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Neka je $D$ točka na stranici $\overline{AC}$ trokuta $ABC$. Neka su $E$ i $F$ točke na dužinama $\overline{BD}$ i $\overline{BC}$ redom, takve da je $\measuredangle BAE = \measuredangle CAF$. Neka su $P$ i $Q$ točke na dužinama $\overline{BC}$ i $\overline{BD}$ redom, takve da je $EP \parallel CD$ i $FQ \parallel CD$. Dokaži da je $\measuredangle BAP = \measuredangle CAQ$.

Unutar trokuta $ABC$ dana je točka $P$ takva da je

$$\measuredangle ABP = \measuredangle PCA = \frac{1}{3} (\measuredangle ABC + \measuredangle BCA).$$

Dokaži da je $\frac{|AB|}{|AC| + |PB|} = \frac{|AC|}{|AB| + |PC|}$.

Na polukružnici s promjerom $\overline{AB}$ dane su točke $K$ i $L$. Simetrala dužine $\overline{AB}$ siječe dužinu $\overline{KL}$ u točki $U$ i pritom su točke $A$ i $K$ s jedne strane te simetrale, a $B$ i $L$ s druge. Neka je $N$ nožište okomice iz sjecišta pravaca $AK$ i $BL$ na pravac $AB$, a $V$ točka na pravcu $KL$ takva da je $\measuredangle VAU = \measuredangle VBU$.

Dokaži da su pravci $NV$ i $KL$ međusobno okomiti.

Neka je $k$ upisana kružnica šiljastokutnog trokuta $ABC$ sa središtem u točki $I$, a $k_c$ pripisana kružnica istog trokuta nasuprot kuta $\angle BCA$. Ako je točka $D$ diralište stranice $\overline{AB}$ i kružnice $k_c$, a točka $S$ sjecište pravca $DI$ s kružnicom $k_c$ (različito od točke $D$), dokaži da je pravac $DI$ simetrala kuta $\angle ASB$.

Unutar šiljastokutnog trokuta $ABC$ dana je točka $S$ takva da je $\measuredangle SAB = \measuredangle SBC = \measuredangle SCA$. Pravci $AS$, $BS$, $CS$ sijeku redom kružnice opisane trokutima $SBC$, $SCA$, $SAB$ u točkama $A_1$, $B_1$, $C_1$. Dokaži nejednakost $$P(A_1CB) + P(B_1AC) + P(C_1BA) \geq 3P(ABC).$$

U trokutu $ABC$ kut pri vrhu $B$ iznosi $120°$. Neka su $A_1, B_1, C_1$ redom točke na stranicama $\overline{BC}$, $\overline{CA}$, $\overline{AB}$, takve da su $AA_1, BB_1, CC_1$ simetrale kutova trokuta $ABC$. Odredi kut $\measuredangle A_1B_1C_1$.

Dan je trokut $ABC$ u kojem je $|AB| > |AC|$. Neka je $P$ polovište stranice $\overline{BC}$, a $S$ točka u kojoj simetrala kuta $\measuredangle BAC$ sijeće tu stranicu. Paralela s pravcem $AS$ kroz točku $P$ sijeće pravce $AB$ i $AC$ redom u točkama $X$ i $Y$. Neka je $Z$ točka takva da je $Y$ polovište dužine $\overline{XZ}$ te neka se pravci $BY$ i $CZ$ sijeku u točki $D$.

Dokaži da je simetrala kuta $\measuredangle BDC$ paralelna s pravcem $AS$.

U četverokutu $ABCD$ je $\measuredangle DAB = 110^\circ$, $\measuredangle ABC = 50^\circ$, $\measuredangle BCD = 70^\circ$. Neka su $M$ i $N$ polovišta dužina $\overline{AB}$ i $\overline{CD}$ redom. Za točku $P$ na dužini $\overline{MN}$ vrijedi $|AM| : |CN| = |MP| : |NP|$ i $|AP| = |CP|$. Odredi veličinu $\measuredangle APC$.

Točka $O$ je središte kružnice opisane šiljastokutnom trokutu $ABC$. Točke $E$ i $F$ redom su odabrane na dužinama $\overline{OB}$ i $\overline{OC}$ tako da je $|BE| = |OF|$. Ako su $M$ i $N$ redom polovišta kružnih lukova $\widehat{EOA}$ i $\widehat{AOF}$, dokaži da je $\measuredangle ENO + \measuredangle OMF = 2\measuredangle BAC$.

U trokutu $ABC$ vrijedi $|AB| < |BC|$. Točka $I$ je središte kružnice upisane tom trokutu. Neka je $M$ polovište stranice $\overline{AC}$, a $N$ polovište luka $\widehat{AC}$ opisane kružnice tog trokuta koji sadrži točku $B$. Dokaži da je

$$\measuredangle IMA = \measuredangle INB.$$

Dana je kružnica promjera $\overline{AB}$. Na toj kružnici, s različitih strana pravca $AB$, nalaze se točke $C$ i $D$ takve da vrijedi $|AC| < |BC|$ i $|AC| < |AD|$. Točka $P$ pripada dužini $\overline{BC}$ te vrijedi $\measuredangle CAP = \measuredangle ABC$. Okomica iz točke $C$ na pravac $AB$ siječe pravac $BD$ u točki $Q$. Pravci $PQ$ i $AD$ sijeku se u točki $R$, a pravci $PQ$ i $CD$ u točki $T$.

Ako je $|AR| = |RQ|$, dokaži da su pravci $AT$ i $PQ$ međusobno okomiti.

Neka je $ABC$ raznostraničan šiljastokutan trokut. Točka $N$ je polovište duljeg luka $\widehat{BC}$ kružnice opisane trokutu $ABC$. Neka je $k$ kružnica promjera $\overline{BC}$.

Simetrala kuta $\measuredangle BAC$ siječe kružnicu $k$ u točkama $D$ i $E$, a $D_1$ i $E_1$ su točke takve da su $\overline{DD_1}$ i $\overline{EE_1}$ promjeri kružnice $k$.

Dokaži da polovište dužine $\overline{BC}$ pripada kružnici opisanoj trokutu $NE_1D_1$.

Neka je $M$ polovište stranice $\overline{AB}$ trokuta $ABC$ u kojem je $|BC| > |AC|$, te neka je $N$ nožište okomice iz točke $A$ na dužinu $\overline{CM}$. Neka je $P$ točka na pravcu $AN$ takva da je $PB$ okomito na $CB$.

Ako vrijedi $\measuredangle CPB = \measuredangle CBA$, dokaži da je $\measuredangle BAC = 90°$.

An isosceles trapezoid with bases $a$ and $c$ and altitude $h$ is given.

(a) On the axis of symmetry of this trapezoid, find all points $P$ such that both legs of the trapezoid subtend right angles at $P$.

(b) Calculate the distance of $P$ from either base.

(c) Determine under what conditions such points $P$ actually exist. (Discuss various cases that might arise.)

Point $A$ and segment $BC$ are given. Determine the locus of points in space which are vertices of right angles with one side passing through $A$, and the other side intersecting the segment $BC$.

In an $n$-gon all of whose interior angles are equal, the lengths of consecutive sides satisfy the relation $$a_1 \geq a_2 \geq \cdots \geq a_n.$$ Prove that $a_1 = a_2 = \cdots = a_n$.

Prove that there is one and only one triangle whose side lengths are consecutive integers, and one of whose angles is twice as large as another.

On the sides of an arbitrary triangle $ABC$, triangles $ABR, BCP, CAQ$ are constructed externally with $\angle CBP = \angle CAQ = 45°$, $\angle BCP = \angle ACQ = 30°$, $\angle ABR = \angle BAR = 15°$. Prove that $\angle QRP = 90°$ and $QR = RP$.

Let $A$ be one of the two distinct points of intersection of two unequal coplanar circles $C_1$ and $C_2$ with centers $O_1$ and $O_2$, respectively. One of the common tangents to the circles touches $C_1$ at $P_1$ and $C_2$ at $P_2$, while the other touches $C_1$ at $Q_1$ and $C_2$ at $Q_2$. Let $M_1$ be the midpoint of $P_1Q_1$, and $M_2$ be the midpoint of $P_2Q_2$. Prove that $\angle O_1AO_2 = \angle M_1AM_2$.

A circle with center $O$ passes through the vertices $A$ and $C$ of triangle $ABC$ and intersects the segments $AB$ and $BC$ again at distinct points $K$ and $N$, respectively. The circumscribed circles of the triangles $ABC$ and $EBN$ intersect at exactly two distinct points $B$ and $M$. Prove that angle OMB is a right angle.

A triangle $A_{1}A_{2}A_{3}$ and a point $P_{0}$ are given in the plane. We define $A_{s} = A_{s-3}$ for all $s \geq 4$. We construct a set of points $P_{1}, P_{2}, P_{3}, \ldots$, such that $P_{k+1}$ is the image of $P_{k}$ under a rotation with center $A_{k+1}$ through angle $120^{\circ}$ clockwise (for $k = 0, 1, 2, \ldots$). Prove that if $P_{1986} = P_{0}$, then the triangle $A_{1}A_{2}A_{3}$ is equilateral.

In an acute-angled triangle $ABC$ the internal bisector of angle $A$ meets the circumcircle of the triangle again at $A_1$. Points $B_1$ and $C_1$ are defined similarly. Let $A_0$ be the point of intersection of the line $AA_1$ with the external bisectors of angles $B$ and $C$. Points $B_0$ and $C_0$ are defined similarly. Prove that:

(i) The area of the triangle $A_0B_0C_0$ is twice the area of the hexagon $AC_1BA_1CB_1$.

(ii) The area of the triangle $A_0B_0C_0$ is at least four times the area of the triangle $ABC$.

Let $ABC$ be a triangle and $P$ an interior point of $ABC$. Show that at least one of the angles $\angle PAB$, $\angle PBC$, $\angle PCA$ is less than or equal to $30°$.

Let $D$ be a point inside acute triangle $ABC$ such that $\angle ADB = \angle ACB + \pi/2$ and $AC \cdot BD = AD \cdot BC$.

(a) Calculate the ratio $(AB \cdot CD)/(AC \cdot BD)$.

(b) Prove that the tangents at $C$ to the circumcircles of $\triangle ACD$ and $\triangle BCD$ are perpendicular.

Let $ABCDEF$ be a convex hexagon with $AB = BC = CD$ and $DE = EF = FA$, such that $\angle BCD = \angle EFA = \pi/3$. Suppose $G$ and $H$ are points in the interior of the hexagon such that $\angle AGB = \angle DHE = 2\pi/3$. Prove that $AG + GB + GH + DH + HE \geq CF$.

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.

The angle at $A$ is the smallest angle of triangle $ABC$. The points $B$ and $C$ divide the circumcircle of the triangle into two arcs. Let $U$ be an interior point of the arc between $B$ and $C$ which does not contain $A$. The perpendicular bisectors of $AB$ and $AC$ meet the line $AU$ at $V$ and $W$, respectively. The lines $BV$ and $CW$ meet at $T$. Show that

$$AU = TB + TC.$$

Let $I$ be the incenter of triangle $ABC$. Let the incircle of $ABC$ touch the sides $BC$, $CA$, and $AB$ at $K$, $L$, and $M$, respectively. The line through $B$ parallel to $MK$ meets the lines $LM$ and $LK$ at $R$ and $S$, respectively. Prove that angle $RIS$ is acute.

Let $ABC$ be an acute-angled triangle with circumcentre $O$. Let $P$ on $BC$ be the foot of the altitude from $A$.

Suppose that $\angle BCA \geq \angle ABC + 30^{\circ}$.

Prove that $\angle CAB + \angle COP < 90^{\circ}$.

In a triangle $ABC$, let $AP$ bisect $\angle BAC$, with $P$ on $BC$, and let $BQ$ bisect $\angle ABC$, with $Q$ on $CA$.

It is known that $\angle BAC = 60^{\circ}$ and that $AB + BP = AQ + QB$.

What are the possible angles of triangle $ABC$?

$BC$ is a diameter of a circle center $O$. $A$ is any point on the circle with $\angle AOC>60^{\circ}$. $EF$ is the chord which is the perpendicular bisector of $AO$. $D$ is the midpoint of the minor arc $AB$. The line through $O$ parallel to $AD$ meets $AC$ at $J$. Show that $J$ is the incenter of triangle $CEF$.

$ABCD$ is cyclic. The feet of the perpendicular from $D$ to the lines $AB, BC, CA$ are $P, Q, R$ respectively. Show that the angle bisectors of $ABC$ and $CDA$ meet on the line $AC$ iff $RP = RQ$.

Let $ABC$ be an acute-angled triangle with $AB \neq AC$. The circle with diameter $BC$ intersects the sides $AB$ and $AC$ at $M$ and $N$ respectively. Denote by $O$ the midpoint of the side $BC$. The bisectors of the angles $\angle BAC$ and $\angle MON$ intersect at $R$. Prove that the circumcircles of the triangles $BMR$ and $CNR$ have a common point lying on the side $BC$.

In a convex quadrilateral $ABCD$ the diagonal $BD$ does not bisect the angles $\angle ABC$ and $\angle CDA$. The point $P$ lies inside $ABCD$ and satisfies $$\angle PBC = \angle DBA \text{ and } \angle PDC = \angle BDA.$$

Prove that $ABCD$ is a cyclic quadrilateral if and only if $AP = CP$.

Let $ABC$ be a triangle with incentre $I$. A point $P$ in the interior of the triangle satisfies $$\angle PBA + \angle PCA = \angle PBC + \angle PCB.$$

Show that $AP \geq AI$, and that equality holds if and only if $P = I$.

Consider five points $A, B, C, D$ and $E$ such that $ABCD$ is a parallelogram and $BCED$ is a cyclic quadrilateral. Let $\ell$ be a line passing through $A$. Suppose that $\ell$ intersects the interior of the segment $DC$ at $F$ and intersects line $BC$ at $G$. Suppose also that $EF = EG = EC$. Prove that $\ell$ is the bisector of angle $DAB$.

Let $ABC$ be a triangle with $AB = AC$. The angle bisectors of $\angle CAB$ and $\angle ABC$ meet the sides $BC$ and $CA$ at $D$ and $E$, respectively. Let $K$ be the incentre of triangle $ADC$. Suppose that $\angle BEK = 45^{\circ}$. Find all possible values of $\angle CAB$.

Let $I$ be the incentre of triangle $ABC$ and let $\Gamma$ be its circumcircle. Let the line $AI$ intersect $\Gamma$ again at $D$. Let $E$ be a point on the arc $\widehat{BDC}$ and $F$ a point on the side $BC$ such that $$\angle BAF = \angle CAE < \frac{1}{2}\angle BAC.$$ Finally, let $G$ be the midpoint of the segment $IF$. Prove that the lines $DG$ and $EI$ intersect on $\Gamma$.

Points $P$ and $Q$ lie on side $BC$ of acute-angled triangle $ABC$ so that $\angle PAB = \angle BCA$ and $\angle CAQ = \angle ABC$. Points $M$ and $N$ lie on lines $AP$ and $AQ$, respectively, such that $P$ is the midpoint of $AM$, and $Q$ is the midpoint of $AN$. Prove that lines $BM$ and $CN$ intersect on the circumcircle of triangle $ABC$.

A convex quadrilateral $ABCD$ satisfies $AB \cdot CD = BC \cdot DA$. Point $X$ lies inside $ABCD$ so that $$\angle XAB = \angle XCD \quad \text{and} \quad \angle XBC = \angle XDA.$$ Prove that $\angle BXA + \angle DXC = 180°$.

In triangle $ABC$, point $A_1$ lies on side $BC$ and point $B_1$ lies on side $AC$. Let $P$ and $Q$ be points on segments $AA_1$ and $BB_1$, respectively, such that $PQ$ is parallel to $AB$. Let $P_1$ be a point on line $PB_1$, such that $B_1$ lies strictly between $P$ and $P_1$, and $\angle PP_1C = \angle BAC$. Similarly, let $Q_1$ be a point on line $QA_1$, such that $A_1$ lies strictly between $Q$ and $Q_1$, and $\angle CQ_1Q = \angle CBA$.

Prove that points $P$, $Q$, $P_1$, and $Q_1$ are concyclic.

Consider the convex quadrilateral $ABCD$. The point $P$ is in the interior of $ABCD$. The following ratio equalities hold: $$\angle PAD : \angle PBA : \angle DPA = 1 : 2 : 3 = \angle CBP : \angle BAP : \angle BPC.$$

Prove that the following three lines meet in a point: the internal bisectors of angles $\angle ADP$ and $\angle PCB$ and the perpendicular bisector of segment $AB$.

Let $D$ be an interior point of the acute triangle $ABC$ with $AB > AC$ so that $\angle DAB = \angle CAD$. The point $E$ on the segment $AC$ satisfies $\angle ADE = \angle BCD$, the point $F$ on the segment $AB$ satisfies $\angle FDA = \angle DBC$, and the point $X$ on the line $AC$ satisfies $CX = BX$. Let $O_1$ and $O_2$ be the circumcentres of the triangles $ADC$ and $EXD$, respectively. Prove that the lines $BC$, $EF$, and $O_1O_2$ are concurrent.

Let $ABCDE$ be a convex pentagon such that $BC = DE$. Assume that there is a point $T$ inside $ABCDE$ with $TB = TD$, $TC = TE$ and $\angle ABT = \angle TEA$. Let line $AB$ intersect lines $CD$ and $CT$ at points $P$ and $Q$, respectively. Assume that the points $P, B, A, Q$ occur on their line in that order. Let line $AE$ intersect lines $CD$ and $DT$ at points $R$ and $S$, respectively. Assume that the points $R, E, A, S$ occur on their line in that order. Prove that the points $P, S, Q, R$ lie on a circle.

Let $ABC$ be a triangle with $AB < AC < BC$. Let the incentre and incircle of triangle $ABC$ be $I$ and $\omega$, respectively. Let $X$ be the point on line $BC$ different from $C$ such that the line through $X$ parallel to $AC$ is tangent to $\omega$. Similarly, let $Y$ be the point on line $BC$ different from $B$ such that the line through $Y$ parallel to $AB$ is tangent to $\omega$. Let $AI$ intersect the circumcircle of triangle $ABC$ again at $P \neq A$. Let $K$ and $L$ be the midpoints of $AC$ and $AB$, respectively.

Prove that $\angle KIL + \angle YPX = 180^{\circ}$.

Let $ABCD$ be a convex quadrilateral such that $AB$ and $CD$ are not parallel and $AB = CD$. The midpoints of the diagonals $AC$ and $BD$ are $E$ and $F$. The line $EF$ meets segments $AB$ and $CD$ at $G$ and $H$, respectively. Show that $\measuredangle AGH = \measuredangle DHG$.

Suppose that $ABCD$ is a cyclic quadrilateral and $CD = DA$. Points $E$ and $F$ belong to the segments $AB$ and $BC$ respectively, and $\measuredangle ADC = 2\measuredangle EDF$. Segments $DK$ and $DM$ are height and median of the triangle $DEF$, respectively. $L$ is the point symmetric to $K$ with respect to $M$. Prove that the lines $DM$ and $BL$ are parallel.

Let $A$, $B$, $C$, $D$, $E$ be points such that $ABCD$ is a cyclic quadrilateral and $ABDE$ is a parallelogram. The diagonals $AC$ and $BD$ intersect at $S$ and the rays $AB$ and $DC$ intersect at $F$. Prove that $\angle AFS = \angle ECD$.

Let $ABCDE$ be a convex pentagon with all five sides equal in length. The diagonals $AD$ and $EC$ meet in $S$ with $\angle ASE=60^{\circ}$. Prove that $ABCDE$ has a pair of parallel sides.

Let $ABC$ be an isosceles triangle with $AC = BC$. Let $N$ be a point inside the triangle such that $2\angle ANB = 180° + \angle ACB$. Let $D$ be the intersection of the line $BN$ and the line parallel to $AN$ that passes through $C$. Let $P$ be the intersection of the angle bisectors of the angles $CAN$ and $ABN$.

Show that the lines $DP$ and $AN$ are perpendicular.