belarusian MO A 11.3
Three distinct points $A$, $B$, and $N$ are marked on the line $l$, with $B$ lying between $A$ and $N$. For an arbitrary angle $\alpha \in (0,\frac{\pi}{2})$, points $C$ and $D$ are marked in the plane on the same side of $l$ such that $N$, $C$, and $D$ are collinear; $\angle NAD = \angle NBC = \alpha$; and $A$, $B$, $C$, and $D$ are concyclic. Find the locus of the intersection points of the diagonals of $ABCD$ as $\alpha$ varies between $0$ and $\frac{\pi}{2}$.
2001 Belarusian MO A 11.7
2002 Belarusian MO B 10.7
2002 Belarusian MO A 11.6
2003 Belarusian MO C 9.7
2003 Belarusian MO B 10.5
2003 Belarusian MO B 10.7
1999 Belarusian MO A 11.7
Let $O$ be the center of circle $ W$. Two equal chords $AB$ and $CD $ of $ W $ intersect at $L $ such that $AL>LB $ and $DL>LC$. Let $M $ and $ N $ be points on $AL$ and $DL$ respectively such that $\angle ALC=2 \angle MON$. Prove that the chord of $W$ passing through $M $and $N$ is equal to $AB$ and $CD$.
Points $M$ and $K$ are marked on the sides $BC$ and $CD$ of a square $ABCD$, respectively.The segments $MD$ and $BK$ intersect at $P$.Prove that $AP{\perp}MK$ if and only if $MC=KD$.
On the side $AB$ of a triangle $ABC$ with $BC < AC < AB$, points $B_1$ and $C_2$ are marked so that $AC_2 = AC$ and $BB_1 = BC$. Points $B2$ on side $AC$ and $C_1$ on the extension of $CB$ are marked so that $CB_2 = CB$ and $CC_1 = CA$. Prove that the lines $C_1C_2$ and $B_1B_2$ are parallel.
In a triangle $ABC$ with a right angle at $C$, the altitude $CD$ intersects the angle bisector $AE$ at $F$. Lines $ED$ and $BF$ meet at $G$. Prove that the area of the quadrilateral $CEGF$ is equal to the area of the triangle $BDF$.
The equilateral triangles $ABF$ and $CAG$ are constructed in the extirior of a right-angled triangle $ABC$ with ${\angle}C=90^\circ$.Let $M$ be the midpoint of $BC$.Given that $MF=11$ and $MG=7$.Find the length of $BC$.
Find the locus of the points $ M$ in the plane $ Oxy$ such that the tangents from $ M$ to $ y=x^2$ are perpendicular.
The diagonals $AC$ and $BD$ of a convex quadrilateral $ABCD$ intersect in point $M$. The angle bisector of $\angle ACD$ intersects the ray $\overrightarrow{BA}$ in point $K$. If $MA.MC+MA.CD=MB.MD$, prove that $\angle BKC=\angle CDB$.
A rectangle $ABCD$ and a point $X$ are given on plane.
(a) Prove that among the segments $XA,XB,XC,XD$, some three are sides of a triangle.
(b) Does (a) necessarily hold if $ABCD$ is a parallelogram?
2000 Belarusian MO A 11.6
A vertex of a tetrahedron is called perfect if the three edges at this vertex are sides of a certain triangle. How many perfect vertices can a tetrahedron have?
In an isosceles triangle $ABC$ ($AB= AC$), the angle at the vertex $A$ is $30^o$. On the side $AB$ , a point $Q$ is marked, which is different from $B$, and on the median $AD$, point $P$ is marked so that $PC = PQ$. Find the angle $PQC$.
The five straight lines intersect as shown. Can the segments obtained when crossing, have the lengths indicated in the figure?
In rhombus $ABCD$, the apex angle $A$ is $60^o$. Points $F, H$ and $G$ are marked on the sides $AD, DC$ and diagonal $AC$, respectively, so that the quadrilateral $DFGH$ is a parallelogram. Prove that triangle $FBH$ is equilateral.
A quadrilateral $ABCD$ is inscribed in a circle. Points $C_1$ and $A_1$ are marked on rays $BA$ and $DC$, respectively, such that $DA = DA_1$ and $BC =BC_1$. Prove that the diagonal $BD$ divides the segment $A_1 C_1$ in half.
Circles $S_1$ and $S_2$ intersect at points $A$ and $B$. Through point $P$ of circle $S_2$, lying inside $S_1$, a chord $AD$ of circle $S_1$ is drawn. Chord $BC$ of circle $S_1$ passes through point $P$. A straight line passing through points $D$ and $B$ intersects circle $S_2$ at a point $Q$ different from point $B$. Let $P_1$ be a point symmetric to point $P$ wrt point $B$. Prove that $CD$ is the diameter of the circle $S_1$ if and only if the points $D, C, Q$ and $P_1$ lie on the same circle.
Inside a right-angled triangle $ABC$ with a right angle at the vertex $C$, a point $X$ is marked, such that $\angle XAB = \angle XBC$. Prove that $$AC \cdot BC^2 \le AC \cdot CX^2 +CX \cdot AB^2.$$
A circle can be inscribed in the quadrilateral $ABCD$. Extensions of side $AB$ beyond point $B$ and side $DC$ beyond point $C$ intersect at point $E$. Extensions of side $DA$ beyond point $A$ and side $CB$ beyond point $B$ intersect at point $F$. Let $I_1,I_2$ and $I_3$ denote the centers of the circles inscribed in triangles $AFB, BEC$ and $ABC$, respectively. Let the straight line $I_1I_3$ intersect the lines $EA$ and $ED$ at the points $K$ and $L$, respectively, and the line $I_2I_3$ intersect the lines $FC$ and $FD$ at the points $M$ and $N$, respectively. Prove that $EK = EL$ if and only if $FM = FN$.
On the Cartesian coordinate plane, the graph of the parabola $y = x^2$ is drawn. Three distinct points $A$, $B$, and $C$ are marked on the graph with $A$ lying between $B$ and $C$. Point $N$ is marked on $BC$ so that $AN$ is parallel to the y-axis. Let $K_1$ and $K_2$ are the areas of triangles $ABN$ and $ACN$, respectively. Express $AN$ in terms of $K_1$ and $K_2$.
Three distinct points $A$, $B$, and $N$ are marked on the line $l$, with $B$ lying between $A$ and $N$. For an arbitrary angle $\alpha \in (0,\frac{\pi}{2})$, points $C$ and $D$ are marked in the plane on the same side of $l$ such that $N$, $C$, and $D$ are collinear; $\angle NAD = \angle NBC = \alpha$; and $A$, $B$, $C$, and $D$ are concyclic. Find the locus of the intersection points of the diagonals of $ABCD$ as $\alpha$ varies between $0$ and $\frac{\pi}{2}$.
2001 Belarusian MO A 11.7
The convex quadrilateral $ABCD$ is inscribed in the circle $S_1$. Let $O$ be the intersection of $AC$ and $BD$. Circle $S_2$ passes through $D$ and $ O$, intersecting $AD$ and $CD$ at $ M$ and $ N$, respectively. Lines $OM$ and $AB$ intersect at $R$, lines $ON$ and $BC$ intersect at $T$, and $R$ and $T$ lie on the same side of line $BD$ as $ A$. Prove that $O$, $R$,$T$, and $B$ are concyclic.
Points $B_1$ and $C_1$ are marked on the bisector of $\angle A$ of the triangle $ABC$ so that $BB_1 \perp AB, CC_1 \perp AC$. Let $M$ be the midpoint of $B_1C_1$. Prove that $MB = MC$.
Let $AM$ and $BN$ be the altitudes of an acute-angled triangle $ABC $($\angle ACB\ne 45^o$). Points $K$ and $T$ are marked on the rays $MA$ and $NB$ so that $MK = MB$ and $NT = NA$. Prove that $KT\parallel MN$.
Points $M$ and $N$ are marked on the sides $AB$ and $AC$ of the triangle $ABC$ respectively so that $MN\parallel BC$. The segment $BN$ intersects the segment $CM$ at $K$. The circumference through $A,K,B$ intersects $BC$ at $P$, and the circumference through $A,K,C$ intersects $BC$ at $Q$. Let $T$ be the point of intersection of the lines $PM$ and $QN$. Prove that $P$ lies on the line $AK$.
Points $A$ and $B$ are marked on the circumference $S$ ($AB$ is not a diameter of $S$). Let $X$ be a variable point on $S$. Let $Y$ be the point of intersection the line $XA$ and the perpendicular to $XB$ at $B$. (If $X$ coincides with $A$ then we consider the tangent to $S$ at $A$ as the line $XA$.) Find the locus of the midpoints of $XY$, when $X$ moves along $S$ without $B$.
Given a rhombus $ABCD$ with the $\angle B= 60^o$. Point M is marked inside $\vartriangle ADC$ so that $\angle AMC = 120^o$. Let $P$ and $Q$ be the intersection points of the lines $BA$ and $CM$, and $BC$ and $AM$, respectively. Prove that $D$ lies on the line $PQ$.
Points $M, L, K$ are marked on the side $BC$ of $\vartriangle ABC$ (the order of the points is $B, M, L, K, C$), so that $BM=ML=LK=LC$. It is known that $\angle ACB= \angle MAB$.
a) Prove that $\angle KAL > 1,5 \angle CAK$.
b) Prove that the coefficient $1,5$ in a) is the largest possible.
Prove or disprove:
There exists a solid such that, for all positive integers $n$ with $n \geq 3$, there exists a "parallel projection" such that the image of the solid under this projection is a convex $n$-gon.
The altitude $CH$ of a right triangle $ABC$, with $\angle{C}=90$, cut the angles bisectors $AM$ and $BN$ at $P$ and $Q$, and let $R$ and $S$ be the midpoints of $PM$ and $QN$. Prove that $RS$ is parallel to the hypotenuse of $ABC$
2003 Grade D 8. not available
Prove that a right-angled triangle can be inscribed in the parabola $y=x^2$ so that its hypotenuse is parallel to the axis of abscissae if and only if the altitude from the right angle is equal to $1$. (A triangle is inscribed in a parabola if all three vertices of the triangle belong the parabola.)
The diagonals $A_1A_4$, $A_2A_5$, and $A_3A_6$ of the convex hexagon $A_1A_2A_3A_4A_5A_6$ meet at point $K$. Given $A_2A_1= A_2A_3 = A_2K$, $A_4A_3 = A_4A_5 = A_4K$, $A_6A_5 = A_6A_1= A_6K$, prove that the hexagon is cyclic.
Different points $A_0,A_1,...,A_{1000}$ are marked on one side of an angle and different points $B_0,B_1,...,B_{1000}$ are marked on its other side so that
$$A_0A_1=A_1A_2=...=A_{999}A_{1000} , B_0B_1= B_1B_2=...= B_{999}B_{1000}.$$ Find the area of quadrilateral $A_{999}A_{1000}B_{1000}B_{999}$ if the areas of quadrilaterals $A_0A_1B_1B_0$ and $A_1A_2B_2B_1$ are equal to $5$ and $7$, respectively.
A quadrilateral $ABCD$ is cyclic and $AB = 2AD, BC = 2CD$. Given that $\angle BAD=\alpha$ and diagonal $AC = d$, find the area of the triangle $ABC$.
The diagonals $AC$ and $BD$ of the convex quadrilateral $ABCD$ are perpendicular and intersect at point $O$. Let circles $S_1, S_2, S_3, S_4$ with centers $O_1, O_2, O_3, O_4$ be inscribed in the triangles $AOB, BOC, COD, DOA$, respectively.
Prove that
a) the sum of the diameters of $S_1, S_2, S_3, S_4$ is less than or equal to $(2-\sqrt2 )(AC + BD)$
b) $O_1O_2 + O_2O_3 + O_3O_4 +O_4O_1< 2(\sqrt2-1) (AC + BD)$.
The quadrilateral $ABCD$ is cyclic and $AB = BC = AD + CD$. Given that $\angle BAD = \alpha$, and the diagonal $AC = d$, find the area of the triangle $ABC$.
We say that a triangle and a rectangle are twin if they have the same perimeters and the same areas. Prove that for a given rectangle there exists a twin triangle if the rectangle is not a square and the ratio of the bigger side of the rectangle to its smaller side is at least $\lambda -1 +\sqrt{\lambda (\lambda -2)}$ where $\lambda = \frac{3\sqrt3}{2}$.
Two triangles are said to be twins if one of them is an image of the other one under a parallel projection. Prove that two triangles are twins if and only if either at least a side of one of them equals a side of another or both the triangles have equal segments that connect the corresponding vertices with some points on the opposite sides which divide these sides in the same ratio.
Given a convex pentagon $ABCDE$ with $AB=BC, CD=DE, \angle ABC=150^o, \angle CDE=30^o, BD=2$. Find the area of $ABCDE$.
Points $N$ and $K$ are marked on the sides $AB$ and $AC$ of triangle $ABC$, respectively, so that $AN = NB$ and $AK = 2 KC$. It turned out that $KN \perp AB$. Find $NC$ if you know that $CB = 8$.
In a convex quadrilateral $ABCD$, the diagonals $AC$ and $BD$ are perpendicular, with $AC = KL = 2$, where $K$ and $L$ are the midpoints of sides $AB$ and $CD$, respectively. Find the length of the diagonal $BD$ and the angle between straight lines $BD$ and $KL$.
Let ABCD be a convex quadrilateral, and let K, L, M, N be the midpoints of its sides AB, BC, CD, DA, respectively. Let the lines NL and KM intersect at a point T. Prove that $\frac83\left|DNTM\right|<\left|ABCD\right|<8\left|DNTM\right|$, where $\left|P_1P_2...P_n\right|$ denotes the area of an arbitrary polygon $P_1P_2...P_n$.
Circles $S_1$ and $S_2$ meet at points $A$ and $B$. A line through $A$ is parallel to the line through the centers of $S_1$ and $S_2$ and meets $S_1$ again at $C$ and $S_2$ again at $D$. The circle $S_3$ with diameter $CD$ meets $S_1$ and $S_2$ again at $P$ and $Q$, respectively. Prove that lines $CP,DQ$, and $AB$ are concurrent.
Let be given two similar triangles such that the altitudes of the first triangle are equal to the sides of the other. Find the largest possible value of the similarity ratio of the triangle.
The diagonals $AD, BE, CF$ of a convex hexagon meet at a point $P$. Find the least possible area of $ABCDEF$, if $[APB] = 4$, $[CPD] = 6$ and $[EPF] = 9$.
Let $ABCD$ be a cyclic quadrilateral such that $AB \cdot BC=2 \cdot AD \cdot DC$. Prove that its diagonals $AC$ and $BD$ satisfy the inequality $8BD^2 \le 9AC^2$.
a) Suppose that there is a point $X$ in the plane of a given convex quadrilateral $ABCD$ such that the perimeters of triangles $ABX,BCX,CDX, DAX$ are equal. Show that $ABCD$ is a tangential quadrilateral.
(b) If a convex $ABCD$ is tangential, does there necessarily exist a point $X$ such that the perimeters of triangles $ABX,BCX,CDX,DAX$ are equal?
2004 Belarusian MO A 11.2
Let $C$ be a semicircle with diameter $AB$. Circles $S$, $S_1$, $S_2$ with radii $r$, $r_1$, $r_2$, respectively, are tangent to $C$ and the segment $AB$, and moreover $S_1$ and $S_2$ are externally tangent to $S$. Prove that $\frac{1}{\sqrt{r_1}}+\frac{1}{\sqrt{r_2}}=\frac{2\sqrt{2}}{\sqrt{r}}$
2004 Belarusian MO A 11.7
A cube $ABCDA_1B_1C_1D_1$ is given. Find the locus of points $E$ on the face $A_1B_1C_1D_1$ for which there exists a line intersecting the lines $AB$, $A_1D_1$, $B_1D$, and $EC$.
2005 Belarusian MO B 10.8
A line parallel to the side $AC$ of a triangle $ABC$ with $\angle C = 90$ intersects side $AB$ at $M$ and side $BC$ at $N$, so that $CN/BN = AC/BC = 2/1$. The segments $CM$ and $AN$ meet at $O$. Let $K$ be a point on the segment $ON$ such that $MO+OK = KN$. The bisector of $\angle ABC$ meets the line through $K$ perpendicular to $AN$ at point $T$. Determine $\angle MTB$.
2005 Belarusian MO A 11.8
Does there exist a convex pentagon such that for any of its inner angles, the angle bisector contains one of the diagonals?
2006 Belarusian MO C 9.5 10.7
2006 Belarusian MO B 10.5
2006 Belarusian MO A 11.4
Let $C$ be a semicircle with diameter $AB$. Circles $S$, $S_1$, $S_2$ with radii $r$, $r_1$, $r_2$, respectively, are tangent to $C$ and the segment $AB$, and moreover $S_1$ and $S_2$ are externally tangent to $S$. Prove that $\frac{1}{\sqrt{r_1}}+\frac{1}{\sqrt{r_2}}=\frac{2\sqrt{2}}{\sqrt{r}}$
2004 Belarusian MO A 11.7
A cube $ABCDA_1B_1C_1D_1$ is given. Find the locus of points $E$ on the face $A_1B_1C_1D_1$ for which there exists a line intersecting the lines $AB$, $A_1D_1$, $B_1D$, and $EC$.
2005 Grade D 8. not available
Let $K$ and $M$ be points on the sides $AB$ and $BC$ respectively of a triangle $ABC$, and $N$ be the intersection point of $AM$ and $CK$.Assume that the quadrilaterals $AKMC$ and $KBMN$ are cyclic with the same circumradius. Find ${\angle}ABC$
Suppose that there is a point $K$ on the side $CD$ of a trapezoid $ABCD$ with $AD \parallel BC$ such that $ABK$ is an equilateral triangle. Show that there is a point $L$ on the line $AB$ such that $CDL$ is also an equilateral triangle.
Let $BE$, $CF$ be two altitudes of a triangle $ABC$, and $H$ is its orthocenter. Let $l$ be the perpendicular to $CA$, (passing $A$). Show that $~$ $BC$, $EF$, $l$ are concurrent if and only if $H$ is the midpoint of $BE$
Does there exist a convex heptagon such that for any of its inner angles, the angle bisector contains one of the diagonals?
2005 Belarusian MO A 11.8
Does there exist a convex pentagon such that for any of its inner angles, the angle bisector contains one of the diagonals?
2006 Grade D 8. not available
Points $X,Y,Z$ are marked on the sides $AB,BC,CD$ of the rhombus $ABCD,$ respectively, so that $XY\parallel AZ.$ Prove that $XZ,AY$ and $BD$ are concurrent.
Given triangle $ABC$ with $\angle A = 60^o$, $AB = 2005$, $AC = 2006$. Bob and Bill in turn (BoB is the first) cut the triangle along any straight line so that two new triangles with area more than or equal to $1$ appear. After that an obtuse-angled triangle (or any of two right-angled triangles) is deleted and the procedure is repeated with the remained triangle. The player losses if he cannot do the next cutting.
Determine, who of the players wins if both play in the best way.
Given real numbers $a, b, k (k > 0)$. The circle with the center $(a, b)$ has at least three common points with the parabola $y=kx^2$ one of them is the origin $(0,0)$ and two of the others lie on the line $y = kx+b$. Prove that $b\ge 2$.
2006 Belarusian MO B 10.1
Given a convex quadrilateral $ABCD$ with $DC = a, BC = b$, $\angle DAB = 90^o$, $\angle DCB = < \phi $, $AB = AD$, find the length of the diagonal $AC$.
Different points $A,B,C$ lie on the parabola $y=x^2$. Let R be the circumradius of the triangle $ABC$.
a) Prove that $R>\frac12$
b) Does there exist a constant $c>\frac12$ so that for any different points $A,B,C$ the inequality $R \ge c$ holds?
Given a quadrilateral $ABCD$ with $\angle ABC = \angle ADC$. Let $BM$ be the altitude of the triangle $ABC$, and $M$ belongs to $AC$. Point $M'$ is marked on the diagonal $AC$ so that $$\frac{AM \cdot CM'}{ AM' \cdot CM}= \frac{AB \cdot CD }{ BC \cdot AD}$$ Prove that the intersection point of $DM'$ and $BM$ coincides with the orthocenter of the triangle $ABC$.
A convex quadrilateral $ABCD$ Is placed on the Cartesian plane. Its vertices $A$ and $D$ belong to the negative branch of the graph of the hyperbola $y= 1/x$, the vertices $B$ and $C$ belong to the positive branch of the graph and point $B$ lies at the left of $C$, the segment $AC$ passes through the origin $(0,0)$. Prove that $\angle BAD = \angle BCD$.
Let $AH_A, BH_B, CH_C$ be altitudes and $BM$ be a median of the acute-angled triangle $ABC$ ($AB > BC$). Let $K$ be a point of intersection of $BM$ and $AH_A$, $T$ be a point on $BC$ such that $KT \parallel AC$, $H$ be the orthocenter of $ABC$. Prove that the lines passing through the pairs of the points $(H_c, H_A), (H, T)$ and $(A, C)$ are concurrent.
Two triangles $ABC$ and $A_1B_1C_1$ are circumscribed around a circle, whose perimeters have ratio $1: 2$. Side $A_1B_1$ intersects the sides $BC$ and $AC$ at points $A_2$ and $B_3$, respectively, side $A_1C_1$ intersects the sides $AB$ and $BC$ at points $C_2$ and $A_3$, respectively, side $B_1C_1$ intersects the sides $AC$ and $AB$ at points $B_2$ and $C_3$, respectively.
Find the ratio of the sum of the perimeters of the triangles $A_1A_2A_3,B_1B_2B_3, C_1C_2C_3$ to the sum of the perimeters of the triangles $AB_2C_3, BC_2A_3, CA_2B_3$.
Point $D$ is marked on the side $AC$ of triangle $ABC$ so that $AD = AB$. Point $F$ is marked on the side $AB$ so that the midpoint of segment $CF$ lies on $BD$. Prove that $BF = CD$.
Give quadrilateral $ ABCD$ with $ \angle CAD = 45^o$, $ \angle ACD = 30^o$, $ \angle BAC =\angle BCA =15^o$, find the value of $ \angle DBC$.
Let $ O$ be the point of intersection of the diagonals $ AC$ and $ BD$ of the quadrilateral $ ABCD$ with $ AB =BC$ and $ CD = DA$. Let $ N$ and $ K$ be the feet of perpendiculars from $ D$ and $ B$ to $ AB$ and $ BC$, respectively. Prove that the points $ N$, $ O$, and $ K$ are colinear.
Three beetles are at the same point of the table. Suddenly they begin to crawl and after a while they are at the vertices of a triangle with the inradius equal to $2$ Prove that at least one of the beetles crawls the distance which is greater than $3$.
Given a convex quadrilateral $ ABCD$ with $ \angle ACB = \angle ADB$, and $ AB = AD$. Let $ N$ and $ K$ be the feet of perpendiculars from $ A$ onto the lines $ CB$ and $ DB$, respectively. Prove that $ NK \perp AC$.
2007 Belarusian MO A 11.2
Circles $S_1$ and $S_2$ with centers $O_1$ and $O_2$, respectively, pass through the centers of each other. Let $A$ be one of their intersection points. Two points $M_1$ and $M_2$ begin to move simultaneously starting from $A$. Point $M_1$ moves along $S_1$ and point $M_2$ moves along $S_2$. Both points move in clockwise direction and have the same linear velocity $v$.
(a) Prove that all triangles $AM_1M_2$ are equilateral.
(b) Determine the trajectory of the movement of the center of the triangle $AM_1M_2$ and find its linear velocity.
2007 Belarusian MO A 11.5
Let $O$ be the intersection point of the diagonals of the convex quadrilateral $ABCD$, $AO = CO$. Points $P$ and $Q$ are marked on the segments $AO$ and $CO$, respectively, such that $PO = OQ$. Let $N$ and $K$ be the intersection points of the sides $AB$, $CD$, and the lines $DP$ and $BQ$ respectively. Prove that the points $N$, $O$, and $K$ are colinear.
2008 Belarusian MO B 10.3 (12)
Circles $S_1$ and $S_2$ with centers $O_1$ and $O_2$, respectively, pass through the centers of each other. Let $A$ be one of their intersection points. Two points $M_1$ and $M_2$ begin to move simultaneously starting from $A$. Point $M_1$ moves along $S_1$ and point $M_2$ moves along $S_2$. Both points move in clockwise direction and have the same linear velocity $v$.
(a) Prove that all triangles $AM_1M_2$ are equilateral.
(b) Determine the trajectory of the movement of the center of the triangle $AM_1M_2$ and find its linear velocity.
2007 Belarusian MO A 11.5
Let $O$ be the intersection point of the diagonals of the convex quadrilateral $ABCD$, $AO = CO$. Points $P$ and $Q$ are marked on the segments $AO$ and $CO$, respectively, such that $PO = OQ$. Let $N$ and $K$ be the intersection points of the sides $AB$, $CD$, and the lines $DP$ and $BQ$ respectively. Prove that the points $N$, $O$, and $K$ are colinear.
In an isosceles triangle $ABC$ ($AC = BC$) on point $D$ is marked on the side $AC$ so that triangle $ADK$ is isosceles, where $K$ is the point of intersection of the segment $BD$ and the altitude $AH$. Find the angle $DBA$.
In triangle $ABC$, the altitude $AK$ and the median $BM$ intersect at point $Q$, with $AK = BM$. Ray $QC$ is is the bisector of the angle $MQK$. Find the angles of triangle $ABC$.
In triangle $ABC$, segments $A_1B_2, B_1C_2$ and $C_1A_2$ touch the inscribed in this triangle circle and are parallel to sides $AB, BC$ and $CA$, respectively (see fig.). Find the value of the sum $\frac{A_1B_2}{AB} + \frac{B_1C_2}{BC}+ \frac{C_1A_2}{CA}$.
The altitude $BH$ and the medians $AM$ and $CN$ are drawn in triangle $ABC$. It turned out that $HM = MN$. Prove that triangle $ABC$ is isosceles.
Through the center $I$ of the inscribed in the triangle $ABC$ circle, pass the segments $A_1B_2$, $B_1C_2$ and $C_1A_2$, with sides parallel to $AB, BC$ and $CA$ respectively(see fig.). Find the value of the sum $\frac{A_1B_2}{AB}+\frac{B_1C_2}{BC}+\frac{C_1A_2}{CA}$.
The lengths of all altitudes in some non-isosceles triangles are expressed in integer numbers. Find the smallest possible value of the inscribed circle radius of this triangle if it is known to be an integer as well.
On sides $BC$ and $AC$ of triangle $ABC$ you draw points $A_1$ and $B_1$, respectively, so that $\frac{AB_1}{CB_1} = \lambda \frac{AB}{CB}$, $ \frac{BA_1}{CA_1} = \lambda \frac{BA}{CA}$ where $\lambda$ is some positive number. Let $M$ be an arbitrary point on the segment $A_1B_1$, and $x, y, z$ be the distances from point M to the sides of the triangle $BC, AC, AB$, respectively. Prove that $z = \lambda (x + y)$.
Two adjacent sides of a quadrilateral $ABCD$ are equal, $BC = CD$, and the other two are not, $AB \ne AD$. Also $\angle BAC = \angle DAC$. A circle is drawn through points $A$ and $C$, which intersects for the second time the segment $AB$ at point $N$, and the line $AD$ at point $M$. Find the length of the segment $DM$ if it is known that $BN = a$.
The pentagon $ABCDE$ is inscribed in the circle, whose side $BC = \sqrt{10}$ Its diagonals $EC$ and $AC$ recut the diagonal $BD$ at points $L$ and $K$, respectively. It turned out that around the quadrilateral $AKLE$ you can circumscribe a circle. Find the length of the tangent from point $C$ to this circle.
2008 Belarusian MO A 11.2
$ABCD$ - quadrilateral inscribed in circle, and $AB=BC,AD=3DC$ . Point $R$ is on the $BD$ and $DR=2RB$. Point $Q$ is on $AR$ and $\angle ADQ = \angle BDQ$. Also $\angle ABQ + \angle CBD = \angle QBD$ . $AB$ intersect line $DQ$ in point $P$. Find $\angle APD$
Point $O$ - center of circle $\omega$. Point $A$ is outside $\omega$. Secant goes through $A$ and intersect circle in points $X$ and $Y$. Point $X'$ is symmetric for point $X$ with respect to line $OA$. Prove, that point of intersection of $OA$ and $X'Y$ is independent from the choice of secant.
$a,b,c$ - are sides of triangle $T$. It is known, that if we increase any one side by $1$, we get new
a) triangle
b) acute triangle
Find minimal possible area of triangle $T$ in case of a) and in case b)
On the coordinate plane $Oxy$, two straight lines are drawn, parallel to the abscissa axis .The distance between the those straight lines is $1$. Point $A$ is one of the points of intersection of the parabola $y = x^2$ with the one drawn straight lines, which is located closer to the abscissa axis, $B$ is the point of intersection of the second straight line with the parabola axis, $O$ is the origin of coordinates. Find the value of the angle $OAB$.
In trapezoid $ABCD$ ($BC \parallel AD$) the length of the side $BD$ is equal to the average of the lengths of the bases of trapezoid and $\angle CAD =30^o$. Find the value of the angle between diagonals $AC$ and $BD$.
On the parabola $y = x^2$, points $A, B, C$ are marked ($A$ - to the left of all) so that the bisector of the angle $ABC$ is parallel to it's axis. It is known that the projection of the segment $AC$ on the abscissa axis is $4$. Find the abscissa of the midpoint of the segment $BC$.
On the sides $AB, AC, BC$ of the triangle $ABC$, points $X, X_1, X_2$ are marked, respectively, so that $XX_1 \perp AC$, $X_1X_2 \perp BC$, $X_2X \perp AB$. Let $Y, Y_1, Y_2$ be points, respectively, on the sides $BC, AC, AB$ of the triangle $ABC$, so that $YY_1 \perp AC$, $Y_1Y_2 \perp AB$. Prove that $Y_2Y \perp BC$, if $XY$ and $AC$ are parallel.
Point $T$ the point of intersection of the two diagonals $AC$ and $BD$ of the convex quadrangular $ABCD$. The orthocenter of the triangle $ABT$ coincides with the center of the circumscribed circle of the triangle $CDT$. Prove that:
a) a circle can be drawn around the quadrilateral $ABCD$
b) the center of the circumscribed circle of thetriangle $CDT$ lies on the circumcircle of $ABCD$.
2009 grade B 10. missing
Let $AB$ be a chord on parabola $y=x^2$ and $AB||Ox$. For each point C on parabola different from $A$ and $B$ we are taking point $C_1$ lying on the circumcircle of $\triangle ABC$ such that $CC_1||Oy$. Find a locus of points $C_1$.
2009 Belarusian MO A 11.2
In the trapezoid $ABCD$, ($BC||AD$) $\angle BCD=72^{\circ}$, $AD=BD=CD$. Let point $K$ be a point on $BD$ such that $AK=AD$. $M$ is a midpoint of $CD$. $N$ is an intersection point of $AM$ and $BD$. Prove that $BK=ND$.
2009 Belarusian MO A 11.5
In acute triangle $\triangle ABC$ $\angle C=60^{\circ}$. Let $B_1$ and $A_1$ be the points on sides $AC$ and $BC$ respectively. Circumcircles of $\triangle BCB_1$ and $\triangle ACA_1$ intersect at the points $C$ and $D$. Prove that $D$ is a point on side $AB$ if and only if $\frac{CB_1}{CB}+\frac{CA_1}{CA}=1$
Given a trapezoid $ABCD$ ($AD\parallel BC$), the bisectors of angles $BAD$ and $CDA$ intersect on the perpenducular bisector of one of it's bases. Prove that $AB=CD$ if $AB + CD = AD$.
In the triangle $ABC$ to the side $AC$ , an angle bisector $BK$ is drawn. Find the angles of the triangle $ABC$, if $AK = 1$, and $BK=KC= 2$.
In triangle $ABC$, in which side $AB$ is the smallest, on the side $CA$ there is a point $M$ such that $CM = MB$, on the side $CB$, point $N$ such that $CN = NA$. Prove that point $A, B, N , M$ and center $O$ of the circumscribed circle of the triangle $ABC$ lie on the same circle
Quadrangle $ABCD$ is incribed in a circle. Prove that $C D \cdot B D> A B \cdot AC $ if $\frac{CD}{AB}>\frac{AB}{AC}$.
Given a trapezoid $ABCD$ ($AD\parallel BC$) $AD = 3BC$. Circle $\Gamma_1$ with center at point $B$ passes through the midpoint of diagonal $BD$, and the circle $\Gamma_2$ with center at point at point $C$ passes through the midpoint of diagonal $AC$. Prove that the straight line passing through the intersection pointcs of the circles $\Gamma_1$ and $\Gamma_1$ , intersects the base $AD$ at it's midpoint.
Circumcircles $\Gamma_1$ anδ $\Gamma_ 2$ touch each other externally at the same time at the point $M_3$ and tangent internally to the circle $\Gamma_3$ at the points $M_1$ and $M_2$, respectively. Let $S$ be the center of the circumscribed circle around the triangle $M_1M_2M_3$. Prove that the line $SM_1$ is tangent to the circle $\Gamma_3$.
2010 Belarusian MO A 11.1
Let $M$ be the point of intersection of the diagonals $AC$ and $BD$ of trapezoid $ABCD$ ($BC||AD$), $AD>BC$. Circle $w_1$ passes through the point $M$ and tangents $AD$ at the point $A$. Circle $w_2$ passes through the point $M$ and tangents $AD$ at the point $D$. Point $S$ is the point of intersection of lines $AB$ and $DC$. Line $AS$ intersects $w_1$ at the point $X$. Line $DS$ intersects $w_2$ at the point $Y$. $O$ is a center of a circumcircle of $\triangle ASD$. Prove that $SO\perp XY$
Let $O_1$ and $O_2$ be the centers of circles $w_1,w_2$ respectively. Circle $w_1$ intersects circle $w_2$ at points $C$ and $D$. Line $O_1O_2$ intersects circle $w_2$ at the point $A$. Line $DA$ intersects circle $w_1$ at the point $S$. Line $O_1O_2$ intersects line $SC$ at the point $F$. $E$ is an intersection point of circle $w_1$ and circumcircle $w_3$ of $\triangle ADF$. Prove that line $O_1E$ tangents circle $w_3$
a) Prove that a quadrilateral is a parallelogram if its diagonals and the two segments connecting the midpoints of opposite sides intersect at one point.
b) Can it be argued that the convexity of a quadrilateral is a parallelogram, if it corresponds to its dnagopalei in two segments, connecting the midpoints of opposite sides all intersect at one point?
On the side $AB$ triangle $ABC$ mark the point $K, L$ so that $\angle ACK = \angle KCL = \angle LCB$. The point $M$ lies on the side of the $BC$ such that $\angle MKC= \angle BKM$. Find the valus of the angle $MLC$, if it is known that the point $L$ lies on the bisector of the angle $KMB$.
The points $A_1 , B_1$ are marked respectively on the sides of the $AC, BC$ of triangle $ABC$ so that $A_1B_1 \parallel AB$. Point $A_2, B_2$ are the feet of the perpendiculars, respectively, dropped from points $A_1 , B_1$ on the side $AB$. Prove that $AC = AB_2 + CB_1$ if only if $BC = BA_2 + CA_1$.
Let $P$ be the point of intersection of the diagonals of the cyclic quadrilateral $ABCD$. On the bisectors of angles $APD, BPC$ are marked points $K , L$ respectively such that $AP=PK$ and $BP = PL$. Let us denote by $M$ the point of intersection of lines $AK$ and $BL$, and by $N$ the instersection of lines $KD$ and $LC$. Prove that lines $KL$ and $MN$ are perpendicular.
The points $M , N$ are the midpoints of the sides $AC$ , $BC$ of the triangle ABC respectively. Prove that the circle passing through the points $C, M$ and $N$, is tangent to the side $AB$ if and only if $AB =\frac{AC+BC}{\sqrt2}$
Point $M$ is the midpoint of the side $AB$ of an acute-angled non-isosceles triangle $ABC, H$ is the orthocenter of this triangle, and $I$ is the center of the circumference inscribed into the triangle. Prove that if points $M, I$ and $H$ lie on one straight line, then the length of the segment $CH$ is equal to the radius of the inscribed circle of the triangle $ABC$.
On the parabola $y = x^2$ mark four points $A, B, C, D$, so that the quadrangle $ABCD$ is a trapezoid ($AD\parallel BC$, $AD> BC$). Let $m, n$ be the distances from the point of intersection of the diagonals of this trapezoid to the midpoints of its sides $AD, BC$ respectively. Calculate the area of the trapezoid $ABCD$.
2011 Belarusian MO A 11.3
Let $M$ be a midpoint of the side $AB$ of the oxygon ${\triangle ABC}$, points $P$ and $Q$ are bases of altitudes $AP$ and $BQ$ of this triangle. It is known that circumcircle of ${\triangle BMP}$ tangents side ${AC}$. Prove that circumcircle of ${\triangle AMQ}$ tangents line ${BC}$.
Let $B$ and $C$ be the points on hyperbola $y=1/x$ $(x>0)$ and abscissa of point $C$ is greater than abscissa of point $B$. Line $OA$ ($O$ is an origin) intersects hyperbola $y=1/x$ $(x<0)$ at point $A$. Prove that the angle $BAC$ equals one frome the angles between line $BC$ and tangent to hyperbola at point $B$
Let $I$ be an incenter of non-isosceles oxygon $\triangle ABC$ and $Q$ is a tangent point lying on $AB$. Point $T$ belongs to side $AB$ and $IT||CQ$. Line $TK$ tangents inscribed circle at the point $K$ (different from the point $Q$ and intersects lines $CA$ and $CB$ at points $L$ and $N$ respectively. Prove that $T$ is a midpoint of $LN$.
On the hypotenuse $AB$ of a right-angled triangle $ABC$ in the outer side is a congruent triangle $AM N$, $\angle ANM = 90^o$, $AN = BC$ (see fig.). The circle $\Gamma_1$, inscribed in a triangle $AM N$, touches the hypotenuse $AM$ at point $P$, and the circle $\Gamma_2$ inscribed in triangle $ABC$, touches leg $BC$ at point $Q$. Prove that the segment $PQ$, the hypotenuse $AB$ and the segment connecting the centers of the circles $\Gamma_1$ and $\Gamma_2$, intersect at one point.
In trapezoid $ABCD$ ($BC\parallel AD$) diagonal $CA$ is bisector of the angle $BCD, CD = AO$ and $BC = OD$, where $O$ is the point of intersection of the diagonals. Find the angles of the trapezoid $ABCD$.
Inside the convex quadrilateral $ABCD$, a point $M$ is marked that is different from the point of intersection of its diagonals $AC$ and $BD$, so that the ratio of the areas of the triangles $AMC$ and $BMD$ is equal to the ratio of the tangents of the angles $AMC$ and $BMD$, i.e. $S (AM C): S (BM D) = tan (\angle AMC): tan (\angle BMD)$. Prove that $AM^2 + MC^2 + BD^2 = AC^2 + BM^2 + M D^2$.
The diagonal $BD$ is drawn in the quadrilateral $ABCD$. Find the maximum possible value of the area of this quadrilateral, if the length of the broken line $ABDC$ is equal to $L$.
In acute-angled $\vartriangle ABC$ on sides $AB$ and $AC$ outwardly, squares with centers $C_1$ and $B_1$ respectively. Square $C_1B_1DE$ is constructed on segment $C_1B_1$, so that points $A$ and $D$ lie in different half-planes relative to $C_1B_1$. Prove that the center of the square $C_1B_1DE$ lies on $BC$.
Some three sides of a quadrangle have lengths $2, 7$ and $11$. Find the area of this quadrilateral if it is known to have the largest area of all quadrangles with the indicated sides.
Let $AB$ and $CD$ be two parallel chordes on hyperbola $y=1/x$. Lines $AC$ and $BD$ intersect axis $Oy$ at points $A_1$ and $D_1$ respectively, and axis $Ox$ - at points $C_1$ and $B_1$ respectively. Prove that the area of $\triangle A_1OC_1$ equals the area of $\triangle D_1OB_1$
Let point $I$ be an incenter of $\triangle ABC$. Ray $AI$ intersects circumcircle of $\triangle ABC$ at point $D$. Circumcircle of $\triangle CDI$ intersects ray $BI$ at ponts $I$ and $K$. Prove that $BK=CK$.
year 2013 missing
Points $X,Y$, and $Z$ are marked on the sides $AD,AB$ and $BC$ of the rectangular $ABCD$ respectively. Given $AX=CZ$, prove $XY+YZ \ge AC$.
Let $H$ he an intersection point of the altitudes $AA_1, BB_1, CC_1$ of an acute-angled triangle $ABC$. Let $M$ and $N$ be the midpoints of the segments $BC$ and $AH$, respectively. Prove that $MN$ is the perpendicular bisector of the segment $B_1C_1$.
Points $B_1$ and $A_1$ are marked on the sides $AC$ and $BC$ of a triangle $ABC$, respectively. Let $X$ be the intersection point of the segments $AA_1$ and $BB_1$. Let $x,y$ and $z$ be the areas of the triangles $B_1CA_1, B_1XA_1$ and $AXB$, respectively (see the fig.). Prove that :
a) $y<z$
b) $y<x$
The graph of the hyperbola $y =\frac{1}{x}$ is drawn on the Cartesian plane $Oxy$. Three snails start simultaneously from the origin $O$ and move along the abscissa axis $Ox$ (each snail has its constant speed)! Let $A(t)$, $B(t)$ and $C(t)$ be points on the graph of the hyperbola such that their abscissae are equal to the abscissae of the first, the second and the third snails at the moment $t$, respectively.
Prove that the area of the triangle $ABC$ is independent of time.
Points $X,Y$, and $Z$ are marked on the sides $AD,AB$ and $BC$ of the parallelogram $ABCD$ respectively. It is known that $AX=CZ$.
a) Prove that at least one of the inequalities holds: $XY+YZ \ge AC$ or $XY+YZ \ge BD$.
b) Is it true that $XY+YZ \ge \frac{AC +BD}{2}$ ?
Points $B_1$ and $A_1$ are marked on the sides $AC$ and $BC$ of a triangle $ABC$, respectively. Let $X$ be the intersection point of the segments $AA_1$ and $BB_1$. Let $x,y$ and $z$ be the areas of the triangles $B_1CA_1, B_1XA_1$ and $AXB$, respectively (see the fig.). Find tha area of the triangle $ABC$.
Let $\Omega$ be the circumcircle of a triangle $ABC$. A circle passing through the vertex $A$ and touching $BC$ at point $X$ meets $\Omega$ at point $Y$ (different from $A$). Let point $Z$ (different from $Y$) be the intersection point of the ray $YX$ and $\Omega$. Prove that $\angle CAX = \angle ZAB$.
The graph of the parabola $y = x^2$ is drawn on the Cartesian plane $Oxy$. The vertices of a triangle $ABC$ belong to the parabola. The median $BM$ of the triangle is parallel to the ordinate axis and is equal to $2$. Find the area of the triangle $ABC$.
Given triangle $ABC$ with $AB = c, BC = a, CA = b$. The pairs of points $C_1$ and $C_2, A_1$ and $A_2, B_1$ and $B_2$ are marked on the sides $AB, BC, CA$, respectively so that the following equalities are valid: $\frac{CA_1}{a}=\frac{CB_2}{b}=\frac{a+b}{a+b+c}$, $\frac{AB_1}{b}=\frac{AC_2}{c}=\frac{b+c}{a+b+c}$, $ \frac{BC_1}{c}=\frac{BA_2}{a}=\frac{a+c}{a+b+c}.$
Prove that the points of intersection of the lines $A_1C_2, C_1B_2$ and $B_1A_2$ belong to the circumcircle of the triangle $ABC$.
Points $B_1$ and $A_1$ are marked on the sides $AC$ and $BC$ of a triangle $ABC$, respectively. Let $X$ be the intersection point of the segments $AA_1$ and $BB_1$. Let $x,y$ and $z$ be the areas of the triangles $B_1CA_1, B_1XA_1$ and $AXB$, respectively (see the fig.). Prove that :
a) $y<\frac{1}{\sqrt5}\sqrt{xz}$
b) $y<\frac{1}{3}\sqrt{xz}$
Let $ABC$ be a triangle inscribed in the parabola $y=x^2$ such that the line $AB \parallel$ the axis $Ox$. Also point $C$ is closer to the axis $Ox$ than the line $AB$. Given that the length of the segment $AB$ is 1 shorter than the length of the altitude $CH$ (of the triangle $ABC$). Determine the angle $\angle{ACB}$ .
The angles at the vertices $A$ and $C$ in the convex quadrilateral $ABCD$ are not acute. Points $K, L, M$ and $N$ are marked on the sides $AB, BC, CD$ and $DA$ respectively. Prove that the perimeter of $KLMN$ is not less than the double length of the diagonal $AC$.
Points $C_1, A_1$ and $B_1$ are marked on the sides $AB, BC$ and $CA$ of a triangle $ABC$ so that the segments $AA_1, BB_1$, and $CC_1$ are concurrent (see the fig.). It is known that the area of the white part of the triangle $ABC$ is equal to the area of its black part. Prove that at least one of the segments $AA_1, BB_1, CC_1$ is a median of the triangle $ABC$.
2015 Grades D 8. C.9 B.10 not available
Line intersects hyperbola $H_1$, given by the equation $y=1/x$ at points $A$ and $B$, and hyperbola $H_2$, given by the equation $y=-1/x$ at points $C$ and $D$. Tangents to hyperbola $H_1$ at points $A$ and $B$ intersect at point $M$, and tangents to hyperbola $H_2$ at points $C$ and $D$ intersect at point $N$. Prove that points $M$ and $N$ are symmetric about the origin.
Let $A_1$ be a midmoint of $BC$, and $G$ is a centroid of the non-isosceles triangle $\triangle ABC$. $GBKL$ and $GCMN$ are the squares lying on the left with respect to rays $GB$ and $GC$ respectively. Let $A_2$ be a midpoint of a segment connecting the centers of the squares $GBKL$ and $GCMN$. Circumcircle of triangle $\triangle A_{1}A_{2}G$ intersects $BC$ at points $A_1$ and $X$. Find $\frac{A_{1}X}{XH}$, where $H$ is a base of altitude $AH$ of the triangle $\triangle ABC$.
Let $I$ be an incenter of a triangle $\triangle ABC$. Points $A_1, B_1, C_1$ are the tangent points of the inscribed circle on sides $BC$, $CA$ and $AB$ respectively. Circumcircle of $\triangle BC_1B_1$ intersects line $BC$ at points $B$ and $K$ and Circumcircle of $\triangle CB_1C_1$ intersects line $BC$ at points $C$ and $L$. Prove that lines $LC_1$, $KB_1$ and $IA_1$ are concurrent.
year 2016 missing
Let M be the midpoint of the hypotenuse $AB$ of the right triangle $ABC$. Point $P$ is chosen on the cathetus $CB$ so that $CP : PB = 1 : 2$. The straight line passing through $B$ meets the segments $AC, AP$ and $PM$ at points $X, Y$, and $Z$ respectively. Prove that the bisector of the angle $PZY$ passes through point $C$ if and only if the bisector of the angle $PYX$ also passes through $C$.
Point $M$ is marked inside a convex quadrilateral $ABCD$. It appears that $AM = BM, CM = DM$ and $\angle AMB =\angle CMD = 60^o$. Let $K, L$, and $N$ be the midpoints of the segments $BC, AM$, and $DM$, respectively. Find the value of the angle $LKN$.
Given a convex hexagon $H$ with obtuse inner angles and pairwrise parallel opposite sides.
a) Prove that there exists a pair of the opposite sides of $H$ which, possesses the following property; there exists a straight line that is perpendicular to these sides and intersects each of them.
b) Is it true that there exist two pairs of the opposite sides of $H$ , each of which possesses the same property, as described, in item a) ?
In an isosceles triangle $ABC$ with $AB= BC$, points $K$ and $M$ are the midpoints of the sides $AB$ and $AC$ , respectively. The circumscribed circle ot the triangle $CKB$ meets the line $BM$ at point $N$ different from $M$. The line passing through $N$ parallel to the side $AC$ meets the circumscribed circle of the triangle $ABC$ at points $A_1$ and $C_1$. Prove that the triangle $A_1BC_1$ is equilateral.
Points $K$ and $M$ are the midpoints of the sides $AB$ and $AC$ of triangle $ABC$ respectively. The equilateral triangles $AMN$ and $BKL$ are constructed on the sides $AM$ and $BK$ to the exterior of the triangle $ABC$. Point $F$ is the midpoint of the segment $LN$. Find the value of the angle $KFM$.
Given a convex $2n$-gon $H$ with pairwise parallel opposite sides.
a) Prove that there: exists a pair of the opposite sides of $H$ which possesses the following property: there exists a straight line that is perpendicular to these sides and intersects each of them.
b) Are there any values of $n$ such that for aconvex $2n$-gon there exist two pairs of its opposite sides for each of which the property described in a) holds?
Point $D$ is marked on the side $AB$ of triangle $ABC$. The bisectors of the angles $ABC$ and $ADC$ meet at point $U$, and the bisectors of the angles $BAC$ and $BDC$ meet at point $V$. Let $S$ be the midpoint of the segment $UV$. Prove that the lines $SD$ and $AB$ are perpendicular if and only if the inscribed circles of the triangles $ADC$ and $BDC$ are tangent.
A parabola $y = x^2-a$ meets the right branch of the hyperbola $y = 1/x$ at point $A$, and meets its left branch at points $B$ and $C$.
a ) Find all possible values of a if the triangle $ABC$ is a right triangle,
b) Find the area of this right triangle for all possible values of $a$.
Let $AA_1, BB_1$ and $CC_1$ be the altitudes of the acute triangle $ABC$ ($A_1\in BG, B_1\in CA$ and $C_1\in AB$). Let $J_a, J_b$, and $J_c$ be the centers of the inscribed circles of the triangles $AC_1B_1, BA_1C_1$ and $CB_1A_1$ respectively. Prove that the orthocenter of the triangle $J_aJ_bJ_c$ coincides with the incenter of the triangle $ABC$.
2017 Belarusian MO A 11.2
Let $M$ - be a midpoint of side $BC$ in triangle $ABC$. A cricumcircle of $ABM$ intersects segment $AC$ at points $A$ and $B_1$ ($B_1 \neq A$). A circumcircle of $AMC$ intersects segment $AB$ at points $A$ and $C_1$ ($C_1 \neq A$). Let $O$ be a circumcircle of $AC_1B_1$. Prove that $OB=OC$
2017 Belarusian MO A 11.6
Let $AA_1, BB_1, CC_1$ be altitudes of an acute-angeled triangle $ABC$ ($A_1 \in BC, B_1 \in AC, C_1 \in AB$). Let $J_a, J_b, J_c$ be centers of inscribed circles of $AC_1B_1$, $BA_1C_1$ and $CB_1A_1$ respectively. Prove that radius of circumecircle of triangle $J_aJ_bJ_c$ equals radius of inscribed circle of triangle $ABC$
2018 Belarusian MO A 11.2
The altitudes $AA_1$, $BB_1$ and $CC_1$ are drawn in the acute triangle $ABC$. The bisector of the angle $AA_1C$ intersects the segments $CC_1$ and $CA$ at $E$ and $D$ respectively. The bisector of the angle $AA_1B$ intersects the segments $BB_1$ and $BA$ at $F$ and $G$ respectively. The circumcircles of the triangles $FA_1D$ and $EA_1G$ intersect at $A_1$ and $X$.
Prove that $\angle BXC=90^{\circ}$.
2018 Belarusian MO A 11.5
The circle $S_1$ intersects the hyperbola $y=\frac1x$ at four points $A$, $B$, $C$, and $D$, and the other circle $S_2$ intersects the same hyperbola at four points $A$, $B$, $F$, and $G$. It's known that the radii of circles $S_1$ and $S_2$ are equal. Prove that the points $C$, $D$, $F$, and $G$ are the vertices of the parallelogram.
2018 Belarusian MO A 11.6
The point $X$ is marked inside the triangle $ABC$. The circumcircles of the triangles $AXB$ and $AXC$ intersect the side $BC$ again at $D$ and $E$ respectively. The line $DX$ intersects the side $AC$ at $K$, and the line $EX$ intersects the side $AB$ at $L$. Prove that $LK\parallel BC$.
2017 Belarusian MO A 11.2
Let $M$ - be a midpoint of side $BC$ in triangle $ABC$. A cricumcircle of $ABM$ intersects segment $AC$ at points $A$ and $B_1$ ($B_1 \neq A$). A circumcircle of $AMC$ intersects segment $AB$ at points $A$ and $C_1$ ($C_1 \neq A$). Let $O$ be a circumcircle of $AC_1B_1$. Prove that $OB=OC$
Let $AA_1, BB_1, CC_1$ be altitudes of an acute-angeled triangle $ABC$ ($A_1 \in BC, B_1 \in AC, C_1 \in AB$). Let $J_a, J_b, J_c$ be centers of inscribed circles of $AC_1B_1$, $BA_1C_1$ and $CB_1A_1$ respectively. Prove that radius of circumecircle of triangle $J_aJ_bJ_c$ equals radius of inscribed circle of triangle $ABC$
Let $ABCD$ be a cyclic quadrilateral with the circumcircle $\omega$ . The points $B_1$ and $D_1$ are symmetric to $A$ with respect to the midpoints of $BC$ and $CD$. The circumcircle of the triangle $CB_1D_1$ intersects $\omega$ at $C$ and $G$. Prove that $AG$ is the diameter of $\omega$ .
In the parallelogram $ABCD$ $(AB//CD)$, the side $AB$ is a half length of the side $BC$. The bisector of the angle $ABC$ intersects the side $AD$ at $K$ and the diagonal $AC$ at $L$. The bisector of the angle $ADC$ intersects the extension of the side $AB$ beyond $B$ at point $M$. The line $ML$ intersects the side $AD$ at $F$. Find the ratio $AF:AD$.
2018 Belarusian MO C 9.3
The bisector of angle $CAB$ of triangle $ABC$ intersects the side $CB$ at $L$. The point $D$ is the foot of the perpendicular from $C$ to $AL$ and the point $E$ is the foot of perpendicular from $L$ to $AB$. The lines $CB$ and $DE$ meet at $F$. Prove that $AF$ is an altitude of triangle $ABC$.
2018 Belarusian MO C 9.5
The quadrilateral $ABCD$ is inscribed in the parabola $y=x^2$. It is known that angle $BAD=90$, the dioganal $AC$ is parallel to the axis $Ox$ and $AC$ is the bisector of the angle BAD. Find the area of the quadrilateral $ABCD$ if the length of the dioganal $BD$ is equal to $p$.
2018 Belarusian MO C 9.7
A point $O$ is choosen inside a triangle $ABC$ so that the length of segments $OA$, $OB$ and $OC$ are equal to $15$,$12$ and $20$, respectively. It is known that the feet of the perpendiculars from $O$ to the sides of the triangle $ABC$ are the vertices of an equilateral triangle. Find the value of the angle $BAC$.
The bisector of angle $CAB$ of triangle $ABC$ intersects the side $CB$ at $L$. The point $D$ is the foot of the perpendicular from $C$ to $AL$ and the point $E$ is the foot of perpendicular from $L$ to $AB$. The lines $CB$ and $DE$ meet at $F$. Prove that $AF$ is an altitude of triangle $ABC$.
2018 Belarusian MO C 9.5
The quadrilateral $ABCD$ is inscribed in the parabola $y=x^2$. It is known that angle $BAD=90$, the dioganal $AC$ is parallel to the axis $Ox$ and $AC$ is the bisector of the angle BAD. Find the area of the quadrilateral $ABCD$ if the length of the dioganal $BD$ is equal to $p$.
2018 Belarusian MO C 9.7
A point $O$ is choosen inside a triangle $ABC$ so that the length of segments $OA$, $OB$ and $OC$ are equal to $15$,$12$ and $20$, respectively. It is known that the feet of the perpendiculars from $O$ to the sides of the triangle $ABC$ are the vertices of an equilateral triangle. Find the value of the angle $BAC$.
The extension of the median $AM$ of the triangle $ABC$ intersects its circumcircle at $D$. The circumcircle of triangle $CMD$ intersects the line $AC$ at $C$ and $E$.The circumcircle of triangle $AME$ intersects the line $AB$ at $A$ and $F$. Prove that $CF$ is the altitude of triangle $ABC$.
2018 Belarusian MO B 10.6
2018 Belarusian MO B 10.7
The vertices of the convex quadrilateral $ABCD$ lie on the parabola $y=x^2$. It is known that $ABCD$ is cyclic and $AC$ is a diameter of its circumcircle. Let $M$ and $N$ be the midpoints of the diagonals of $AC$ and $BD$ respectively. Find the length of the projection of the segment $MN$ on the axis $Oy$.
The square $A_1B_1C_1D_1$ is inscribed in the right triangle $ABC$ (with $C=90$) so that points $A_1$, $B_1$ lie on the legs $CB$ and $CA$ respectively ,and points $C_1$, $D_1$ lie on the hypotenuse $AB$. The circumcircle of triangles $B_1A_1C$ an $AC_1B_1$ intersect at $B_1$ and $Y$. Prove that the lines $A_1X$ and $B_1Y$ meet on the hypotenuse $AB$.
The altitudes $AA_1$, $BB_1$ and $CC_1$ are drawn in the acute triangle $ABC$. The bisector of the angle $AA_1C$ intersects the segments $CC_1$ and $CA$ at $E$ and $D$ respectively. The bisector of the angle $AA_1B$ intersects the segments $BB_1$ and $BA$ at $F$ and $G$ respectively. The circumcircles of the triangles $FA_1D$ and $EA_1G$ intersect at $A_1$ and $X$.
Prove that $\angle BXC=90^{\circ}$.
2018 Belarusian MO A 11.5
The circle $S_1$ intersects the hyperbola $y=\frac1x$ at four points $A$, $B$, $C$, and $D$, and the other circle $S_2$ intersects the same hyperbola at four points $A$, $B$, $F$, and $G$. It's known that the radii of circles $S_1$ and $S_2$ are equal. Prove that the points $C$, $D$, $F$, and $G$ are the vertices of the parallelogram.
2018 Belarusian MO A 11.6
The point $X$ is marked inside the triangle $ABC$. The circumcircles of the triangles $AXB$ and $AXC$ intersect the side $BC$ again at $D$ and $E$ respectively. The line $DX$ intersects the side $AC$ at $K$, and the line $EX$ intersects the side $AB$ at $L$. Prove that $LK\parallel BC$.
2019 grade D 8. missing
2019 Belarusian MO C 9.2
The rhombus $ABCD$ is given. Let $E$ be one of the points of intersection of the circles $\Gamma_B$ and $\Gamma_C$, where $\Gamma_B$ is the circle centered at $B$ and passing through $C$, and $\Gamma_C$ is the circle centered at $C$ and passing through $B$. The line $ED$ intersects $\Gamma_B$ at point $F$.Find the value of angle $\angle AFB$.
The point $M$ is the midpoint of the side $BC$ of triangle $ABC$. A circle is passing through $B$, is tangent to the line $AM$ at $M$, and intersects the segment $AB$ secondary at the point $P$.
Prove that the circle, passing through $A$, $P$, and the midpoint of the segment $AM$, is tangent to the line $AC$.
A point $P$ is chosen in the interior of the side $BC$ of triangle $ABC$. The points $D$ and $C$ are symmetric to $P$ with respect to the vertices $B$ and $C$, respectively. The circumcircles of the triangles $ABE$ and $ACD$ intersect at the points $A$ and $X$. The ray $AB$ intersects the segment $XD$ at the point $C_1$ and the ray $AC$ intersects the segment $XE$ at the point $B_1$. Prove that the lines $BC$ and $B_1C_1$ are parallel.
The tangents to the circumcircle of the acute triangle $ABC$, passing through $B$ and $C$, meet at point $F$. The points $M$, $L$, and $N$ are the feet of perpendiculars from the vertex $A$ to the lines $FB$, $FC$, and $BC$, respectively. Prove the inequality $AM+AL\ge 2AN$.
a) Find all real numbers $a$ such that the parabola $y=x^2-a$ and the hyperbola $y=1/x$ intersect each other in three different points.
b) Find the locus of centers of circumcircles of such triples of intersection points when $a$ takes all possible values.
The altitudes $CC_1$ and $BB_1$ are drawn in the acute triangle $ABC$. The bisectors of angles $\angle BB_1C$ and $\angle CC_1B$ intersect the line $BC$ at points $D$ and $E$, respectively, and meet each other at point $X$. Prove that the intersection points of circumcircles of the triangles $BEX$ and $CDX$ lie on the line $AX$.
The diagonals of the inscribed quadrilateral $ABCD$ intersect at the point $O$. The points $P$, $Q$, $R$, and $S$ are the feet of the perpendiculars from $O$ to the sides $AB$, $BC$, $CD$, and $DA$, respectively. Prove the inequality $BD\ge SP+QR$.
The rhombus $ABCD$ is given. Let $E$ be one of the points of intersection of the circles $\Gamma_B$ and $\Gamma_C$, where $\Gamma_B$ is the circle centered at $B$ and passing through $C$, and $\Gamma_C$ is the circle centered at $C$ and passing through $B$. The line $ED$ intersects $\Gamma_B$ at point $F$.Find the value of angle $\angle AFB$.
(S. Mazanik)
2019 Belarusian MO C 9.6The point $M$ is the midpoint of the side $BC$ of triangle $ABC$. A circle is passing through $B$, is tangent to the line $AM$ at $M$, and intersects the segment $AB$ secondary at the point $P$.
Prove that the circle, passing through $A$, $P$, and the midpoint of the segment $AM$, is tangent to the line $AC$.
(A. Voidelevich)
2019 Belarusian MO B 10.2A point $P$ is chosen in the interior of the side $BC$ of triangle $ABC$. The points $D$ and $C$ are symmetric to $P$ with respect to the vertices $B$ and $C$, respectively. The circumcircles of the triangles $ABE$ and $ACD$ intersect at the points $A$ and $X$. The ray $AB$ intersects the segment $XD$ at the point $C_1$ and the ray $AC$ intersects the segment $XE$ at the point $B_1$. Prove that the lines $BC$ and $B_1C_1$ are parallel.
(A. Voidelevich)
2019 Belarusian MO B 10.8The tangents to the circumcircle of the acute triangle $ABC$, passing through $B$ and $C$, meet at point $F$. The points $M$, $L$, and $N$ are the feet of perpendiculars from the vertex $A$ to the lines $FB$, $FC$, and $BC$, respectively. Prove the inequality $AM+AL\ge 2AN$.
(V. Karamzin)
2019 Belarusian MO A 11.1a) Find all real numbers $a$ such that the parabola $y=x^2-a$ and the hyperbola $y=1/x$ intersect each other in three different points.
b) Find the locus of centers of circumcircles of such triples of intersection points when $a$ takes all possible values.
(I. Gorodnin)
2019 Belarusian MO A 11.4The altitudes $CC_1$ and $BB_1$ are drawn in the acute triangle $ABC$. The bisectors of angles $\angle BB_1C$ and $\angle CC_1B$ intersect the line $BC$ at points $D$ and $E$, respectively, and meet each other at point $X$. Prove that the intersection points of circumcircles of the triangles $BEX$ and $CDX$ lie on the line $AX$.
(A. Voidelevich)
2019 Belarusian MO A 11.6The diagonals of the inscribed quadrilateral $ABCD$ intersect at the point $O$. The points $P$, $Q$, $R$, and $S$ are the feet of the perpendiculars from $O$ to the sides $AB$, $BC$, $CD$, and $DA$, respectively. Prove the inequality $BD\ge SP+QR$.
(A. Naradzetski)
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