Zhautykov 2005-22 (IZhO) (Kazakhstan) 37p

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

with aops links in the names

Zhautykov geometry problems 2005-17 EN in pdf with aops links

Zhautykov all 2005-21 EN in pdf, with solutions 2013-21

2005 - 2022

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

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 ${X \in \{ A,B,C\} }$.

IZhO 2005 Senior 5

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.

IZhO 2006.2

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.

IZhO 2006.6

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 .

IZhO 2007.2

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.

IZhO 2007.6

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.

IZhO 2008.1

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.

IZhO 2008.5

Let A₁A₂ be the external tangent line to the nonintersecting circles ω₁(O₁) and ω₂(O₂), A₁∈ω₁,  A₂∈ω₂. 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.

IZhO 2009.3

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

IZhO 2009.5

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.

IZhO 2010.2

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 .

IZhO 2010.6

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

IZhO 2011.1

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?

IZhO 2011.6

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.)

IZhO 2012.1

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.

IZhO 2012.5

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.

IZhO 2013.1

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.

IZhO 2013.5

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.

IZhO 2014.1

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)

IZhO 2014.6

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)

IZhO 2015.2

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)

IZhO 2015.6

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)

IZhO 2016.1

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) 

IZhO 2016.5

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)

IZhO 2017.1

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) 

IZhO 2017.6

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 $IQ=JP$

IZhO 2018.6

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. $$r_1+r_2+r_3+r_4+r_5+r_6\leq R\sqrt{3}$$ .

IZhO 2019.3

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.

IZhO 2019.4

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 $FO_1=O_1O_2$.

IZhO 2020.3

Given convex hexagon $ABCDEF$, inscribed in the circle. Prove that $AC*BD*DE*CE*EA*FB \geq 27 AB * BC * CD * DE * EF * FA$

IZhO 2020.4

In a scalene triangle $ABC$ $I$ is the incentr and $CN$ is the bisector of angle $C$. The line $CN$ meets the circumcircle of $ABC$ again at $M$. The line $l$ is parallel to $AB$ and touches the incircle of $ABC$. The point $R$ on $l$ is such. That $CI \bot IR$. The circumcircle of $MNR$ meets the line $IR$ again at S. Prpve that $AS=BS$.

IZhO 2021.2

In a convex cyclic hexagon $ABCDEF$, $BC=EF$ and $CD=AF$. Diagonals $AC$ and $BF$ intersect at point $Q,$ and diagonals $EC$ and $DF$ intersect at point $P.$ Points $R$ and $S$ are marked on the segments $DF$ and $BF$ respectively so that $FR=PD$ and $BQ=FS.$ The segments $RQ$ and $PS$ intersect at point $T.$ Prove that the line $TC$ bisects the diagonal $DB$.

IZhO 2021.4

Let there be an incircle of triangle $ABC$, and 3 circles each inscribed between incircle and angles of $ABC$. Let $r, r_1, r_2, r_3$ be radii of these circles ($r_1, r_2, r_3 < r$). Prove that $$r_1+r_2+r_3 \geq r$$

IZhO 2022.3

A ten-level $2$-tree is drawn in the plane: a vertex $A_1$ is marked, it is connected by segments with two vertices $B_1$ and $B_2$, each of $B_1$ and $B_2$ is connected by segments with two of the four vertices $C_1, C_2, C_3, C_4$ (each $C_i$ is connected with one $B_j$ exactly); and so on, up to $512$ vertices $J_1, \ldots, J_{512}$. Each of the vertices $J_1, \ldots, J_{512}$ is coloured blue or golden. Consider all permutations $f$ of the vertices of this tree, such that (i) if $X$ and $Y$ are connected with a segment, then so are $f(X)$ and $f(Y)$, and (ii) if $X$ is coloured, then $f(X)$ has the same colour. Find the maximum $M$ such that there are at least $M$ permutations with these properties, regardless of the colouring.

IZhO 2022.4

In triangle $ABC$, a point $M$ is the midpoint of $AB$, and a point $I$ is the incentre. Point $A_1$ is the reflection of $A$ in $BI$, and $B_1$ is the reflection of $B$ in $AI$. Let $N$ be the midpoint of $A_1B_1$. Prove that $IN > IM$.

sources:

artofproblemsolving.com

izho.kz

olympiads.kz