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Trapez $ABCD$ s duljom osnovicom $\overline{AB}$ upisan je u kružnicu $k$. Neka su $A_0$, $B_0$ redom polovišta dužina $\overline{BC}$, $\overline{CA}$. Neka je $N$ nožište visine iz vrha $C$ na $AB$, a $G$ težište trokuta $ABC$. Kružnica $k_1$ prolazi točkama $A_0$ i $B_0$ te dodiruje kružnicu $k$ u točki $X$, različitoj od $C$. Dokaži da su točke $D$, $G$, $N$ i $X$ kolinearne.

Kružnice $k_1$ i $k_2$ sijeku se u točkama $M$ i $N$. Pravac $l$ siječe kružnicu $k_1$ u točkama $A$ i $C$, a kružnicu $k_2$ u točkama $B$ i $D$ tako da se točke $A$, $B$, $C$ i $D$ na pravcu $l$ nalaze u tom poretku. Neka je $X$ točka na pravcu $MN$ takva da se točka $M$ nalazi između točaka $X$ i $N$. Neka je $P$ sjecište pravaca $AX$ i $BM$, a $Q$ sjecište pravaca $DX$ i $CM$.

Ako je $K$ polovište dužine $\overline{AD}$, a $L$ polovište dužine $\overline{BC}$, dokaži da se pravci $XK$ i $ML$ sijeku na pravcu $PQ$.

U ravnini je dan skup $A$ koji sadrži $2016$ točaka tako da nikoje četiri točke iz skupa $A$ ne leže na istom pravcu. Dokaži da je moguće odabrati podskup $B \subset A$ koji sadrži barem $63$ točke tako da nikoje tri točke iz skupa $B$ ne leže na istom pravcu.

Neka je $\overline{AD}$ visina šiljastokutnog trokuta $ABC$. Na pravcu $AD$ nalaze se međusobno različite točke $E$ i $F$ takve da vrijedi $|DE| = |DF|$ i pritom je točka $E$ u unutrašnjosti trokuta $ABC$. Kružnica opisana trokutu $BEF$ siječe dužine $\overline{BC}$ i $\overline{AB}$ redom u točkama $K$ i $M$. Kružnica opisana trokutu $CEF$ siječe dužine $\overline{BC}$ i $\overline{CA}$ redom u točkama $L$ i $N$.

Dokaži da se pravci $AD$, $KM$ i $LN$ sijeku u jednoj točki.

Dan je trokut $ABC$ takav da je $|AC| = |BC|$ i točka $D$ na stranici $\overline{AB}$ takva da je $|AD| < |BD|$. Točke $P$ i $Q$ su redom nožišta okomica iz točke $D$ na stranice $\overline{AC}$ i $\overline{BC}$. Simetrala dužine $\overline{PQ}$ siječe $\overline{CP}$ u točki $E$. Kružnice opisane trokutima $ABC$ i $PQC$ sijeku se u točkama $C$ i $F$.

Ako su točke $E$, $F$ i $Q$ kolinearne, dokaži da je $\measuredangle ACB$ pravi kut.

Neka je $ABCD$ paralelogram takav da je $|AC| = |BC|$. Neka je $P$ točka na pravcu $AB$ takva da $B$ leži između $A$ i $P$. Opisana kružnica trokuta $ACD$ siječe dužinu $\overline{PD}$ u točki $Q$, $Q \neq D$. Opisana kružnica trokuta $APQ$ siječe dužinu $\overline{PC}$ u točki $R$, $R \neq P$.

Dokaži da se pravci $CD$, $AQ$ i $BR$ sijeku u jednoj točki.

Dan je šiljastokutni trokut $ABC$ u kojem vrijedi $|BC| : |AC| = 3 : 2$. Neka je $D$ polovište stranice $\overline{AC}$, a $P$ polovište dužine $\overline{BD}$. Na pravcu $AC$ dana je točka $X$ tako da je $|AX| = |BC|$, pri čemu je $A$ između $X$ i $C$. Pravac $XP$ siječe stranicu $\overline{BC}$ u $E$. Pravac $DE$ siječe pravac $AP$ u $Y$. Dokaži da točke $A, X, Y, E$ leže na jednoj kružnici ako i samo ako je $|AB| = |BC|$.

Neka je $ABCD$ tetivni četverokut. Neka su $M$ i $N$ redom polovišta dužina $\overline{BC}$ i $\overline{AD}$. Pretpostavimo da točke $Q, A, B, P$ leže na pravcu u tom poretku, da je $AC$ tangenta opisane kružnice trokuta $ADQ$ te da je $BD$ tangenta opisane kružnice trokuta $BCP$. Dokaži da se pravac $CD$, tangenta opisane kružnice trokuta $ANQ$ u točki $A$ i tangenta opisane kružnice trokuta $BMP$ u točki $B$ sijeku u jednoj točki.

A non-isosceles triangle $A_1A_2A_3$ is given with sides $a_1, a_2, a_3$ ($a_i$ is the side opposite $A_i$). For all $i = 1, 2, 3, M_i$ is the midpoint of side $a_i$, and $T_i$ is the point where the incircle touches side $a_i$. Denote by $S_i$ the reflection of $T_i$ in the interior bisector of angle $A_i$. Prove that the lines $M_1S_1$, $M_2S_2$, and $M_3S_3$ are concurrent.

The diagonals $AC$ and $CE$ of the regular hexagon $ABCDEF$ are divided by the inner points $M$ and $N$, respectively, so that

$$\frac{AM}{AC} = \frac{CN}{CE} = r.$$

Determine $r$ if $B$, $M$, and $N$ are collinear.

Let $n$ and $k$ be positive integers and let $S$ be a set of $n$ points in the plane such that

(i) No three points of $S$ are collinear, and

(ii) For any point $P$ of $S$ there are at least $k$ points of $S$ equidistant from $P$.

Prove that:

$$k < \frac{1}{2} + \sqrt{2n}.$$

$ABC$ is an isosceles triangle with $AB = AC$. Suppose that

1. $M$ is the midpoint of $BC$ and $O$ is the point on the line $AM$ such that $OB$ is perpendicular to $AB$;
2. $Q$ is an arbitrary point on the segment $BC$ different from $B$ and $C$;
3. $E$ lies on the line $AB$ and $F$ lies on the line $AC$ such that $E, Q, F$ are distinct and collinear.

Prove that $OQ$ is perpendicular to $EF$ if and only if $QE = QF$.

Let $A, B, C, D$ be four distinct points on a line, in that order. The circles with diameters $AC$ and $BD$ intersect at $X$ and $Y$. The line $XY$ meets $BC$ at $Z$. Let $P$ be a point on the line $XY$ other than $Z$. The line $CP$ intersects the circle with diameter $AC$ at $C$ and $M$, and the line $BP$ intersects the circle with diameter $BD$ at $B$ and $N$. Prove that the lines $AM, DN, XY$ are concurrent.

Let $P$ be a point inside triangle $ABC$ such that

$$\angle APB - \angle ACB = \angle APC - \angle ABC.$$

Let $D, E$ be the incenters of triangles $APB, APC$, respectively. Show that $AP, BD, CE$ meet at a point.

$AB$ is tangent to the circles $CAMN$ and $NMBD$. $M$ lies between $C$ and $D$ on the line $CD$, and $CD$ is parallel to $AB$. The chords $NA$ and $CM$ meet at $P$; the chords $NB$ and $MD$ meet at $Q$. The rays $CA$ and $DB$ meet at $E$. Prove that $PE = QE$.

Let $\mathcal{S}$ be a finite set of at least two points in the plane. Assume that no three points of $\mathcal{S}$ are collinear. A windmill is a process that starts with a line $\ell$ going through a single point $P \in \mathcal{S}$. The line rotates clockwise about the pivot $P$ until the first time that the line meets some other point belonging to $\mathcal{S}$. This point, $Q$, takes over as the new pivot, and the line now rotates clockwise about $Q$, until it next meets a point of $\mathcal{S}$. This process continues indefinitely.

Show that we can choose a point $P$ in $\mathcal{S}$ and a line $\ell$ going through $P$ such that the resulting windmill uses each point of $\mathcal{S}$ as a pivot infinitely many times.

Let $ABC$ be an acute-angled triangle with orthocentre $H$, and let $W$ be a point on the side $BC$, lying strictly between $B$ and $C$. The points $M$ and $N$ are the feet of the altitudes from $B$ and $C$, respectively. Denote by $\omega_1$ the circumcircle of $BWN$, and let $X$ be the point on $\omega_1$ such that $WX$ is a diameter of $\omega_1$. Analogously, denote by $\omega_2$ the circumcircle of $CWM$, and let $Y$ be the point on $\omega_2$ such that $WY$ is a diameter of $\omega_2$. Prove that $X$, $Y$ and $H$ are collinear.

Triangle $ABC$ has circumcircle $\Omega$ and circumcentre $O$. A circle $\Gamma$ with centre $A$ intersects the segment $BC$ at points $D$ and $E$, such that $B$, $D$, $E$ and $C$ are all different and lie on line $BC$ in this order. Let $F$ and $G$ be the points of intersection of $\Gamma$ and $\Omega$, such that $A$, $F$, $B$, $C$ and $G$ lie on $\Omega$ in this order. Let $K$ be the second point of intersection of the circumcircle of triangle $BDF$ and the segment $AB$. Let $L$ be the second point of intersection of the circumcircle of triangle $CGE$ and the segment $CA$.

Suppose that the lines $FK$ and $GL$ are different and intersect at the point $X$. Prove that $X$ lies on the line $AO$.

We are given a cyclic quadrilateral $ABCD$ with a point $E$ on the diagonal $AC$ such that $AD = AE$ and $CB = CE$. Let $M$ be the center of the circumcircle $k$ of the triangle $BDE$. The circle $k$ intersects the line $AC$ in the points $E$ and $F$. Prove that the lines $FM$, $AD$, and $BC$ meet at one point.

Let $ABC$ be a triangle. The internal bisector of $\angle ABC$ intersects the side $AC$ at $L$ and the circumcircle of triangle $ABC$ again at $W \neq B$. Let $K$ be the perpendicular projection of $L$ onto $AW$. The circumcircle of triangle $BLC$ intersects line $CK$ again at $P \neq C$. Lines $BP$ and $AW$ meet at point $T$. Prove that $AW = WT$.

Let $ABC$ be a right-angled triangle with its right angle at $B$ and circumcircle $c$. Denote by $D$ the midpoint of the shorter arc $AB$ of $c$. Let $P$ be the point on the side $AB$ such that $CP = CD$ and let $X$ and $Y$ be two distinct points on $c$ satisfying $AX = AY = PD$. Prove that the points $X$, $Y$, and $P$ are collinear.

Let $ABCD$ be a parallelogram with $\angle DAB < 90^{\circ}$. Let $E \neq B$ be the point on the line $BC$ such that $AE = AB$ and let $F \neq D$ be the point on the line $CD$ such that $AF = AD$. The circumcircle of the triangle $CEF$ intersects the line $AE$ again in $P$ and the line $AF$ again in $Q$. Let $X$ be the reflection of $P$ over the line $DE$ and $Y$ the reflection of $Q$ over the line $BF$. Prove that $A, X$ and $Y$ lie on the same line.

Neka je trokut $ABC$ s tupim kutom kod vrha $B$, neka su $D$ i $E$ polovišta stranica $\overline{AB}$ i $\overline{AC}$ redom, $F$ točka na stranici $\overline{BC}$ takva da je $\measuredangle BFE$ pravi, te $G$ točka na dužini $\overline{DE}$ takva da je kut $\measuredangle BGE$ pravi.

Dokaži da točke $A$, $F$ i $G$ leže na istom pravcu ako i samo ako je $2|BF| = |CF|$.

Kružnice $k_1$ i $k_2$ sijeku se u točkama $A$ i $B$. Pravac $l$ siječe kružnicu $k_1$ u točkama $C$ i $E$, a kružnicu $k_2$ u točkama $D$ i $F$ tako da se točka $D$ nalazi između $C$ i $E$, a točka $E$ između $D$ i $F$. Pravci $CA$ i $BF$ sijeku se u točki $G$, a pravci $DA$ i $BE$ u točki $H$. Dokaži da je $CF \parallel HG$.

Neka je $ABC$ šiljastokutni trokut. Točka $B'$ je osnosimetrična slika točke $B$ s obzirom na pravac $AC$, a točka $C'$ je osnosimetrična slika točke $C$ s obzirom na pravac $AB$. Kružnice opisane trokutima $ABB'$ i $ACC'$ sijeku se u točkama $A$ i $P$. Dokaži da središte kružnice opisane trokutu $ABC$ leži na pravcu $AP$.

U ravnini je dano pet točaka $P_1, P_2, P_3, P_4, P_5$ sa cjelobrojnim koordinatama. Pokažite da postoji bar jedan par $(P_i, P_j)$ za $i \neq j$ tako da pravac $P_iP_j$ sadrži neku točku $Q$ sa cjelobrojnim koordinatama koja leži između $P_i$ i $P_j$.

U ravnini su dane dvije različite točke $O$ i $P$. Odaberimo paralelogram $ABCD$ kojem je točka $O$ središte. Označimo s $M$ i $N$ redom polovišta dužina $\overline{AP}$ i $\overline{BP}$. Točka $Q$ je presjek dužina $\overline{MC}$ i $\overline{ND}$. Dokažite da točke $O$, $Q$ i $P$ leže na istom pravcu i da točka $Q$ ne ovisi o izboru paralelograma $ABCD$.

U ravnini je dano pet točaka $P_1, P_2, P_3, P_4, P_5$ sa cjelobrojnim koordinatama. Pokažite da postoji bar jedan par $(P_i, P_j)$ za $i \neq j$ tako da pravac $P_iP_j$ sadrži neku točku $Q$ sa cjelobrojnim koordinatama koja leži između $P_i$ i $P_j$.

Neka je $I$ točka na simetrali kuta $\measuredangle BAC$ trokuta $ABC$, a $M$ i $N$ redom točke na stranicama $\overline{AB}$ i $\overline{AC}$, takve da je $\measuredangle ABI = \measuredangle NIC$ i $\measuredangle ACI = \measuredangle MIB$. Dokažite da je $I$ središte upisane kružnice trokuta $ABC$ ako i samo ako su točke $M$, $N$ i $I$ kolinearne.

Upisana kružnica šiljastokutnog trokuta $ABC$ dodiruje stranice $\overline{BC}$, $\overline{CA}$ i $\overline{AB}$ redom u točkama $D$, $E$ i $F$. Središte te kružnice je točka $S$, a pravac $DS$ siječe dužinu $\overline{EF}$ u točki $P$. Ako je $M$ polovište stranice $\overline{BC}$, dokaži da su točke $A$, $P$ i $M$ kolinearne.