geometry problems from Estonian IMO Team Selection Tests (TST) and IMO Training Tests
with aops links in the names
1994 Estonia TST p1
2001 Estonia TST p2
Point $X$ is taken inside a regular $n$-gon of side length $a$. Let $h_1,h_2,...,h_n$ be the distances from $X$ to the lines defined by the sides of the $n$-gon. Prove that $\frac{1}{h_1}+\frac{1}{h_2}+...+\frac{1}{h_n}>\frac{2\pi}{a}$
2001 Estonia TST p6
Let $C_1$ and $C_2$ be the incircle and the circumcircle of the triangle $ABC$, respectively. Prove that, for any point $A'$ on $C_2$, there exist points $B'$ and $C'$ such that $C_1$ and $C_2$ are the incircle and the circumcircle of triangle $A'B'C'$, respectively.
2002 Estonia TST p2
Consider an isosceles triangle $KL_1L_2$ with $|KL_1|=|KL_2|$ and let $KA, L_1B_1,L_2B_2$ be its angle bisectors. Prove that $\cos \angle B_1AB_2 < \frac35$
2002 Estonia TST p4
Let $ABCD$ be a cyclic quadrilateral such that $\angle ACB = 2\angle CAD$ and $\angle ACD = 2\angle BAC$. Prove that $|CA| = |CB| + |CD|$.
2003 Estonia TST p6
Let $ABC$ be an acute-angled triangle, $O$ its circumcenter and $H$ its orthocenter. The orthogonal projection of the vertex $A$ to the line $BC$ lies on the perpendicular bisector of the segment $AC$. Compute $\frac{CH}{BO}$ .
Let $O$ be the circumcentre of the acute triangle $ABC$ and let lines $AO$ and $BC$ intersect at point $K$. On sides $AB$ and $AC$, points $L$ and $M$ are chosen such that $|KL|= |KB|$ and $|KM| = |KC|$. Prove that segments $LM$ and $BC$ are parallel.
2004 Estonia TST p6
Call a convex polyhedron a footballoid if it has the following properties.
(1) Any face is either a regular pentagon or a regular hexagon.
(2) All neighbours of a pentagonal face are hexagonal (a neighbour of a face is a face that has a common edge with it).
2005 Estonia TST p1
On a plane, a line $\ell$ and two circles $c_1$ and $c_2$ of different radii are given such that $\ell$ touches both circles at point $P$. Point $M \ne P$ on $\ell$ is chosen so that the angle $Q_1MQ_2$ is as large as possible where $Q_1$ and $Q_2$ are the tangency points of the tangent lines drawn from $M$ to $c_i$ and $c_2$, respectively, differing from $\ell$ . Find $\angle PMQ_1 + \angle PMQ_2$·
2006 Estonia TST p2
The center of the circumcircle of the acute triangle $ABC$ is $O$. The line $AO$ intersects $BC$ at $D$. On the sides $AB$ and $AC$ of the triangle, choose points $E$ and $F$, respectively, so that the points $A, E, D, F$ lie on the same circle. Let $E'$ and $F'$ projections of points $E$ and $F$ on side $BC$ respectively. Prove that length of the segment $E'F'$ does not depend on the position of points $E$ and $F$.
2006 Estonia TST p4
The side $AC$ of an acute triangle $ABC$ is the diameter of the circle $c_1$ and side $BC$ is the diameter of the circle $c_2$. Let $E$ be the foot of the altitude drawn from the vertex $B$ of the triangle and $F$ the foot of the altitude drawn from the vertex $A$. In addition, let $L$ and $N$ be the points of intersection of the line $BE$ with the circle $c_1$ (the point $L$ lies on the segment $BE$) and the points of intersection of $K$ and $M$ of line $AF$ with circle $c_2$ (point $K$ is in section $AF$). Prove that $K LM N$ is a cyclic quadrilateral.
2007 Estonia TST p2
Let $D$ be the foot of the altitude of triangle $ABC$ drawn from vertex $A$. Let $E$ and $F$ be points symmetric to $D$ w.r.t. lines $AB$ and $AC$, respectively. Let $R_1$ and $R_2$ be the circumradii of triangles $BDE$ and $CDF$, respectively, and let $r_1$ and $r_2$ be the inradii of the same triangles. Prove that $|S_{ABD} - S_{ACD}| > |R_1r_1 - R_2r_2|$
2007 Estonia TST p4
In square $ABCD,$ points $E$ and $F$ are chosen in the interior of sides $BC$ and $CD$, respectively. The line drawn from $F$ perpendicular to $AE$ passes through the intersection point $G$ of $AE$ and diagonal $BD$. A point $K$ is chosen on $FG$ such that $|AK|= |EF|$. Find $\angle EKF.$
2008 Estonia TST p2
Let $ABCD$ be a cyclic quadrangle whose midpoints of diagonals $AC$ and $BD$ are $F$ and $G$, respectively.
a) Prove the following implication: if the bisectors of angles at $B$ and $D$ of the quadrangle intersect at diagonal $AC$ then $\frac14 \cdot |AC| \cdot |BD| = | AG| \cdot |BF| \cdot |CG| \cdot |DF|$.
b) Does the converse implication also always hold?
Points $A$ and $B$ are fixed on a circle $c_1$. Circle $c_2$, whose centre lies on $c_1$, touches line $AB$ at $B$. Another line through $A$ intersects $c_2$ at points $D$ and $E$, where $D$ lies between $A$ and $E$. Line $BD$ intersects $c_1$ again at $F$. Prove that line $EB$ is tangent to $c_1$ if and only if $D$ is the midpoint of the segment $BF$.
2009 Estonia TST p3
Find all natural numbers $n$ for which there exists a convex polyhedron satisfying the following conditions:
(i) Each face is a regular polygon.
(ii) Among the faces, there are polygons with at most two different numbers of edges.
(iii) There are two faces with common edge that are both $n$-gons.
2009 Estonia TST p4
Points $A', B', C'$ are chosen on the sides $BC, CA, AB$ of triangle $ABC$, respectively, so that $\frac{|BA'|}{|A'C|}=\frac{|CB'|}{|B'A|}=\frac{|AC'|}{|C'B|}$. The line which is parallel to line $B'C'$ and goes through point $A$ intersects the lines $AC$ and $AB$ at $P$ and $Q$, respectively. Prove that $\frac{|PQ|}{|B'C'|} \ge 2$
2010 Estonia TST p3
Let the angles of a triangle be $\alpha, \beta$, and $\gamma$, the perimeter $2p$ and the radius of the circumcircle $R$. Prove the inequality $\cot^2 \alpha + \cot^2 \beta + \cot^2 \gamma \ge 3 \left(\frac{9R^2}{p^2}-1\right)$. When is the equality achieved?
2010 Estonia TST p4
In an acute triangle $ABC$ the angle $C$ is greater than the angle $A$. Let $AE$ be a diameter of the circumcircle of the triangle. Let the intersection point of the ray $AC$ and the tangent of the circumcircle through the vertex $B$ be $K$. The perpendicular to $AE$ through $K$ intersects the circumcircle of the triangle $BCK$ for the second time at point $D$. Prove that $CE$ bisects the angle $BCD$.
2011 Estonia TST p1
Two circles lie completely outside each other.Let $A$ be the point of intersection of internal common tangents of the circles and let $K$ be the projection of this point onto one of their external common tangents.The tangents,different from the common tangent,to the circles through point $K$ meet the circles at $M_1$ and $M_2$.Prove that the line $AK$ bisects angle $M_1 KM_2$.
2012 Estonia TST p3
In a cyclic quadrilateral $ABCD$ we have $|AD| > |BC|$ and the vertices $C$ and $D$ lie on the shorter arc $AB$ of the circumcircle. Rays $AD$ and $BC$ intersect at point $K$, diagonals $AC$ and $BD$ intersect at point $P$. Line $KP$ intersects the side $AB$ at point $L$. Prove that $\angle ALK$ is acute.
2012 Estonia TST p4
Let $ABC$ be a triangle where $|AB| = |AC|$. Points $P$ and $Q$ are different from the vertices of the triangle and lie on the sides $AB$ and $AC$, respectively. Prove that the circumcircle of the triangle $APQ$ passes through the circumcenter of $ABC$ if and only if $|AP| = |CQ|$.
2013 Estonia TST p4
Let $D$ be the point different from $B$ on the hypotenuse $AB$ of a right triangle $ABC$ such that $|CB| = |CD|$. Let $O$ be the circumcenter of triangle $ACD$. Rays $OD$ and $CB$ intersect at point $P$, and the line through point $O$ perpendicular to side AB and ray $CD$ intersect at point $Q$. Points $A, C, P, Q$ are concyclic. Does this imply that $ACPQ$ is a square?
2014 Estonia TST p3
Three line segments, all of length $1$, form a connected figure in the plane. Any two different line segments can intersect only at their endpoints. Find the maximum area of the convex hull of the figure.
2014 Estonia TST p4
In an acute triangle the feet of altitudes drawn from vertices $A$ and $B$ are $D$ and $E$, respectively. Let $M$ be the midpoint of side $AB$. Line $CM$ intersects the circumcircle of $CDE$ again in point $P$ and the circumcircle of $CAB$ again in point $Q$. Prove that $|MP| = |MQ|$.
2017 Estonia TST p4
Let $ABC$ be an isosceles triangle with apex $A$ and altitude $AD$. On $AB$, choose a point $F$ distinct from $B$ such that $CF$ is tangent to the incircle of $ABD$. Suppose that $\vartriangle BCF$ is isosceles. Show that those conditions uniquely determine:
a) which vertex of $BCF$ is its apex,
b) the size of $\angle BAC$
2017 Estonia TST p10
Let $ABC$ be a triangle with $AB = \frac{AC}{2 }+ BC$. Consider the two semicircles outside the triangle with diameters $AB$ and $BC$. Let $X$ be the orthogonal projection of $A$ onto the common tangent line of those semicircles. Find $\angle CAX$.
2018 Estonia TST p7
Let $AD$ be the altitude $ABC$ of an acute triangle. On the line $AD$ are chosen different points $E$ and $F$ so that $|DE |= |DF|$ and point $E$ is in the interior of triangle $ABC$. The circumcircle of triangle $BEF$ intersects $BC$ and $BA$ for second time at points $K$ and $M$ respectively. The circumcircle of the triangle $CEF$ intersects the $CB$ and $CA$ for the second time at points $L$ and $N$ respectively. Prove that the lines $AD, KM$ and $LN$ intersect at one point.
with aops links in the names
(only those not in IMO Shortlist)
IMO TST 1993 - 2021
Let $\alpha, \beta$ and $\gamma$ be the angles of a triangle, $R$ be the radius of the circle of this triangle and $S$ its area. Prove that $$\tan \frac{\alpha}{2}+\tan \frac{\beta}{2}+\tan \frac{\gamma}{2} \le \frac{9R^2}{4S}$$
Define the lines $L_A, L_B, L_C$ respective to the vertices $A, B, C$ of the acute triangle $ABC$ as follows:
Let $H$ be the foot of the altitude drawn from vertex $A$ on the side $BC$ and let $S_A$ be the circle with diameter $AH$, that intersects sides $AB$ and $AC$ at $M$ and $N$, respectively ($M \ne A$ and $N \ne A$); then define $L_A$ as the line passing through point $A$ perpendicular to $MN$. Lines $L_B$ and $L_C$ are constructed respectively.
Prove that the lines $L_A, L_B$ and $L_C$ intersect at one point.
Draw from the vertex $B$ of the triangle $ABC$ $n$ rays that intersect the side $AC$ at points $X_1, X_2,..., X_n$ respectively. (points $A, X_1,..., X_n, C$ lie on the line $AC$ in this order). Let $r_0, r_1,... , r_n$ and $R_0, R_1,... , R_n$ be the radii of the inscribed and exscribed circles triangles $ABX_1, X_1BX_2,..., X_nBC$ respectivley (see figure). Prove that the number $\frac{r_0r_1... r_n} {R_0R_1... R_n}$ does not depend on the number $n$ of rays or the selection of the points $X_1, X_2,..., X_n$ .
The two circles touch externally. One of them has an inscribed equilateral triangle whose vertex is not at the point of contact with the circles. From each vertex of this triangle a tangent is drawn to the second circle. Prove that the length of one resulting tangent is equal to the sum of the lengths of the two tangents.
From the point of intersection of the altitudes of the acute triangle $ABC$ is drawn by a line $\ell$ which does not pass the vertices of this triangle. Prove that three lines that are summetric to the line $\ell$ wrt the sidelines of the triangle $ABC$ intersect at one point and that point lies on the circumcircle of triangle $ABC$.
1996 Estonia TST p2
Let $a,b,c$ be the sides of a triangle, $\alpha ,\beta ,\gamma$ the corresponding angles and $r$ the inradius. Prove that$$a\cdot sin\alpha+b\cdot sin\beta+c\cdot sin\gamma\geq 9r$$
1996 Estonia TST p5
Let $H$ be the orthocenter of an obtuse triangle $ABC$ and $A_1B_1C_1$ arbitrary points on the sides $BC,AC,AB$ respectively.Prove that the tangents drawn from $H$ to the circles with diametrs $AA_1,BB_1,CC_1$ are equal.
1997 Estonia TST p1
In a triangle $ABC$ points $A_1,B_1,C_1$ are the midpoints of $BC,CA,AB$ respectively,and $A_2,B_2,C_2$ are the midpoints of the altitudes from $A,B,C$ respectively. Show that the lines $A_1A_2,B_1B_2,C_1,C_2$ are concurrent.
1997 Estonia TST p5
A quadrilateral $ABCD$ is inscribed in a circle. On each of the sides $AB,BC,CD,DA$ one erects a rectangle towards the interior of the quadrilateral, the other side of the rectangle being equal to $CD,DA,AB,BC,$ respectively. Prove that the centers of these four rectangles are vertices of a rectangle.
1998 Estonia TST p1
Let $a,b,c$ be the sides of a triangle, $\alpha ,\beta ,\gamma$ the corresponding angles and $r$ the inradius. Prove that$$a\cdot sin\alpha+b\cdot sin\beta+c\cdot sin\gamma\geq 9r$$
1996 Estonia TST p5
Let $H$ be the orthocenter of an obtuse triangle $ABC$ and $A_1B_1C_1$ arbitrary points on the sides $BC,AC,AB$ respectively.Prove that the tangents drawn from $H$ to the circles with diametrs $AA_1,BB_1,CC_1$ are equal.
In a triangle $ABC$ points $A_1,B_1,C_1$ are the midpoints of $BC,CA,AB$ respectively,and $A_2,B_2,C_2$ are the midpoints of the altitudes from $A,B,C$ respectively. Show that the lines $A_1A_2,B_1B_2,C_1,C_2$ are concurrent.
A quadrilateral $ABCD$ is inscribed in a circle. On each of the sides $AB,BC,CD,DA$ one erects a rectangle towards the interior of the quadrilateral, the other side of the rectangle being equal to $CD,DA,AB,BC,$ respectively. Prove that the centers of these four rectangles are vertices of a rectangle.
On the side $AB$ of the triangle $ABC$, the points $C_1$ and $C_2$ are chosen so that the point $C_2$ is located between points $C_1$ and $B$. Similarly, points $A_1, A_2$ on side BC and points $B_1, B_2$ on side $CA$ are chosen such that point $A_2$ is located between points $A_1$ and $C$ and point $B_2$ is located between points $B_1$ and $A$. Intersect the rays $C_2A_1$ and $B_1A_2$ at point $D$, the rays $A_2B_1$ and $C_1B_2$ at point $E$ and rays $B_2C_1$ and $A_1C_2$ at point $F$. Prove that equations $$ \frac{| A_1A_2 |}{| BC |}= \frac{|B_1B_2|}{| CA |}= \frac{|C_1C_2|}{|AB |}$$ are valid if and only if $$ |\frac{|B_2C_1|}{|EF|}=\frac{|C_2A_1|}{|F D|}=\frac{|A_2B_1|}{|DE|}$$
The incircle of triangle $ABC$ with center $O$ is tangent to side $BC$ of the triangle $ABC$ at point $K$. Let $N$ and $M$ be the midpoints of the segments $BC$ and $AK$, respectively. Prove that the points $M, O$ and $N$ lie on the same line.
Let $M, N$ and $K$ be the points of tangency of the incircle of the triangle $ABC$ with the sides. Let $Q$ be the center of a circle passing through the midpoints of the segments $MN, NK$ and $KM$. Prove that the centers of the circumscribed and inscribed circle of the triangle $ABC$ and the point $Q$ are on the same line.
The triangle $ABC$ has $| AC | \ne| BC |$. Let's choose inside the triangle a point $X$ and denote by $\angle A = \alpha, \angle B = \alpha, \angle ACX = \phi $ and $\angle BCX = \psi$. Prove that the equation
$$\frac{\sin \alpha \sin \beta}{\sin (\alpha - \beta)}=\frac{\sin \phi \sin \psi.}{\sin (\phi - \psi)}$$
is valid if and only if the point $X$ lies on the on the median of the triangle $ABC$ drawn from vertex $C$ .
2001 Estonia TST p2
Point $X$ is taken inside a regular $n$-gon of side length $a$. Let $h_1,h_2,...,h_n$ be the distances from $X$ to the lines defined by the sides of the $n$-gon. Prove that $\frac{1}{h_1}+\frac{1}{h_2}+...+\frac{1}{h_n}>\frac{2\pi}{a}$
2001 Estonia TST p6
Let $C_1$ and $C_2$ be the incircle and the circumcircle of the triangle $ABC$, respectively. Prove that, for any point $A'$ on $C_2$, there exist points $B'$ and $C'$ such that $C_1$ and $C_2$ are the incircle and the circumcircle of triangle $A'B'C'$, respectively.
Consider an isosceles triangle $KL_1L_2$ with $|KL_1|=|KL_2|$ and let $KA, L_1B_1,L_2B_2$ be its angle bisectors. Prove that $\cos \angle B_1AB_2 < \frac35$
Let $ABCD$ be a cyclic quadrilateral such that $\angle ACB = 2\angle CAD$ and $\angle ACD = 2\angle BAC$. Prove that $|CA| = |CB| + |CD|$.
Let $ABC$ be an acute-angled triangle, $O$ its circumcenter and $H$ its orthocenter. The orthogonal projection of the vertex $A$ to the line $BC$ lies on the perpendicular bisector of the segment $AC$. Compute $\frac{CH}{BO}$ .
(J. Willemson)
2004 Estonia TST p2Let $O$ be the circumcentre of the acute triangle $ABC$ and let lines $AO$ and $BC$ intersect at point $K$. On sides $AB$ and $AC$, points $L$ and $M$ are chosen such that $|KL|= |KB|$ and $|KM| = |KC|$. Prove that segments $LM$ and $BC$ are parallel.
Call a convex polyhedron a footballoid if it has the following properties.
(1) Any face is either a regular pentagon or a regular hexagon.
(2) All neighbours of a pentagonal face are hexagonal (a neighbour of a face is a face that has a common edge with it).
2005 Estonia TST p1
On a plane, a line $\ell$ and two circles $c_1$ and $c_2$ of different radii are given such that $\ell$ touches both circles at point $P$. Point $M \ne P$ on $\ell$ is chosen so that the angle $Q_1MQ_2$ is as large as possible where $Q_1$ and $Q_2$ are the tangency points of the tangent lines drawn from $M$ to $c_i$ and $c_2$, respectively, differing from $\ell$ . Find $\angle PMQ_1 + \angle PMQ_2$·
2006 Estonia TST p2
The center of the circumcircle of the acute triangle $ABC$ is $O$. The line $AO$ intersects $BC$ at $D$. On the sides $AB$ and $AC$ of the triangle, choose points $E$ and $F$, respectively, so that the points $A, E, D, F$ lie on the same circle. Let $E'$ and $F'$ projections of points $E$ and $F$ on side $BC$ respectively. Prove that length of the segment $E'F'$ does not depend on the position of points $E$ and $F$.
2006 Estonia TST p4
The side $AC$ of an acute triangle $ABC$ is the diameter of the circle $c_1$ and side $BC$ is the diameter of the circle $c_2$. Let $E$ be the foot of the altitude drawn from the vertex $B$ of the triangle and $F$ the foot of the altitude drawn from the vertex $A$. In addition, let $L$ and $N$ be the points of intersection of the line $BE$ with the circle $c_1$ (the point $L$ lies on the segment $BE$) and the points of intersection of $K$ and $M$ of line $AF$ with circle $c_2$ (point $K$ is in section $AF$). Prove that $K LM N$ is a cyclic quadrilateral.
2007 Estonia TST p2
Let $D$ be the foot of the altitude of triangle $ABC$ drawn from vertex $A$. Let $E$ and $F$ be points symmetric to $D$ w.r.t. lines $AB$ and $AC$, respectively. Let $R_1$ and $R_2$ be the circumradii of triangles $BDE$ and $CDF$, respectively, and let $r_1$ and $r_2$ be the inradii of the same triangles. Prove that $|S_{ABD} - S_{ACD}| > |R_1r_1 - R_2r_2|$
In square $ABCD,$ points $E$ and $F$ are chosen in the interior of sides $BC$ and $CD$, respectively. The line drawn from $F$ perpendicular to $AE$ passes through the intersection point $G$ of $AE$ and diagonal $BD$. A point $K$ is chosen on $FG$ such that $|AK|= |EF|$. Find $\angle EKF.$
2008 Estonia TST p2
Let $ABCD$ be a cyclic quadrangle whose midpoints of diagonals $AC$ and $BD$ are $F$ and $G$, respectively.
a) Prove the following implication: if the bisectors of angles at $B$ and $D$ of the quadrangle intersect at diagonal $AC$ then $\frac14 \cdot |AC| \cdot |BD| = | AG| \cdot |BF| \cdot |CG| \cdot |DF|$.
b) Does the converse implication also always hold?
Find all natural numbers $n$ for which there exists a convex polyhedron satisfying the following conditions:
(i) Each face is a regular polygon.
(ii) Among the faces, there are polygons with at most two different numbers of edges.
(iii) There are two faces with common edge that are both $n$-gons.
2009 Estonia TST p4
Points $A', B', C'$ are chosen on the sides $BC, CA, AB$ of triangle $ABC$, respectively, so that $\frac{|BA'|}{|A'C|}=\frac{|CB'|}{|B'A|}=\frac{|AC'|}{|C'B|}$. The line which is parallel to line $B'C'$ and goes through point $A$ intersects the lines $AC$ and $AB$ at $P$ and $Q$, respectively. Prove that $\frac{|PQ|}{|B'C'|} \ge 2$
Let the angles of a triangle be $\alpha, \beta$, and $\gamma$, the perimeter $2p$ and the radius of the circumcircle $R$. Prove the inequality $\cot^2 \alpha + \cot^2 \beta + \cot^2 \gamma \ge 3 \left(\frac{9R^2}{p^2}-1\right)$. When is the equality achieved?
2010 Estonia TST p4
In an acute triangle $ABC$ the angle $C$ is greater than the angle $A$. Let $AE$ be a diameter of the circumcircle of the triangle. Let the intersection point of the ray $AC$ and the tangent of the circumcircle through the vertex $B$ be $K$. The perpendicular to $AE$ through $K$ intersects the circumcircle of the triangle $BCK$ for the second time at point $D$. Prove that $CE$ bisects the angle $BCD$.
Two circles lie completely outside each other.Let $A$ be the point of intersection of internal common tangents of the circles and let $K$ be the projection of this point onto one of their external common tangents.The tangents,different from the common tangent,to the circles through point $K$ meet the circles at $M_1$ and $M_2$.Prove that the line $AK$ bisects angle $M_1 KM_2$.
In a cyclic quadrilateral $ABCD$ we have $|AD| > |BC|$ and the vertices $C$ and $D$ lie on the shorter arc $AB$ of the circumcircle. Rays $AD$ and $BC$ intersect at point $K$, diagonals $AC$ and $BD$ intersect at point $P$. Line $KP$ intersects the side $AB$ at point $L$. Prove that $\angle ALK$ is acute.
Let $ABC$ be a triangle where $|AB| = |AC|$. Points $P$ and $Q$ are different from the vertices of the triangle and lie on the sides $AB$ and $AC$, respectively. Prove that the circumcircle of the triangle $APQ$ passes through the circumcenter of $ABC$ if and only if $|AP| = |CQ|$.
Let $D$ be the point different from $B$ on the hypotenuse $AB$ of a right triangle $ABC$ such that $|CB| = |CD|$. Let $O$ be the circumcenter of triangle $ACD$. Rays $OD$ and $CB$ intersect at point $P$, and the line through point $O$ perpendicular to side AB and ray $CD$ intersect at point $Q$. Points $A, C, P, Q$ are concyclic. Does this imply that $ACPQ$ is a square?
Three line segments, all of length $1$, form a connected figure in the plane. Any two different line segments can intersect only at their endpoints. Find the maximum area of the convex hull of the figure.
2014 Estonia TST p4
In an acute triangle the feet of altitudes drawn from vertices $A$ and $B$ are $D$ and $E$, respectively. Let $M$ be the midpoint of side $AB$. Line $CM$ intersects the circumcircle of $CDE$ again in point $P$ and the circumcircle of $CAB$ again in point $Q$. Prove that $|MP| = |MQ|$.
2015 Estonia TST p4 (2015 Ukraine MO X p7)
Altitudes $AD$ and $BE$ of an acute triangle $ABC$ intersect at $H$. Let $C_1 (H,HE)$ and $C_2(B,BE)$ be two circles tangent at $AC$ at point $E$. Let $P\ne E$ be the second point of tangency of the circle $C_1 (H,HE)$ with its tangent line going through point $C$, and $Q\ne E$ be the second point of tangency of the circle $C_2(B,BE)$ with its tangent line going through point $C$. Prove that points $D, P$, and $Q$ are collinear.
2015 Estonia TST p9
The orthocenter of an acute triangle $ABC$ is $H$. Let $K$ and $P$ be the midpoints of lines $BC$ and $AH$, respectively. The angle bisector drawn from the vertex $A$ of the triangle $ABC$ intersects with line $KP$ at $D$. Prove that $HD\perp AD$.
2015 Estonia TST p11
Let $M$ be the midpoint of the side $AB$ of a triangle $ABC$. A circle through point $C$ that has a point of tangency to the line $AB$ at point $A$ and a circle through point $C$ that has a point of tangency to the line $AB$ at point $B$ intersect the second time at point $N$. Prove that $|CM|^2 + |CN|^2 - |MN|^2 = |CA|^2 + |CB|^2 - |AB|^2$.
2016 Estonia TST p5 (Slovenia IMO TST 2015)
Let $O$ be the circumcentre of the acute triangle $ABC$. Let $c_1$ and $c_2$ be the circumcircles of triangles $ABO$ and $ACO$. Let $P$ and $Q$ be points on $c_1$ and $c_2$ respectively, such that OP is a diameter of $c_1$ and $OQ$ is a diameter of $c_2$. Let $T$ be the intesection of the tangent to $c_1$ at $P$ and the tangent to $c_2$ at $Q$. Let $D$ be the second intersection of the line $AC$ and the circle $c_1$. Prove that the points $D, O$ and $T$ are collinear
2016 Estonia TST p7
On the sides $AB, BC$ and $CA$ of triangle $ABC$, points $L, M$ and $N$ are chosen, respectively, such that the lines $CL, AM$ and $BN$ intersect at a common point O inside the triangle and the quadrilaterals $ALON, BMOL$ and $CNOM$ have incircles. Prove that
$$\frac{1}{AL\cdot BM} +\frac{1}{BM\cdot CN} +\frac{1}{CN \cdot AL} =\frac{1}{AN\cdot BL} +\frac{1}{BL\cdot CM} +\frac{1}{CM\cdot AN} $$
2016 Estonia TST p12 (Croatia MO 2015)
The circles $k_1$ and $k_2$ intersect at points $M$ and $N$. The line $\ell$ intersects with the circle $k_1$ at points $A$ and $C$ and with circle $k_2$ at points $B$ and $D$, so that points $A, B, C$ and $D$ are on the line $\ell$ in that order. Let $X$ be a point on line $MN$ such that the point $M$ is between points $X$ and $N$. Lines $AX$ and $BM$ intersect at point $P$ and lines $DX$ and $CM$ intersect at point $Q$. Prove that $PQ \parallel \ell $.
Altitudes $AD$ and $BE$ of an acute triangle $ABC$ intersect at $H$. Let $C_1 (H,HE)$ and $C_2(B,BE)$ be two circles tangent at $AC$ at point $E$. Let $P\ne E$ be the second point of tangency of the circle $C_1 (H,HE)$ with its tangent line going through point $C$, and $Q\ne E$ be the second point of tangency of the circle $C_2(B,BE)$ with its tangent line going through point $C$. Prove that points $D, P$, and $Q$ are collinear.
2015 Estonia TST p9
The orthocenter of an acute triangle $ABC$ is $H$. Let $K$ and $P$ be the midpoints of lines $BC$ and $AH$, respectively. The angle bisector drawn from the vertex $A$ of the triangle $ABC$ intersects with line $KP$ at $D$. Prove that $HD\perp AD$.
2015 Estonia TST p11
Let $M$ be the midpoint of the side $AB$ of a triangle $ABC$. A circle through point $C$ that has a point of tangency to the line $AB$ at point $A$ and a circle through point $C$ that has a point of tangency to the line $AB$ at point $B$ intersect the second time at point $N$. Prove that $|CM|^2 + |CN|^2 - |MN|^2 = |CA|^2 + |CB|^2 - |AB|^2$.
2016 Estonia TST p5 (Slovenia IMO TST 2015)
Let $O$ be the circumcentre of the acute triangle $ABC$. Let $c_1$ and $c_2$ be the circumcircles of triangles $ABO$ and $ACO$. Let $P$ and $Q$ be points on $c_1$ and $c_2$ respectively, such that OP is a diameter of $c_1$ and $OQ$ is a diameter of $c_2$. Let $T$ be the intesection of the tangent to $c_1$ at $P$ and the tangent to $c_2$ at $Q$. Let $D$ be the second intersection of the line $AC$ and the circle $c_1$. Prove that the points $D, O$ and $T$ are collinear
2016 Estonia TST p7
On the sides $AB, BC$ and $CA$ of triangle $ABC$, points $L, M$ and $N$ are chosen, respectively, such that the lines $CL, AM$ and $BN$ intersect at a common point O inside the triangle and the quadrilaterals $ALON, BMOL$ and $CNOM$ have incircles. Prove that
$$\frac{1}{AL\cdot BM} +\frac{1}{BM\cdot CN} +\frac{1}{CN \cdot AL} =\frac{1}{AN\cdot BL} +\frac{1}{BL\cdot CM} +\frac{1}{CM\cdot AN} $$
2016 Estonia TST p12 (Croatia MO 2015)
The circles $k_1$ and $k_2$ intersect at points $M$ and $N$. The line $\ell$ intersects with the circle $k_1$ at points $A$ and $C$ and with circle $k_2$ at points $B$ and $D$, so that points $A, B, C$ and $D$ are on the line $\ell$ in that order. Let $X$ be a point on line $MN$ such that the point $M$ is between points $X$ and $N$. Lines $AX$ and $BM$ intersect at point $P$ and lines $DX$ and $CM$ intersect at point $Q$. Prove that $PQ \parallel \ell $.
2017 Estonia TST p4
Let $ABC$ be an isosceles triangle with apex $A$ and altitude $AD$. On $AB$, choose a point $F$ distinct from $B$ such that $CF$ is tangent to the incircle of $ABD$. Suppose that $\vartriangle BCF$ is isosceles. Show that those conditions uniquely determine:
a) which vertex of $BCF$ is its apex,
b) the size of $\angle BAC$
2017 Estonia TST p10
Let $ABC$ be a triangle with $AB = \frac{AC}{2 }+ BC$. Consider the two semicircles outside the triangle with diameters $AB$ and $BC$. Let $X$ be the orthogonal projection of $A$ onto the common tangent line of those semicircles. Find $\angle CAX$.
Let $AD$ be the altitude $ABC$ of an acute triangle. On the line $AD$ are chosen different points $E$ and $F$ so that $|DE |= |DF|$ and point $E$ is in the interior of triangle $ABC$. The circumcircle of triangle $BEF$ intersects $BC$ and $BA$ for second time at points $K$ and $M$ respectively. The circumcircle of the triangle $CEF$ intersects the $CB$ and $CA$ for the second time at points $L$ and $N$ respectively. Prove that the lines $AD, KM$ and $LN$ intersect at one point.
2019 Estonia TST p2
In an acute-angled triangle $ABC$, the altitudes intersect at point $H$, and point $K$ is the base of the altitude dropped from the vertex $A$. Circle $c$ passing through points A and K intersects sides $AB$ and $AC$ at points $M$ and $N$, respectively. The line passing through point $A$ and parallel to line BC intersects the circumcircles of triangles $AHM$ and $AHN$ for second time, respectively, at points $X$ and $Y$. Prove that $ | X Y | = | BC |$.
2019 Estonia TST p7
An acute-angled triangle $ABC$ has two altitudes $BE$ and $CF$. The circle with diameter $AC$ intersects the segment $BE$ at point $P$. A circle with diameter $AB$ intersects the segment $CF$ at point $Q$ and the extension of this altitude at point $Q'$. Prove that $\angle PQ'Q = \angle PQB$.
In an acute-angled triangle $ABC$, the altitudes intersect at point $H$, and point $K$ is the base of the altitude dropped from the vertex $A$. Circle $c$ passing through points A and K intersects sides $AB$ and $AC$ at points $M$ and $N$, respectively. The line passing through point $A$ and parallel to line BC intersects the circumcircles of triangles $AHM$ and $AHN$ for second time, respectively, at points $X$ and $Y$. Prove that $ | X Y | = | BC |$.
2019 Estonia TST p7
An acute-angled triangle $ABC$ has two altitudes $BE$ and $CF$. The circle with diameter $AC$ intersects the segment $BE$ at point $P$. A circle with diameter $AB$ intersects the segment $CF$ at point $Q$ and the extension of this altitude at point $Q'$. Prove that $\angle PQ'Q = \angle PQB$.
The radius of the circumcircle of triangle $\Delta$ is $R$ and the radius of the inscribed circle is $r$.
Prove that a circle of radius $R + r$ has an area more than $5$ times the area of triangle $\Delta$.
Let $M$ be the midpoint of side BC of an acute-angled triangle $ABC$. Let $D$ and $E$ be the center of the excircle of triangle $AMB$ tangent to side $AB$ and the center of the excircle of triangle $AMC$ tangent to side $AC$, respectively. The circumscribed circle of triangle $ABD$ intersects line$ BC$ for the second time at point $F$, and the circumcircle of triangle $ACE$ is at point $G$. Prove that $| BF | = | CG|$.
2021 were all from shortlist
IMO Training Tests
between 1998-2014
We construct outside a triangle $ABC$ on the sides the equilateral triangles $BDC, CEA$ and $AFB$ based on sides $BC$, $CA$ and $AB$ respectively. Let's draw a perpendicular line from vertex $A$ on the line $EF$, a perpendicular line from vertex $B$ on the line $FD$ and a perpendicular line from the vertex $C$ on the line $DE$. Prove that these three perpendicular lines intersect at one point.
We call an operation for a a convex n-gon cutting, where we mark the midpoints of two consecutive sides $AB$ and $BC$ (let $M$ and $N$ respectively) and the sides $AB$ and $BC$ are replaced by the three sides $AM, MN$ and $NC$ (in other words, replace the triangle $MBN$ of the original $n$-gon by cutting to a convex $(n + 1)$-gon ). By cutting a regular hexagon $P_6$ with an area of $1$, we form a convex heptagon $P_7$, and by cutting it (in one of seven possible ways) we get a convex octagon $P_8$, and so on. Prove that by performing the circumscribed cuttings in any way, each time the area of the resulting polygon $P_n$ ($n\ge 6$) is greater than $1/3$.
On the sides $AC$ and $AB$ of the triangle $ABC$, lie the points $M$ and $N$ respectively, so that $BM$ and $CN$ are the bisectors of the triangle. The ray $MN$ intersects the circumcircle of triangle $ABC$ at $D$. Prove the equation $\frac{1}{|BD|}= \frac{1}{|AD|}+\frac{1}{|CD|}$.
In an acute-angled non-equilateral triangle $ABC$ the altitudes $AA_1 , CC_1$ respectively and intersect the angle bisectors respective to the sides $AB , BC$ at points $P,Q$. Let $H$ be the point of intersection of the altitudes $ABC$ of the triangle and $M$ the midpoint of side $AC$. Prove that the lines drawn from points $P,Q$ perpendicular on sides $AB, BC$ respectively and the bisector of the angle $ABC$ intersect at one point, that lies on segment $HM$.
original wording
Olgu teravnurkse mitte-võrdhaarse kolmnurga ABC tippudest A ja C tõmmatud kõrgused vastavalt AA1 ja CC1 ning lõigaku nende lõikumisel tekkiva teravnurga poolitajasirge kolmnurga külgi AB ja BC vastavalt punktides P ja Q. Olgu H kolmnurga ABC kõrguste lõikepunkt ning M külje AC keskpunkt. Tõesta, et punktidest P ja Q vastavalt külgedele AB ja BC tõmmatud ristsirged ja nurga ABC poolitaja lõikuvad ühes punktis, mis asub lõigul HM .
On the side BC of triangle $ABC$, the points $M$ and $N$ are taken such that the point $M$ lies between the points $B$ and $N$, and $| BM | = | CN |$. On segments $AN$ and $AM$, points $P$ and $Q$ are taken so that $\angle PMC = \angle MAB$ and $\angle QNB = \angle NAC$. Prove that $\angle QBC = \angle PCB$.
On the side $AC$ of triangle $ABC$ select point $B_1$ and on side $AB$ select point $C_1$. Segments $BB_1$ and $CC_1$ intersect at $D$. Prove that quadrilateral $AB_1DC_1$ is tagnential if and only if the incircles of the triangles $ABD$ and $ACD$ are tangnent.
In a convex quadrilateral $ABCD$ is $| AB | = | AD | + | BC |$. Inside the quadrilateral, there is a point $P$ whose distance from the line $CD$ is $p$ , $| AP | = | AD | + p$ and $| BP | = | BC | + p$. Prove that $$\frac{1}{\sqrt {p}}\ge \frac{1}{\sqrt {AD}}+\frac{1}{\sqrt {BC}}$$
On the sides $BC, CA$ and $AB$ of the triangle $ABC$ lie the points $D,E$ and $F$ respectively. The rays $AD, BE$ and $CF$ intersect the circumcircle of triangle $ABC$ for second time at points $P, Q$, and $R$ respectively. Prove that for any choice ot the points $D, E$ and $F$ for any choice, the inequality holds $$\frac{AD}{PD}+\frac{BE}{QE}+\frac{CF}{RF}\ge 9$$. In which case does the equality apply here?
On the sides $AB , CD$ of the cyclic $ABCD$, the points $E, F$ are selected respectively such that $| AE | : | EB | = | CF | : | FD |$. Let $P$ be such a point of the segment $EF$, such that $| PE | : | PF | = | AB | : | CD |$. Prove that the ratio of areas of the triangles $APD$ and $BPC$ is the same independent of any choice of the points $E$ and $F$ which satisfie the above condition.
In the interior of triangle $ABC$, choose a point $P$ such that $\angle BPC = 90^o$ and $\angle BAP = \angle BCP$. Let $M$ and $N$ be the midpoints of the sides $AC$ and $BC$ of the triangle, respectively. Prove that the triangle $M N P$ is right-angled.
Let $A',B',C'$ be the feet of altitudes drawn from the vertices $A, B$ and $C$ of the acute triangle $ABC$ respectively. Let $P$ be the projection of $C'$ on the line $A'B'$ and let $Q$ be a point on the line $A'B'$ such that $| AQ | = | BQ |$. Prove that $\angle PAQ = \angle PBQ = \angle PC'C$.
Let $ABC$ be an isosceles acute-angled triangle and let$ A_1$ and $C_1$ be the feet of the altitudes drawn from its vertices $A$ and $C$, respectively. Let $H$ be the point of intersection of the altitudes $ABC$ of the triangle, $O$ the center of its circumcircle, and $B_0$ the midpoint of the side $AC$. Lines $BO$ and $AC$ intersect at point $P$ and lines $B H$ and $A_1C_1$ at point $Q$. Prove that the lines $B_0H$ and $PQ$ are parallel.
Let $ABCD$ be a parallelogram. The bisector of the angle $\angle BAD$ intersects the lines $BC$ and $CD$ at points $K$ and $L$, respectively. Prove that the center of the circumcircle of the triangle $CKL$ is lies on the circumcircle of triangle $BCD$.
Let $ABC$ be an isosceles triangle, with $| AC | = | BC |$. Let $P$ be a point inside the triangle $ABC$ such that $\angle PAB = \angle PBC$. Let $M$ be the midpoint of side $AB$. Prove that $\angle APC + \angle BPM = 180^o$.
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