geometry problems from Italian Team Selection Tests (TST) with aops links in the names
(only those not in IMO BMO Shortlist)
collected inside aops here
IMO TST 1993 - 2018
(pre-IMO)
Let $ABC$ be an isosceles triangle with base $AB$ and $D$ be a point on side $AB$ such that the incircle of triangle $ACD$ is congruent to the excircle of triangle $DCB$ across $C$. Prove that the diameter of each of these circles equals half the altitude of $\vartriangle ABC$ from $A$
Given a circle $\gamma$ and a point $P$ inside it, find the maximum and minimum value of the sum of the lengths of two perpendicular chords of $\gamma$ passing through $P$.
In a triangle $ABC$, $P$ and $Q$ are the feet of the altitudes from $B$ and $A$ respectively. Find the locus of the circumcentre of triangle $PQC$, when point $C$ varies (with $A$ and $B$ fixed) in such a way that $\angle ACB$ is equal to $60^{\circ}$.
Let $A$ and $B$ be two diametrically opposite points on a circle with radius $1$. Points $P_1,P_2,...,P_n$ are arbitrarily chosen on the circle. Let a and b be the geometric means of the distances of $P_1,P_2,...,P_n$ from $A$ and $B$, respectively. Show that at least one of the numbers $a$ and $b$ does not exceed $\sqrt{2}$
Let ABCD be a parallelogram with side AB longer than AD and acute angle $\angle DAB$. The bisector of ∠DAB meets side CD at L and line BC at K. If O is the circumcenter of triangle LCK, prove that the points B,C,O,D lie on a circle.
Let $ABC$ be a triangle with $AB = AC$. Suppose that the bisector of $\angle ABC$ meets the side $AC$ at point $D$ such that $BC = BD+AD$. Find the measure of $\angle BAC$.
In a triangle $ABC$, points $H,M,L$ are the feet of the altitude from $C$, the median from $A$, and the angle bisector from $B$, respectively. Show that if triangle $HML$ is equilateral, then so is triangle $ABC$.
Let $D$ and $E$ be points on sides $AB$ and $AC$ respectively of a triangle $ABC$ such that $DE$ is parallel to $BC$ and tangent to the incircle of $ABC$. Prove that $DE\le\frac{1}{8}(AB+BC+CA)$
Let $ ABC$ be an isosceles right triangle and $M$ be the midpoint of its hypotenuse $AB$. Points $D$ and $E$ are taken on the legs $AC$ and $BC$ respectively such that $AD=2DC$ and $BE=2EC$. Lines $AE$ and $DM$ intersect at $F$. Show that $FC$ bisects the $\angle DFE$.
The diagonals $ AC$ and $ BD$ of a convex quadrilateral $ ABCD$ intersect at point $ M$. The bisector of $ \angle ACD$ meets the ray $ BA$ at $ K$. Given that $ MA \cdot MC +MA \cdot CD = MB \cdot MD$, prove that $ \angle BKC = \angle CDB$.
Given that in a triangle $ABC$, $AB=3$, $BC=4$ and the midpoints of the altitudes of the triangle are collinear, find all possible values of the length of $AC$.
A scalene triangle $ABC$ is inscribed in a circle $\Gamma$. The bisector of angle $A$ meets $BC$ at $E$. Let $M$ be the midpoint of the arc $BAC$. The line $ME$ intersects $\Gamma$ again at $D$. Show that the circumcentre of triangle $AED$ coincides with the intersection point of the tangent to $\Gamma$ at $D$ and the line $BC$.
Let $B\not= A$ be a point on the tangent to circle $S_1$ through the point $A$ on the circle. A point $C$ outside the circle is chosen so that segment $AC$ intersects the circle in two distinct points. Let $S_2$ be the circle tangent to $AC$ at $C$ and to $S_1$ at some point $D$, where $D$ and $B$ are on the opposite sides of the line $AC$. Let $O$ be the circumcentre of triangle $BCD$. Show that $O$ lies on the circumcircle of triangle $ABC$.
The incircle of a triangle $ABC$ touches the sides $AB,BC,CA$ at points $D,E,F$ respectively. The line through $A$ parallel to $DF$ meets the line through $C$ parallel to $EF$ at $G$.
(a) Prove that the quadrilateral $AICG$ is cyclic.
(b) Prove that the points $B,I,G$ are collinear.
Let $\mathcal{P}_0=A_0A_1\ldots A_{n-1}$ be a convex polygon such that $A_iA_{i+1}=2^{[i/2]}$ for $i=0, 1,\ldots ,n-1$ (where $A_n=A_0$). Define the sequence of polygons $\mathcal{P}_k=A_0^kA_1^k\ldots A_{n-1}^k$ as follows: $A_i^1$ is symmetric to $A_i$ with respect to $A_0$, $A_i^2$ is symmetric to $A_i^1$ with respect to $A_1^1$, $A_i^3$ is symmetric to $A_i^2$ with respect to $A_2^2$ and so on. Find the values of $n$ for which infinitely many polygons $\mathcal{P}_k$ coincide with $\mathcal{P}_0$.
Two circles $\gamma_1$ and $\gamma_2$ intersect at $A$ and $B$. A line $r$ through $B$ meets $\gamma_1$ at $C$ and $\gamma_2$ at $D$ so that $B$ is between $C$ and $D$. Let $s$ be the line parallel to $AD$ which is tangent to $\gamma_1$ at $E$, at the smaller distance from $AD$. Line $EA$ meets $\gamma_2$ in $F$. Let $t$ be the tangent to $\gamma_2$ at $F$.
(a) Prove that $t$ is parallel to $AC$.
(b) Prove that the lines $r,s,t$ are concurrent.
(a) Prove that in a triangle the sum of the distances from the centroid to the sides is not less than three times the inradius, and find the cases of equality.
(b) Determine the points in a triangle that minimize the sum of the distances to the sides.
The circle $\Gamma$ and the line $\ell$ have no common points. Let $AB$ be the diameter of $\Gamma$ perpendicular to $\ell$, with $B$ closer to $\ell$ than $A$. An arbitrary point $C\not= A$, $B$ is chosen on $\Gamma$. The line $AC$ intersects $\ell$ at $D$. The line $DE$ is tangent to $\Gamma$ at $E$, with $B$ and $E$ on the same side of $AC$. Let $BE$ intersect $\ell$ at $F$, and let $AF$ intersect $\Gamma$ at $G\not= A$. Let $H$ be the reflection of $G$ in $AB$. Show that $F,C$, and $H$ are collinear.
Let $ABC$ be a triangle, let $H$ be the orthocentre and $L,M,N$ the midpoints of the sides $AB, BC, CA$ respectively. Prove that $HL^{2} + HM^{2} + HN^{2} < AL^{2} + BM^{2} + CN^{2}$ if and only if $ABC$ is acute-angled.
The circles $\gamma_1$ and $\gamma_2$ intersect at the points $Q$ and $R$ and internally touch a circle $\gamma$ at $A_1$ and $A_2$ respectively. Let $P$ be an arbitrary point on $\gamma$. Segments $PA_1$ and $PA_2$ meet $\gamma_1$ and $\gamma_2$ again at $B_1$ and $B_2$ respectively.
a) Prove that the tangent to $\gamma_{1}$ at $B_{1}$ and the tangent to $\gamma_{2}$ at $B_{2}$ are parallel.
b) Prove that $B_{1}B_{2}$ is the common tangent to $\gamma_{1}$ and $\gamma_{2}$ iff $P$ lies on $QR$.
Let $ABC$ an acute triangle.
(a) Find the locus of points that are centers of rectangles whose vertices lie on the sides of $ABC$;
(b) Determine if exist some points that are centers of $3$ distinct rectangles whose vertices lie on the sides of $ABC$.
Let $ABC$ a acute triangle.
(a) Find the locus of all the points $P$ such that, calling $O_{a}, O_{b}, O_{c}$ the circumcenters of $PBC$, $PAC$, $PAB$: $\frac{ O_{a}O_{b}}{AB}= \frac{ O_{b}O_{c}}{BC}=\frac{ O_{c}O_{a}}{CA}$
(b) For all points $P$ of the locus in (a), show that the lines $AO_{a}$, $BO_{b}$ , $CO_{c}$ are cuncurrent (in $X$);
(c) Show that the power of $X$ wrt the circumcircle of $ABC$ is: $-\frac{ a^{2}+b^{2}+c^{2}-5R^{2}}4$ where $a=BC$ , $b=AC$ and $c=AB$.
Let $ABC$ be an acute triangle, let $AM$ be a median, and let $BK$ and $CL$ be the altitudes. Let $s$ be the line perpendicular to $AM$ passing through $A$. Let $E$ be the intersection point of $s$ with $CL$, and let $F$ be the intersection point of $s$ with $BK$.
(a) Prove that $A$ is the midpoint of $EF$.
(b) Let $\Gamma$ be the circumscribed circumference of the MEF triangle, and let $\Gamma_1$ and $\Gamma_2$ be any two circles that have two points $P$ and $Q$ in common, and are tangent to the segment $EF$ and the arc $EF$ of $\Gamma$ not containing the point $M$. Prove that points $M, P, Q$ are collinear.
$ABC$ is a triangle in the plane. Find the locus of point $P$ for which $PA,PB,PC$ form a triangle whose area is equal to one third of the area of triangle $ABC$.
Two circles $O_1$ and $O_2$ intersect at $M,N$. The common tangent line nearer to $M$ of the two circles touches $O_1,O_2$ at $A,B$ respectively. Let $C,D$ be the symmetric points of $A,B$ with respect to $M$ respectively. The circumcircle of triangle $DCM$ intersects circles $O_1$ and $O_2$ at points $E,F$ respectively which are distinct from $M$. Prove that the circumradii of the triangles $MEF$ and $NEF$ are equal.
In a triangle $ABC$ an excircle is tangent to the side $BC$ at point $D$, and is tangent to the extensions of sides $AB,AC$ at points $E ,F$ respectively. Let $P$ the projection of $D$ on $EF, M$ the midpoint of $EF$, and $\Gamma$ the circle circumscribed around $ABC$. Prove that $P$ lies on $\Gamma$ if and only if $M$ lies on $\Gamma$ .
Let $ABCD$ be a cyclic quadrilateral in which the lines $BC$ and $AD$ meet at a point $P$. Let $Q$ be the point of the line $BP$, different from $B$, such that $PQ = BP$. We construct the parallelograms $CAQR$ and $DBCS$. Prove that the points $C, Q, R, S$ lie on the same circle.
Let $ABCD$ be a cyclic quadrilateral. The circle $\Gamma_1$ passes through $A$ and $B$ and is tangent to the side $CD$ in point $E$. The circle $\Gamma_2$ passes through $B$ and $C$ and is tangent to the side $DA$ at point $F$. The circle $\Gamma_3$ passes through $C$ and $D$ and is tangent to the side $AB$ in the point $G$. The circle $\Gamma_4$ passes through $D$ and $A$ and is tangent to the side $BC$ at point $H$. Prove that the lines $EG$ and $FH$ are perpendicular.
Let $ABCD$ be a convex quadrilateral. The lines $AD$ and $BC$ intersect in $E$. The lines $AC$ and $BD$ intersect in $P$. The circles circumscribed around the triangles $AP D$ and $BP C$ intersect again in $Q$. Let $M$ be the midpoint of $AD$, let $N$ be the midpoint of $BC$, and $K$ be the intersection between $AC$ and $MN$, and let $L$ be the intersection between $BD$ and $MN$.
(a) Prove that $EMQN$ is cyclic.
(b) Prove that the circles circumscribed around the triangles $AMK$ and $BNL$ pass through point $Q$.
(c) Prove that the circles circumscribed around the triangles $AMK$ and $BNL$ pass through the same point on the line $AB$.
Let $ABCD$ be a quadrilateral. Suppose there is a point $P$ inside the quadrilateral such that $\angle APD = \angle BPC = 90^o$ and $PA \cdot PD = PB \cdot PC$. Let $O$ be the circumcenter of $CPD$. Prove that the line $OP$ passes through the midpoint of $AB$.
2007 -18 Balkan MO BMO TST
(winter camp) 2010 missing
Let $ABCD$ be a quadrilateral inscribed in a circle $\Gamma$, and let $P$ be an interior point to $\Gamma$. Prove that at least one of the triangles $PAB, PBC, PCD, PDA$ has a radius of circumscribed circle smaller than the radius of $\Gamma$.
2010 missing
ABC an acute triangle. D,E,F the feet of the altitudes from A,B,C respectively. $P_1$ and $P_2$ are the points of intersection between the circumcircle of ABC and EF. $Q_1=DF\cap BP_1$, $Q_2=DF\cap BP_2$. Show that $P_1,P_2,Q_1,Q_2$ are concyclic with centre in A.
Let $ABC$ be a triangle and let $P$ be a point inside it (not on the sides). Let $A_1$ be the the other intersection point of the line $AP$ and the circle circumscribed to $ABC$, let $M_a$ be the midpoint of side $BC$, and let $A_2$ be the symmetric of $A_1$ wrt to $M_a$. We define in a way similar points $B_2$ and $C_2$ (starting from $B_1, C_1, M_b, M_c$).
(a) Determine (if any) all points $P$ for which $A_2, B_2, C_2$ are not three points distinct.
(b) In cases where $A_2, B_2, C_2$ are three distinct points, let K be the circumcenter of the triangle $A_2B_2C_2$. Prove that, as $P$ varies, the midpoint of the segment $PK$ remains fixed.
ABC an acute triangle. $\omega$ is a circumference, with centre L on BC, which is tangent to AB and AC in B' and C' respectively. Suppose that the circumcircle of ABC has its centre on the smallest of the arc B'C'. Show that the circumcircle of ABC and $\omega$ meet in two distinct points.
ABC is a triangle and $\Gamma$ the circumcircle of ABC. $A_0$ , $B_0$ , $C_0$ are the midpoints of the sides BC, AC and AB. D is the feet of the altitude from A and $D_0$ is the projection of $A_0$ on $B_0 C_0$ . G is the centroid of ABC. $\omega$ is the circumference which passes through $B_0$ $C_0$ and is tangent to $\Gamma$ in P (different from A). Show that:
1) $B_0 C_0$ and the tangents to $\Gamma$ in A and P are concurrent;
2) $P,D,G,D_0$ are collinear.
Prove that in the plane there are infinite acute-angled scalene triangles, two by two not congruent, with sides and altitudes of rational length and perimeter $1$.
Let $ABCD$ be a cyclic quadrilateral. Let $M$ be a point of the segment $CD$, and let $N$ be the point of the $BA$ segment such that $\frac{CM}{CD} =\frac{BN}{BA}$. Let $Q$ be the second point of intersection of the circles circumscribed around the triangles $AMD$ and $BMC$. Prove that the circumscribed circle of the triangle $NQB$ is tangent to the line $BC$..
Let $ABC$ be a triangle, right at $C$, and let $H$ be the foot of the altitude from $C$. Let $D$ a point inside the triangle $CBH$ such that the midpoint of $AD$ lies on the segment $CH$. Let $P$ be the intersection point of the lines $BD$ and $CH$, and let $\omega$ be the circle of diameter $BD$. Let $T$ be the further intersection between $\omega$ and the straight line $AD$. Let $Q$ be the further intersection point between $\omega$ and the straight line $CT$. Prove that the line $PQ$ is tangent to $\omega$.
Let $ABC$ be a triangle with $AB <AC <BC$, let $K$ be the foot of the altitude from $A$, and let $D, E, F$ be the midpoints of the sides $BC, CA, AB$, respectively. Let $\omega_1$ and $\omega_2$ the semicircles of diameter $AB$ and $AC$, respectively, external with respect to triangle $ABC$. The line $KE$ intersects $\omega_2$ at $P$, the line $DE$ intersects $\omega_2$ at $Q$, the line $KF$ intersects $\omega_1$ at $R$, the line $DF$ intersects $\omega_1$ at $S$. The line $PQ$ intersects the line $RS$ at $T$.
(a) Prove that the lines $P R$ and $QS$ intersect in $A$.
(b) Prove that the lines $QS$ and $DT$ intersect on the circumscribed circle of $ABC$.
2012-18 EGMO TST
2016, 2018 missing
$ABCD$ is a cyclic quadrangle and $P$ is the point of intersection of diagonals. $K$ is a point on the bisector of $\angle{APD}$ such that AP=KP and $L$ is a point on the bisector of $\angle{BPC}$ such that $BP=PL$.$ M$ is the point of intersection between $AK$ and $BL$ and $N$ is the point of intersection between $DK$ and $CL$. Show that $KL$ and $MN$ are perpendicular.
Let $ABCDE$ be a cyclic (non-entangled) pentagon where $AB = BC$ and $CD = DE$. Be $P$ the intersection between $AD$ and $BE$, let $Q$ be the intersection between $BD$ and $CA$, and let $R$ be the intersection between $BD$ and $CE$.
(a) Prove that the triangle $PQR$ is isosceles.
(b) Prove that the line $CP$ is perpendicular to the line $BD$.
2014 Italy EGMO TST p3 (past a is BMO 2013 G3 extended)
Two circles $\Gamma_1$ and $\Gamma_2$ intersect at points $M,N$. A line $\ell$ is tangent to $\Gamma_1 ,\Gamma_2$ at $A$ and $B$, respectively. The lines passing through $A$ and $B$ and perpendicular to $\ell$ intersects $MN$ at $C$ and $D$ respectively.
(a) Prove that $ABCD$ is a parallelogram.
(b) Let $E$ be the second intersection point between $CA$ and $\Gamma_1$, let $F$ be the second intersection point between $CB$ and $\Gamma_2$, let $X$ be the second intersection point between $DA$ and $\Gamma_1$, and let $Y$ be the second intersection point between $DB$ and $\Gamma_2$,. Prove that the points $E, F, X, Y$ are collinear.
2016 missing
Let $C_1$ and $C_2$ be two externally tangent circles at $S$ such that the radius of $C_2$ is three times the radius of $C_1$. Let us consider a tangent line to $C_1$ at $P \ne S$ and to $C_2$ in $Q \ne S$. Let $T$ be a point in $C_2$ such that $QT$ is a diameter of $C_2$. Finally the bisector of the angle $\angle SQT$ meets $ST$ in $R$. Prove that $QR = RT$.
2018 missing
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