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

with aops links in the names

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.

IZhO 2005 Senior 2

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.

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.

IZhO 2005 Senior 2

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 A

_{1}A_{2}be the external tangent line to the nonintersecting circles ω_{1}(O_{1}) and ω_{2}(O_{2}), A_{1}∈ω_{1}, A_{2}∈ω_{2}. Points K is the midpoint of A_{1}A_{2}. And KB_{1}and KB_{2}are tangent lines to ω_{1}and ω_{2}, respectively (B_{1}≠ A_{1}, B_{2}≠ A_{2}).Lines A_{1}B_{1}and A_{2}B_{2}meet in point L, and lines KL and O_{1}O_{2}meet in point P. Prove that points B_{1},B_{2}, 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 = 90

^{o}. 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 > 90

^{o}
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 H

_{1}and H_{2}be the orthocenters of triangles MNC and MND, respectively. Prove that the area of the quadrilateral AH_{1}BH_{2}does not depend on the position of D on AB.
Equilateral
triangles ACB′ and BDC′ are drawn on the
diagonals of a convex quadrilateral ABCD so that B and B′ are
on the same side of AC, and C and C′ are on the same
sides of BD. Find ÐBAD + ÐCDA if B′C′ =
AB + CD.

Given a
trapezoid ABCD (AD // BC) with ÐABC > 90◦ .
Point M is chosen on the lateral side AB. Let O

_{1}and O_{2}be the circumcenters of the triangles MAD and MBC, respectively. The circumcircles of the triangles MO_{1}D and MO_{2}C meet again at the point N. Prove that the line O_{1}O_{2}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)

(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 R

_{1}, R_{2}, R_{3}, R_{4}, R_{5}. 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 $IQ=JP$

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 $FO_1=O_1O_2$.

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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