Italy 2001-19 (ITAMO) 38p

geometry problems from Italian Mathematical Olympiads  (ITAMO)
with aops links in the names

2001 - 2019

A hexagon has all its angles equal, and the lengths of four consecutive sides are $5$, $3$, $6$ and $7$, respectively. Find the lengths of the remaining two edges.

2001 ITAMO P5
Let $ABC$ be a triangle and $\gamma$ the circle inscribed in $ABC$. The circle $\gamma$ is tangent to side $AB$ at the point $T$. Let $D$ be the point of $\gamma$ diametrically opposite to $T$, and $S$ the intersection point of the line through $C$ and $D$ with side $AB$. Prove that $AT=SB$.

2002 ITAMO P3
Let $A$ and $B$ are two points on a plane, and let $M$ be the midpoint of $AB$. Let $r$ be a line and let $R$ and $S$ be the projections of $A$ and $B$ onto $r$. Assuming that $A$, $M$, and $R$ are not collinear, prove that the circumcircle of triangle $AMR$ has the same radius as the circumcircle of $BSM$.

2003 ITAMO P3
Let a semicircle is given with diameter $AB$ and centre $O$ and let $C$ be a arbitrary point on the segment $OB$. Point $D$ on the semicircle is such that $CD$ is perpendicular to $AB$. A circle with centre $P$ is tangent to the arc $BD$ at $F$ and to the segment $CD$ and $AB$ at $E$ and $G$ respectively. Prove that the triangle $ADG$ is isosceles.

2004 ITAMO P2
Two parallel lines $r,s$ and two points $P \in r$ and $Q \in s$ are given in a plane. Consider all pairs of circles $(C_P, C_Q)$ in that plane such that $C_P$ touches $r$ at $P$ and $C_Q$ touches $s$ at $Q$ and which touch each other externally at some point $T$. Find the locus of $T$.

2004 ITAMO P6
Let $P$ be a point inside a triangle $ABC$. Lines $AP,BP,CP$ meet the opposite sides of the triangle at points $A',B',C'$ respectively. Denote $x =\frac{AP}{PA'}, y = \frac{BP}{PB'}$ and $z = \frac{CP}{PC'}$. Prove that $xyz = x+y+z+2$.

2005 ITAMO P1
Let $ABC$ be a right angled triangle with hypotenuse $AC$, and let $H$ be the foot of the altitude from $B$ to $AC$. Knowing that there is a right-angled triangle with side-lengths $AB, BC, BH$, determine all the possible values ​​of $\frac{AH}{CH}$

2005 ITAMO P6
Two circles $\gamma_1, \gamma_2$ in a plane, with centers $A$ and $B$ respectively, intersect at $C$ and $D$. Suppose that the circumcircle of $ABC$ intersects $\gamma_1$ in $E$ and $\gamma_2$ in $F$, where the arc $EF$ not containing $C$ lies outside $\gamma_1$ and $\gamma_2$. Prove that this arc $EF$ is bisected by the line $CD$.

2006 ITAMO P3
Let $A$ and $B$ be two distinct points on the circle $\Gamma$, not diametrically opposite. The point $P$, distinct from $A$ and $B$, varies on $\Gamma$. Find the locus of the orthocentre of triangle $ABP$.

2007 ITAMO P1
It is given a regular hexagon in the plane. Let P be a point of the plane. Define s(P) as the sum of the distances from P to each side of the hexagon, and v(P) as the sum of the distances from P to each vertex.
a) Find the locus of points P that minimize s(P)
b) Find the locus of points P that minimize v(P)

2007 ITAMO P3
Let ABC be a triangle, G its centroid, M the midpoint of AB, D the point on the line $AG$ such that $AG = GD, A \neq D$, E the point on the line $BG$ such that $BG = GE, B \neq E$. Show that the quadrilateral BDCM is cyclic if and only if $AD = BE$.

2008 ITAMO P1
Let $ABCDEFGHILMN$ be a regular dodecagon, let $P$ be the intersection point of the diagonals $AF$ and $DH$. Let $S$ be the circle which passes through $A$ and $H$, and which has the same radius of the circumcircle of the dodecagon, but is different from the circumcircle of the dodecagon. Prove that:
a. $P$ lies on $S$
b. the center of $S$ lies on the diagonal $HN$
c. the length of $PE$ equals the length of the side of the dodecagon

2008 ITAMO P5
Let $ABC$ be a triangle, all of whose angles are greater than $45^{\circ}$ and smaller than $90^{\circ}$.
(a) Prove that one can fit three squares inside $ABC$ in such a way that: (i) the three squares are equal (ii) the three squares have common vertex $K$ inside the triangle (iii) any two squares have no common point but $K$ (iv) each square has two opposite vertices onthe boundary of $ABC$, while all the other points of the square are inside $ABC$.
(b) Let $P$ be the center of the square which has $AB$ as a side and is outside $ABC$. Let $r_{C}$ be the line symmetric to $CK$ with respect to the bisector of $\angle BCA$. Prove that $P$ lies on $r_{C}$.

2009 ITAMO P2
$ABCD$ is a square with centre $O$. Two congruent isosceles triangle $BCJ$ and $CDK$ with base $BC$ and $CD$ respectively are constructed outside the square. let $M$ be the midpoint of $CJ$. Show that $OM$ and $BK$ are perpendicular to each other.

2009 ITAMO P4
Let $ABC$ be an acute-angled scalene triangle and $\Gamma$ be its circumcircle. $K$ is the foot of the internal bisector of $\angle BAC$ on $BC$. Let $M$ be the midpoint of the arc $BC$ containing $A$. $MK$ intersect $\Gamma$ again at $A'$. $T$ is the intersection of the tangents at $A$ and $A'$. $R$ is the intersection of the perpendicular to $AK$ at $A$ and perpendicular to $A'K$ at $A'$. Show that $T, R$ and $K$ are collinear.

2010 ITAMO P3
Let $ABCD$ be a convex quadrilateral. such that $\angle CAB = \angle CDA$ and $\angle BCA = \angle ACD$. If $M$ be the midpoint of $AB$, prove that $\angle BCM = \angle DBA$.

2010 ITAMO P4
In a trapezium $ABCD$, the sides $AB$ and $CD$ are parallel and the angles $\angle ABC$ and $\angle BAD$ are acute. Show that it is possible to divide the triangle $ABC$ into 4 disjoint triangle $X_1. . . , X_4$ and the triangle $ABD$ into 4 disjoint triangles $Y_1,. . . , Y_4$ such that the triangles $X_i$ and $Y_i$ are congruent for all $i$.

2011 ITAMO P4
A trapezium is given with parallel bases having lengths $1$ and $4$. Split it into two trapeziums by a cut, parallel to the bases, of length $3$. We now want to divide the two new trapeziums, always by means of cuts parallel to the bases, in $m$ and $n$ trapeziums, respectively, so that all the $m + n$ trapezoids obtained have the same area. Determine the minimum possible value for $m + n$ and the lengths of the cuts to be made to achieve this minimum value.

2011 ITAMO P4
$ABCD$ is a convex quadrilateral. $P$ is the intersection of external bisectors of $\angle DAC$ and $\angle DBC$. Prove that $\angle APD = \angle BPC$ if and only if $AD+AC=BC+BD$

2012 ITAMO P1
On the sides of a triangle $ABC$ right angled at $A$ three points $D, E$ and $F$ (respectively $BC, AC$ and $AB$) are chosen so that the quadrilateral $AFDE$ is a square. If $x$ is the length of the side of the square, show that $\frac{1}{x}=\frac{1}{AB}+\frac{1}{AC}$

2012 ITAMO P5
$ABCD$ is a square. Describe the locus of points $P$, different from $A, B, C, D$, on that plane for which $\widehat{APB}+\widehat{CPD}=180^\circ$

2013 ITAMO P2
In triangle $ABC$, suppose we have $a> b$, where $a=BC$ and $b=AC$. Let $M$ be the midpoint of $AB$, and $\alpha, \beta$ are inscircles of the triangles $ACM$ and $BCM$ respectively. Let then $A'$ and $B'$ be the points of tangency of $\alpha$ and $\beta$ on $CM$. Prove that $A'B'=\frac{a - b}{2}$.

2013 ITAMO P5
$ABC$ is an isosceles triangle with $AB=AC$ and the angle in $A$ is less than $60^{\circ}$. Let $D$ be a point on $AC$ such that $\angle{DBC}=\angle{BAC}$. $E$ is the intersection between the perpendicular bisector of $BD$ and the line parallel to $BC$ passing through $A$. $F$ is a point on the line $AC$ such that $FA=2AC$ ($A$ is between $F$ and $C$).
Show that $EB$ and $AC$ are parallel and that the perpendicular from $F$ to $AB$, the perpendicular from $E$ to $AC$ and $BD$ are concurrent.

2014 ITAMO P2
Let $ABC$ be a triangle. Let $H$ be the foot of the altitude from $C$ on $AB$. Suppose that $AH = 3HB$. Suppose in addition we are given that
(a) $M$ is the midpoint of $AB$;
(b) $N$ is the midpoint of $AC$;
(c) $P$ is a point on the opposite side of $B$ with respect to the line $AC$ such that $NP = NC$ and $PC = CB$.
Prove that $\angle APM = \angle PBA$.

2014 ITAMO P4
Let $\omega$ be a circle with center $A$ and radius $R$. On the circumference of $\omega$ four distinct points $B, C, G, H$ are taken in that order in such a way that $G$ lies on the extended $B$-median of the triangle $ABC$, and H lies on the extension of altitude of $ABC$ from $B$. Let $X$ be the intersection of the straight lines $AC$ and $GH$. Show that the segment $AX$ has length $2R$.

2015 ITAMO P1
Let $ABCDA'B'C'D'$ be a rectangular parallelipiped, where $ABCD$ is the lower face and $A, B, C$ and $D'$ are below $A', B', C'$ and $D'$, respectively. The parallelipiped is divided into eight parts by three planes parallel to its faces. For each vertex P, let $V_P$ denote the volume of the part containing $P$. Given that $V_A= 40, V_C = 300 , V_B' = 360$ and $V_C'= 90$, find the volume of $ABCDA'B'C'D'$.

2015 ITAMO P3
Let $ABC$ a triangle, let $K$ be the foot of the bisector relative to $BC$ and $J$ be the foot of the trisectrix relative to $BC$ closer to the side $AC$ ($3 \angle JAC=3 \angle CAB$ ). Let $C'$ and $B'$ be two point on the line $AJ$ on the side of $J$ with respect to $A$, such that $AC'=AC$ and $AB=AB'$. Prove that $ABB'C$ is cyclic if and only if lines $C'K$ and $BB'$ are parallel.

2015 ITAMO P5
Let $AB$ be a chord of a circle $\Gamma$ and let $C$ be a point on the segment $AB$. Let $r$ be a line through $C$ which intersects $\Gamma$ at the points $D,E$; suppose that $D,E$ lie on different sides with respect to the perpendicular bisector of $AB$. Let $\Gamma_D$ be the circumference which is externally tangent to $\Gamma$ at $D$ and touches the line $AB$ at $F$. Let $\Gamma_E$ be the circumference which is externally tangent to $\Gamma$ at $E$ and touches the line $AB$ at $G$. Prove that $CA=CB$ if and only if $CF=CG$.

2016 ITAMO P1
Let $ABC$ be a triangle, and let $D$ and $E$ be the orthogonal projections of $A$ onto the internal bisectors from $B$ and $C$. Prove that $DE$ is parallel to $BC$.

2016 ITAMO P3
Let $\Gamma$ be the excircle of triangle $ABC$ opposite to the vertex $A$ (namely, the circle tangent to $BC$ and to the prolongations of the sides $AB$ and $AC$ from the part $B$ and $C$). Let $D$ be the center of $\Gamma$ and $E$, $F$, respectively, the points in which $\Gamma$ touches the prolongations of $AB$ and $AC$. Let $J$ be the intersection between the segments $BD$ and $EF$. Prove that $\angle CJB$ is a right angle.

2017 ITAMO P4
Let $ABCD$ be a thetraedron with the following propriety: the four lines connecting a vertex and the incenter of opposite face are concurrent. Prove $AB \cdot CD= AC \cdot BD = AD\cdot BC$.

2018 ITAMO P2
Let $ABC$ be an acute-angeled triangle , non-isosceles and with barycentre $G$ (which is , in fact , the intersection of the medians).Let $M$ be the midpoint of $BC$ , and let Ω be the circle with centre $G$ and radius $GM$ , and let $N$ be the point of intersection between Ω and $BC$ that is distinct from $M$.Let $S$ be the symmetric point of $A$ with respect to $N$ , that is , the point on the line $AN$ such that $AN=NS$. Prove that $GS$ is perpendicular to $BC$

2018 ITAMO P6
Let $ABC$ be a triangle with $AB=AC$ and let $I$ be its incenter. Let $\Gamma$ be the circumcircle of $ABC$. Lines $BI$ and $CI$ intersect $\Gamma$ in two new points, $M$ and $N$ respectively. Let $D$ be another point on $\Gamma$ lying on arc $BC$ not containing $A$, and let $E,F$ be the intersections of $AD$ with $BI$ and $CI$, respectively. Let $P,Q$ be the intersections of $DM$ with $CI$ and of $DN$ with $BI$ respectively.
(i) Prove that $D,I,P,Q$ lie on the same circle $\Omega$
(ii) Prove that lines $CE$ and $BF$ intersect on $\Omega$

2019 ITAMO P1
Let $ABCDEF$ be a hexagon inscribed in a circle such that $AB=BC,$ $CD=DE$ and $EF=AF.$ Prove that segments $AD,$ $BE$ and $CF$ are concurrent.

2019 ITAMO P5
Let $ABC$ be an acute angled triangle$.$ Let $D$ be the foot of the internal angle bisector of $\angle BAC$ and let $M$ be the midpoint of $AD.$ Let $X$ be a point on segment $BM$ such that $\angle MXA=\angle DAC.$ Prove that $AX$ is perpendicular to $XC.$