There is a point $M$ inside a circle, at a distance $OM = d$ of the center $O$. Two chords $AB$ and $CD$ are traced through $M$ that form a right angle . Join $A$ with $C$ and $B$ with $D$. Determine the cosine of the angle that must form the chord $AB$ with $OM$ so that the sum of the areas of the triangles $AMC$ and $BMD$ be minimal.

[b]Problem 1.[/b] Points $D$ and $E$ are chosen on the sides $AB$ and $AC$, respectively, of a triangle $\triangle ABC$ such that $DE\parallel BC$. The circumcircle $k$ of triangle $\triangle ADE$ intersects the lines $BE$ and $CD$ at the points $M$ and $N$ (different from $E$ and $D$). The lines $AM$ and $AN$ intersect the side $BC$ at points $P$ and $Q$ such that $BC=2\cdot PQ$ and the point $P$ lies between $B$ and $Q$. Prove that the circle $k$ passes through the point of intersection of the side $BC$ and the angle bisector of $\angle BAC$. [i]Nikolai Nikolov[/i]

A parallelogram of $ABCD$ is given. Line parallel to $AB$ intersects the bisectors of angles $A$ and $C$ at points $P$ and $Q$, respectively. Prove that the angles $ADP$ and $ABQ$ are equal. (A. Hakobyan)

For every four points $P_{1},P_{2},P_{3},P_{4}$ on the plane, find the minimum value of $\frac{\sum_{1\le\ i<j\le\ 4}P_{i}P_{j}}{\min_{1\le\ i<j\le\ 4}(P_{i}P_{j})}$.

Consider the triangle $ABC$ with circumcenter $O$. Let $D$ be the intersection of the angle bisector of $\angle{A}$ with $BC$. Show that $OA$, the perpendicular bisector of $AD$ and the perpendicular to $BC$ passing through $D$ are concurrent.

Among the spatial points $ A,B,C,D, $ at most two of are aparted at a distance greater than $ 1. $ Find the the maximum value of the expression: $$ g(A,B,C,D) =AB+BC+ AD+CA+DB+DC. $$

In the plane, there are $n \geqslant 6$ pairwise disjoint disks $D_{1}, D_{2}, \ldots, D_{n}$ with radii $R_{1} \geqslant R_{2} \geqslant \ldots \geqslant R_{n}$. For every $i=1,2, \ldots, n$, a point $P_{i}$ is chosen in disk $D_{i}$. Let $O$ be an arbitrary point in the plane. Prove that \[O P_{1}+O P_{2}+\ldots+O P_{n} \geqslant R_{6}+R_{7}+\ldots+R_{n}.\] (A disk is assumed to contain its boundary.)

[b]p1.[/b] A light beam shines from the origin into the unit square at an angle of $\theta$ to one of the sides such that $\tan \theta = \frac{13}{17}$ . The light beam is reflected by the sides of the square. How many times does the light beam hit a side of the square before hitting a vertex of the square? [img]https://cdn.artofproblemsolving.com/attachments/5/7/1db0aad33ed9bf82bee3303c7fbbe0b7c2574f.png[/img] [b]p2.[/b] Alex is given points $A_1,A_2,...,A_{150}$ in the plane such that no three are collinear and $A_1$, $A_2$, $...$, $A_{100}$ are the vertices of a convex polygon $P$ containing $A_{101}$, $A_{102}$, $ ...$, $A_{150}$ in its interior. He proceeds to draw edges $A_iA_j$ such that no two edges intersect (except possibly at their endpoints), eventually dividing $P$ up into triangles. How many triangles are there? [img]https://cdn.artofproblemsolving.com/attachments/d/5/12c757077e87809837d16128b018895a8bcc94.png[/img] [b]p3. [/b]The polynomial P(x) has the property that $P(1)$, $P(2)$, $P(3)$, $P(4)$, and $P(5)$ are equal to $1$, $2$, $3$, $4$,$5$ in some order. How many possibilities are there for the polynomial $P$, given that the degree of $P$ is strictly less than $4$? PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

A square $ (n \minus{} 1) \times (n \minus{} 1)$ is divided into $ (n \minus{} 1)^2$ unit squares in the usual manner. Each of the $ n^2$ vertices of these squares is to be coloured red or blue. Find the number of different colourings such that each unit square has exactly two red vertices. (Two colouring schemse are regarded as different if at least one vertex is coloured differently in the two schemes.)

An eighth degree polynomial funtion $ y \equal{} ax^8 \plus{} bx^7 \plus{} cx^6 \plus{} dx^5 \plus{} ex^4 \plus{} fx^3 \plus{} gx^2\plus{}hx\plus{}i\ (a\neq 0)$ touches the line $ y \equal{} px \plus{} q$ at $ x \equal{} \alpha ,\ \beta ,\ \gamma ,\ \delta \ (\alpha < \beta < \gamma <\delta).$ Find the area of the region bounded by these graphs in terms of $ a,\ \alpha ,\ \beta ,\gamma ,\ \delta .$

Let $AB=AC$ in $\triangle ABC$, and let $D$ be a point on segment $AB$. The tangent at $D$ to the circumcircle $\omega$ of $BCD$ hits $AC$ at $E$. The other tangent from $E$ to $\omega$ touches it at $F$, and $G=BF \cap CD$, $H=AG \cap BC$. Prove that $BH=2HC$. [i]Proposed by David Stoner[/i]

Let scalene triangle $ABC$ have orthocentre $H$ and circumcircle $\Gamma$. $AH$ meets $\Gamma$ at $D$ distinct from $A$. $BH$ and $CH$ meet $CA$ and $AB$ at $E$ and $F$ respectively, and $EF$ meets $BC$ at $P$. The tangents to $\Gamma$ at $B$ and $C$ meet at $T$. Show that $AP$ and $DT$ are concurrent on the circumcircle of $AFE$.

A plane intersects a rectangular parallelepiped in a regular hexagon. Prove that the rectangular parallelepiped is a cube.

The altitude from one vertex of a triangle, the bisector from the another one and the median from the remaining vertex were drawn, the common points of these three lines were marked, and after this everything was erased except three marked points. Restore the triangle. (For every two erased segments, it is known which of the three points was their intersection point.) (A. Zaslavsky)

Determine the possible interior angles of isosceles triangles that can be subdivided in two isosceles triangles with disjoint interior.

Prove that $$AB + PQ + QR + RP \le AP + AQ + AR + BP + BQ + BR$$ where $A, B, P, Q$ and $R $ are any five points in a plane.

Suppose that $ABCD$ is a parallelogram such that $\angle DAC = 90^o$. Let $H$ be the foot of perpendicular from $A$ to $DC$, also let $P$ be a point along the line $AC$ such that the line $PD$ is tangent to the circumcircle of the triangle $ABD$. Prove that $\angle PBA = \angle DBH$. Proposed by Iman Maghsoudi

You have some white one-by-one tiles and some black and white two-bye-one tiles as shown below. There are four different color patterns that can be generated when using these tiles to cover a three-by-one rectangoe by laying these tiles side by side (WWW, BWW, WBW, WWB). How many different color patterns can be generated when using these tiles to cover a ten-by-one rectangle? [asy] import graph; size(5cm); real labelscalefactor = 0.5; pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); draw((12,0)--(12,1)--(11,1)--(11,0)--cycle); fill((13.49,0)--(13.49,1)--(12.49,1)--(12.49,0)--cycle, black); draw((13.49,0)--(13.49,1)--(14.49,1)--(14.49,0)--cycle); draw((15,0)--(15,1)--(16,1)--(16,0)--cycle); fill((17,0)--(17,1)--(16,1)--(16,0)--cycle, black); [/asy]

In $\triangle ABC$, we have $AB>AC>BC$. $D,E,F$ are the tangent points of the inscribed circle of $\triangle ABC$ with the line segments $AB,BC,AC$ respectively. The points $L,M,N$ are the midpoints of the line segments $DE,EF,FD$. The straight line $NL$ intersects with ray $AB$ at $P$, straight line $LM$ intersects ray $BC$ at $Q$ and the straight line $NM$ intersects ray $AC$ at $R$. Prove that $PA \cdot QB \cdot RC = PD \cdot QE \cdot RF$.

$ABCD$ is a quadrilateral having both an inscribed circle (one tangent to all four sides) with center $I,$ and a circumscribed circle with center $O$. Let $S$ be the point of intersection of the diagonals of $ABCD$. Show that if any two of $S, I$ and $O$ coincide, then $ABCD$ is a square (and hence all three coincide).

Prove that if all dihedral angles of a tetrahedron are acute, then all its faces are acute-angled triangles.

Let $ABCD$ be a square with side $20$ and $T_1, T_2, ..., T_{2000}$ are points in $ABCD$ such that no $3$ points in the set $S = \{A, B, C, D, T_1, T_2, ..., T_{2000}\}$ are collinear. Prove that there exists a triangle with vertices in $S$, such that the area is less than $1/10$.

In the quadrilateral $ABCD$, we have $AB = AD$ and $\angle B = \angle D = 90 ^ \circ $. The points $P$ and $Q $ lie on $BC$ and $CD$, respectively, so that $AQ$ is perpendicular on $DP$. Prove that $AP$ is perpendicular to $BQ$.

Given triangle $ABC$. Suppose $P$ and $P_1$ are points on $BC, Q$ lies on $CA, R$ lies on $AB$, such that $$\frac{AR}{RB}=\frac{BP}{PC}=\frac{CQ}{QA}=\frac{CP_1}{P_1B}$$ Let $G$ be the centroid of triangle $ABC$ and $K = AP_1 \cap RQ$. Prove that points $P,G$, and $K$ are collinear.

Point $D$ is chosen on the Euler line of triangle $ABC$ and it is inside of the triangle. Points $E,F$ are were the line $BD,CD$ intersect with $AC,AB$ respectively. Point $X$ is on the line $AD$ such that $\angle EXF =180 - \angle A$, also $A,X$ are on the same side of $EF$. If $P$ is the second intersection of circumcircles of $CXF,BXE$ then prove the lines $XP,EF$ meet on the altitude of $A$ Proposed by [i]Alireza Danaie[/i]