Let $A,B,C$ and $D$ be non-coplanar points such that $\angle ABC=\angle ADC$ and $\angle BAD=\angle BCD$. Show that $AB=CD$ and $AD=BC$.

Three concentric circles centered at $O$ have radii of $1$, $2$, and $3$. Points $B$ and $C$ lie on the largest circle. The region between the two smaller circles is shaded, as is the portion of the region between the two larger circles bounded by central angle $BOC$, as shown in the figure below. Suppose the shaded and unshaded regions are equal in area. What is the measure of $\angle{BOC}$ in degrees? [asy] size(100); import graph; draw(circle((0,0),3)); real radius = 3; real angleStart = -54; // starting angle of the sector real angleEnd = 54; // ending angle of the sector label("$O$",(0,0),W); pair O = (0, 0); filldraw(arc(O, radius, angleStart, angleEnd)--O--cycle, lightgray); filldraw(circle((0,0),2),lightgray); filldraw(circle((0,0),1),white); draw((1.763,2.427)--(0,0)--(1.763,-2.427)); label("$B$",(1.763,2.427),NE); label("$C$",(1.763,-2.427),SE); [/asy] $\textbf{(A)}\ 108 \qquad \textbf{(B)}\ 120 \qquad \textbf{(C)}\ 135 \qquad \textbf{(D)}\ 144 \qquad \textbf{(E)}\ 150$

Let $ABCD$ be a square of center $O$. The parallel to $AD$ through $O$ intersects $AB$ and $CD$ at $M$ and $N$ and a parallel to $AB$ intersects diagonal $AC$ at $P$. Prove that $$OP^4 + \left(\frac{MN}{2} \right)^4 = MP^2 \cdot NP^2$$

Find the smallest possible value of a real number $c$ such that for any $2012$-degree monic polynomial \[P(x)=x^{2012}+a_{2011}x^{2011}+\ldots+a_1x+a_0\] with real coefficients, we can obtain a new polynomial $Q(x)$ by multiplying some of its coefficients by $-1$ such that every root $z$ of $Q(x)$ satisfies the inequality \[ \left\lvert \operatorname{Im} z \right\rvert \le c \left\lvert \operatorname{Re} z \right\rvert. \]

In acute-angled triangle $ABC$, a semicircle with radius $r_a$ is constructed with its base on $BC$ and tangent to the other two sides. $r_b$ and $r_c$ are defined similarly. $r$ is the radius of the incircle of $ABC$. Show that \[ \frac{2}{r} = \frac{1}{r_a} + \frac{1}{r_b} + \frac{1}{r_c}. \]

Let $\triangle{ABC}$ be a triangle, $\omega$ its circumcircle and $O$ the center of $\omega$. Let $P$ be a point on the segment $BC$. We denote by $Q$ the second intersection point of the circumcircles of triangles $\triangle{AOB}$ and $\triangle{APC}$. Prove that the line $PQ$ and the tangent to $\omega$ at point $A$ intersect on the circumcircle of triangle $\triangle AOB$.

Let $ABCD$ be a quadrilateral such that $AB = BC = 13$, $CD = DA = 15$ and $AC = 24$. Let the midpoint of $AC$ be $E$. What is the area of the quadrilateral formed by connecting the incenters of $ABE$, $BCE$, $CDE$, and $DAE$?

Find all possibilities: how many acute angles can there be in a convex polygon?

In quadrilateral $ABCD,$ suppose that $\overline{CD}$ is perpendicular to $\overline{BC}$ and $\overline{DA}$. Point $E$ is chosen on segment $\overline{CD}$ such that $\angle AED = \angle BEC$. If $AB = 6$, $AD = 7$, and $\angle ABC = 120^o$ , compute $AE + EB$.

Circles $ \Gamma_1$ and $ \Gamma_2$ are internally tangent to circle $ \Gamma$ at $ P$ and $ Q$, respectively. Let $ P_1$ and $ Q_1$ are on $ \Gamma_1$ and $ \Gamma_2$ respectively such that $ P_1Q_1$ is the common tangent of $ P_1$ and $ Q_1$. Assume that $ \Gamma_1$ and $ \Gamma_2$ intersect at $ R$ and $ R_1$. Define $ O_1,O_2,O_3$ as the intersection of $ PQ$ and $ P_1Q_1$, the intersection of $ PR$ and $ P_1R_1$, and the intersection $ QR$ and $ Q_1R_1$. Prove that the points $ O_1,O_2,O_3$ are collinear. [i]Rudi Adha Prihandoko, Bandung[/i]

Let $ABCDE$ be a convex pentagon such that quadrilateral $ABDE$ is a parallelogram and quadrilateral $BCDE$ is inscribed in a circle. The circle with center $C$ and radius $CD$ intersects the line $BD, DE$ at points $F, G(\neq D)$, and points $A, F, G$ is on line l. Let $H$ be the intersection point of line $l$ and segment $BC$. Consider the set of circle $\Omega$ satisfying the following condition. Circle $\Omega$ passes through $A, H$ and intersects the sides $AB, AE$ at point other than $A$. Let $P, Q(\neq A)$ be the intersection point of circle $\Omega$ and sides $AB, AE$. Prove that $AP+AQ$ is constant.

On a chess table $n\times m$ we call a [i]move [/i] the following succesion of operations (i) choosing some unmarked squares, any two not lying on the same row or column; (ii) marking them with 1; (iii) marking with 0 all the unmarked squares which lie on the same line and column with a square marked with the number 1 (even if the square has been marked with 1 on another move). We call a [i]game [/i]a succession of moves that end in the moment that we cannot make any more moves. What is the maximum possible sum of the numbers on the table at the end of a game?

In a triangle $ABC$ with $\angle ACB = 90^{\circ}$ $D$ is the foot of the altitude of $C$. Let $E$ and $F$ be the reflections of $D$ with respect to $AC$ and $BC$. Let $O_1$ and $O_2$ be the circumcenters of $\triangle {ECB}$ and $\triangle {FCA}$. Show that: $$2O_1O_2=AB$$

Let $ABCD$ be a tetrahedron and suppose that $M$ is a point inside it such that $\angle MAD=\angle MBC$ and $\angle MDB=\angle MCA$. Prove that $$MA\cdot MB+MC\cdot MD<\max(AD\cdot BC,AC\cdot BD).$$

Segment $AD$ is the diameter of the circumscribed circle of an acute-angled triangle $ABC$. Through the intersection of the altitudes of this triangle, a straight line was drawn parallel to the side $BC$, which intersects sides $AB$ and $AC$ at points $E$ and $F$, respectively. Prove that the perimeter of the triangle $DEF$ is two times larger than the side $BC$.

In a right circular cone of wood, the radius of the circumference $T$ of the base circle measures $10$ cm, while every point on said circumference is $20$ cm away. from the apex of the cone. A red ant and a termite are located at antipodal points of $T$. A black ant is located at the midpoint of the segment that joins the vertex with the position of the termite. If the red ant moves to the black ant's position by the shortest possible path, how far does it travel?

Bottom surface of triangular pyramid $S-ABC$ is an isosceles right triangle (hypotenuse is $AB$). $SA=SB=SC=AB=2$, and $S,A,B,C$ are on a sphere with center of $O$. The distance of $O$ to plane $ABC$ is________.

Let \(D\) be a point on segment \(PQ\). Let \(\omega\) be a fixed circle passing through \(D\), and let \(A\) be a variable point on \(\omega\). Let \(X\) be the intersection of the tangent to the circumcircle of \(\triangle ADP\) at \(P\) and the tangent to the circumcircle of \(\triangle ADQ\) at \(Q\). Show that as \(A\) varies, \(X\) lies on a fixed line. [i]Proposed by Elliott Liu and Anthony Wang[/i]

Consider a circle with center $O$ and radius $R,$ and let $A$ and $B$ be two points in the plane of this circle. [b]a.)[/b] Draw a chord $CD$ of the circle such that $CD$ is parallel to $AB,$ and the point of the intersection $P$ of the lines $AC$ and $BD$ lies on the circle. [b]b.)[/b] Show that generally, one gets two possible points $P$ ($P_{1}$ and $P_{2}$) satisfying the condition of the above problem, and compute the distance between these two points, if the lengths $OA=a,$ $OB=b$ and $AB=d$ are given.

Show that ten equal-sized squares cannot be placed on a plane in such a way that no two have an interior point in common and the first touches each of the others.

In $\mathbb{R}^3$ some $n$ points are coloured. In every step, if four coloured points lie on the same line, Vojtěch can colour any other point on this line. He observes that he can colour any point $P \in \mathbb{R}^3$ in a finite number of steps (possibly depending on $P$). Find the minimal value of $n$ for which this could happen.

In the interior of a square $ABCD$ we construct the equilateral triangles $ABK, BCL, CDM, DAN.$ Prove that the midpoints of the four segments $KL, LM, MN, NK$ and the midpoints of the eight segments $AK, BK, BL, CL, CM, DM, DN, AN$ are the 12 vertices of a regular dodecagon.

Let $AH_A, BH_B$ and $CH_C$ be altitudes in $\triangle ABC$. Let $p_A,p_B,p_C$ be the perpendicular lines from vertices $A,B,C$ to $H_BH_C, H_CH_A, H_AH_B$ respectively. Prove that $p_A,p_B,p_C$ are concurrent lines.

Let $ABC$ be a triangle with bisectors $BE$ and $CF$ meet at $I$. Let $D$ be the projection of $I$ on the $BC$. Let M and $N$ be the orthocenters of triangles $AIF$ and $AIE$, respectively. Lines $EM$ and $FN$ meet at $P.$ Let $X$ be the midpoint of $BC$. Let $Y$ be the point lying on the line $AD$ such that $XY \perp IP$. Prove that line $AI$ bisects the segment $XY$. [i]Proposed by Tran Quang Hung - Vietnam[/i]

Two circles $\Gamma_1$ and $\Gamma_2$ intersect at points $A$ and $Z$ (with $A\neq Z$). Let $B$ be the centre of $\Gamma_1$ and let $C$ be the centre of $\Gamma_2$. The exterior angle bisector of $\angle{BAC}$ intersects $\Gamma_1$ again at $X$ and $\Gamma_2$ again at $Y$. Prove that the interior angle bisector of $\angle{BZC}$ passes through the circumcenter of $\triangle{XYZ}$. [i]For points $P,Q,R$ that lie on a line $\ell$ in that order, and a point $S$ not on $\ell$, the interior angle bisector of $\angle{PQS}$ is the line that divides $\angle{PQS}$ into two equal angles, while the exterior angle bisector of $\angle{PQS}$ is the line that divides $\angle{RQS}$ into two equal angles.[/i]