geometry problems from Mathematical Olympiads from Israel (Gillis)
with aops links in the names
with aops links in the names
collected inside aops here
2011 - 2023
Let $\alpha_1,\alpha_2,\alpha_3$ be three congruent circles that are tangent to each other. A third circle $\beta$ is tangent to them at points $A_1,A_2,A_3$ respectively. Let $P$ be a point on $\beta$ which is different from $A_1,A_2,A_3$. For $i=1,2,3$, let $B_i$ be the second intersection point of the line $PA_i$ with circle $\alpha_i$. Prove that $\Delta B_1B_2B_3$ is equilateral.
In the picture below, the circles are tangent to each other and to the edges of the rectangle. The larger circle's radius equals 1. Determine the area of the rectangle.
2012 Israel P6
Let $A,B,C,O$ be points in the plane such that angles $\angle AOB,\angle BOC, \angle COA$ are obtuse. On $OA,OB,OC$ points $X,Y,Z$ respectively are chosen, such that $OX=OY=OZ$. On segments $OX,OY,OZ$ points $K,L,M$ respectively are chosen.
The lines $AL$ and $BK$ intersect at point $R$, which isn't on $XY$. The segment $XY$ intersects $AL,BK$ at points $R_1,R_2$.
The lines $BM$ and $CL$ intersect at point $P$, which isn't on $YZ$. The segment $YZ$ intersects $BM,CL$ at points $P_1,P_2$.
The lines $CK$ and $AM$ intersect at point $Q$, which isn't on $ZX$. The segment $ZX$ intersects $CK,AM$ at points $Q_1,Q_2$.
Suppose that $PP_1=PP_2$ and $QQ_1=QQ_2$. Prove that $RR_1=RR_2$.
2013 Israel P1
In the picture there are six coins, each with radius 1cm. Each coin is tangent to exactly two other coins next to it (as in the picture). Between the coins, there is an empty area whose boundary is a star-like shape. What is the perimeter of this shape?
A point in the plane is called integral if both its $x$ and $y$ coordinates are integers. We are given a triangle whose vertices are integral. Its sides do not contain any other integral points. Inside the triangle, there are exactly 4 integral points. Must those 4 points lie on one line?
2014 Israel P2
Let $\Delta A_1A_2A_3, \Delta B_1B_2B_3, \Delta C_1C_2C_3$ be three equilateral triangles. The vertices in each triangle are numbered clockwise. It is given that $A_3=B_3=C_3$. Let $M$ be the center of mass of $\Delta A_1B_1C_1$, and let $N$ be the center of mass of $\Delta A_2B_2C_2$.
Prove that $\Delta A_3MN$ is an equilateral triangle.
Let $\Delta A_1A_2A_3, \Delta B_1B_2B_3, \Delta C_1C_2C_3$ be three equilateral triangles. The vertices in each triangle are numbered clockwise. It is given that $A_3=B_3=C_3$. Let $M$ be the center of mass of $\Delta A_1B_1C_1$, and let $N$ be the center of mass of $\Delta A_2B_2C_2$.
Prove that $\Delta A_3MN$ is an equilateral triangle.
Let $ABCDEF$ be a convex hexagon. In the hexagon there is a point $K$, such that $ABCK,DEFK$ are both parallelograms. Prove that the three lines connecting $A,B,C$ to the midpoints of segments $CE,DF,EA$ meet at one point.
A triangle is given whose altitudes' lengths are $\frac{1}{5},\frac{1}{5},\frac{1}{8}$. Evaluate the triangle's area.
2015 Israel P5
Let $ABCD$ be a tetrahedron. Denote by $S_1$ the inscribed sphere inside it, which is tangent to all four faces. Denote by $S_2$ the outer escribed sphere outside $ABC$, tangent to face $ABC$ and to the planes containing faces $ABD,ACD,BCD$. Let $K$ be the tangency point of $S_1$ to the face $ABC$, and let $L$ be the tangency point of $S_2$ to the face $ABC$. Let $T$ be the foot of the perpendicular from $D$ to the face $ABC$.
Prove that $L,T,K$ lie on one line.
2016 Israel P2
We are given a cone with height 6, whose base is a circle with radius $\sqrt{2}$. Inside the cone, there is an inscribed cube: Its bottom face on the base of the cone, and all of its top vertices lie on the cone. What is the length of the cube's edge?
Points $A_1,A_2,A_3,...,A_{12}$ are the vertices of a regular polygon in that order. The 12 diagonals $A_1A_6,A_2A_7,A_3A_8,...,A_{11}A_4,A_{12}A_5$ are marked, as in the picture below. Let $X$ be some point in the plane. From $X$, we draw perpendicular lines to all 12 marked diagonals. Let $B_1,B_2,B_3,...,B_{12}$ be the feet of the perpendiculars, so that $B_1$ lies on $A_1A_6$, $B_2$ lies on $A_2A_7$ and so on.
Evaluate the ratio $\frac{XA_1+XA_2+\dots+XA_{12}}{B_1B_6+B_2B_7+\dots+B_{12}B_5}$.
a) In the right picture there is a square with four congruent circles inside it. Each circle is tangent to two others, and to two of the edges of the square. Evaluate the ratio between the blue part and white part of the square's area.
b) In the left picture there is a regular hexagon with six congruent circles inside it. Each circle is tangent to two others, and to one of the edges on the hexagon in its midpoint. Evaluate the ratio between the blue part and white part of the hexagon's area.
A large collection of congruent right triangles is given, each with side length 3,4,5. Find the maximal number of such triangles you can place inside a 20x20 square, with no two triangles intersecting (in their interiors).
2017 Israel P5
A regular pentagon $ABCDE$ is given. The point $X$ is on his circumcircle, on the arc $AE$. Prove that $|AX|+|CX|+|EX|=|BX|+|DX|$.
Remark:
Here's a more general version of the problem: Prove that for any point $X$ in the plane, $|AX|+|CX|+|EX|\ge|BX|+|DX|$, with equality only on the arc $AE$.
2018 Israel P6
In the corners of triangle $ABC$ there are three circles with the same radius. Each of them is tangent to two of the triangle's sides. The vertices of triangle $MNK$ lie on different sides of triangle $ABC$, and each edge of $MNK$ is also tangent to one of the three circles. Likewise, the vertices of triangle $PQR$ lie on different sides of triangle $ABC$, and each edge of $PQR$ is also tangent to one of the three circles (see picture below). Prove that triangles $MNK,PQR$ have the same inradius.
Six congruent isosceles triangles have been put together as described in the picture below. Prove that points M, F, C lie on one line.
2019 Israel P7
In the plane points $A,B,C$ are marked in blue and points $P,Q$ are marked in red (no 3 marked points lie on a line, and no 4 marked points lie on a circle). A circle is called separating if all points of one color are inside it, and all points of the other color are outside of it. Denote by $O$ the circumcenter of $ABC$ and by $R$ the circumradius of $ABC$.
Prove that there exists a separating circle if and only if for each point on segment $PQ$ which also lies inside the triangle $ABC$, we have $PX\cdot XQ \neq R^2-OX^2$.
In a convex hexagon $ABCDEF$ the triangles $BDF, ACE$ are equilateral and congruent. Prove that the three lines connecting the midpoints of opposite sides are concurrent.
Two triangles $ACE, BDF$ are given which intersect at six points: $G, H, I, J, K, L$ as in the picture. It is known that in each of the quadrilaterals
\[ABIK ,BCJL ,CDKG ,DELH ,EFGI\] it is possible to inscribe a circle. Is it possible for the quadrilateral $FAHJ$ is also circumscribed around a circle?
Let $P$ be a point inside a triangle $ABC$, $d_a$, $d_b$ and $d_c$ be distances from $P$ to the lines $BC$, $AC$ and $AB$ respectively, $R$ be a radius of the circumcircle and $r$ be a radius of the inscribed circle for $\Delta ABC.$ Prove that:
$$\sqrt{d_a}+\sqrt{d_b}+\sqrt{d_c}\leq\sqrt{2R+5r}.$$
Let $ABC$ be a triangle. Let $X$ be the tangency point of the incircle with $BC$. Let $Y$ be the second intersection point of segment $AX$ with the incircle.
Prove that
\[AX+AY+BC>AB+AC\]
Triangle $ABC$ is given.
The circle $\omega$ with center $I$ is tangent at points $D,E,F$ to segments $BC,AC,AB$ respectively.
When $ABC$ is rotated $180$ degrees about point $I$, triangle $A'B'C'$ results.
Lines $AD, B'C'$ meet at $U$, lines $BE, A'C'$ meet at $V$, and lines $CF, A'B'$ meet at $W$.
Line $BC$ meets $A'C', A'B'$ at points $D_1, D_2$ respectively.
Line $AC$ meets $A'B', B'C'$ at $E_1, E_2$ respectively.
Line $AB$ meets $B'C', A'C'$ at $F_1,F_2$ respectively.
Six (not necessarily convex) quadrilaterals were colored orange:
\[AUIF_2 , C'FIF_2 , BVID_1 , A'DID_2 , CWIE_1 , B'EIE_2\]Six other quadrilaterals were colored green:
\[AUIE_2 , C'FIF_1 , BVIF_2 , A'DID_1 , CWID_2 , B'EIE_1\]Prove that the sum of the green areas equals the sum of the orange areas.
Let $w$ be a circle of diameter $5$. Four lines were drawn dividing $w$ into $5$ "strips", each of width $1$. The strips were colored orange and purple alternatingly, as depicted. Which area is larger: the orange or the purple?
A paper convex quadrilateral will be called folding if there are points $P,Q,R,S$ on the interiors of segments $AB,BC,CD,DA$ respectively so that if we fold in the triangles $SAP, PBQ, QCR, RDS$, they will exactly cover the quadrilateral $PQRS$. In other words, if the folded triangles will cover the quadrilateral $PQRS$ but won't cover each other.
Prove that if quadrilateral $ABCD$ is folding, then $AC\perp BD$ or $ABCD$ is a trapezoid.
A triangle $ABC$ is given together with an arbitrary circle $\omega$. Let $\alpha$ be the reflection of $\omega$ with respect to $A$, $\beta$ the reflection of $\omega$ with respect to $B$, and $\gamma$ the reflection of $\omega$ with respect to $C$. It is known that the circles $\alpha, \beta, \gamma$ don't intersect each other.
Let $P$ be the meeting point of the two internal common tangents to $\beta, \gamma$ (see picture). Similarly, $Q$ is the meeting point of the internal common tangents of $\alpha, \gamma$, and $R$ is the meeting point of the internal common tangents of $\alpha, \beta$.
Prove that the triangles $PQR, ABC$ are congruent.
Let $ABC$ be an equilateral triangle whose sides have length $1$. The midpoints of $AB,BC$ are $M,N$ respectively. Points $K,L$ were chosen on $AC$ so that $KLMN$ is a rectangle. Inside this rectangle are three semi-circles with the same radius, as in the picture (the endpoints are on the edges of the rectangle, and the arcs are tangent).
Find the minimum possible value of the radii of the semi-circles.
Can you add the 26 clock even triangle problem?
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