Through a point $P$ exterior to a given circle pass a secant and a tangent to the circle. The secant intersects the circle at $A$ and $B$, and the tangent touches the circle at $C$ on the same side of the diameter through $P$ as the points $A$ and $B$. The projection of the point $C$ on the diameter is called $Q$. Prove that the line $QC$ bisects the angle $\angle AQB$.

Let $ABCD$ be a convex quadrilateral, the incenters of $\triangle ABC$ and $\triangle ADC$ are $I,J$, respectively. It is known that $AC,BD,IJ$ concurrent at a point $P$. The line perpendicular to $BD$ through $P$ intersects with the outer angle bisector of $\angle BAD$ and the outer angle bisector $\angle BCD$ at $E,F$, respectively. Show that $PE=PF$.

The hexagon $ABCDEF$ has all angles equal . We know that four consecutive sides of the hexagon have lengths $7, 6, 3$ and $5$ in this order. What is the sum of the lengths of the two remaining sides?

Divide a $ 2\times 4 $ rectangle into $ 8 $ unit squares to obtain a set of $ 15 $ vertices denoted by $ \mathcal{M} . $ Find the points $ A\in\mathcal{M} $ that have the property that the set $ \mathcal{M}\setminus \{ A\} $ can form $ 7 $ pairs $ \left( A_1,B_1\right) ,\left( A_2,B_2\right) ,\ldots ,\left( A_7,B_7\right)\in\mathcal{M}\times\mathcal{M} $ such that $$ \overrightarrow{A_1B_1} +\overrightarrow{A_2B_2} +\cdots +\overrightarrow{A_7B_7} =\overrightarrow{O} . $$

Betal marks $2021$ points on the plane such that no three are collinear, and draws all possible segments joining these. He then chooses any $1011$ of these segments, and marks their midpoints. Finally, he chooses a segment whose midpoint is not marked yet, and challenges Vikram to construct its midpoint using [b]only[/b] a straightedge. Can Vikram always complete this challenge? [i]Note.[/i] A straightedge is an infinitely long ruler without markings, which can only be used to draw the line joining any two given distinct points. [i]Proposed by Prithwijit De and Sutanay Bhattacharya[/i]

Let $ABC$ be a triangle with side lengths $AB = 7$, $BC = 8$, and $CA = 9$. Let $O$ be the circumcenter of $\triangle ABC$, and let $AO$, $BO$, $CO$ intersect the circumcircle of $\triangle ABC$ again at $D$, $E$, and $F$, respectively. The area of convex hexagon $AFBDCE$ can be expressed as $m\sqrt{n}$, where $m$ and $n$ are positive integers and $n$ is square-free. Find $m + n$. [i]Individual #10[/i]

Let $\triangle ABC$ be a triangle with $\angle A \neq 60^\circ$. Let $I_B, I_C$ be the $B, C$-excenters of triangle $ABC$, let $B^\prime$ be the reflection of $B$ with respect to $AC$, and let $C^\prime$ be the reflection of $C$ with respect to $AB$. Let $P$ be the intersection of $I_C B^\prime$ and $I_B C^\prime$. Denote by $P_A, P_B, P_C$ the reflections of the point $P$ with respect to $BC, CA, AB$. Show that the three lines $A P_A, B P_B, C P_C$ meet at a single point.

Let $TABCD$ be a pyramid with top vertex $T$, such that its base $ABCD$ is a square of side length 4. It is given that, among the triangles $TAB$, $TBC$, $TCD$ and $TDA$, one can find an isosceles triangle and a right-angled triangle. Find all possible values for the volume of the pyramid.

In $\triangle AEF$, let $B$ and $D$ be on segments $AE$ and $AF$ respectively, and let $ED$ and $FB$ intersect at $C$. Define $K,L,M,N$ on segments $AB,BC,CD,DA$ such that $\frac{AK}{KB}=\frac{AD}{BC}$ and its cyclic equivalents. Let the incircle of $\triangle AEF$ touch $AE,AF$ at $S,T$ respectively; let the incircle of $\triangle CEF$ touch $CE,CF$ at $U,V$ respectively. Prove that $K,L,M,N$ concyclic implies $S,T,U,V$ concyclic.

Let $O$ and $\Omega$ denote the circumcenter and circumcircle, respectively, of scalene triangle $\triangle ABC$. Furthermore, let $M$ be the midpoint of side $BC$. The tangent to $\Omega$ at $A$ intersects $BC$ and $OM$ at points $X$ and $Y$, respectively. If the circumcircle of triangle $\triangle OXY$ intersects $\Omega$ at two distinct points $P$ and $Q$, prove that $PQ$ bisects $\overline{AM}$. [i]Proposed by Andrew Wen[/i]

There are two identical clocks on the wall, one showing the current Moscow time and the other showing current local time. The minimum distance between the ends of their hour hands equals $m$ and the maximum distance equals $M$. Find the distance between the centres of the clocks. (S Fomin, Leningrad)

In a nonisosceles triangle $ABC$, point $I$ is its incentre and $\Gamma$ is its circumcircle. Points $E$ and $D$ lie on $\Gamma$ and the circumcircle of triangle $BIC$ respectively such that $AE$ and $ID$ are both perpendicular to $BC$. Let $M$ be the midpoint of $BC$, $N$ be the midpoint of arc $BC$ on $\Gamma$ containing $A$, $F$ is the point of tangency of the $A-$excircle on $BC$, and $G$ is the intersection of line $DE$ with $\Gamma$. Prove that lines $GM$ and $NF$ intersect at a point located on $\Gamma$. (Possibly proposed by Farras Faddila)

Let $ABCD$ be a trapezoid, with $AD \parallel BC$, let $M$ be the midpoint of $AD$, and let $C_1$ be symmetric point to $C$ with respect to line $BD$. Segment $BM$ meets diagonal $AC$ at point $K$, and ray $C_1K$ meets line $BD$ at point $H$. Prove that $\angle{AHD}$ is a right angle. [i]Proposed by Giorgi Arabidze, Georgia[/i]

The triangle $ABC$ is right in $A$ and $R$ is the midpoint of the hypotenuse $BC$ . On the major leg $AB$ the point $P$ is marked such that $CP = BP$ and on the segment $BP$ the point $Q$ is marked such that the triangle $PQR$ is equilateral. If the area of triangle $ABC$ is $27$, calculate the area of triangle $PQR$ .

In a triangle $ABC$, let $a,b,c$ be its sides and $m,n,p$ be the corresponding medians. For every $\alpha>0$, let $\lambda(\alpha)$ be the real number such that $$a^\alpha+b^\alpha+c^\alpha=\lambda(\alpha)^\alpha\left(m^\alpha+n^\alpha+p^\alpha\right)^\alpha.$$ (a) Compute $\lambda(2)$. (b) Find the limit of $\lambda(\alpha)$ as $\alpha$ approaches $0$. (c) For which triangles $ABC$ is $\lambda(\alpha)$ independent of $\alpha$?

Let $ABCD$ be a quadrilateral inscribed in the circle $O$. For a point $E\in O$, its projections $K,L,M,N$ on the lines $DA,AB,BC,CD$, respectively, are considered. Prove that if $N$ is the orthocentre of the triangle $KLM$ for some point $E$, different from $A,B,C,D$, then this holds for every point $E$ of the circle.

[b]p1.[/b] A man is walking due east at $2$ m.p.h. and to him the wind appears to be blowing from the north. On doubling his speed to $4$ m.p.h. and still walking due east, the wind appears to be blowing from the nortl^eas^. What is the speed of the wind (assumed to have a constant velocity)? [b]p2.[/b] Prove that any triangle can be cut into three pieces which can be rearranged to form a rectangle of the same area. [b]p3.[/b] An increasing sequence of integers starting with $1$ has the property that if $n$ is any member of the sequence, then so also are $3n$ and $n + 7$. Also, all the members of the sequence are solely generated from the first nummber $1$; thus the sequence starts with $1,3,8,9,10, ...$ and $2,4,5,6,7...$ are not members of this sequence. Determine all the other positive integers which are not members of the sequence. [b]p4.[/b] Three prime numbers, each greater than $3$, are in arithmetic progression. Show that their common difference is a multiple of $6$. [b]p5.[/b] Prove that if $S$ is a set of at least $7$ distinct points, no four in a plane, the volumes of all the tetrahedra with vertices in $S$ are not all equal. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Three line segments divide a triangle into five triangles. The area of these triangles is called $u, v, x,$ yand $z$, as in the figure. (a) Prove that $uv = yz$. (b) Prove that the area of the great triangle is at most $ \frac{xz}{y}$ [img]https://cdn.artofproblemsolving.com/attachments/9/4/2041d62d014cf742876e01dd8c604c4d38a167.png[/img]

Pawns are arranged on an $8 \times 8$ chessboard such that: Each $2 \times 1$ or $1 \times 2$ rectangle has at least $1$ pawn; Each $7 \times 1$ or $1 \times 7$ rectangle has at least $1$ pair of adjacent pawns. What is the minimum number of pawns in such an arrangement?

The graphs of the equations \[ y=k, \qquad y=\sqrt{3}x+2k, \qquad y=-\sqrt{3}x+2k, \] are drawn in the coordinate plane for $k=-10,-9,-8,\ldots,9,10.$ These 63 lines cut part of the plane into equilateral triangles of side $2/\sqrt{3}.$ How many such triangles are formed?

For which values of n does there exist a polyhedron having $n$ edges?

A strip of width $w$ is the set of all points which lie on, or between, two parallel lines distance $w$ apart. Let $S$ be a set of $n$ ($n \ge 3$) points on the plane such that any three different points of $S$ can be covered by a strip of width $1$. Prove that $S$ can be covered by a strip of width $2$.

A circle which is tangent to sides $AB$ and $BC$ of triangle $ABC$ is also tangent to its circumcircle at point $T$. If $I$ in the incenter of triangle $ABC$, show that $\angle ATI=\angle CTI$.

Prove that if a diagonal is drawn in a quadrilateral inscribed in a circle, the sum of the radii of the circles inscribed in the two triangles thus formed is the same, no matter which diagonal is drawn.

The circle inscribed in the triangle $ABC$ touches the side BC at point $K$. Prove that the segment $AK$ is longer than the diameter of the circle.