Zhautykov 2005-19 (Kazakhstan) 31p

geometry problems from International Zhautykov Olympiad (also known as IZhO)

2005 - 2019

IZhO 2005  Junior 5
Let the circle (I, r) be inscribed in the triangle ABC. Let D be the point of contact of this circle with BC. Let E and F be the midpoints of BC and AD, respectively. Prove that the three points I, E, F are collinear.

Let ${SABC}$. be a regular triangular pyramid (SA=SB=SC. and AB=BC=CA). Find the locus of all points ${D \, (D\ne S)}$. in the space that satisfy the equation ${ |cos \delta_A -2cos \delta_B - 2cos \delta_C | = 3 }$. where the angle ${\delta_X=\angle XSD}$ for each ${X \in \{ A,B,C\} }$.

The inner point X of a quadrilateral is observable from the side YZ if the perpendicular to the line Y Z meet it in the closed interval [Y Z]. The inner point of a quadrilateral is a k−point if it is observable from the exactly k sides of the quadrilateral. Prove that if a convex quadrilateral has a 1-point then it has a k − point for each k = 2, 3, 4.

Let ABC be a triangle and K and L be two points on (AB), (AC) such that BK = CL and let P = CK $\cap$  BL. Let the parallel through P to the interior angle bisector of ÐBAC intersect AC in M. Prove that CM = AB.

Let ABCDEF be a convex hexagon such that AD = BC + EF, BE = AF + CD, CF = DE + AB. Prove that: AB / DE = CD / AF = EF / BC .

Let ABCD be a convex quadrilateral, with ÐBAC = ÐDAC and M a point inside such that ÐMBA = ÐMCD and ÐMBC = ÐMDC. Show that the angle ÐADC is equal to ÐBMC or ÐAMB.

Let ABCDEF be a convex hexagon and it‘s diagonals have one common point M. It is known that the circumcenters of triangles MAB, MBC, MCD, MDE, MEF, MFA lie on a circle. Show that the quadrilaterals ABDE, BCEF, CDFA have equal areas.

Points K,L,M,N are respectively the midpoints of sides AB,BC,CD,DA in a convex quadrilateral ABCD. Line KM meets diagonals AC and BD at points P and Q, respectively .Line LN meets diagonals AC and BD at points R and S, respectively. Prove that if AP·PC = BQ·QD, then AR·RC = BS·SD.

Let A1A2 be the external tangent line to the nonintersecting circles ω1(O1) and ω2(O2), A1ω1,  A2ω2. Points K is the midpoint of A1A2. And KB1 and KB2 are tangent lines to ω1 and ω2, respectively (B1 ≠ A1, B2 ≠ A2).Lines A1B1 and A2B2 meet in point L, and lines KL and O1O2 meet in point P. Prove that points B1,B2, P and L are concyclic.

For a convex hexagon ABCDEF with an area S, prove that:
AC·(BD+BF − DF)+CE·(BD+DF −BF)+AE·(BF +DF − BD) ≥ 2√3 S

Given a quadrilateral ABCD with ÐB = ÐD = 90o. Point M is chosen on segment AB so that AD=AM. Rays DM and CB intersect at point N. Points H and K are feet of perpendiculars from points D and C to lines AC and AN, respectively. Prove that ÐMHN = ÐMCK.

In a cyclic quadrilateral ABCD with AB = AD points M,N lie on the sides BC and CD respectively so that MN = BM +DN . Lines AM and AN meet the circumcircle of ABCD again at points P and Q respectively. Prove that the orthocenter of the triangle APQ lies on the segment MN .

Let ABC arbitrary triangle (AB ≠ BC ≠AC ≠AB) and O,I,H it’s circumcenter, incenter and orthocenter . Prove, that
1) ÐOIH  >  90o
2) ÐOIH > 135 o

Given is trapezoid ABCD, M and N being the midpoints of the bases of AD and BC, respectively.
a) Prove that the trapezoid is isosceles if it is known that the intersection point of perpendicular bisectors of the lateral sides belongs to the segment MN.
b) Does the statement of part a) remain true if it is only known that the intersection point of perpendicular bisectors of the lateral sides belongs to the line MN?

Diagonals of a cyclic quadrilateral ABCD intersect at point K. The midpoints of diagonals AC and BD are M and N, respectively. The circumscribed circles ADM and BCM intersect at points M and L. Prove that the points K,L,M, and N lie on a circle. (all points are supposed to be different.)

An acute triangle ABC is given. Let D be an arbitrary inner point of the side AB. Let M and N be the feet of the perpendiculars from D to BC and AC, respectively. Let H1 and H2 be the orthocenters of triangles MNC and MND, respectively. Prove that the area of the quadrilateral AH1BH2 does not depend on the position of D on AB.

Equilateral triangles ACBand BDCare drawn on the diagonals of a convex quadrilateral ABCD so that B and Bare on the same side of AC, and C and Care on the same sides of BD. Find ÐBAD + ÐCDA if BC= AB + CD.

Given a trapezoid ABCD (AD // BC) with ÐABC > 90. Point M is chosen on the lateral side AB. Let O1 and O2 be the circumcenters of the triangles MAD and MBC, respectively. The circumcircles of the triangles MO1D and MO2C meet again at the point N. Prove that the line O1O2 passes through the point N.

Given convex hexagon ABCDEF with AB // DE, BC // EF, and CD // FA . The distance between the lines AB and DE is equal to the distance between the lines BC and EF and to the distance between the lines CD and FA. Prove that the sum AD+BE+CF does not exceed the perimeter of hexagon ABCDEF.

Points M, N, K lie on the sides BC, CA, AB of a triangle ABC, respectively, and are different from its vertices. The triangle MNK is called beautiful if ÐBAC = ÐKMN and ÐABC = ÐKNM. If in the triangle ABC there are two beautiful triangles with a common vertex, prove that the triangle ABC is right-angled.
(Nairi M. Sedrakyan, Armenia)

Four segments divide a convex quadrilateral into nine quadrilaterals. The points of intersections of these segments lie on the diagonals of the quadrilateral (see figure). It is known that the quadrilaterals 1, 2, 3, 4 admit inscribed circles. Prove that the quadrilateral 5 also has an inscribed circle.

(Nairi M. Sedrakyan, Armenia)
Inside the triangle ABC a point M is given. The line BM meets the side AC at N. The point K is symmetrical to M with respect to AC. The line BK meets AC at P. If ÐAMP = ÐCMN, prove that ÐABP = ÐCBN.

(Nairi M. Sedrakyan, Armenia)
The area of a convex pentagon ABCDE is S, and the circumradii of the triangles ABC, BCD, CDE, DEA, EAB are R1, R2, R3, R4, R5. Prove the inequality
$R_{1}^{4}+R_{2}^{4}+R_{3}^{4}+R_{4}^{4}+R_{5}^{4}\ge \frac{4}{5{{\sin }^{2}}{{108}^{o}}}{{S}^{2}}$

(Nairi M. Sedrakyan, Armenia)
A quadrilateral ABCD is inscribed in a circle with center O. It’s diagonals meet at M. The circumcircle of ABM intersects the sides AD and BC at N and K respectively. Prove that areas of NOMD and KOMC are equal.

(Sava Grozdev)
A convex hexagon ABCDEF is given such that AB||DE, BC||EF, CD||FA. The point M,N,K are common points of the lines BD and AE, AC and DF, CE and BF respectively. Prove that perpendiculars drawn from M,N,K to lines AB,CD,EF respectively concurrent.

(Nairi M. Sedrakyan, Armenia)
Let ABC be a non-isosceles triangle with circumcircle ω and let H,M be orthocenter and midpoint of AB respectively. Let P,Q be points on the arc AB of ω not containing C such that ÐACP = ÐBCQ  <  ÐACQ. Let R, S be the foot of altitudes from H to CQ,CP respectively. Prove that the points P,Q,R, S are concyclic and M is the center of this circle.

(M. Kungojin
Let ABCD be the regular tetrahedron, and M,N points in space. Prove that:
AM · AN + BM · BN + CM · CN ≥ DM · DN

(Nairi M. Sedrakyan, Armenia)
IZhO 2018.2
Let $N,K,L$ be points on $AB,BC,CA$ such that $CN$ bisector of angle $\angle ACB$ and $AL=BK$.Let $BL\cap AK=P$.If $I,J$ be incenters of triangles $\triangle BPK$ and $\triangle ALP$ and $IJ\cap CN=Q$ prove that $IQ=JP$

In a circle with a radius $R$ a convex hexagon is inscribed. The diagonals $AD$ and $BE$,$BE$ and $CF$,$CF$ and $AD$ of the hexagon intersect at the points $M$,$N$ and$K$, respectively. Let $r_1,r_2,r_3,r_4,r_5,r_6$ be the radii of circles inscribed in triangles $ABM,BCN,CDK,DEM,EFN,AFK$ respectively. Prove that.$$r_1+r_2+r_3+r_4+r_5+r_6\leq R\sqrt{3}$$ .

Triangle $ABC$ is given. The median $CM$ intersects the circumference of $ABC$ in $N$. $P$ and $Q$ are chosen on the rays $CA$ and $CB$ respectively, such that $PM$ is parallel to $BN$ and $QM$ is parallel to $AN$. Points $X$ and $Y$ are chosen on the segments $PM$ and $QM$ respectively, such that both $PY$ and $QX$ touch the circumference of $ABC$. Let $Z$ be intersection of $PY$ and $QX$. Prove that, the quadrilateral $MXZY$ is circumscribed.

Triangle $ABC$ with $AC=BC$ given and point $D$ is chosen on the side $AC$. $S1$ is a circle that touches $AD$ and extensions of $AB$ and $BD$ with radius $R$ and center $O_1$. $S2$ is a circle that touches $CD$ and extensions of $BC$ and $BD$ with radius $2R$ and center $O_2$. Let $F$ be intersection of the extension of $AB$ and tangent at $O_2$ to circumference of $BO_1O_2$. Prove that $FO_1=O_1O_2$.

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