geometry problems from Lithuanian Team Selection Tests (TST) with aops links in the names
(only those not in IMO BMO Shortlist)
collected inside aops here
MEMO + IMO TST 2005-2019, 2022
2007 missing
Let $ABCD$ be a convex quadrilateral, and write $\alpha=\angle DAB$; $\beta=\angle ADB$; $\gamma=\angle ACB$; $\delta= \angle DBC$; and $\epsilon=\angle DBA$. Assuming that $\alpha<\pi/2$, $\beta+\gamma=\pi /2$, and $\delta+2\epsilon=\pi$, prove that
$(DB+BC)^2=AD^2+AC^2$
Inside a convex quadrilateral $ABCD$ there is a point $P$ such that the triangles $PAB, PBC, PCD, PDA$ have equal areas. Prove that the area of $ABCD$ is bisected by one of the diagonals.
Prove that in every polygon there is a diagonal that cuts off a triangle and lies within the polygon.
2007 missing
Let O be the center of circumcircle c of acuted-triangle ABC. Another circle c* with center O* tangents internally circle c at point A and side BC at point D. Also circle c* intersect the lines AB and AC respectively at points E and F, and lines OO* and EO* respectively at points K and A, G and E. Lines BO and KG intersect at H. Prove DF^2=AF x GH.
In triangle ABC, angle A=30, points S and I are ABC circumcentre and incentre respectively. Points D and E are on sides BA and CA respectively such that BD=CE=BC. Prove that SI is perpendicular and equal to DE.
Point $I$ is the center of the circle inscribed in the triangle $ABC$ with $AB\ne AC$. The lines $BI$ and $CI$ intersect the sides $AC$ and $AB$ of that triangle at points $D$ and $E$, respectively. What is the measure of the angle $BAC$ of the triangle at which $DI=EI$?
In non equilateral triangle ABC with incentre I and cerntroid M, the lines AM and BI intersects at P and BM and AI at Q. Also 2AB=AC+BC. Prove that the line PQ bisects IM.
The points $M$ and $N$ are marked on the diagonal $AC$ and the side $BC$ of the rhombus $ABCD$, respectively such that $DM = MN$. The lines AC and $DN$ intersect at the point $P$, and the lines $AB$ and $DM$ intersect at point R. Prove that $RP = PD$.
The angle bisectors of an acute triangle $ABC$ intersect at point $I$ and the atlitudes at point $H$. Let $M$ be the midpoint of the shorter arc $AC$ of the circle circumscribed to the triangle $ABC$ . Find the angle $ABC$ if $MI = MH$ is known.
The points $A_0, B_0, C_0$ bisect the sides $BC, CA, AB$, and points $A_1, B_1, C_1$ bisect the broken lines $BAC, CBA, ACB$ respectively. Prove that the lines $A_0A_1, B_0B_1, C_0C_1$ intersect at one point.
The triangle $ABC$ is acute. The points $B_1$ and $C_1$ are the feet of altitudes of the triangle from vertices $B$ and $C$, respectively. Given points $P$ and $Q$ such that $B_1$ and $C_1$ are the internal points of the segments $BP$ and $CQ$, respectively and the angle $PAQ$ is right. Point $F$ is the foot of the altitude of the triangle $APQ$ drawn from the vertex $A$. Prove that the angle $BFC$ is right.
Inside a rectangle with sides $4$ and $5$, there are six points. Prove that there are two points with distance less than $3$.
The diagonals of the rectangle $ABCD$ intersect at point $O$. Point $E$ belongs to the line $AB$ and $AE = AO$ (point $A$ is between points $E$ and $B$). Point $F$ belongs to the line $DB$ and $BF = BO$ ($B$ is between $F$ and $D$). Triangle $EFC$ is equilateral. Prove that the line $EO$ is perpendicular to the line $DB$.
The base of the perpendicular from the point of intersection of the medians of the acute triangle $ABC$ (where $AB> AC$) on the side $BC$ is the point $D$. Find the ratio $BD: BA$ if $AD$ is a bisector of the angle $BAC$.
Circle touches parallelogram‘s $ABCD$ borders $AB, BC$ and $CD$ respectively at points $K, L$ and $M$. Perpendicular is drawn from vertex $C$ to $AB$ . Prove, that the line $KL$ divides this perpendicular into two equal parts (with the same length).
Circles ω_1 and ω_2 have no common point. Where is outerior tangents a and b, interior tangent c. Lines a, b and c touches circle ω_1 respectively on points A_1, B_1 and C_1, and circle ω_2 – respectively on points A_2, B_2 and C_2. Prove that triangles A_1B_1C_1 and A_2B_2C_2 have area ratio the same as ratio of radii of ω_1 and ω_2 .
Two tangent circles $c_1$ and $c_2$ are given. A common tangent to the circles touches $c_1$ and $c_2$ at (different) points $A$ and $B$, respectively. Segment $AP$ is the diameter of the circle $c_1$, and the line passing through the point P is tangent to $c_2$ at the point $Q$. Prove that $AP = PQ$.
Given a parallelogram $ABCD$ with center $S$. Inscribed circles of triangle $ABD$ has center $O$ and touches $BD$ at point $T$. Prove that the lines $OS$ and $CT$ are parallel.
The altitude $CK$ of an acute-angled triangle $ABC$ intersects with the other two altitudes at point $H$. The point $D$ is marked on the side $AB$, that the segment $CD$ passes through the center $O$ of the circumscribed circle of triangle $ABC$ . Point $E$ divides the segment $CD$ in half. Prove that the line $EK$ divides the segment $OH$ in half.
The altitude $AD$ and the angle bisector $AL$ of the triangle $ABC$ are drawn. Line $AL$ intersects it's circumcircle $\omega$ at the point $M \ne A$, the line $MD$ intersects $\omega$ at the point $N \ne M$, and the line $NL$ intersects $\omega$ at $E \ne N$. Prove that $AE$ is the diameter of $\omega$.
On the sides $AB$ and $AC$ of the triangle $ABC$ lie the points $K$ and $N$, respectively such that $KB = KN$. The angle bisector of $ACB$ intersects the circumcircle of the triangle $ABC$ at points $C$ and $R$. From point $R$ the perperndicular line on $AB$ intersects the segment $BN$ at point $D$. Prove that the points $A, K, D$, and $N$ lie on one circle.
A quadrilateral $ABCD$ is inscribed in a circle. The extensions of the lines $AB$ and $DC$ intersects at point $K$. The midpoints of the segments $AC$ and $CK$ are $M$ and $N$ respectively. The points $B, D, M, N$ belong to one circle. Find all possible values of the $\angle ADC$ .
The midpoint $M$ is marked on the side $AB$ of the square $ABCD$. From the point $B$, the perpendicular onthe line $CM$ intersects the line $CM$ at point $P$. The point $N$ divides segment $CP$ in half. The angle bisector of $DAN$ and the segment $DP$ intersect at point $Q$. Prove that the quadrilateral $BMQN$ is parallelogram.
The angle bisectors of triangle $ABC$ intersect at point $I$. Points $M$ and $N$ divide the sides $AB$ and $AC$ in half, respectively. The lines $MN$ and $CI$ intersect at point $P$. The point $Q$ is marked such that the lines $MN$ and $PQ$ are perpendicular, and the lines $BI$ and $NQ$ are parallel. Find the angle between the lines $AC$ and $IQ$.
The circles $\omega_1$ and $\omega_2$ intersect at two points $A$ and $B$. The lines $\ell_1$ and $\ell_2$ pass through $B$ and intersects the circle $\omega_1$ at points $C$ and $E$, respectively, and the circle $\omega_2$ at points $D$ and $F$, respectively (where $C, E, D, F \ne B$). The segment $CF$ intersects $\omega_1$ and $\omega_2$ at points $P \ne C$ and $Q \ne F$ respectively. The midpoints of the arcs $BP$ and $BQ$ inside $\omega_2$ and $\omega_1$, respectively, are marked $M$ and $N$. Prove that if $CD = EF$, then the points $C, F, M, N$ belong to one circle.
Given a triangle $ABC$ in which $\angle BAC = 60^o$. On the extension of side $AB$ beyond the vertex $B$ and the extension of side $AC$ beyond the vertex $C$, points $D$ and $E$ are respectively marked such that $BD = BC = CE$. The circumscribed circle of the triangle $ACD$ intersects the segment $DE$ at the point $P \ne D$. Prove that segment $AP$ is the angle bisector of triangle $ADE$.
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