Latvia TST 2015-22 (BW) 26p

geometry problems from Latvian Team Selection Tests (TST) for Baltic Way (BW) with aops links in the names

(only those not in any Shortlist)

collected inside aops here

all 2016 geo problems came from BW shortlist

                                                                 2015 - 2022 

2015 Latvia BW TST p5

$BE$ is the altitude of acute triangle $ABC$. The line $\ell$ touches the circumscribed circle of the triangle $ABC$ at point $B$. A perpendicular $CF$ is drawn from $C$ on line $\ell$. Prove that the lines $EF$ and $AB$ are parallel.

2015 Latvia BW TST p6

$AM$ is the median of triangle $ABC$. A perpendicular $CC_1$ is drawn from point $C$ on the bisector of angle $\angle CMA$, a perpendicular $BB_1$ is drawn from point $B$ on the bisector of angle $\angle BMA$. Prove that line $AM$ intersects segment $B_1C_1$ at its midpoint.

2015 Latvia BW TST p7

Two circle $\Gamma_1$ and $\Gamma_2$ intersect at points $A$ and $B$, point $P$ is not on the line $AB$. Line $AP$ intersects again $\Gamma_1$ and $\Gamma_2$ at points $K$ and $L$ respectively, line $BP$ intersects again $\Gamma_1$ and $\Gamma_2$ at points $M$ and $N$ respectively and all the points mentioned so far are different. The centers of the circles circumscribed around the triangles $KMP$ and $LNP$ are $O_1$ and $O_2$ respectively. Prove that $O_1O_2$ is perpendicular to $AB$.

2017 Latvia BW TST p9

In an isosceles triangle $ABC$ in which $AC = BC$ and $\angle ABC < 60^o$, $I$ and $O$ are the centers of the inscribed and circumscribed circles, respectively. For the triangle $BIO$, the circumscribed circle intersects the side $BC$ again at $D$. Prove that:

i) lines $AC$ and $DI$ are parallel,

ii) lines $OD$ and $IB$ are perpendicular.

2017 Latvia BW TST p10

In an obtuse triangle $ABC$, for which $AC < AB$, the radius of the inscribed circle is $R$, the midpoint of its arc $BC$ (which does not contain $A$) is $S$. A point $T$ is placed on the extension of altitude $AD$ such that $D$ is between $A$ and $T$ and $AT = 2R$. Prove that $\angle AST = 90^o$.

2017 Latvia BW TST p11

On the extension of the angle bisector $AL$ of the triangle $ABC$, a point $P$ is placed such that $P L = AL$. Prove that the perimeter of triangle $PBC$ does not exceed the perimeter of triangle $ABC$.

2017 Latvia BW TST p12

A diameter $AK$ is drawn for the circumscribed circle $\omega$ of an acute-angled triangle $ABC$, an arbitrary point $M$ is chosen on the segment $BC$, the straight line $AM$ intersects $\omega$ at point $Q$. The foot of the perpendicular drawn from $M$ on $AK$ is $D$, the tangent drawn to the circle $\omega$ through the point $Q$, intersects the straight line $MD$ at $P$. A point $L$ (different from $Q$) is chosen on $\omega$ such that $PL$ is tangent to $\omega$. Prove that points $L$, $M$ and $K$ lie on the same line.

2018  Latvia BW TST p9

Acute triangle $\triangle ABC$ with $AB<AC$, circumcircle $\Gamma$ and circumcenter $O$ is given. Midpoint of side $AB$ is $D$. Point $E$ is chosen on side $AC$ so that $BE=CE$. Circumcircle of triangle $BDE$ intersects $\Gamma$ at point $F$ (different from point $B$). Point $K$ is chosen on line $AO$ satisfying $BK \perp AO$ (points $A$ and $K$ lie in different half-planes with respect to line $BE$). Prove that the intersection of lines $DF$ and $CK$ lies on $\Gamma$.

2018  Latvia BW TST p10

Let $ABC$ be an obtuse triangle with obtuse angle $\angle B$ and altitudes $AD, BE, CF$. Let $T$ and $S$ be the midpoints of $AD$ and $CF$, respectively. Let $M$ and $N$ and be the symmetric images of $T$ with respect to lines $BE$ and $BD$, respectively. Prove that $S$ lies on the circumcircle of triangle $BMN$.

2018  Latvia BW TST p11

Let $ABC$ be a triangle with angles $\angle A = 80^\circ, \angle B = 70^\circ, \angle C = 30^\circ$. Let $P$ be a point on the bisector of $\angle BAC$ satisfying $\angle BPC =130^\circ$. Let $PX, PY, PZ$ be the perpendiculars drawn from $P$ to the sides $BC, AC, AB$, respectively. Prove that the following equation with segment lengths is satisfied

$$AY^3+BZ^3+CX^3=AZ^3+BX^3+CY^3.$$

2018  Latvia BW TST p12

Let $ABCD$ be a parallelogram. Let $X$ and $Y$ be arbitrary points on sides $BC$ and $CD$, respectively. Segments $BY$ and $DX$ intersect at $P$. Prove that the line going through the midpoints of segments $BD$ and $XY$ is either parallel to or coincides with line $AP$.

2019  Latvia BW TST p9

Let $ABCD$ be a rhombus with the condition $\angle ABC > 90^o$. The circle $\Gamma_B$ with center at $B$ goes through $C$, and the circle $\Gamma_C$ with center at $C$ goes through $B$. Denote by $E$ one of the intersection points of $\Gamma_B$ and $\Gamma_C$. The line $ED$ intersects intersects $\Gamma_B$ again at $F$. Find the value of $\angle  AFB$.

2019  Latvia BW TST p10

Let $\triangle ABC$ be an acute angled triangle with orthocenter $H$ and let $M$ be a midpoint of $BC$. Circle with diameter $AH$ is $\omega_1$ and circle with center $M$ is $\omega_2$. If $\omega_2$ is tangent to circumcircle of $\triangle ABC$, then prove that circles $\omega_1$ and $\omega_2$ are tangent to each other.

2019  Latvia BW TST p11

Let $A_1A_2...A_{2018}$ be regular $2018$-gon. Radius of it's circumcircle is $R$. Prove that:

$$A_1A_{1008}-A_1A_{1006}+A_1A_{1004}-A_1A_{1002} + ... + A_1A_4 -A_1A_2=R$$

2019  Latvia BW TST p12

Let $AX$, $AY$ be tangents to circle $\omega$ from point $A$. Le $B$, $C$ be points inside $AX$ and $AY$ respectively, such that perimeter of $\triangle ABC$ is equal to length of $AX$. $D$ is reflection of $A$ over $BC$. Prove that circumcircle $\triangle BDC$ and $\omega$ are tangent to each other.

2020  Latvia BW TST p9

Given $\triangle ABC$, whose all sides have different length. Point $P$ is chosen on altitude $AD$. Lines $BP$ and $CP$ intersect lines $AC, AB$ respectively and point $X, Y$.It is given that $AX=AY$. Prove that there is circle, whose centre lies on $BC$ and is tangent to sides $AC$ and $AB$ at points $X,Y$.

2020  Latvia BW TST p10

Given $\triangle ABC$ and it's orthocenter $H$. Point $P$ is arbitrary chosen on the side $BC$. Let $Q$ and $R$ be reflections of point $P$ over sides $AB, AC$. It is given that points $Q,H,R$ are collinear. Prove that $\triangle ABC$ is right angled.

2020  Latvia BW TST p11

Circle centred at point $O$ intersects sides $AC, AB$ of triangle $\triangle ABC$ at points $B_1$ and $C_1$ respectively and passes through points $B,C$. It is known that lines $AO, CC_1, BB_1$ are concurrent. Prove that $\triangle ABC$ is isosceles.

2020  Latvia BW TST p12

There are rhombus $ABCD$ and circle $\Gamma_B$, which is centred at $B$ and has radius $BC$, and circle $\Gamma_C$, which is centred at $C$ and has radius $BC$. Circles $\Gamma_B$ and $\Gamma_C$ intersect at point $E$. The line $ED$ intersects $\Gamma_B$ at point $F$. Find all possible values of $\angle AFB$.

2021 Latvia BW TST p9

Pentagon $ABCDE$ with $CD\parallel BE$ is inscribed in circle $\omega$. Tangent to $\omega$ through $B$ intersects line $AC$ at $F$ in a way that $A$ lies between $C$ and $F$. Lines $BD$ and $AE$ intersect at $G$. Prove that $FG$ is tangent to the circumcircle of $\triangle ADG$.

2021 Latvia BW TST p10

Circle $\omega$ with centre $M$ and diameter $XY$ is given. Point $A$ is chosen on $\omega$ so that $AX<AY$. Points $B, C$ are chosen on segments $XM, YM$, respectively, in a way that $BM=CM$. A parallel line to $AB$ is constructed through $C$; the line intersects $\omega$ at $P$ so that $P$ lies on the smaller arc $\widehat{AY}$. Similarly, a parallel line to $AC$ is constructed through $B$; the line intersects $\omega$ at $Q$ so that $Q$ lies on the smaller arc $\widehat{XA}$. Lines $PQ$ and $XY$ intersect at $S$. Prove that $AS$ is tangent to $\omega$.

2021 Latvia BW TST 11

Incircle of $\triangle ABC$ has centre $I$ and touches sides $AC, AB$ at $E,F$, respectively. The perpendicular bisector of segment $AI$ intersects side $AC$ at $P$. On side $AB$ a point $Q$ is chosen so that $QI \perp FP$. Prove that $EQ \perp AB$.

2021 Latvia BW TST 12

Five points $A,B,C,P,Q$ are chosen so that $A,B,C$ aren't collinear. The following length conditions hold: $\frac{AP}{BP}=\frac{AQ}{BQ}=\frac{21}{20}$ and $\frac{BP}{CP}=\frac{BQ}{CQ}=\frac{20}{19}$. Prove that line $PQ$ goes through the circumcentre of $\triangle ABC$.

2022 Latvia BW TST 9

Let $ABCD$ be a cyclic quadrilateral inscribed in circle $\Omega$. Let the lines $AB$ and $CD$ intersect at $P$, and the lines $AD$ and $BC$ intersect at $Q$. Let then the circumcircle of the triangle $\triangle APQ$ intersect $\Omega$ at $R \neq A$. Prove that the line $CR$ goes through the midpoint of the segment $PQ$.

2022 Latvia BW TST 10

Let $\triangle ABC$ be a triangle satisfying $AB<AC$. Let $D$ be a point on the segment $AC$ such that $AB=AD$. Let then $X$ be a point on the segment $BC$ satisfying $BD^2=BX\cdot BC$. Let the circumcircles of the triangles $\triangle XDC$ and $\triangle ABC$ intersect at $M \neq C$. Prove that the line $MD$ goes through the midpoint of the arc $\widehat{BAC}$ of the circumcircle of $\triangle ABC$.

2022 Latvia BW TST 11

Let $\triangle ABC$ be an acute triangle. Point $D$ is arbitrarily chosen on the side $BC$. Let the circumcircle of the triangle $\triangle ADB$ intersect the segment $AC$ at $M$, and the circumcircle of the triangle $\triangle ADC$ intersect the segment $AB$ at $N$. Prove that the tangents of the circumcircle of the triangle $\triangle AMN$ at $M$ and $N$ intersect at a point that lies on the line $BC$.

2022 Latvia BW TST 12

Let $\triangle ABC$ be a triangle with incentre $I$. Points $E$ and $F$ are the tangency points of the incircle and the sides $AC$ and $AB$, respectively. Suppose that the lines $BI$ and $CI$ intersect the line $EF$ at $Y$ and $Z$, respectively. Let $M$ denote the midpoint of the segment $BC$, and $N$ denote the midpoint of the segment $YZ$. Prove that $AI \parallel MN$.