### India TST (IMO Training Camp) 2001-19 (IMOTC) 49p (-08)

geometry problems from Indian Team Selection Tests (TST), IMO Training Camp (IMOTC)
with aops links in the names
(only those not in IMO Shortlist)

2001-
(missing 2008)

In a triangle $ABC$ with incircle $\omega$ and incenter $I$ , the segments $AI$ , $BI$ , $CI$ cut $\omega$ at $D$ , $E$ , $F$ , respectively. Rays $AI$ , $BI$ , $CI$ meet the sides $BC$ , $CA$ , $AB$ at $L$ , $M$ , $N$ respectively. Prove that: $AL+BM+CN \leq 3(AD+BE+CF)$. When does equality occur?

If on $\triangle ABC$, trinagles $AEB$ and $AFC$ are constructed externally such that $\angle AEB=2 \alpha$, $\angle AFB= 2 \beta$. $AE=EB$, $AF=FC$. COnstructed externally on $BC$ is triangle $BDC$ with $\angle DBC= \beta$ , $\angle BCD= \alpha$. Prove that
1. $DA$ is perpendicular to $EF$.
2. If $T$ is the projection of $D$ on $BC$, then prove that $\frac{DA}{EF}= 2 \frac{DT}{BC}$.

Let $ABCD$ be a rectangle, and let $\omega$ be a circular arc passing through the points $A$ and $C$. Let $\omega_{1}$ be the circle tangent to the lines $CD$ and $DA$ and to the circle $\omega$, and lying completely inside the rectangle $ABCD$. Similiarly let $\omega_{2}$ be the circle tangent to the lines $AB$ and $BC$ and to the circle $\omega$, and lying completely inside the rectangle $ABCD$. Denote by $r_{1}$ and $r_{2}$ the radii of the circles $\omega_{1}$ and $\omega_{2}$, respectively, and by $r$ the inradius of triangle $ABC$.
(a) Prove that $r_{1}+r_{2}=2r$.
(b) Prove that one of the two common internal tangents of the two circles $\omega_{1}$ and $\omega_{2}$ is parallel to the line $AC$ and has the length $\left|AB-AC\right|$.

Let $A,B$ and $C$ be three points on a line with $B$ between $A$ and $C$. Let $\Gamma_1,\Gamma_2, \Gamma_3$ be semicircles, all on the same side of $AC$ and with $AC,AB,BC$ as diameters, respectively. Let $l$ be the line perpendicular to $AC$ through $B$. Let $\Gamma$ be the circle which is tangent to the line $l$, tangent to $\Gamma_1$ internally, and tangent to $\Gamma_3$ externally. Let $D$ be the point of contact of $\Gamma$ and $\Gamma_3$. The diameter of $\Gamma$ through $D$ meets $l$ in $E$. Show that $AB=DE$.

Given two distinct circles touching each other internally, show how to construct a triangle with the inner circle as its incircle and the outer circle as its nine point circle.

Let $ABC$ and $PQR$ be two triangles such that
(a) $P$ is the mid-point of $BC$ and $A$ is the midpoint of $QR$.
(b) $QR$ bisects $\angle BAC$ and $BC$ bisects $\angle QPR$
Prove that $AB+AC=PQ+PR$.

Let $A',B',C'$ be the midpoints of the sides $BC, CA, AB$, respectively, of an acute non-isosceles triangle $ABC$, and let $D,E,F$ be the feet of the altitudes through the vertices $A,B,C$ on these sides respectively. Consider the arc $DA'$ of the nine point circle of triangle $ABC$ lying outside the triangle. Let the point of trisection of this arc closer to $A'$ be $A''$. Define analogously the points $B''$ (on arc $EB'$) and $C''$(on arc $FC'$). Show that triangle $A''B''C''$ is equilateral.

Let $ABC$ be a triangle, and let $r, r_1, r_2, r_3$ denoted its inradius and the exradii opposite the vertices $A,B,C$, respectively. Suppose $a>r_1, b>r_2, c>r_3$. Prove that
(a) triangle $ABC$ is acute,
(b) $a+b+c>r+r_1+r_2+r_3$.

Let $ABCD$ be a cyclic quadrilateral. Let $P$, $Q$, $R$ be the feet of the perpendiculars from $D$ to the lines $BC$, $CA$, $AB$, respectively. Show that $PQ=QR$ if and only if the bisectors of $\angle ABC$ and $\angle ADC$ are concurrent with $AC$.

Let $ABC$ be an acute-angled triangle and $\Gamma$ be a circle with $AB$ as diameter intersecting $BC$ and $CA$ at $F ( \not= B)$ and $E (\not= A)$ respectively. Tangents are drawn at $E$ and $F$ to $\Gamma$ intersect at $P$. Show that the ratio of the circumcentre of triangle $ABC$ to that if $EFP$ is a rational number.

Let $ABC$ be a triangle with all angles $\leq 120^{\circ}$. Let $F$ be the Fermat point of triangle $ABC$, that is, the interior point of $ABC$ such that $\angle AFB = \angle BFC = \angle CFA = 120^\circ$. For each one of the three triangles $BFC$, $CFA$ and $AFB$, draw its Euler line - that is, the line connecting its circumcenter and its centroid. Prove that these three Euler lines pass through one common point.

Remark: The Fermat point $F$ is also known as the  first Fermat point  or the first Toricelli point of triangle $ABC$.

Floor van Lamoen
Let $ABCD$ be a convex quadrilateral. The lines parallel to $AD$ and $CD$ through the orthocentre $H$ of $ABC$ intersect $AB$ and $BC$ Crespectively at $P$ and $Q$. prove that the perpendicular through $H$ to th eline $PQ$ passes through the orthocentre of triangle $ACD$

For a given triangle $ABC$, let $X$ be a variable point on the line $BC$ such that the point $C$ lies between the points $B$ and $X$. Prove that the radical axis of the incircles of the triangles $ABX$ and $ACX$ passes through a point independent of $X$.

Let $ABC$ be a triangle and let $P$ be a point in the plane of $ABC$ that is inside the region of the angle $BAC$ but outside triangle $ABC$.
(a) Prove that any two of the following statements imply the third.
(i) the circumcentre of triangle $PBC$ lies on the ray $\stackrel{\to}{PA}$.
(ii) the circumcentre of triangle $CPA$ lies on the ray $\stackrel{\to}{PB}$.
(iii) the circumcentre of triangle $APB$ lies on the ray $\stackrel{\to}{PC}$.
(b) Prove that if the conditions in (a) hold, then the circumcentres of triangles $BPC,CPA$ and $APB$ lie on the circumcircle of triangle $ABC$.

Let $ABC$ be an equilateral triangle, and let $D,E$ and $F$ be points on $BC,BA$ and $AB$ respectively. Let $\angle BAD= \alpha, \angle CBE=\beta$ and $\angle ACF =\gamma$. Prove that if $\alpha+\beta+\gamma \geq 120^\circ$, then the union of the triangular regions $BAD,CBE,ACF$ covers the triangle $ABC$.

Show that in a non-equilateral triangle, the following statements are equivalent:
a) The angles of the triangle are in arithmetic progression.
b) The common tangent to the Nine-point circle and the Incircle is parallel to the Euler Line.

2008 problems are missing from aops

Let $ABC$ be a triangle with $\angle A = 60^{\circ}$.Prove that if $T$ is point of contact of Incircle And Nine-Point Circle, Then $AT = r$, $r$ being inradius.

Let $\gamma$ be circumcircle of $\triangle ABC$.Let $R_a$ be radius of circle touching $AB,AC$&$\gamma$ internally.Define $R_b,R_c$ similarly. Prove That $\frac {1}{aR_a} + \frac {1}{bR_b} +\frac {1}{cR_c} = \frac {r^2}{sabc}$.

Let $ABC$ be a triangle in which $BC<AC$. Let $M$ be the mid-point of $AB$, $AP$ be the altitude from $A$ on $BC$, and $BQ$ be the altitude from $B$ on to $AC$. Suppose that $QP$ produced meets $AB$ (extended) at $T$. If $H$ is the orthocenter of $ABC$, prove that $TH$ is perpendicular to $CM$.

Let $ABCD$ be a cyclic quadrilaterla and let $E$ be the point of intersection of its diagonals $AC$ and $BD$. Suppose $AD$ and $BC$ meet in $F$. Let the midpoints of $AB$ and $CD$ be $G$ and $H$ respectively.  If $\Gamma$ is the circumcircle of triangle $EGH$, prove that $FE$ is tangent to $\Gamma$.

Let $ABC$ be a triangle each of whose angles is greater than $30^{\circ}$. Suppose a circle centered with $P$ cuts segments $BC$ in $T,Q; CA$ in $K,L$ and $AB$ in $M,N$ such that they are on a circle in counterclockwise direction in that order.Suppose further $PQK,PLM,PNT$ are equilateral. Prove that:
a) The radius of the circle is $\frac{2abc}{a^2+b^2+c^2+4\sqrt{3}S}$ where $S$ is area.
b)  $a\cdot AP=b\cdot BP=c\cdot PC.$

The cirumcentre of the cyclic quadrilateral $ABCD$ is $O$. The second intersection point of the circles $ABO$ and $CDO$, other than $O$, is $P$, which lies in the interior of the triangle $DAO$. Choose a point $Q$ on the extension of $OP$ beyond $P$, and a point $R$ on the extension of $OP$ beyond $O$. Prove that $\angle QAP=\angle OBR$ if and only if $\angle PDQ=\angle RCO$.

A quadrilateral $ABCD$ without parallel sides is circumscribed around a circle with centre $O$. Prove that $O$ is a point of intersection of middle lines of quadrilateral $ABCD$ (i.e. barycentre of points $A,\,B,\,C,\,D$) iff $OA\cdot OC=OB\cdot OD$.

Let $ABC$ be an isosceles triangle with $AB=AC$. Let $D$ be a point on the segment $BC$ such that $BD=2DC$. Let $P$ be a point on the segment $AD$ such that $\angle BAC=\angle BPD$. Prove that $\angle BAC=2\angle DPC$.

Let $ABCD$ be a trapezium with $AB\parallel CD$. Let $P$ be a point on $AC$ such that $C$ is between $A$ and $P$; and let $X, Y$ be the midpoints of $AB, CD$ respectively. Let $PX$ intersect $BC$ in $N$ and $PY$ intersect $AD$ in $M$. Prove that $MN\parallel AB$.

In a triangle $ABC$, with $\widehat{A} > 90^\circ$, let $O$ and $H$ denote its circumcenter and orthocenter, respectively. Let $K$ be the reflection of $H$ with respect to $A$. Prove that $K, O$ and $C$ are collinear if and only if $\widehat{A} - \widehat{B} = 90^\circ$.

In a triangle $ABC$, let $I$ denote its incenter. Points $D, E, F$ are chosen on the segments $BC, CA, AB$, respectively, such that $BD + BF = AC$ and $CD + CE = AB$. The circumcircles of triangles $AEF, BFD, CDE$ intersect lines $AI, BI, CI$, respectively, at points $K, L, M$ (different from $A, B, C$), respectively. Prove that $K, L, M, I$ are concyclic.

In a triangle $ABC$, with $AB \ne BC$, $E$ is a point on the line $AC$ such that $BE$ is perpendicular to $AC$. A circle passing through $A$ and touching the line $BE$ at a point $P \ne B$ intersects the line $AB$ for the second time at $X$. Let $Q$ be a point on the line $PB$ different from $P$ such that $BQ = BP$. Let $Y$ be the point of intersection of the lines $CP$ and $AQ$. Prove that the points $C, X, Y, A$ are concyclic if and only if $CX$ is perpendicular to $AB$.

Let $ABCD$ by a cyclic quadrilateral with circumcenter $O$. Let $P$ be the point of intersection of the diagonals $AC$ and $BD$, and $K, L, M, N$ the circumcenters of triangles $AOP, BOP$, $COP, DOP$, respectively. Prove that $KL = MN$.
In a triangle $ABC$ with $B = 90^\circ$, $D$ is a point on the segment $BC$ such that the inradii of triangles $ABD$ and $ADC$ are equal. If $\widehat{ADB} = \varphi$ then prove that $\tan^2 (\varphi/2) = \tan (C/2)$.

In a triangle $ABC$, let $I$ be its incenter; $Q$ the point at which the incircle touches the line $AC$; $E$ the midpoint of $AC$ and $K$ the orthocenter of triangle $BIC$. Prove that the line $KQ$ is perpendicular to the line $IE$.

In a triangle $ABC$, with $AB\neq AC$ and $A\neq 60^{0},120^{0}$, $D$ is a point on line $AC$ different from $C$. Suppose that the circumcentres and orthocentres of triangles $ABC$ and $ABD$ lie on a circle. Prove that $\angle ABD=\angle ACB$.

In a triangle $ABC$, points $X$ and $Y$ are on $BC$ and $CA$ respectively such that $CX=CY$,$AX$ is not perpendicular to $BC$ and $BY$ is not perpendicular to $CA$.Let $\Gamma$ be the circle with $C$ as centre and $CX$ as its radius.Find the angles of triangle $ABC$ given that the orthocentres of triangles $AXB$ and $AYB$ lie on $\Gamma$.

Let $ABCD$ be a convex quadrilateral and let the diagonals $AC$ and $BD$ intersect at $O$. Let $I_1, I_2, I_3, I_4$ be respectively the incentres of triangles $AOB, BOC, COD, DOA$. Let $J_1, J_2, J_3, J_4$ be respectively the excentres of triangles $AOB, BOC, COD, DOA$ opposite $O$. Show that $I_1, I_2, I_3, I_4$ lie on a circle if and only if $J_1, J_2, J_3, J_4$ lie on a circle.

2015 India TST2 P1
In a triangle $ABC$, a point $D$ is on the segment $BC$, Let $X$ and $Y$ be the incentres of triangles $ACD$ and $ABD$ respectively. The lines $BY$ and $CX$ intersect the circumcircle of triangle $AXY$ at $P\ne Y$ and $Q\ne X$, respectively. Let $K$ be the point of intersection of lines $PX$ and $QY$. Suppose $K$ is also the reflection of $I$ in $BC$ where $I$ is the incentre of triangle $ABC$. Prove that $\angle BAC=\angle ADC=90^{\circ}$.

Let $ABC$ be a triangle in which $CA>BC>AB$. Let $H$ be its orthocentre and $O$ its circumcentre. Let $D$ and $E$ be respectively the midpoints of the arc $AB$ not containing $C$ and arc $AC$ not containing $B$. Let $D'$ and $E'$ be respectively the reflections of $D$ in $AB$ and $E$ in $AC$. Prove that $O, H, D', E'$ lie on a circle if and only if $A, D', E'$ are collinear.

2016 India TST4 P1
Let $ABC$ be an acute triangle with circumcircle $\Gamma$. Let $A_1,B_1$ and $C_1$ be respectively the midpoints of the arcs $BAC,CBA$ and $ACB$ of $\Gamma$. Show that the inradius of triangle $A_1B_1C_1$ is not less than the inradius of triangle $ABC$.

2016 India TST Practice Test1 P1
An acute-angled $ABC \ (AB<AC)$ is inscribed into a circle $\omega$. Let $M$ be the centroid of $ABC$, and let $AH$ be an altitude of this triangle. A ray $MH$ meets $\omega$ at $A'$. Prove that the circumcircle of the triangle $A'HB$ is tangent to $AB$.

2018 India TST4 P1
Let $ABC$ be an acute angled triangle with incenter $I$. Line perpendicular to $BI$ at $I$ meets $BA$ and $BC$ at points $P$ and $Q$ respectively. Let $D, E$ be the incenters of $\triangle BIA$ and $\triangle BIC$ respectively. Suppose $D,P,Q,E$ lie on a circle. Prove that $AB=BC$.

2018 India TST Practice Test1 P3
Let $ABCD$ be a cyclic quadrilateral inscribed in circle $\Omega$ with $AC \perp BD$. Let $P=AC \cap BD$ and $W,X,Y,Z$ be the projections of $P$ on the lines $AB, BC, CD, DA$ respectively. Let $E,F,G,H$ be the mid-points of sides $AB, BC, CD, DA$ respectively.
(a) Prove that $E,F,G,H,W,X,Y,Z$ are concyclic.
(b) If $R$ is the radius of $\Omega$ and $d$ is the distance between its centre and $P$, then find the radius of the circle in (a) in terms of $R$ and $d$.

In an acute triangle $ABC$, points $D$ and $E$ lie on side $BC$ with $BD<BE$. Let $O_1, O_2, O_3, O_4, O_5, O_6$ be the circumcenters of triangles $ABD, ADE, AEC, ABE, ADC, ABC$, respectively. Prove that $O_1, O_3, O_4, O_5$ are con-cyclic if and only if $A, O_2, O_6$ are collinear.

2018 India TST1 P2
Let $A,B,C$ be three points in that order on a line $\ell$ in the plane, and suppose $AB>BC$. Draw semicircles $\Gamma_1$ and $\Gamma_2$ respectively with $AB$ and $BC$ as diameters, both on the same side of $\ell$. Let the common tangent to $\Gamma_1$ and $\Gamma_2$ touch them respectively at $P$ and $Q$, $P\ne Q$. Let $D$ and $E$ be points on the segment $PQ$ such that the semicircle $\Gamma_3$ with $DE$ as diameter touches $\Gamma_2$ in $S$ and $\Gamma_1$ in $T$.
Prove that $A,C,S,T$ are concyclic.
Prove that $A,C,D,E$ are concyclic.

Let $ABC$ be a triangle and $AD,BE,CF$ be cevians concurrent at a point $P$. Suppose each of the quadrilaterals $PDCE,PEAF$ and $PFBD$ has both circumcircle and incircle. Prove that $ABC$ is equilateral and $P$ coincides with the center of the triangle.

2018 India TST Practice Test1 P1
Let $\Delta ABC$ be an acute triangle. $D,E,F$ are the touch points of incircle with $BC,CA,AB$ respectively. $AD,BE,CF$ intersect incircle at $K,L,M$ respectively. If,$\sigma = \frac{AK}{KD} + \frac{BL}{LE} + \frac{CM}{MF}$, $\tau = \frac{AK}{KD}.\frac{BL}{LE}.\frac{CM}{MF}$. Then prove that $\tau = \frac{R}{16r}$. Also prove that there exists integers $u,v,w$ such that, $uvw \neq 0$, $u\sigma + v\tau +w=0$.

2018 India TST Practice Test2 P1
Let $ABCD$ be a convex quadrilateral inscribed in a circle with center $O$ which does not lie on either diagonal. If the circumcentre of triangle $AOC$ lies on the line $BD$, prove that the circumcentre of triangle $BOD$ lies on the line $AC$.

In an acute angled triangle $ABC$ with $AB < AC$, let $I$ denote the incenter and $M$ the midpoint of side $BC$. The line through $A$ perpendicular to $AI$ intersects the tangent from $M$ to the incircle (different from line $BC$) at a point $P$> Show that $AI$ is tangent to the circumcircle of triangle $MIP$.

by Tejaswi Navilarekallu

Let $ABC$ be an acute-angled scalene triangle with circumcircle $\Gamma$ and circumcenter $O$. Suppose $AB < AC$. Let $H$ be the orthocenter and $I$ be the incenter of triangle $ABC$. Let $F$ be the midpoint of the arc $BC$ of the circumcircle of triangle $BHC$, containing $H$. Let $X$ be a point on the arc $AB$ of $\Gamma$ not containing $C$, such that $\angle AXH = \angle AFH$. Let $K$ be the circumcenter of triangle $XIA$. Prove that the lines $AO$ and $KI$ meet on $\Gamma$.

by Anant Mudgal
Let $ABC$ be a triangle with $\angle A=\angle C=30^{\circ}.$ Points $D,E,F$ are chosen on the sides $AB,BC,CA$ respectively so that $\angle BFD=\angle BFE=60^{\circ}.$ Let $p$ and $p_1$ be the perimeters of the triangles $ABC$ and $DEF$, respectively. Prove that $p\le 2p_1.$

Let the points $O$ and $H$ be the circumcenter and orthocenter of an acute angled triangle $ABC.$ Let $D$ be the midpoint of $BC.$ Let $E$ be the point on the angle bisector of $\angle BAC$ such that $AE\perp HE.$ Let $F$ be the point such that $AEHF$ is a rectangle. Prove that $D,E,F$ are collinear.