### Tuymaada 1994 - 2019 (Russia) 87p

with aops links in the names

1994 - 2019

Point $M$ lies inside triangle $ABC$. Prove that for any other point $N$ lying inside the triangle $ABC$, at least one of the following three inequalities is fulfilled: $AN>AM, BN>BM, CN>CM$.

Let a convex polyhedron be given with volume $V$ and full surface $S$.
Prove that inside a polyhedron it is possible to arrange a ball of radius $\frac{V}{S}$.

In three houses $A,B$ and $C$, forming a right triangle with the legs $AC=30$ and $CB=40$, live three beetles $a,b$ and $c$, capable of moving at speeds of $2, 3$ and $4$, respectively. Suppose that you simultaneously release these bugs from point $M$ and mark the time after which beetles reach their homes. Find on the plane such a point $M$, where is the last time to reach the house a bug would be minimal.

Prove that in space there is a sphere containing exactly $1994$ points with integer coordinates.

Give a geometric proof of the statement that the fold line on a sheet of paper is straight.

Given a circle of radius $r= 1995$. Show that around it you can describe exactly $16$ primitive Pythagorean triangles. The primitive Pythagorean triangle is a right-angled triangle, the lengths of the sides of which are expressed by coprime integers.
Inside the triangle $ABC$ a point $M$ is given . Find the points  $P,Q$ and $R$ lying on the sides $AB,BC$ and  $AC$ respectively and such so that the sum $MP+PQ+QR+RM$ is the smallest.

Given a segment of length $7\sqrt3$ .
Is it possible to use only compass to construct a segment of length $\sqrt7$?

Given a tetrahedron $ABCD$, in which $AB=CD= 13 , AC=BD=14$ and $AD=BC=15$.
Show that the centers of the inscribed sphere and sphere around it coincide, and find the radii of these spheres.

Using only angle with angle $\frac{\pi}{7}$ and a ruler, constuct angle $\frac{\pi}{14}$

Find a right triangle that can be cut into $365$ equal triangles.

The segment of length  $\ell$ with the ends on the border of a triangle divides the area of that triangle in half. Prove that $\ell >r\sqrt2$, where  $r$ is the radius of the inscribed circle of the triangle.

Given the tetrahedron $ABCD$, whose opposite edges are equal, that is, $AB=CD, AC=BD$ and $BC=AD$. Prove that exist exactly $6$ planes intersecting the triangular angles of the tetrahedron and dividing the total surface and volume of this tetrahedron in half.

A right triangle is inscribed in parabola  $y=x^2$. Prove that it's hypotenuse is not less than $2$.

Given the pyramid $ABCD$. Let $O$ be the middle of edge $AC$. Given that $DO$ is the height of the pyramid, $AB=BC=2DO$ and the angle $ABC$ is right. Cut this pyramid into $8$ equal and similar to it pyramids.

In the triangle ABC we have ÐABC = 100 o, ÐACB = 65o, M ä AB, N ä AC, and ÐMCB = 55o, ÐNBC = 80o. Find ÐNMC.
(St.Petersburg folklore)

Let $O$ be the center of the circle described around the the triangle $ABC$.  The centers of the circles described around the squares $OAB,OBC,OCA$ lie at the vertices of a regular triangle. Prove that the triangle $ABC$ is right.

A tangent l to the circle inscribed in a rhombus meets its sides AB and BC at points E and F respectively. Prove that the product AE · CF is independent of the choice of l.

Let ABC be an acute isosceles triangle  ($AB=BC$) inscribed in a circle with center $O$ .  The line through the midpoint of the chord $AB$ and point $O$ intersects the line $AC$ at $L$ and the circle at the point $P$. Let the bisector of angle $BAC$ intersects the circle at point $K$.  Lines $AB$ and  $PK$ intersect at point $D$. Prove that the points $L,B,D$  and $P$ lie on the same circle.

On the side $AB$ of an isosceles triangle $AB$ ($AC=BC$) lie points $P$ and $Q$ such that  $\angle PCQ \le \frac{1}{2} \angle ACB$.  Prove that $PQ \le \frac{1}{2} AB$.

ABCD is a convex quadrilateral, half-lines DA and CB meet at point Q, half-lines BA and CD meet at point P. It is known that ÐAQB = ÐAPD. The bisector of angle ÐAQB meets the sides AB and CD of the quadrilateral at points X and Y , respectively, the bisector of angle ÐAPD meets the sides AD and BC at points Z and T, respectively. The circumcircles of triangles ZQT and XPY meet at point K inside the quadrilateral. Prove that K lies on the diagonal AC.

( S. Berlov)
Points on the sides $BC$, $CA$ and $AB$ of the triangle $ABC$ are respectively $A_1$, $B_1$ and $C_1$ such that $AC_1: C_1B = BA_1: A_1C = CB_1: B_1A = 2: 1$. Prove that if triangle $A_1B_1C_1$ is equilateral, then triangle $ABC$ is also equilateral.

The circle with the center of $O$ touches the sides of the corner with the vertex $A$ at the points of $K$ and $M$. The tangent to the circle intersects the segments $AK$ and $AM$ at points $B$ and $C$ respectively, and the line $KM$ intersects the segments $OB$ and $OC$ at the points $D$ and $E$. Prove that the area of the triangle $ODE$ is equal to a quarter of the area of a triangle $BOC$ if and only if the angle $A$ is $60^\circ$.

A circle having common centre with the circumcircle of triangle ABC meets the sides of the triangle at six points forming convex hexagon A1A2B1B2C1C2 (A1 and A2 lie on BC, B1 and B2 lie on AC, C1 and C2 lie on AB). If A1B1 is parallel to the bisector of angle B, prove that A2C2 is parallel to the bisector of angle C.

(S. Berlov)
The points D and E on the circumcircle of an acute triangle ABC are such that AD = AE = BC. Let H be the common point of the altitudes of triangle ABC. It is known that AH2 = BH2 + CH2. Prove that H lies on the segment DE.

(D. Shiryaev)
In the acute triangle $ABC$, the point $I$ is the center of the inscribed the circle, the point $O$ is the center of the circumscribed circle and the point $I_a$ is the center the excircle tangent to the side $BC$ and the extensions of the sides $AB$ and $AC$. Point $A'$ is symmetric to vertex $A$ with respect to the line $BC$. Prove that $\angle IOI_a = \angle IA'I_a$.

Through the point $K$ lying outside the circle $\omega$, the tangents are drawn $KB$ and $KD$ to this circle ($B$ and $D$ are tangency points) and a line intersecting a circle at points $A$ and $C$. The bisector of angle $ABC$ intersects the segment $AC$ at the point $E$ and circle $\omega$ at $F$. Prove that $\angle FDE = 90^\circ$.

In a quadrilateral ABCD sides AB and CD are equal, ÐA = 150o, ÐB = 44o, ÐC = 72o. Perpendicular bisector of the segment AD meets the side BC at point P. Find ÐAPD.

(F. Bakharev)
In a convex quadrilateral ABCD we have AB · CD = BC· DA and 2ÐA + ÐC = 180o. Point P lies on the circumcircle of triangle ABD and is the midpoint of the arc BD not containing A. It is known that the point P lies inside the quadrilateral ABCD. Prove that ÐBCA = ÐDCP.

(S. Berlov)
Point $O$ is the center of the circumscribed circle of an acute triangle $Abc$. A certain circle passes through the points $B$ and $C$ and intersects sides $AB$ and $AC$ of a triangle. On its arc lying inside the triangle, points $D$ and $E$ are chosen so that the segments $BD$ and $CE$ pass through the point $O$. Perpendicular $DD_1$ to $AB$ side and perpendicular $EE_1$ to $AC$ side intersect at $M$. Prove that the points $A$, $M$ and $O$ lie on the same straight line.

The incircle of triangle ABC touches its sides AB and BC at points P and Q. The line PQ meets the circumcircle of triangle ABC at points X and Y. Find ÐXBY if ÐABC = 90 o.

(A. Smirnov)

An acute triangle ABC is inscribed in a circle of radius 1 with centre O, all the angles of ABC are greater than 45o. B1 is the foot of perpendicular from B to CO, B2 is the foot of perpendicular from B1 to AC. Similarly, C1 is the foot of perpendicular from C to BO, C2 is the foot of perpendicular from C1 to AB. The lines B1B2 and C1C2 intersect at A3. The points B3 and C3 are defined in the same way. Find the circumradius of triangle A3B3C3.

(F.Bakharev & F.Petrov)
Points $X$ and $Y$ are the midpoints of the sides $AB$ and $AC$ of the triangle $ABC$, $I$ is the center of its inscribed circle, $K$ is the point of tangency of the inscribed circles with side $BC$. The external angle bisector at the vertex $B$ intersects the line $XY$ at the point $P$, and the external angle bisector at the vertex of $C$ intersects $XY$ at $Q$. Prove that the area of the quadrilateral $PKQI$ is equal to half the area of the triangle  $ABC$.

The point $I$ is the center of the inscribed circle of the triangle $ABC$. The points $B_1$ and $C_1$ are the midpoints of the sides $AC$ and $AB$, respectively. It is known that $\angle BIC_1 + \angle CIB_1 = 180^\circ$. Prove the equality $AB + AC = 3BC$

In a triangle ABC, let A1, B1, C1 be the points where the excircles touch the sides BC, CA and AB respectively. Prove that AA1, BB1 and CC1 are the sidelenghts of a triangle.

(L. Emelyanov)
Let I be the incentre of triangle ABC. A circle containing the points B and C meets the segments BI and CI at points P and Q respectively. It is known that BP · CQ = PI · QI. Prove that the circumcircle of the triangle PQI is tangent to the circumcircle of ABC.

(S. Berlov)
On the  equal $AC$ and $BC$ of an isosceles right triangle $ABC$ ,  points $D$ and $E$ are marked respectively, so that $CD = CE$. Perpendiculars on the straight line $AE$, passing through the points $C$ and $D$, intersect the side $AB$ at the points $P$ and $Q$.Prove that $BP = PQ$.

The median $BM$ of a triangle $ABC$ intersects the circumscribed circle at point $K$. The circumcircle of the triangle $KMC$ intersects the segment $BC$ at point $P$, and the circumcircle of $AMK$ intersects the extension of $BA$ at $Q$. Prove that $PQ> AC$.

A line d is given in the plane. Let B äd and A another point, not on d, and such that AB is not perpendicular on d. Let ω be a variable circle touching d at B and letting A outside, and X and Y the points on ω such that AX and AY are tangent to the circle. Prove that the line XY passes through a fixed point.

(F. Bakharev)
Let ABC be a triangle, G its centroid, H its orthocenter, and M the midpoint of the arc AC (not containing B). It is known that MG = R, where R is the radius of the circumcircle. Prove that BG≥ BH.

(F. Bakharev)
An acute-angle non-isosceles triangle $ABC$ is given. The point $H$ is its orthocenter, the points $O$ and $I$ are the centers of its circumscribed and inscribed circles, respectively. The circumcircle of the triangle $OIH$ passes through the vertex  $A$. Prove that one of the angles of the triangle is $60^\circ$.

On the $AB$ side of the triangle $ABC$, points $X$ and $Y$ are chosen, on the side of $AC$ is a point of $Z$, and on the side of $BC$ is a point of $T$. Wherein $XZ \parallel BC$, $YT \parallel AC$. Line $TZ$ intersects the circumscribed circle of triangle $ABC$ at points $D$ and $E$. Prove that points $X$, $Y$, $D$ and $E$ lie on the same circle.

AA1, BB1, CC1 are altitudes of an acute triangle ABC. A circle passing through A1 and B1 touches the arc AB of its circumcircle at C2. The points A2, B2 are defined similarly. Prove that the lines AA2, BB2, CC2 are concurrent.

(R. Sakhipov)
Point D is chosen on the side AB of triangle ABC. Point L inside the triangle ABC is such that BD = LD and ÐLAB = ÐLCA = ÐDCB. It is known that ÐALD + ÐABC = 180o. Prove that ÐBLC = 90o.

(R. Sakhipov)
Point I1 is the reflection of incentre I of triangle ABC across the side BC. The circumcircle of BCI1 intersects the line II1 again at point P. It is known that P lies outside the incircle of the triangle ABC. Two tangents drawn from P to the latter circle touch it at points X and Y . Prove that the line XY contains a medial line of the triangle ABC.

(L. Emelyanov)
Let ABCD be an isosceles trapezoid with AD k BC. Its diagonals AC and BD intersect at point M. Points X and Y on the segment AB are such that AX = AM, BY = BM. Let Z be the midpoint of XY and N is the point of intersection of the segments XD and Y C. Prove that the line ZN is parallel to the bases of the trapezoid.

(A. Akopyan &A. Myakishev)
A convex hexagon is given. Let s be the sum of the lengths of the three segments connecting the midpoints of its opposite sides. Prove that there is a point in the hexagon such that the sum of its distances to the lines containing the sides of the hexagon does not exceed s.

(N. Sedrakyan)
In a cyclic quadrilateral ABCD the sides AB and AD are equal, CD > AB + BC. Prove that ÐABC > 120o.

M is the midpoint of base BC in a trapezoid ABCD. A point P is chosen on the base AD. The line PM meets the line CD at a point Q such that C lies between Q and D. The perpendicular to the bases drawn through P meets the line BQ at K. Prove that ÐQBC = ÐKDA.

(S. Berlov)
On the side AB of a cyclic quadrilateral ABCD there is a point X such that diagonal BD  bisects CX and diagonal AC bisects DX. What is the minimum possible value of AB / CD?

(S. Berlov)
A triangle ABC is given. Let B1 be the reflection of B across the line AC, C1 the reflection of C across the line AB, and O1 the reflection of the circumcentre of ABC across the line BC. Prove that the circumcentre of AB1C1 lies on the line AO1.

(A. Akopyan)
Let ABC be an acute triangle, H its orthocentre, D a point on the side [BC], and P a point such that ADPH is a parallelogram. Show that ÐBPC > ÐBAC.

(S. Berlov)
Let ABC be a triangle, I its incenter, ω its incircle, P a point such that PI $\perp$ BC and PA // BC, Qä(AB), R ä (AC) such that QR // BC and QR tangent to ω. Show that ÐQPB = ÐCPR.

(V. Smykalov)
In acute triangle ABC, let H denote its orthocenter and let D be a point on side BC. Let P be the point so that ADPH is a parallelogram. Prove that ÐDCP < ÐBHP.

(S. Berlov)
In a cyclic quadrilateral ABCD, the extensions of sides AB and CD meet at point P, and the extensions of sides AD and BC meet at point Q. Prove that the distance between the orthocenters of triangles APD and AQB is equal to the distance between the orthocenters of triangles CQD and BPC.

(L. Emelyanov)
An excircle of triangle ABC touches the side AB at P and the extensions of sides AC and BC at Q and R, respectively. Prove that if the midpoint of PQ lies on the circumcircle of ABC, then the midpoint of PR also lies on that circumcircle.

(S. Berlov)
A circle passing through the vertices A and B of a cyclic quadrilateral ABCD intersects diagonals AC and BD at E and F, respectively. The lines AF and BC meet at a point P, and the lines BE and AD meet at a point Q. Prove that PQ is parallel to CD.

(A. Akopyan)
Circles ω1 and ω2 intersect at points A and B, and M is the midpoint of AB. Points S1 and S2 lie on the line AB (but not between A and B). The tangents drawn from S1 to ω1 touch it at X1 and Y1, and the tangents drawn from S2 to ω2 touch it at X2 and Y2. Prove that if the line X1X2 passes through M, then line Y1Y2 also passes through M.

(A. Akopyan)
In a convex hexagon AC΄BA΄CB΄, every two opposite sides are equal. Let A1 denote the point of intersection of BC with the perpendicular bisector of AA΄. Define B1 and C1 similarly. Prove that A1, B1, and C1 are collinear.

(A. Akopyan)
A rectangle ABCD is given. Segment DK is equal to BD and lies on the half-line DC. M is the midpoint of BK. Prove that AM is the angle bisector of ÐBAC.

(S. Berlov)
A circle is contained in a quadrilateral with successive sides of lengths 3, 6, 5 and 8. Prove that the length of its radius is less than 3.

(K. Kokhas)
Point P is taken in the interior of the triangle ABC, so that ÐPAB = ÐPCB =1/4  (ÐA + ÐC). Let L be the foot of the angle bisector of ÐB. The line PL meets the circumcircle of APC at point Q. Prove that QB is the angle bisector of ÐAQC.

(S. Berlov)
Quadrilateral ABCD is both cyclic and circumscribed. Its incircle touches its sides AB and CD at points X and Y , respectively. The perpendiculars to AB and CD drawn at A and D, respectively, meet at point U, those drawn at X and Y meet at point V , and finally, those drawn at B and C meet at point W. Prove that points U, V and W are collinear.

(A. Golovanov)
ABCDEF is a convex hexagon, such that in it AC // DF, BD // AE and CE // BF. Prove that
AB2+ CD2 + EF2 = BC2 + DE2 + AF2.

(N. Sedrakyan)
The point A1 on the perimeter of a convex quadrilateral ABCD is such that the line AA1 divides the quadrilateral into two parts of equal area. The points B1, C1, D1 are defined similarly.  Prove that the area of the quadrilateral A1B1C1D1 is greater than a quarter of the area of ABCD.

(L. Emelyanov)
Points X and Y inside the rhombus ABCD are such that Y is inside the convex quadrilateral BXDC and 2ÐXBY = 2ÐXDY = ÐABC. Prove that the lines AX and CY are parallel.

(S. Berlov)
Points A1, A2, A3, A4 are the vertices of a regular tetrahedron of edge length 1. The points B1 and B2 lie inside the figure bounded by the plane A1A2A3 and the spheres of radius 1 and centres A1, A2, A3. Prove that B1B2 < max{B1A1,B1A2,B1A3,B1A4}.

(A.  Kupavsky)
The points K and L on the side BC of a triangle △ABC are such that ÐBAK = ÐCAL = 90o.  rove that the midpoint of the altitude drawn from A, the midpoint of KL and the circumcentre of △ABC are collinear.

(A. Akopyan, S. Boev, P. Kozhevnikov)
Radius of the circle ωA with centre at vertex A of a triangle △ABC is equal to the radius of the  excircle tangent to BC. The circles ωB and ωC are defined similarly. Prove that if two of these circles are tangent then every two of them are tangent to each other.

(L. Emelyanov)
A parallelogram ABCD is given. The excircle of triangle △ABC touches the sides AB at L and the extension of BC at K. The line DK meets the diagonal AC at point X, the line BX meets the median CC1 of triangle △ABC at Y . Prove that the line Y L, median BB1 of triangle △ABC and its bisector CC′ have a common point.

(A. Golovanov)
D is midpoint of AC for ABC. Bisectors of ÐACB, ÐABD are perpendicular. Find max value for ÐBAC.

(S. Berlov)
CL is bisector of ÐC of ABC and intersect circumcircle at K. I - incenter of ABC. IL = LK. Prove, that CI = IK.

(D. Shiryaev)
In ABC points M,O are midpoint of AB and circumcenter. It is true, that OM = R - r. Bisector of external ÐA intersect BC at D and bisector of external ÐC intersect AB at E. Find possible values of ÐCED

(D. Shiryaev)
The point D on the altitude AA1 of an acute triangle ABC is such that \BDC = 90◦; H is the orthocentre of ABC. A circle with diameter AH is constructed. Prove that the tangent drawn from B to this circle is equal to BD.

(L. Emelyanov)
The numbers a, b, c, d satisfy 0 < a ≤ b ≤ d ≤ c and a + c = b + d. Prove that for every internal point P of a segment with length a this segment is a side of a circumscribed quadrilateral with consecutive sides a, b, c, d, such that its incircle contains P.

(L. Emelyanov)
Altitudes AA1, BB1, CC1 of an acute triangle ABC meet at H. A0, B0, C0 are the midpoints of BC, CA, AB respectively. Points A2, B2, C2 on the segments AH, BH, HC1 respectively are such that ÐA0B2A2 = ÐB0C2B2 = ÐC0A2C2 = 90o. Prove that the lines AC2, BA2, CB2 are concurrent.

(A.Pastor)
The diagonals AC and BD of a cyclic quadrilateral ABCD are perpendicular and meet at point P. The point Q on the segment PC is such that AP=QC. Prove that the perimeter of the triangle BQD is at least 2AC.

(A. Kuznetsov)
BL is the bisector of an isosceles triangle ABC. A point D is chosen on the base BC and a point E is chosen on the lateral side AB so that AE=1/2 AL=CD . Prove that LE=LD.

(A. Kuznetsov)
A point E lies on the extension of the side AD of the rectangle ABCD over D. The ray EC meets the circumcircle ω of ABE at the point F ≠ E. The rays DC and AF meet at P. H is the foot of the perpendicular drawn from C to the line l going through E and parallel to AF. Prove that the line PH is tangent to ω.

(A. Kuznetsov)
Two points A and B are given in the plane. A point X is called their preposterous midpoint if there is a Cartesian coordinate system in the plane such that the coordinates of A and B in this system are non-negative, the abscissa of X is the geometric mean of the abscissae of A and B, and the ordinate of X is the geometric mean of the ordinates  of A and B. Find the locus of all the preposterous midpoints of A and B.

(K. Tyschuk)
A circle touches the side $AB$ of the triangle $ABC$ at $A$, touches the side $BC$ at $P$ and intersects the side $AC$ at $Q$. The line symmetrical to $PQ$ with respect to $AC$ meets the line $AP$ at $X$. Prove that $PC=CX$.

(S. Berlov)
Quadrilateral $ABCD$ with perpendicular diagonals is inscribed in a circle with centre $O$. The tangents to this circle at $A$ and $C$ together with line $BD$ form the triangle $\Delta$. Prove that the circumcircles of $BOD$ and $\Delta$ are tangent.

Show that this point lies belongs to $\omega$, the circumcircle of $OAC$

(A. Kuznetsov)
A point $P$ on the side $AB$ of a triangle $ABC$ and points $S$ and $T$ on the sides $AC$ and $BC$ are such that $AP=AS$ and $BP=BT$. The circumcircle of $PST$ meets the sides $AB$ and $BC$ again at $Q$ and $R$, respectively. The lines $PS$ and $QR$ meet at $L$. Prove that the line $CL$ bisects the segment $PQ$.

(A. Antropov)
A triangle $ABC$ with $AB < AC$ is inscribed in a circle $\omega$. Circles  $\gamma_1$ and $\gamma_2$ touch the lines $AB$ and $AC$, and their centres lie on the circumference of $\omega$. Prove that $C$ lies on a common external tangent to  $\gamma_1$ and $\gamma_2$.

(A. Kuznetsov)
A circle $\omega$ touches the sides $A$B and $BC$ of a triangle $ABC$ and intersects its side $AC$ at $K$. It is known that the tangent to $\omega$ at $K$ is symmetrical to the line $AC$ with respect to the line $BK$. What can be the difference $AK -CK$ if $AB = 9$ and $BC = 11$?

(S. Berlov)
A trapezoid $ABCD$ with $BC // AD$ is given. The points $B'$ and $C'$ are symmetrical to $B$ and $C$ with respect to $CD$ and $AB$, respectively. Prove that the midpoint of the segment joining the circumcentres of $ABC'$ and $B'CD$ is equidistant from $A$ and $D$.
(A. Kuznetsov)

In $\triangle ABC$ $\angle B$ is obtuse and $AB \ne BC$. Let $O$ is the circumcenter and $\omega$ is the circumcircle of this triangle. $N$ is the midpoint of arc $ABC$. The circumcircle of $\triangle BON$ intersects $AC$ on points $X$ and $Y$. Let $BX \cap \omega = P \ne B$ and  $BY \cap \omega = Q \ne B$. Prove that $P, Q$ and reflection of $N$ with respect to line $AC$ are collinear.
(A. Kuznetsov)