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Mathematical Multiathlon Juniors - geometry
3
Mathematical Multiathlon Tournament (Russia) / Математическое многоборье
Reggata
2008.1.2
Inside the rectangle $ABCD$, whose sides $AB = CD = 15$ and $BC = AD = 10$, a point $P$ is given such that $AP = 9$, $BP = 12$. Find $CP$.
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2008.2.2
Given a convex pentagon $ABCDE$ such that $AB = AE = DC = BC + DE = 1$ and $\angle ABC = \angle DEA = 90^o$. What is the area of this pentagon?
parmenides51
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2008.3.2
An isosceles triangle $ABC$ with a base $AC$ is inscribed in circle $\omega$. It turned out that the radius of the inscribed circle $ABC$ is equal to the radius of the circle tangent to the smaller arc $BC$ of the circle $\omega$ and the side of the $BC$ in its midpoint (see fig.). Find the ratio of the sides of the triangle $ABC$.
https://cdn.artofproblemsolving.com/attachments/8/7/5272e7556b7cc85b14bb01345200d2db5c12ab.png
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2008.4.2
In a convex polygon $A_1A_2...A_{2006}$ opposite sides are parallel ($A_1A_2\parallel A_{1004}A_{1005}, ...$). Prove that the diagonals $A_1A_{1004},A_2A_{1005},...,A_{1003}A_{2006}$ intersect at one point if and only if every two opposite sides are equal.
parmenides51
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2009.1.2
In a triangle $ABC$ with an angle $BAC$ equal to $24^o$, the points $X$ and $Y$ are taken on the sides $AB$ and $AC$, respectively. In this case, a circle centered at $Y$ passing through $A$ also passes through $X$, and a circle centered at $X$ passing through $B$ also passes through $C$ and $Y$. Find $\angle ABC$.
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2009.2.2
$AL, BM, CN$ are medians of triangle $ABC$, intersecting at point $K$. It is known that the quadrilateral $CLKM$ is cyclic and $AB = 2$. Find the length of the median $CN$.
parmenides51
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2009.3.2
Can a right isosceles triangle be split into $6$ different right isosceles triangles?
parmenides51
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2009.4.2
Let $E$ be the intersection point of the diagonals of the convex quadrilateral $ABCD$. It is known that the perimeters of triangles $ABE, BCE, CDE, DAE$ are the same, and the radii of the inscribed circles of triangles $ABE, CE, CDE$ are equal to $3, 4, 6$, respectively. Find the radius of the inscribed circle of triangle $DAE$.
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2020.1.2
An equilateral triangle $ABC$ is given. Point $D$ is such that $\angle BDC = 90^o$ and points $B, A$ lie in different half-planes wrt line $BC$. Point $M$ is the midpoint of side $AB$. Find the angle $BDM$.
parmenides51
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2010.2.2
In triangle $ABC$, point $M$ is the midpoint of side $AB$ and $BD$ is the angle bisector. Prove that $\angle MDB = 90^o$ if and only if $AB=3BC$.
parmenides51
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2010.3.2
In the triangle $ABC$ from the vertex $A$, the altitude $AH$ was drawn. It turned out that $CH:HB=CA^2: AB^2 \ne 1$. What values can the angle $A$ take?
parmenides51
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2010.4.2
In a right-angled triangle $ABC$ with a right angle $A$, bisectors $BB_1$ and $CC_1$ are drawn. From points $B_1$ and $C_1$, perpendiculars $B_1B_2$ and $C_1C_2$ are drawn on the hypotenuse $BC$. What is the angle $B_2AC_2$ ?
parmenides51
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2011.1.2
Point $P$ lies on side $BC$ of square $ABCD$. A square $APRS$ was built with side the segment $AP$ . Prove that the angle $RCD$ is $45^o$. (The vertices of both squares are labeled clockwise.)
parmenides51
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2011.2.2
In an isosceles triangle $ABC$ ($AB = AC$), the angle $BAC$ is $40^o$. Points $S$ and $T$ lie on sides $AB$ and $BC$ respectively, such that $\angle BAT = \angle BCS = 10^o$. Segments $AT$ and $CS$ meet at $P$. Prove that $BT = 2PT$.
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2011.3.2
The angle bisector $BD$ is drawn in an isosceles triangle $ABC$ with base $BC$. It turned out that $BD + DA = BC$. Find the angles of triangle $ABC$.
parmenides51
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2011.4.2
The $h_a$ and $h_b$ are drawn on the adjacent sides $a$ and $b$ of parallelogram $ABCD$, respectively. It is known that $a + h_a = b + h_b$. Consider segments $AB$, $AC$, $AD$, $BC$, $BD$, $CD$. What is the largest number of different ones among them?
parmenides51
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2012.1.2
On the sides $AB$ and $BC$ of triangle $ABC$, there are points $X$ and $Y$, respectively, such that $\angle BYX = \angle AYC$ and $\frac{BY}{Y C} = \frac{2BX}{XA}$ . Prove that triangle $ABC$ is right-angled.
parmenides51
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2012.2.2
On the lateral side $AB$ of a right trapezoid $ABCD$ ($AB\perp BC$), a semicircle is constructed (having it as diameter) that touches the side $CD$ at point $K$. The diagonals of the trapezoid meet at point $O$. Find the length of the segment $OK$ if the lengths of the bases of the trapezoid $ABCD$ are equal to $2$ and $3$.
parmenides51
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2012.3.2
In trapezoid $ABCD$ with bases $AB$ and $CD$, it turned out that $AD = DC = CB <AB$. Points $E$ and $F$ lie on the sides $CD$ and $BC$ are respectively such that $\angle ADE = \angle AEF$. Prove that $4CF \le BC$.
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2012.4.2
The quadrilateral $ABCD$ is inscribed in a circle. It is known that $AB = AC$ and $BC = CD$. The diagonals of the quadrilateral $ABCD$ meet at the point $O$, and the point $X$ is the midpoint of the arc $CD$ not containing the point $A$. Prove that $XO \perp AB$.
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2013.1.2
Points $M, N, P$ are the midpoints of sides $AB, CD$ and $DA$ of the inscribed quadrilateral $ABCD$. It is known that $\angle MPD=150^o$, $\angle BCD=140^o$. Find the angle $\angle PND$.

(D. Maksimov)
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2013.2.2
There are $2013$ line segments of unit length on the plane, each intersecting with everyone. Prove that all of them can be covered with a circle of radius $1.5$.

(A. Shapovalov)
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2013.3.2
A circle center $O$ is inscribed in the quadrilateral $ABCD$. $AB$ is parallel to and longer than $CD$ and has midpoint $M$. The line $OM$ meets $CD$ at $F$. $CD$ touches the circle at $E$. Show that $DE = CF$ iff $AB = 2CD$.
Megus
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2013.3.2
A circle center $O$ is inscribed in the quadrilateral $ABCD$. $AB$ is parallel to and longer than $CD$ and has midpoint $M$. The line $OM$ meets $CD$ at $F$. $CD$ touches the circle at $E$. Show that $DE = CF$ iff $AB = 2CD$.
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2013.4.2
In parallelogram $ABCD$, $BD=BC$. A point $M$ on $AC$ is such that $3AM=AC$. Prove that $AM=BM$.

(13th Ural Tournament, Major League)
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2014.1.2
A circle with center $O$ is inscribed in the triangle $ABC$. The point $L$ lies on the extension of the side $AB$ beyond $A$. The tangent from $L$ intersects the side $AC$ in the point $K$. Find $\angle KOL$ if $\angle BAC = 50^o$.

(revision of the problem of the Soros Olympiad 1995)
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2014.2.2
From the point $A_0$ black and red rays are drawn with an angle $7^o$ between them. Then a polyline $A_0A_1...A_{20}$ is drawn (possibly self-intersecting, but with all vertices different), in which all segments have length $1$, all vertices with even numbers lie on the black ray and the ones with odd numbers - on the red ray. What is the index of the vertex that is farthest from $A_0$?

(old Mathematical Kangaroo)
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2014.3.2
In the convex quadrilateral $ABCD$, the bisectors of the angles $A$ and $C$ are parallel and intersect the diagonal $BD$ in two distinct points $P$ and $Q$, so that $BP = DQ$. Prove that the quadrilateral $ABCD$ is a parallelogram.

(A.Chapovalov)
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2014.4.2
A paper rectangle $ABCD$ ($AB = 3$, $BC = 9$) is folded in such way that vertices $A$ and $C$ coincide. What is the area of the obtained pentagon?

(additional problems from Mathematical Kangaroo 2014)
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2015.1.2
Different points $A,B$ and $C$ are marked on the straight line in such a way that $AB=AC=1$. On the segments $AB$ and $AC$, a square $ABDE$ and an equilateral triangle $ACF$ are constructed in one half-plane, respectively. Find the angle between lines $BF$ and $CE$.
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2015.2.2
In a quadrilateral $ABCD$, point $M$ is the midpoint of side $AB$. Prove that if the angle $DMC$ is right , then $AD+BC \ge CD .$
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2015.3.2
An angle bisector $AD$ is drawn in an acute-angled triangle $ABC$. The perpendicular drawn from point $B$ on line $AD$ intersects the circumscribed circle of the triangle $ABD$ at a point $E$ other than $B$. Prove that points $A,E$ and the center of the circumcircle $O$ of triangle $ABC$ are collinear.
parmenides51
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2015.4.2
On side $AB$ of triangle $ABC$, points $K$ and $L$ are selected in such a way that $\angle ACK = \angle KCL = \angle LCB$. The point $M$ on the side $BC$ is such that $\angle BKM =\angle MKC$ . It turned out that $ML$ is the bisector of the angle $KMB$. Find $\angle CLM$.
parmenides51
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2016.1.2
Given a parallelogram $ABCD$. Points $E ,F$ are marked on lines $AB ,BC$ respectively such that $AF=AB$ and $CE=CB$, and the points $E$ and $F$ do not coincide with $B$. Prove that $DE=DF$.
parmenides51
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2016.2.2
The vertices of a convex polygon are located at the nodes of an lattice points, moreover, none of its sides passes along the lattice lines. Prove that the sum of the lengths of the vertical segments of the lattice lines enclosed within the polygon, equal to the sum of the horizontal lines
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2016.3.2
In a regular hexagon $ABCDEF$, the point $M$ is the midpoint of the diagonal $AC$, $N$ is the midpoint of side $DE$. Prove that triangle $FMN$ is equilateral.
parmenides51
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2016.4.2
$P$ and $Q$ are the midpoints of sides $BC$ and $AD$ of rectangle $ABCD$ , respectively. Diagonals of rectangle PQDC meet at point $R$. It turned out that $AP$ is a bisector of the angle $BAR$. Find the length of the side $BC$ if $AB=1$.
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2017.1.2
Is it possible to mark points $A, B, C, D, E$ on a straight line so that the distances between them in centimeters are equal: $AB = 6$, $BC = 7$, $CD = 10$, $DE = 9$, $AE = 12$ ?
parmenides51
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2017.2.2
On the sides $BC$ and $AB$ of the triangle $ABC$, there are points $L$ and $K$, respectively, such that $AL$ is the bisector of the angle $BAC$, $\angle ACK = \angle ABC$, $\angle CLK = \angle BKC$. Prove that $AC = KB$
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2017.3.2
The diagonals of the convex quadrilateral $ABCD$ are perpendicular and intersect at the point $O$, and $BC = AO$. Point $F$ is such that $CF \perp CD$ and $CF = BO$. Prove that triangle $ADF$ is isosceles.
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2017.4.2
In an acute-angled triangle $ABC$ on the side $AC$, a point P is chosen such that $2AP = BC$. Points $X$ and $Y $are symmetric to point $P$ wrt vertices $A$ and $C$. It turned out that $BX = BY$. What is the angle $C$ of the original triangle?
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Geometry Individual Round
2008.1
Inside the triangle $ABC$, a point $M$ is taken such that $\angle CMB = 100^o $. Perpendicular bisectors of $BM$ and $CM$ intersect the sides $AB$ and $AC$, respectively, at points $P$ and $Q$. Points $P, Q$ and $M$ lie on one straight line. Find the value of $\angle CAB$ .
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2008.2
Given a quadrilateral $ABCD$ inscribed in a semicircle $\omega$ with diameter $AB$. Lines $AC$ and $BD$ intersect at point $E$, lines $AD$ and $BC$ intersect at point $F$. Line $EF$ intersects semicircle $\omega$ at point $G$, and line $AB$ at point $H$. Prove that $E$ is the midpoint of segment $GH$ if and only if $G$ is midpoint of $FH$.
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2008.3
An acute-angled triangle $ABC$ was drawn on the plane, in which the angle is $A\ne 60^o$. The point of intersection of the altitudes $H$ and the center of the circumscribed circle $O$ was marked in it, then straight lines $m = BH$, $n = CH$ were drawn. After that, all but the straight lines were erased from the drawing, those $m$ and $n$, points $O$ (that is, there were two straight lines and one point). Reconstruct triangle $ABC$ using a compass and ruler.

original wording
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2008.4
Let $ABCD$ be a convex quadrilateral with $\angle DAC = 30^o$, $\angle BDC = 50^o$, $\angle CBD = 15^o$, and $\angle BAC = 75^o$. The diagonals of the quadrilateral intersect at point $P$. Find the value of $\angle APD$ .
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2009.1
$ABC$ is an acute-angled triangle, $AD$ is its angle bisector, and $BM$ is its altitude . Prove that $\angle DMC> 45^o$.
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2009.2
$ABCD$ is trapezoid with bases $BC$ and $AD$. Equilateral triangles $ADK$ and $BCL$ are built on the bases ouside $ABCD$. Prove that $AC, BD$ and $KL$ meet at one point.
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2009.3
Cut a square into $5$ pieces, from which you can make $3$ pairs of different squares.
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2009.4
In $\vartriangle ABC$, $AB = AC$, $\angle BAC = 100^o$. Inside $\vartriangle ABC$, a point $M$ is taken such that $\angle MCB = 20^o$, $\angle MBC = 30^o$. Find $\angle BAM$.
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2010.1
The median $BD$ is drawn in the triangle $ABC$, and the point of intersection of the medians $G$ is marked on it. A straight line parallel to $BC$ and passing through point $G$ intersects $AB$ at point $E$. It turned out that $\angle AEC = \angle DGC$. Prove that $\angle ACB = 90^o$.
parmenides51
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2010.2
Points $X$ and $Y$ are selected on the sides $AB$ and $CD$ of rectangle $ABCD$, respectively. The segments $AY$ and $DX$ intersect at the point $P$, and the segments $CX$ and $BY$ intersect at the point $Q$. Prove that $PQ\ge \frac12 AB$.
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2010.3
In an acute-angled triangle $ABC$, the altitude $AD$ is drawn. Points $M$ and $N$ are symmetrical to point $D$ wrt lines $AC$ and $AB$, respectively. Ray $AO$ (where $O$ is the center of the circumscribed circle of triangle $ABC$) intersects $BC$ at point $E$. Prove that $\angle CME = \angle BNE$.
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2010.4
The bisector of angle $A$ of triangle $ABC$ intersects the circumscribed circle of triangle $ABC$ at point $M$, and the side $BC$ at point $A_1$. A circle $\omega$ was drawn through points $B$ and $C$ with center at point $M$. Chord $XY$ of circle $\omega$ passes through point $A_1$. Prove that the centers of the inscribed circles of triangles $ABC$ and $AXY$ coincide.
parmenides51
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2011.1
In triangle $ABC$, the median $AA_1$ is drawn and point $M$, the point of intersection of the medians, is marked on it . Point $K$ lies on side $AB$ is such that $MK \parallel AC$. It turned out that $AM = CK$. Find the angle $ACB$ .
parmenides51
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2011.2
$2011$ points are marked on the plane. A pair of marked points $A$ and $B$ is said to be isolated if all other points are strictly outside the circle constructed on $AB$ as the diameter. What is the smallest number of isolated pairs possible?
parmenides51
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2011.3
In triangle $ABC$, points $M$ and $L$ on side $BC$ are the feet of the median and angle bisector, respectively, drawn from vertex $A$. Points $P$ and $Q$ are the feet of perpendiculars drawn from point $L$ on sides $AB$ and $AC$, respectively. Point $X$ lies on the median $AM$ such that $XL\perp BC$. Prove that points $P, X$ and $Q$ are collinear.
parmenides51
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2011.4
In the convex quadrilateral $ABCD$, it turned out that $AB + CD =\sqrt2 AC$ and $BC + DA = \sqrt2 BD$. Prove that $ABCD$ is a parallelogram.
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2012.1
In a parallelogram $ABCD$, the angle bisector at $A$ meets side $BC$ in its midpoint $M$. Assume that $\angle BDC = 90^o$. Find the angles of the parallelogram $ABCD$.
parmenides51
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2012.2
In a trapezoid $ABCD$ with the parallel sides $AD$ and $BC$, the diagonals are orthogonal. The line parallel to $AD$ and passing through the intersection of the diagonals meets the lateral sides $AB$ and $CD$ at points $K$ and $L$ respectively. Point $M$ on side $AB$ is such that $AM = BK$. Prove that $LM = AB$.
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2012.3
The incircle of an isosceles triangle $ABC$ with $AB = BC$ is tangent to $BC$ and $AB$ at $E$ and $F$ respectively. A half-line trough $A$ inside the angle $EAB$ intersects the incircle at points $P$ and $Q$. The lines $EP$ and $EQ$ meet the line $AC$ at $P'$ and $Q'$. Prove that $P'A = Q'C$.
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2012.4
Prove or disprove that any triangle of area $3$ can be covered by an axially symmetric convex polygon of area $5$.
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2013.1
Points $D$ on the side $AC$ and $E$ on the side $BC$ of triangle ABC are such that $\angle ABD=\angle CBD=\angle CAE$ and $\angle ACB=\angle BAE$. Let $F$ be the intersection point of segments $BD$ and $AE$. Prove that $AF=DE$.

(Ф.Ивлев, Ф.Бахарев)
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2013.2
A hexagon $ABCDEF$ is inscribed into a circle. $X$ is the intersection point of the segments $AD$ and $BE$, $Y$ is the intersection of $AD$ and $CF$, and $Z$ is theintersection of $BE$ and $CF$. Given $AX=DY$ and $CY=FZ$, prove that $BX=EZ$.

(Д.Максимов, Ф.Петров)
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2013.3
A quadrilateral $ABCD$ is inscribed into a circle, given $AB>CD$ and $BC>AD$. Points $K$ and $M$ are chosen on the rays $AB$ and $CD$ respectively in such a way that $AK=CM=\frac 12 (AB+CD)$. Points $L$ and $N$ are chosen on the rays $BC$ and $DA$ respectively in such a way that $BL=DN=\frac 12 (BC+AD)$. Prove that the $KLMN$ is a rectangle of the same area as $ABCD$.

(Eisso J.Atzema, proposed by В.Дубровский)
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2013.4
Point $O$ is the circumcenter and point $H$ is the orthocenter in an acute non-isosceles triangle $ABC$. Circle $\omega_A$ is symmetric to the circumcircle of $AOH$ with respect to $AO$. Circles $\omega_B$ and $\omega_C$ are defined similarly. Prove that circles $\omega_A$, $\omega_B$ and $\omega_C$ have a common point, which lies on the circumcircle of $ABC$.

(Ф.Бахарев, inspired by Iran TST 2013)
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2014.1
In a right triangle $ABC$ with the right angle $B$, the angle bisector $CL$ is drawn. The point $L$ is equidistant from the points $B$ and the midpoint of the hypotenuse $AC$. Find the angle $BAC$.

(F.Nilov)
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2014.2
In a convex quadrilateral $ABCD$ the equality $\angle BCA +\angle CAD = 180^o$ holds. Prove that $AB + CD \ge AD + BC$.

(A. Smirnov inspired by Serbian regional olympiad 2014)
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2015.1
You are given a circle and its chord $AB$. At the ends of the chord to the circle are drawn tangent and equal segments $AK$ and $BL$, lying on different sides wrt line $AB$. Prove that line $AB$ divides the segment $KL$ in half.
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2015.2
Consider a pentagonal star formed by diagonals of an arbitrary convex pentagon.
Let's circle $10$ segments of its outer contour one by one solid and dotted lines (see figure).
Prove that the product of the lengths of solid segments is equal to the product of the lengths of the dotted segments .
https://cdn.artofproblemsolving.com/attachments/b/c/ee74d0b00209f0916394a2479dfa3f3b5d7dca.png
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2015.3
On the bisectors of angles $A, B, C, D$ of the convex quadrilateral $ABCD$ taken points $A ', B', C ', D'$, respectively, so that line $A'B'$ is parallel to $AB$, line $B'C'$ is parallel to $BC$ and line $C'D'$ is parallel to $CD$. Prove that
a) the line $D'A'$ is parallel to $DA$;
b) $B'D '|| AC$, if additionally it is known that $A'C '|| BD$ .
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2015.4
Angle $B$ of triangle $ABC$ is twice the angle $C$. Circle of radius $AB$ with the center $A$ intersects the perpendicular bisector of the segment $BC$ at point $D$, lying inside the angle $BAC$. Prove that $\angle DAC = \frac13 \angle A$.
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2016.1
Is it possible to construct a triangle (with a compass and a unmarked ruler) given two given angles $\alpha$ and $\beta$ and
a) known perimeter $P$
b) any any altitude of the triangle?
If it is possible, give the construction algorithm (sequence of actions) and indicate the number of different triangles in each case; if not ,justify your answer.
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2016.2
Find the sides of a right triangle if it's perimeter $P$ and it's area $S$ are known
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2016.3
In the sea off the coast of Kamchatka, three suspicious fishing vessels are regularly recorded, traditionally located at the tops of one isosceles triangle with the angle $120^o$ and lengths of the sides $20$ in km. Find the locus of points (GMT) in the space where it is possible to place technical means of control (with a range of no more than $30$ km) for illegal catch of Kamchatka crab, so that all points of this GMT are equidistant from vessels - potential violators of environmental legislation.
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2016.4
Two straight-line railways intersect at point $N$ at an angle of $60^o$. Inside this angle there is an airfield (point A) at distances of $10$ km and $20$ km from these roads. Find the locations of points $B$ and $C$ (places of loading and unloading cargo) on these roads so that the cost of cyclic transportation along the highway from $A$ to $B$, then to $C$ with a return to $A$ is minimal. Considering that the transportation of $1$ ton of cargo along the highway $ABCA$ costs $100$ rubles for $1$ km, determine
a) is it possible for a cargo of $10$ t to keep within the amount of $50$ thousand rubles, excluding the costs of loading and unloading cargo?
b) the same question for the amount of 60 thousand rubles taking into account the cost of a full reloading of goods ($2,500$ rubles per $10$ t) at $2$ points out of $4$? in $3$ points out of $4$? at all $4$ points?
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2017.1
The bisector $BD$ of angle $B$ is drawn in an isosceles triangle $ABC$ ($AB = AC$). The perpendicular to $BD$ at point $D$ intersects line $BC$ at point $E$. Find $BE$ if $CD = d$.

(A. A. Egorov)
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2017.2
The billiard table has a parallelogram shape. Two balls, placed in the middle of one of the sides, hit so that they bounced off different adjacent sides, after which both hit the same point on the opposite side. One ball traveled twice the distance before bouncing than after. Find the ratio of the lengths of the path segments before and after the bounce for the other ball.

(I.N.Sergeev)
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2017.3
Equilateral triangle $ABC$ and right-angled triangle $ABD$ are constructed on segment $AB$ on opposite sides of it, in which $\angle ABD = 90^o$, $\angle BAD = 30^o$. The circumcircle of the first triangle intersects the median $DM$ of the second at point $K$. Find the ratio $AK: KB$.

(A variation of the Nguyen Dung Thanh problem from Cut-the-knot)
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2017.4
Points $K, L, M$ and $N$ are taken on the sides $AB, BC, CD$, and $AD$ of the convex quadrilateral $ABCD$, respectively, so that $KLMN$ is a rectangle and $AK <KB$, $BL> LC$ and $CM <MD$. Can the area of this rectangle be more than half the area of the quadrilateral $ABCD$?

(I.N.Sergeev)
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Team Round
2008.4
Find the locus of the points of intersection of the medians of the triangles, all the vertices of each of which lie on different sides of the given square.
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2008.7
Two unequal circles $\omega_1$ and $\omega_2$ of radii $r_1$ and $r_2$, respectively, intersect at points $A$ and $B$. On the plane, a point $O$ is taken, for which $\angle OAB$ is $90^o$, and a circle $\omega$ with center $O$ is drawn, tangent to the internally to the circles $\omega_1$ and $\omega_2$. Find the radius of the circle $\omega$.
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2009.5
Point $D$ is selected inside the triangle $ABC$. The circumscribed circles of triangles $CAD$ and $CBD$ intersect the segments $CB$ and $CA$, at points $E$ and $F$ , respectively. It turned out that $BE = AF$. Prove that $CD$ is the bisector of the angle $\angle ACB$.
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2009.7
You are given a parallelogram $ABCD$. On rays $DB$ and $AC$, there were such points $K$ and $L$, respectively, that $KL \parallel BC$, $\angle BCD = 2 \angle KLD$. Prove that $AK \perp DL$.
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2010.3
In an acute-angled triangle $ABC$, a square is inscribed with side $m$ so that its two vertices lie on the side $AB$, and one vertex on the sides $BC$ and $AC$ . Denote $h_c $ the altitude of the triangle $ABC$ drawn from the vertex $C$ , and $c$ the side $AB$. Prove that $\frac{1}{h_c}=\frac{1}{m}+\frac{1}{c}$.
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2010.6
In the non-isosceles triangle $ABC$, let $M$ be the point of intersection of the medians and $I$ be the point of intersection of the angle bisectors . It turned out that line $MI$ is perpendicular to the side $BC$ . Prove that $AB + AC = 3BC$.
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2011.2
The angle bisector $AL$ is drawn in triangle $ABC$. It is known that $AB = 2007$, $BL = AC$. Find the sides of triangle $ABC$ if known to be integers.
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2011.8
Points $P$ and $Q$ on side$AB$ of convex quadrilateral $ABCD$ are such that $AP = QB$. Point $X$ is a non-$D$ intersection point of the circumscribed circles of triangles $APD$ and $DQB$, and point $Y$ is a non-$C$ intersection point of the circumscribed circles of triangles $ACP$ and $QCB$. Prove that points $C, D, X$ and $Y$ lie on the same circle.
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2012.3
A paper triangle with sides $a, b, c$ bent in a straight line so that the vertex opposite side of length $c$, hit this side. It is known that in the resulting quadrilateral two angles are equal, to the fold line. Find the lengths of the segments into which the vertex that gets there divides side $c$.
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2012.6
Given an isosceles triangle $ABC$ ($AB = BC$). On the side $AB$, point $K$ is selected, and on the side $BC$ point $L$ so that $AK + CL = \frac12AB$. Find the locus of the midpoints of the line segments $KL$.
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2012.10
In a regular heptagon $ABCDEFG$, the sides are $1$. The diagonals $AD$ and $CG$ meet at a point $H$. Prove that $FH = \sqrt2$.
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2013.5
Inside the triangle $ABC$ is chosen point $P$ such that $\angle ABP=\angle CPM$, where $M$ is the midpoint of the segment $AC$. The line $MP$ intersects the circumcircle of the triangle $APB$ at points $P$ and $Q$. Prove that $QA=PC$.
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2014.4
Several chords are drawn in a circle so that every pair of them intersects inside the circle. Prove that all the drawn chords can be intersected by the same diameter.

(A.Chapovalov)
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2014.6
The point $I_b$ is the center of an excircle of the triangle $ABC$, that is tangent to the side $AC$. Another excircle is tangent to the side $AB$ in the point $C_1$. Prove that the points $B, C, C_1$ and the midpoint of the segment $BI_b$ lie on the same circle.

(inspired by olympiad of the Faculty of Mathematics and Mechanics of SPBU)
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2015.5
Let $AB$ be a diameter of circle $\omega$. $\ell$ is the tangent line to $\omega$ at $B$. Given two points $C, D$ on $\ell$ such that $B$ is between $C$ and $D. E, F$ are the intersections of $\omega$ and $AC, AD$, respectively, and $G, H$ are the intersections of $\omega$ and $CF, DE$, respectively. Prove that $AH = AG$.
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2015.9
Let $ABC$ be an isosceles triangle ($AB = AC$). On the extensions of the sides $BC, AB$ and $AC$, points $P, X, Y$ are selected in such a way that $PX \parallel AC$ and $PY \parallel AB$ and point $P$ lies on the ray $CB$ . Point $T$ is the midpoint of the arc $BC$ of the circumscribed circle of the triangle $ABC$ ($T \ne A$). Prove that $PT \perp XY$.
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2016.1
Construct a right-angled triangle for a given hypotenuse $c$, if it is known that the median drawn to $c$ is the geometric mean of its legs.
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2016.9
A quadrilateral $ABCD$ is inscribed in a circle. Lines$ AB$ and $CD$ meet at point $E$, lines $AD$ and $BC$ met at point $F$. The bisector of angle $AEC$ intersects side $BC$ at point $M$ and side $AD$ at point $N$, and the bisector of angle $BFD$ intersects side $AB$ at point $P$ and side $CD$ at point $Q$. Prove that the quadrilateral $MNPQ$ is a rhombus.
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2017.7
In a right-angled triangle $ABC$, the altitude $CH$ is drawn to the hypotenuse. The bisector $BD$ of angle $B$ intersects the altitude at point $E$. Let $K$ be the intersection point of line segments $AE$ and $HD$. Prove that the quadrilateral $CDKE$ and the triangle $AHK$ have equal areas.

(Peru Geometrico)
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